The structure of intercrystalline interfaces

Progress in Materials Science 45 (2000) 339±568 www.elsevier.com/locate/pmatsci

The structure of intercrystalline interfaces Witold Lojkowski a,b,*, Hans-JoÈrg Fecht b a

High Pressure Research Center (UNIPRESS), Polish Academy of Sciences, Sokolowska 29, 01-142 Warsaw, Poland b Materials Science Division, University of Ulm, Albert Einstein Alee 47, D-89081 Ulm, Germany Received 11 February 1999; accepted 9 July 1999

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

2.

The structure of intercrystalline interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. What is an interface and what is its structure. . . . . . . . . . . . . . . . . . . . . . 2.1.1. De®nition of interface and ``good matching'' . . . . . . . . . . . . . . . . 2.1.2. The structure and the structure elements . . . . . . . . . . . . . . . . . . . 2.1.3. Interfacial interactions relevant for structure elements . . . . . . . . . 2.1.4. Evolution of the concepts of structural units and structure elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Relation between geometry and properties of interfaces . . . . . . . . . . . . . . 2.2.1. De®nition of interface periodicity . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Periodicity of global and local properties . . . . . . . . . . . . . . . . . . 2.2.3. One-dimensional model of an interface . . . . . . . . . . . . . . . . . . . . 2.2.4. The two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5. The three-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Special crystallographic orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. CSL and dislocations models of grain boundaries . . . . . . . . . . . . 2.3.2. Angular distance between coincidence grain boundaries . . . . . . . . 2.3.3. Universal equation of state for interfaces . . . . . . . . . . . . . . . . . . 2.3.4. Hierarchy of special grain boundaries according to the spacing of planes parallel to the interface . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5. Special 4 to general grain boundary transformation . . . . . . . . . .

* Corresponding author. Tel.: +48-22-632-4302; fax: +48-22-632-4218. E-mail address: [email protected] (W. Lojkowski). 0079-6425/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 7 9 - 6 4 2 5 ( 9 9 ) 0 0 0 0 8 - 0

350 350 351 355 355 364 373 373 374 381 388 403 405 408 410 414 415 417

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2.3.6.

Static distortion wave analysis of periodic and incommensurate interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. The localisation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. The Peierls model of a dislocation . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Surface tension, surface stress and interfacial non-equilibrium. . . . 2.4.3. Interface stress at triple junctions . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Interface shear modulus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5. Interface shear modulus and the core width of interfacial dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6. Relaxation of interface mis®t strains by increasing interface width or its dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7. Core width of a dislocation in an interface and the localisation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.8. Adjustment of the static distortion waves theory to include the localisation parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.9. The collectivity coecient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.10. Relation between localisation parameter and the degree of localisation of mis®t dislocations cores . . . . . . . . . . . . . . . . . . . . 2.4.11. Calculation of the localisation parameter p . . . . . . . . . . . . . . . . . 2.4.12. E€ect of the p value on properties of interfaces and the coherent 4 non-coherent transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.13. The localisation parameter and criteria for low energy interfaces. . 2.4.14. E€ect of localisation parameter on the mode of growth of epitaxial layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Work of adhesion and interface structure . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Surface and adhesion determined energy minima . . . . . . . . . . . . . 2.5.2. Estimation of interfacial bonding energy and the p value . . . . . . . 2.5.3. Grain and phase boundaries in metals and covalent materials . . . . 2.5.4. Importance of polarity on the example of the SiC/Al interface . . . 2.5.5. Oxides and nitrides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6. E€ect of interface kinetics on its structure . . . . . . . . . . . . . . . . . . 2.6. Sintering of spheres experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1. Sintering metal spheres to metal substrates . . . . . . . . . . . . . . . . . 2.6.2. Sintering metal spheres to ionic substrates. . . . . . . . . . . . . . . . . . 2.6.3. Experiments with single rotating sphere . . . . . . . . . . . . . . . . . . . 2.6.4. ``Smoke'' experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5. Summary of the sintering of spheres experiments . . . . . . . . . . . . . 3.

Grain boundary kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Grain boundary phase transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Grain boundary energy in bicrystals . . . . . . . . . . . . . . . . . . . . . . 3.1.2. Relation between grain boundary energy and free volume . . . . . . 3.1.3. Energy of grain boundaries in tricrystals and grain boundary phase transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4. In¯uence of a small additional disorientation component on GB energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5. In¯uence of a small additional disorientation component on GB di€usion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

417 420 420 422 423 426 429 430 432 434 434 437 438 438 445 446 448 450 453 457 464 464 471 471 473 479 483 483 484 485 485 485 487 489 491 491

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3.1.6.

3.2.

3.3. 3.4.

3.5.

3.6.

4.

In¯uence of impurities on the special 4 general GB transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.7. Segregation e€ects on GB kinetics . . . . . . . . . . . . . . . . . . . . . . . Di€usion and stresses in grain boundaries . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Relaxation of dislocation strain ®elds in grain boundaries . . . . . . 3.2.2. Distribution function for GB di€usivity determined based on spreading experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Compensation relationship and the activation vector . . . . . . . . . . Reactive di€usion at an interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grain boundary pre-melting, pre-wetting and wetting. . . . . . . . . . . . . . . . 3.4.1. Basic thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. Grain boundary pre-melting and pre-wetting . . . . . . . . . . . . . . . . 3.4.3. TEM studies of pre-melting in thin foils of strained aluminium and copper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4. The pre-wetting transition in grain boundaries in Fe±Si bicrystals . 3.4.5. Solidi®cation of grain boundaries with increasing temperature . . . 3.4.6. Grain boundary wetting and segregation in the Cu±Bi system. . . . 3.4.7. Grain boundary wetting and grain growth in the Ga±Al system . . 3.4.8. Grain boundary wetting by amorphous phase and solid state amorphisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nanocrystalline materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1. Production of nanocrystalline materials by mechanical alloying. . . 3.5.2. Thermal properties of nanocrystalline metals . . . . . . . . . . . . . . . . 3.5.3. Phase stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4. Instability of nanocrystalline materials with respect to melting or amorphisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5. Mechanical properties of nanocrystals. . . . . . . . . . . . . . . . . . . . . 3.5.6. Reactive sintering of nanocrystalline SiC-diamond composites under pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.7. Grain boundaries harder than diamond . . . . . . . . . . . . . . . . . . . High pressure investigation of grain boundaries . . . . . . . . . . . . . . . . . . . . 3.6.1. Grain boundary migration in aluminium bicrystals . . . . . . . . . . . 3.6.2. The compensation pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3. Pressure e€ect on grain growth in polycrystalline aluminium . . . . 3.6.4. Pressure e€ect on the migration rate of the amorphous phase/ crystalline phase in covalent solids and the high pressure catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

341

497 497 500 500 504 507 517 519 519 524 524 527 530 530 533 534 538 538 540 541 542 543 544 544 546 546 549 549 551

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558

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Nomenclature o S a CB aC p pE f t H Na k R O r R rij r0 x d d b s a L L0 T A s k du, u um x, y, z e D h l {x} LD

dimensionless density of possible reaction sites at the interface inverse fraction of coincidence points fraction of good bonds broken bonds compensation factor covalence coecient localisation parameter localisation parameter for thin ®lms stochastic factor characteristic strain relaxation time Hamaker constant Avogadro constant Boltzman constant gas constant atomic volume interatomic distance crystal size, grain size distance i±j equilibrium interatomic distance scaling parameter spacing of atoms, interplanar spacing dimensionless interplanar spacing Burgers vector core width of a dislocation lattice constant periodicity domain length Rigid Body Translation (RBT) vector the amplitude of atomic oscillations stress force or spring constant bending of planes, displacement of atoms maximum u value space co-ordinates, z is perpendicular to the interface strain, relative displacement di€erence of lattice constants thickness of the thin ®lm thickness of a growing intermetallic layer small neighbourhood around the point x length of a periodic domain

W. Lojkowski, H.-J. Fecht / Progress in Materials Science 45 (2000) 339±568

n, m n nA, nB S(x ) F(x ) F D D M r j q {t} y j c b c, C cx , c1, c' Dcp cGBO g gADH gA gI gSLO gWaals DG DGAB DGB DG IN DZ m, Dm EAB

343

natural numbers co-ordination number number of electrons associated with the cation and anion, respectively structure propriety force di€usion coecient chemical di€usion coecient GB mobility dislocation density ¯ux of dislocations or atoms in a di€usion process wavevector set of vectors of the reciprocal lattice misorientation angle wetting angle, dihedral angle lattice mis®t reaction rate at the interface constant concentration heat capacity GB solubility limit surface energy adhesion energy surface energy of the crystal A interface surface energy the free surface energy of the S/L interface van der Waals surface adhesion energy Gibbs free energy Gibbs free energy of the A±B compound formation Gibbs free energy of dissociation of a gas molecule into atoms Gibbs free energy di€erence of a nanocrystal with respect to the single crystal di€erence of the Gibbs free energy of oxide formation and metal±oxide formation, for a metal layer on an oxide chemical potential, di€erence of chemical potentials bonding energy between atoms A and B or total bonding energy between crystals A and B,

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EA ECOH EWaals VC VI DHf Ey G B hGI0 i hGI i GI0 GI vSL vGB V VK …z†, Vz …K †

G Q S A Ss Ss SS Qs Vs P T PC TC Tm misorientation disorientation wrong bond lock-in

cohesion energy of the element A cohesion energy van der Waals bonding energy/atom covalent bonding energy ionic bonding energy heat of fusion Young modulus bulk shear modulus bulk modulus average low frequency shear modulus average high frequency shear modulus low frequency shear modulus for good matching sectors high frequency shear modulus for good matching sectors free volume of the solid/liquid interface free volume of the GB activation volume Fourier transforms of the interatomic potential parallel to the interface at distance z, where K is the vector of the inverse lattice and z the spatial co-ordinate activation free energy activation energy activation entropy activation vector segregation vector segregation factor segregation entropy segregation energy segregation volume pressure temperature compensation pressure compensation temperature melting point rotation angle of one crystal relative to the other crystal forming the interface di€erence of misorientations between the given GB and the closest low energy coincidence GB the B±B or A±A bond in a GB in an AB compound structural feature of an interface, where closely

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packed rows of atoms of one crystal lock into energy valleys of the other crystal plane-matching situation when crystallographic planes are continuous across the interface work of adhesion energy needed to separate two crystals adhesion energy work of adhesion per unit surface vicinal plane crystal plane such that the interface energy is low surface type energy minimum an energy minimum of the interface caused by the fact that the free surface of the crystal for the given OR has a minimum adhesion type energy cusp an energy minimum of the interface caused by the fact that there is a good atomic matching at the interface combined type energy cusp an energy minimum of the interface caused by the fact that at the same time there is the surface type minimum and adhesion type minimum free interface volume the increase of the volume of the material when the interface area is increased by unit surface. Is measured in nm, since it is a volume per unit area tricrystal a material with three GBs joining along a triple junction bicrystal a material with only one GB MD Mis®t Dislocation GB Grain Boundary S/L Solid±Liquid interface DPM Displacement Position Map SGBD Structural Grain Boundary Dislocation EGBD Extrinsic Grain Boundary Dislocation SSGBD Statistically Stored GBD GNGBD Geometrically Necessary GBD TLD Trapped Lattice Dislocation RBT Rigid Body Translation 2D, 3D two-dimensional, three-dimensional SE Structural Element SU Structural Unit OR Orientation Relationship LEOR Low Energy OR TEM Transmission Electron Microscopy LEED Low Energy Electron Di€raction HREM High Resolution TEM

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FIM CSL DSC DOF IV SIGBPM

Field Emission Microscope Coincidence Site Lattice Displacement Shift Complete: lattice of Burgers vectors in the bicrystal lattice Degree of Freedom Independent Variable Strain Induced Grain Boundary Pre-melting

1. Introduction Internal interfaces in materials are extended defects including grain boundaries and interphase boundaries, found in almost every natural or arti®cially produced material. Consider for example today's integrated-circuit technology where several di€erent materials are combined. Interfaces play a crucial role for the performance of these devices. Extreme service conditions sometimes prevail such as high temperatures, aggressive surrounding media, electromigration at high current densities. It follows that the structure of interfaces must be regarded from the point of view of the whole complexity of the thermodynamic conditions during fabrication and exploitation of devices, frequently in thermodynamic nonequilibrium. Therefore, interface engineering is an important ®eld of materials engineering. For many technological applications, the control of internal interfaces, including the orientation relationship, the interfacial mis®t, and of interface kinetics play a crucial role. In that respect, interfaces o€er a considerable challenge. This is caused by their complexity, which results from the large number of independent variables upon which the properties of interfaces depend. There are considerable diculties already in the de®nition of what the structure of an interface is and how it depends on the independent variables. One of the major topics is the relation between the atomic structure of interfaces and their properties [1±9]. Ballu and Sutton [10] pointed out that a major challenge in that respect is to bridge the length scales in the study of interfaces, so that information acquired in atomic scale investigations is not lost when macroscopic properties are considered. The question especially becomes complex if the interface kinetics and thermodynamics are included. Interfaces in metals are relatively insensitive to the thermodynamic parameters. This follows from the character of the metallic bond, which is not signi®cantly a€ected by temperature unless the temperature approaches the boiling point. The situation is di€erent for various ceramic materials. The Gibbs free energy of the bond formation for such interfaces

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strongly depends on the temperature. In addition, the interfaces in real materials are frequently in thermodynamic non-equilibrium conditions. Thermodynamic non-equilibrium introduces selection rules where it is not the energy but the fast reaction rate [11±13] that determines the structure. Such structures may be related to the maximisation of entropy dissipation rate [14] instead of energy. Comparing with the situation in the ®eld up to the mid-70s [15±17], the increasing complexity of the question is recognised, as can be seen from the proceedings of recent interface conferences [18±23]. In our opinion, a solution to the question of bridging atomic scale structure of interfaces and properties is to identify parameters, which may serve to construct maps of interface properties and interface phase diagrams. These maps should permit to categorise interfaces and give simple guidelines to their properties. In its essence, this is a problem of engineering, when owing to the complexity of the problem, science cannot provide solutions but just engineering guidelines. Fig. 1 shows the routes, presented by Ashby [24], to an engineering solution for a complete design problem involving materials. The same scheme can be used for interfaces: the diagonal route corresponds to introduction of a set of structural parameters and development of property maps, which might be helpful for interface engineering. The development of such maps and parameters is a process

Fig. 1. Three possible routes to the solution of an engineering problem [24]. In the case of interfaces, the diagonal route would correspond to developing phase diagrams and ``structure maps'' with a limited number of parameters. Courtesy M. Ashby. Published with permission from the Royal Society.

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of synthesis of a large number of data, ®nding out the general rules, adjusting them according to coming new experimental results, and introducing new maps or parameters as new phenomena are discovered. Having that in mind, it seems reasonable to us to make an attempt to provide a theory that shows how the main features of interfaces: dislocations, structure elements, low energy orientation relationships, etc., change depending on the type of crystals that are joined and on the thermodynamic conditions. Such a theory should end up with ``maps'' of interfacial properties, which may serve as guidelines for constructing devices and for interpretation of experimental results. The conclusions concerning the properties of interfaces should be based on the most basic and general principles. An important consequence of the above approach is that the importance of the geometrical criteria for low energy interfaces, as for example mis®t, is a relative question. Thermodynamic conditions may cause a change as far as which geometrical criteria are important. The ®rst part of the present review (Section 2) is concerned with the structure of intercrystalline interfaces. In Section 2.1, we present an attempt to give a de®nition, what is an interface and what is its structure. It is obvious that the term ``interface structure'' means something di€erent for a scientist working with high TC superconductors, semiconductors, magnetic materials, or high strength steels. We show that the term ``interface structure'' is relative, it depends on the type of interactions important for the given application. Furthermore, we discuss the concepts of ``atomic matching'' (Section 2.1.1) and structural elements (Section 2.1.2). We show that there are several ways of interpretation of the structural elements, depending on the type of interatomic interactions crucial for interface properties. In Sections 2.2 and 2.3, we discuss the criteria for special and low energy interfaces. In Section 2.2, the geometrical properties of interfaces are analysed and it is shown that a number of properties of interfaces can be derived from very basic considerations of the interplay of the two lattices. Since the geometrical properties alone are not sucient for understanding the interfaces, we introduce in Section 2.4 physical parameters which may help to understand what geometrical factors may play an important role for a given interface. The main one is the localisation parameter, a parameter that permits to predict up to what extent mis®t dislocation cores are localised (Section 2.5). It is shown that the degree of mis®t localisation is crucial for a number of interface properties. The degree of mis®t localisation and the properties of interfaces depend on the cohesive forces and elastic properties of interfaces. The factors in¯uencing interfacial cohesion are discussed and an attempt is presented to give general rules concerning the relation between the energy of free surfaces of crystals, the type of interfacial bonding and the interface energy. The main conclusion from this section is that mis®t at the real interface is not an absolute value. It is an absolute value in geometrical terms. But in physical terms, mis®t depends on the shallowness of the interatomic potentials and/or on the thermodynamic conditions such as: temperature, pressure, chemical compositions, partial pressures of impurities in the atmosphere, and the crystallographic parameters.

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In the second part of the present review (Section 3), we present some recent and interesting experimental results concerning interface energy and kinetics. We stress the e€ect of small impurity content on the properties of interfaces. What small means depends on the material studied, for instance, semiconductor: 10ÿ17, metal: 10ÿ9. The question ``How do you know it is not an impurity e€ect?'' is almost standard in interface studies. Special attention is given to the problems of HREM and computer simulation studies, some aspects of interface di€usion and migration, wetting, pre-wetting and pre-melting, amorphisation, nanocrystallisation. The richness of interface phenomena that escape the attempts to organise the ®eld is shown. In the light of such results, questions about ``scienti®c economy'' arise; since, for instance, the thoroughly investigated structures of GBs of pure metals are of limited validity in real materials. One section is devoted to a short review of recent results in producing nanocrystalline and amorphous materials by mechanical alloying and heavy deformation (Section 3.5). This section leads in a natural way to consider the e€ect of high pressures on interfaces. High pressures are particularly important for the compaction of nanocrystalline powders, and results obtained during sintering of nanocrystalline diamond and silicon carbide are reported (Section 3.5.6). Some recent results of the e€ect of high pressure on interfaces are reported in more detail, since such investigations may provide a valuable insight on the mechanisms of atomic movements and on the factors important for the interface energy (Section 3.6). The present work was considerably facilitated by the recent publication of a number of extensive reviews and textbooks on grain boundaries and interfaces. A deep treatment of the relation between geometry and properties of interfaces can be found in the book of Sutton and Ballu [1]. The book of Howe [2] presents a good introduction to the interfaces basics. Recent advancement in research of the

Fig. 2. The deformation ®eld u(x) within a neighbourhood of atoms as a function of the RBT vector within one crystal. x0 is the centre of the neighbourhood and {x} is the set of points that belong to the given neighbourhood. Two neighbourhoods {x} and {x '} are related: x ' = x + T + u(T).

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electronic structure of interfaces in semiconductors are given in the work of Fionova and Artemyev [4]. Randle [3] presents a review of the interface geometry and the e€ect of interface plane on their properties [8]. The book of Murr [6] was of considerable help. Wolf and Yip edited an advanced review of recent progress in interface study [5]. Interfaces in electronic ceramics are reviewed in the book of Hozer [25].

2. The structure of intercrystalline interfaces 2.1. What is an interface and what is its structure In this section, we will attempt to provide a de®nition of an interface in terms of a transformation operator A(T). The operator depends on the Rigid Body Translation (RBT) vector T, which connects two neighbourhoods. The form of A is di€erent if both neighbourhoods are on the same side of the interface, or on the two di€erent sides. In this way, the T vectors pointing at the interface are de®ned. The de®nition will be useful for further considerations of mis®t localisation in interfaces. Let us consider two equivalent neighbourhoods fx0 g and fx00 g: A neighbourhood of x0 is a small space around x0 , where the physical parameters are approximately constant. In a crystal without strain and interface, they are related by a RBT vector T. In the case of strain ®elds, neighbourhoods are related to the operator O (Fig. 2): ÿ
 0 …1† x0 ˆ O x00 : The operation O can be divided into four di€erent stages: (1) RBT translation T of the neighbourhood fx0 g; (2) elastic translation u(T) to bring the centre of the neighbourhood to the new position fx00 g, where u is within the unit cell; (3), an elastic distortion of the neighbourhood and (4) its rotation. In the following text, we omit from the equations, the symmetry operators and translation operators. The local strain u is: u…T † ˆ x 0 ÿ x:

…2†

The deformation ®eld is a function of T: @ ui 6ˆ0: @Tj

…3†

If the RBT vector crosses the interface, then in addition there is the ®fth stage: the A operator. In a non-strained material, we may de®ne an idealised in®nitely thin interface as a plane separating two sets of points such that for fx0 g and fx00 g on one side of the plane (Fig. 3):

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Fig. 3. E€ect of RBT on u(x) for a neighbourhood translated to another crystal. Both crystals are nonstrained. Each point of the neighbourhood {x} in one crystal corresponds to a point in the neighbourhood {x0} in the other crystal: x0 = AT(x), where A is a linear operator. When the translation T is within the same crystal, {x} c {x'} and u(T)00. If translation crosses the interface, the deformation ®eld u(T) becomes a function of T.

x 0 ˆ T
x,

…4†

while for fx0 g and fx00 g on two sides of the plane: x 00 ˆ A…x†:

…5†

Let us consider what conditions A must ful®l to de®ne an interface. We assume that the two crystals are composed of the same atoms. In the opposite case, the ``chemical operator'' C has to be introduced which replaces atoms of one type by atoms of the other type: AC ˆ C
A,

…6†

where AC is the transformation operator which contains the chemical information and A is the ``geometrical operator'', transforming nodes of one lattice into nodes of the other lattice. As shown in the next paragraph, for a coherent interface, only this chemical operator de®nes the interface. 2.1.1. De®nition of interface and ``good matching'' In order to discuss the subject of ``good matching'' of the crystals, it is essential that the operator A is a function of the RBT vector T. Since A is a function of T, then the co-ordinates in the transformed neighbourhood are a function of T, as

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Fig. 4. Deformation ®eld as a function of the translation vector T for various interfaces.

well as the deformation ®eld u. Consider the function u(T) for a strained material or a material with an interface (Fig. 4). The following situations can be envisaged. (a) Translation within a crystal. Directions and planes are continuous (Fig. 4a) for any x: uL …T † ˆ uR …T †,

…7†

@ ui @ui ˆ , @Tj L @ Tj R

…8†

where the suxes L, R mean the limits when approaching T from the left of right side. (b) There is a coherent interface. Directions and planes are continuous (Fig. 4b): uL …T † ˆ uR …T †, T 2 Interface @ ui @ui ˆ , @Tj L @ Tj R

…9† …10†

where L, R mean the limits when approaching the interface from the left and right side. It is seen that unless the chemical composition changes, the two cases are indistinguishable. The curves representing the strain in Fig. 4a and b are

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353

Fig. 5. Schematic presentation of the deformation ®eld for a semicoherent interface.

qualitatively identical. The position of the interface plane can be de®ned only by the chemical composition. At the same time, the gradient of composition induces the deformation ®eld, since the lattice constants depend on the chemical composition of the phases. There are situations where the chemical composition changes gradually, as in di€usion fronts. In the further text, we will consider only sharp chemical interfaces, where the composition changes over a few lattice constants. (c) There is a discontinuity of directions but planes are continuous (Fig. 4c): uL …T † ˆ uR …T †,

…11†

@ ui @ui ˆ 6 : @Tj L @ Tj R

…12†

Such a con®guration corresponds to the coherent twin grain boundary. Calling this interface coherent is an exception of case (b), since the directions are not continuous. (d) Planes are discontinuous (Fig. 4d): uL 6ˆuR :

…13†

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(e) Semicoherent interface. In some sectors of the interface case (b) is valid and in other sectors case (d) holds.

In a semicoherent interface, the value of derivative @u=@T will depend on the path across the interface, since coherent and incoherent regions alternate. Fig. 5 shows schematically a semicoherent interface. Eq. (14) describes the incoherent interface (Fig. 4c and d). @ ui ˆ Dud…T †: …14† @Tj T2Interface Fig. 4d shows the function u(T) for an idealised incoherent interface. In a real system, the interface has a ®nite thickness and there are deformation ®elds. Fig. 4e shows the incoherent interface, where Eq. (14) takes the form: @ ui Du , ˆ @Tj d

…15†

where d is the thickness of the interface and Du is the discontinuity of the deformation ®eld. Good matching between two crystals cannot be de®ned without observing the structure and energy of the interface as a function of some variables: misorientation of the crystals or their relative translation or thermodynamic variables. A meaningful de®nition of good matching is that: (a) if by some operation the good matching is destroyed, the crystals restore the good matching regions at the expense of some elastic strain. (b) planes tend to bend to increase the area of ``good matching sectors''. Observation of ``good geometrical matching'' as expressed by the equation: Du10, which means geometrical continuity or: @ ui @ui ˆ , @Tj L @ Tj R

…16†

…17†

i.e., continuity of the directions of atomic planes or rows, is not a sucient criterion. According to (b), an indication of the importance of good matching is that atomic planes locally bend to extend the good matching sectors. Bending of planes would be re¯ected by the derivative du=dTj…T† at the interface. Let us consider one component of this derivative. In general, it takes the following form: @Dui ˆ rd…T †, @Ti

…18†

where r(T) is the discontinuity of atomic planes across the interface at point T. (i) If r = const(T) and r > 0, there is no atomic matching and the interface is

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fully incoherent. (ii) If r = 0 for all T, the interface is fully coherent. (iii) If r = r(T), atomic planes tend to bend and generate sections of good atomic matching. In summary, it seems that an interface exists if one of the following conditions are ful®lled: (a) there is a discontinuity of chemical composition changes, (b) there is a discontinuity of atomic rows or planes or of their directions.

2.1.2. The structure and the structure elements Before discussing the interface structure, let us see what does the term ``structure'' means. According to the Encyclopaedia [26], there is a broader and narrower de®nition. Structure is in a broader sense, the arrangement of a set of elements and the set of relations between these elements, characteristic for the given system. In a narrower sense, this is the method of (or rules for) connecting the elements in an entity. Another formulation is the relationships between the elements and between the elements and the entity. It is seen that in its essence, a structure is a set of elements and their interactions. In all the above de®nitions, the interactions played the crucial role for the description of the structure. From the above, the following de®nition of the microstructure follows. The microstructure of a material can be de®ned as the set of defects in the material and of their interactions [27]. According to the above de®nition to describe the structure of interfaces, one has to de®ne the ``Structural Elements'' (SEs). The de®nition depends on the interactions, which are considered relevant for the given problem. If a SE is added or replaced, the interface energy changes. Therefore, energy can be assigned to SEs. Let us assume that there are interfaces built from low energy SEs. Hence, one can consider segments of low energy interface as SEs themselves. However, the SEs can be de®ned only in relation to the kind of interaction controlling the structure. First, we will review shortly some interactions that may be relevant for interfaces and the corresponding SEs. Further, we will discuss the evolution of the SE concept. 2.1.3. Interfacial interactions relevant for structure elements 2.1.3.1. Ab-initio calculations of interface structure. Consider interactions between electrons and nuclei in an interface. In such a case, the interface structure is calculated ab initio and SEs are electrons and atomic nuclei [28,29]. This approach has particular advantage for intercrystalline interfaces, since the interatomic potentials for interactions in most cases are not known a priori. The crucial question is what kind of bonding of the two adjoining crystals determines the structure? Assuming a compound crystal 1 with atoms A and B joined to a compound crystal 2 with

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atoms C and D, there are several possible types of bonds across the interfaces. The crystal 1 can be A terminated, B terminated or terminated by a plane with both A and B atoms. The crystal 2 can be C or D terminated or terminated by a mixed plane. In the case of mixed planes, the two crystals can be situated one opposite to the other in such a way that the A±C and B±D or the A±D and B±C bonds form. Furthermore, the type of bond may vary along the interface. To limit the necessary calculations, it is useful to combine ab-initio calculations and microscopic observations. High Resolution Electron Microscopy (HREM) combined with Electron Energy Loss Spectrometry (EELS) and computer simulation techniques may lead to identi®cation of the atomic bond across the interface, as for instance, identi®cation of the V±Mg bond at the V/MgO interface by Ikuhara et al. [30]. However, more direct is the Z-contrast technique, which permits the chemical analysis of single atoms at the interface [31]. The alternative method is also the Field Emission Microscope (FIM) combined with a mass spectrometer [32]. In the case of ab-initio calculations, the concept of structural elements is of limited help for understanding the interface structure. 2.1.3.2. Electronic structure of interfaces. The electronic structure of interfaces is the electronic structure of the thin slab of material along it or its e€ect on the electronic properties of the material. Recent results are reviewed in Refs. [1,4]. Tung [33] reviews some aspects of the relation between microstructure of interfaces and their electronic properties such as Schotky barrier heights. Niles and Margaritondo review the electronic properties of interfaces in semiconductors as a function of interface chemistry [34]. Magnetic, electronic and superconducting properties of interfaces are also considered in Ref. [35]. For instance, it is known that there is a correlation between the superconducting properties and crystallography of interfaces in high TC superconductors [36]. In the present case, the basic SEs are the defects and their interactions, higher level SEs are groups of atoms, each with distinct electronic properties, interacting with each other via electromagnetic ®elds. 2.1.3.3. E€ect of electromagnetic ®elds on interfaces. Two interfaces can interact with each other via electromagnetic ®elds. Electromagnetic ®elds play a crucial role for the design of interfaces in microchips. Such interaction leads to a decrease of performance as the system gets miniaturised [37]. There are also examples of magnetic interactions across a grain boundary. Rabkin et al. [38] have shown that close to the Curie point, wetting of GBs in Fe± 6at%Si bicrystals is suppressed (Fig. 6). This result was attributed to the magnetic interactions of the two grains and their e€ect on GB energy. There are also examples of indirect e€ect of the magnetic ®eld on GBs. Molodov et al. have shown that a magnetic ®eld drives GB migration in magnetically anisotropic bismuth [39]. Electromigration of grain boundaries in thin ®lms under strong electric ®elds is

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Fig. 6. The thermal dependence of the Zn concentration close to a GB in a Fe±5at%Si bicrystal [40± 43]. Zn di€usion along the GB causes the GB pre-wetting transition (cf. Section 3.4). The GB solubility limit [43] has a minimum at the Curie temperature. The solubility limit is a function of pressure.

a well-known phenomenon, and is attributed to the drag e€ect of the electric current on impurities segregated to grain boundaries (see review [44]). In all the above cases, the system consists of both the interfaces and the crystals. These are larger systems than the interface itself, and this topic is out of the scope of the present review. A more detailed treatment of semiconductor, superconductors and ceramic interfaces in electric ®elds is given in Refs. [4,5,25]. 2.1.3.4. Electromagnetic interactions between impurities and vacancies in interfaces. Point defects segregating at interfaces may lead to steady electrostatic ®elds. They arise in ionic materials or semiconductors owing to non-stoichiometry, segregation of impurities and vacancies or unsaturated bonds. This subject is outside the scope of the present review and is partly covered in text books [1,5,25]. However, it is clear that the electronic properties of SEs in such materials will have to be considered. Another type of interaction represents isolated magnetic impurities interacting with each other. Fig. 7 shows the MoÈssbauer spectrum of a Fe±Zr±B alloy containing both bcc-Fe nanocrystals and the amorphous phase. Its decomposition into contributions from bcc-Fe crystallites (a), crystal/amorphous interface (b) and amorphous phase (c, d) is shown. The spectrum from the interface is clearly di€erent from either the bulk or the amorphous material. Slawska-Waniewska and

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Fig. 7. MoÈssbauer spectra of a nanocrystalline Fe±Zr±B alloy acquired at 77 and 550 K. The decomposition into contributions from bcc-Fe nanocrystals (a), crystal/amorphous interfaces (b) and amorphous phase (c, d). Courtesy A. Slawska-Waniewska and J.M. Greneche.

Greneche [45] have thus shown that the interface represents a new magnetic phase. It was shown that the spectrum is in¯uenced by both the structural and the spin disorder of Fe atoms. It follows that in the case of the bcc-Fe/amorphous Fe±Zr± B alloy, SEs will be characterised by their magnetic and structural properties, and that they will interact with each other via magnetic interactions. 2.1.3.5. Image forces and van der Waals interactions. van der Waals attractive forces result from the summation of: Kesson interactions between two oscillating dipoles, Debaye interactions between a dipole and an induced dipole and London forces, between induced dipoles and induced dipoles [46]. The sum of the above forces for two interacting blocks separated by a distance d leads to the following potential energy term: Vˆ

ÿH 12pd

…19†

where H is the Hamaker constant that describes the attraction between the two materials in vacuum. Recently, the scratch technique was developed further by Venkataraman et al. [47] to measure the van der Waals adhesion energies of thin evaporated ®lms on ionic substrates. One may consider as a special case of van der Waals interaction, the image forces between a charge on the surface of an ionic crystal and a metal. In ionic crystals, atomic planes or rows can be neutrally charged or have a static charge, depending on the orientation of the plane. It can, therefore, be expected that the

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359

Fig. 8. Image forces between a NaCl crystal and a metal. (a) Faceting parallel to the {111} planes increases the image forces. (b) For the low energy {001} plane, the bonding energy by means of image forces is decreased owing to cross interactions of the same charges. For transparency, only the repulsion of the positive charges is shown.

interface plane will facet in such a way that its electrostatic energy takes a minimum. NaCl single crystals in air cleave parallel to the electrically neutral {100} planes. However, if the crystals are grown in a conducting medium, it is possible that the crystal surface facet parallel to charged planes (Fig. 8). Stoneham and Tasker [48] calculated the energy of interaction between the image charge and the charge in an ionic crystal and its contribution to interfacial adhesion. They estimated that the adhesion energy of the interface bonded by image forces might reach 1 J/m2, i.e., a value close to that of GBs in metals. They proposed the following formula for the interaction energy between a charge and its image:  2   q e1 ÿ e2 , …20† EAB ˆ 4e1 r e1 ‡ e2 where EAB is the additional bonding energy, q is the charge, r is the distance between the charges and e1 , e2 are the dielectric constants. It is seen that the image forces increase as the charge and its image approach. Fig. 8 shows that bonding is increased if the ionic crystal plane is electrically charged. The image or electrostatic forces have been studied in detail by several authors [49,50]. For a charged surface, adhesion is determined by a sum of terms as in Eq. (20), where all the A±B interaction terms have the same sign. For the (001) surface, the sum is of terms with alternate sign and must be considerably smaller. It follows that for image forces, the interface structure may optimise the energy by increasing the fraction of segments of surface where one type of ions is present and the two surfaces can approach as much as possible (Fig. 8). Therefore, segments of facets parallel to charged planes seem to be plausible SEs.

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2.1.3.6. Interatomic potentials and bond energy calculations. Extensive research is being undertaken to explain the structures and properties of interfaces using computer simulations. However, most of the studies concerned special interfaces for which the calculations are a€ordable. The computer simulations resemble the experiments to some extent. Their interpretation requires carrying out studies of interfaces as a function of some variable parameters and ultimately developing physical models that may explain the observed relationships. From the point of view of SEs, the simulation starts from atoms as elements of their structure and interatomic potentials as their interactions. The resulting structures are then interpreted as build from higher order SEs composed from several atoms. Interface structure calculations based on interatomic potentials permit to identify topologically identical groups of atoms that can be identi®ed as Structural Elements or Units. However, the information whether the SEs are independent elements of the interface ``construction'' cannot be easily obtained (cf. Section 2.1.4.1) because the correlation between topology and properties is not clear. The bond-valence sum calculations. Here we would like to draw attention to some simpli®ed models useful for the description of bonding in partially covalent materials. This is important; since for covalent bonding, the SEs may have a clear physical meaning. Pauling [51] proposed that a fraction of the valence of each atom can be assigned to each of its bonds. The valence of an atom is expressed by the equation: X  r0 ÿ rij  , …21† exp Val ˆ 0:37 i where Val is the valence, rij are the bonds lengths and the summation is over the number of bonds and r0 is a constant characteristic of the elements in the bond [52±54]. Browning et al. [55] have shown the usefulness of the bond valence sum calculations for describing the structure of interfaces in SrTiO3. The structure of the GB was simulated in such a way that Val, according to Eq. (21), is as close as possible to the average of the bulk. Tight binding approximation and the degree of covalence. For a compound with mixed ionic±covalent bonding, the degree of covalence of the bond, the interplanar spacing and elastic constants can be derived quite simply [56] based on the tight binding approximation [57,58] where only the interactions of the sp3 hybrids are considered: p 1 3 jpi jh2 i ˆ jsi ÿ 2 2 p 1 3 hpj, hh1 j ˆ hsj ‡ 2 2

…22†

Let us denote the energies of s and p electrons of a free atom by es and ep , respectively. The energy of a hybridised electron of the atom A is:

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eA h ˆ

A eA s ‡ 3ep : 4

361

…23†

The analogous equation holds for the B atom. The covalent energy VC is the energy of interaction of the sp3 hybrids along their longer axis. VC ˆ ÿhh1 jHjh2 i ˆ ÿ

  p 1 Vsss ÿ 2 3Vsps ÿ 3Vpps , 4

…24†

where Vklm ˆ Zklm …hÿ =md 2 † are the matrix elements for interactions between the orbitals of the coupled atoms and Zsss ˆ ÿ1:4; Zsps ˆ 1:84; Zpps ˆ 3:24 and Zppp ˆ ÿ0:81 [58]. According to Baranowski [56]: VC ˆ cjheh ijSo ,

…25†

where c is an empirical parameter, So is the overlap term hajbi and heh i is the weighted average of the cation and anion hybrid energy: heh i ˆ


1ÿ A nA eh ‡ nB eBh , 8

…26†

where nA and nB are the number of electrons associated with the cation and anion, respectively. The c value for C, Si, Ge and Sn rows is 2.3, 1.45, 1.33 and 1.12, respectively. For compounds cij ˆ …ci cj †1=2 : The bonding energy is: EAB ˆ

B ÿ
1=2 eA h ‡ eh ‡ V C2 ‡ V I2 ‡So VC : 2

…27†

The ®rst term is the average cation and anion hybrid energy. The second term depends on the covalent energy (Eq. (25)) and ionic energy VI: VI ˆ

B eA h ÿ eh : 2

…28†

The third term is the repulsive overlap energy. The covalent interaction is proportional to VC 11=d 2 , while the repulsive term depends on the distance as follows: VC S11=d 4 , where d is the interatomic spacing. The hybrid covalency is: aC ˆ ÿ

VC V C2 ‡ V I2


1=2 :

…29†

Therefore, the degree of covalence is the ratio of the covalent bonding energy to the geometrical average of the polar and covalent bonding energy. It can be easily calculated if the energies of the p and s electrons in free atoms are known. Minimising the bond energy, Baranowski [56] obtained the following simple term for the bond length of all tetrahedral compounds:

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ÿ dˆ ÿ

2Z0 hÿ 2 =m


1=2

k 2 heh2 i ÿ 4V I2


1=4 ,

…30†

where Z0 ˆ 4:373 for the sp3 bond. For non-tetrahedral compounds, the appropriate value of Z0 has to be chosen. m is the electron mass and for hÿ 2 =m ˆ 7:62 eV AÊ2, d in AÊ is obtained. One of the important properties of covalent materials is that in interfaces, the co-ordination number is preserved. This leads to the concept that bonds are equally important members of the SEs as atoms (Section 2.1.4.2). The reason of the preservation of the co-ordination number is the high energy necessary to bend the bonds. The small shear distortion e in the zinc-blend structure is connected p with bending of bonds by y ˆ 2=3e: The resulting change of the hybrid covalent energy is: dVC ˆ …1 ‡ l †VC y 2 ,

…31†

where l ˆ 0:738 is a constant depending on the Harrisson matrix elements [58]. The ®nal expression for the change of the bond energy is [56]: 2 dEAB ˆ …1 ‡ l †VC aC2 e 2 : 3

…32†

Further, it is possible to correlate the covalency factor with the shear force constant C11 ±C12 : p   3 1 3 3… VC aC 1 ‡ l † ÿ jVppp j 3 , …33† C11 ±C12 ˆ 4 d 2 The bulk modulus B is [56]: p   2 3 7:8 1 VC a3C ‡ 2 : Bˆ 3 d d3

…34†

All these values can be calculated starting only from the Harrisson's matrix elements and energies of the s and p electrons as well as the empirical k constant. The relation between the shear modulus and energy to stretch and bending interatomic bonds is useful for the considerations of mis®t localisation in interfaces (Section 2.4) and makes it theoretically possible to calculate the shear, Young and bulk modulus of interfaces. 2.1.3.7. Packing principle. In metals, there is a clear correlation between the GB free volume and its energy [59]. The GB free volume or excess volume is the di€erence in volume of a body with the GB and without it for a constant number of atoms. The crucial role of free volume as a structural parameter of GBs was experimentally demonstrated by Lojkowski and Otsuki using high pressure techniques (Section 3.1.2). The free volume concept has lead Bernal to propose a Structural Units model

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for liquids [60]. He studied hard sphere models of liquids and identi®ed a ®nite number of clusters (polyhedra) of hard spheres such that between the spheres there is no empty space for another sphere. The spacing between atoms is not strictly de®ned. This concept was extended to the case of GBs [61]. A number of polyhedra were identi®ed for GBs in metals and called Structural Units [62]. In the case of metals, the above SUs may be regarded from two points of view. They are ``packing SUs'' as they help to densely pack the atoms in the GB. On the other hand, if the atoms are packed in dense clusters, the system has less energy, so such polyhedra may serve as a ``glue'' for the adjoining grains. However, the free volume issue may lead to another approach to SUs. It is known that GBs parallel to low index planes (or planes of large interplanar spacing) are more densely packed than GBs on random planes or high index planes [63]. The same argument holds for the comparison of tilt and twist GBs, the latter having higher free volumes. Therefore, the packing principle leads to the conclusion that SUs are segments of GBs parallel to low index planes and with low index atomic rows parallel to each other. 2.1.3.8. Strain ®elds. Strain ®elds describe interactions of groups of atoms. In fact, a strain ®eld u is a continuous function, and a continuous function is de®ned over some space. This implies that if strain ®eld is the main interaction in the system, the SEs are groups of atoms. If the concept of a SE has to be meaningful, there must be some border between SEs where the derivative du=dx has a higher value than within the SE. Consider, for example, a dislocation. The strain ®eld rises continuously up to the dislocation core where it levels up. We may say that the matter around the core is build of SEs of the perfect lattice. At the border of the core, there is a change of the du=dx gradient and a new type of SE starts, characteristic for the core. 2.1.3.9. Sti€ness of structural elements. As discussed above, if SEs are elements of the structure, they must possess a given sti€ness in relation to the acting force. Otherwise, the SE is solely a topological concept. Let us de®ne the sti€ness of an SE as follows: 2nDw ˆ GSE , e2

…35†

where GSE is the sti€ness of the SE, e is the relative displacement of one of the members of the SE, Dw is the change in density of energy in the SE and n is the number of members in the SE. The above equation was obtained simply by calculating the work to pull one member at a distance u from the equilibrium position. The strain is e ˆ u=O1=3 : One can assume that the sti€ness of the SE is a function of the co-ordination number of the displaced atoms. GSE ˆ

n G, n0

…36†

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Fig. 9. The structural units of Bishop and Chalmers [64]. Reprinted from [64], copyright 1968, with permission from Elsevier Science.

where G is the bulk shear modulus, n is the co-ordination number within the SE, n0 is the co-ordination number in the bulk. The above reasoning may lead to the conclusion that when SEs are sti€, the atoms may participate in interactions only in forms of groups or segments of closely packed structures: atomic rows, planes. This leads to an idea of collectivity coecient, which is related to the sti€ness of the SEs. 2.1.4. Evolution of the concepts of structural units and structure elements 2.1.4.1. Structural units as compact groups of atoms. Bishop and Chalmers [64] analysed theoretically the structure of non-relaxed tilt GBs; and found out that for short period GBs, the steps on compact atomic planes of the two crystals form a repeating pattern of interacting ledges, as shown in Fig. 9. They were ®rst to call the repeating patterns Structural Units and used the terms Structural Units-ledges. It was shown that all the tilt GBs for a constant tilt axis are built from mixture of SUs of the closest periodic GBs and the proportion of SUs varied continuously with misorientation (Fig. 9). The above approach tacitly assumes that relaxation of the GB structure introduces only a small correction to the GB energy, so that the SUs for non-relaxed GBs are relevant for

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the relaxed GB structure. The geometrical theory permitting a development of that concept was presented by Pond [65]. Computer simulations of GB structure permitted to consider separately two types of GB relaxation: rigid body relaxation leading to relative positioning of the two crystals and local relaxation of atoms to minimum energy positions. Weins et al. [66±68] have shown that the rigid body translation accounts for a considerable relaxation of the GB energy and that the relaxed structure of short period boundaries displays topologically identical repeating groups of atoms. GBs of larger period can be regarded as a mixture of such groups from the nearest short period GBs. The model also included description of GB faceting so that each facet contains SUs from low energy GBs. Sutton and Vitek [69] analysed the calculated relaxed structures of GBs for a constant tilt or twist axis and a wide range of misorientation angles. They found delimiting or favoured GBs, where only one type of polyhedra formed by atoms can be found. They called these polyhedra as Structural Units (SUs) as well. As far as the question whether SUs are also SEs is concerned, the answer is not obvious. The conceptual di€erence is that SUs can be treated just as a description of the interface topology. The SEs, on the other hand, are structure elements that interact with each other. This is the strict de®nition of SEs as members of a structure (Section 2.1.2). Since the sti€ness (Section 2.1.3.9) of the topological SUs was not analysed, their role as SEs is not clear. It is not clear whether to consider these units as ``packing elements'' or ``gluing elements''. The polyhedra embedded between the two crystals are necessarily stretched. Distortion does not change the topology of the SU but changes its energy. If SUs can distort easily, they represent just a way to pack the atoms, without relevance for the GB energy (Section 2.1.3.8). Such SUs cannot be treated as SEs. SUs of packing type are relevant only from the point of view of paths for di€usion of atoms and sites for segregating atoms. Recent research of SUs includes GBs in alloys and intermetallic materials, where the chemical composition of SUs has to be taken into account as well as their topology. In such a case, the richness of possible topological SEs is greatly increased. For example, Shamasuzoha et al. [70] have found in the S5 symmetrical tilt GB of an Al±5wt%Mg alloy new SUs compared to pure Al, which are stabilised by Mg. Yan and Vitek [71] have studied the SUs in weakly and strongly ordered intermetallics Cu3Au and Ni3Al. It is clear that in compound materials the topological richness of possible polyhedra is larger than that in metals, and a wide research ®eld is open. Champion and Hagege extended the SUs study to heterophase interfaces [72], where they used the ledge-structural units approach as introduced by Bishop and Chalmers [64]. According to the ledge-units approach to SUs, the asymmetric GBs parallel to low index planes are in a natural way special interfaces, since there is no principal topological di€erence between ledges in a symmetric and asymmetric GB. The asymmetric interfaces are just di€erent from the symmetric ones by the fact that in the ®rst case ledges are equally numerous on both surfaces, whereas in the

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Fig. 10. SEs in a symmetric S9 398 {221} h110i GB in silicon. The six-membered ring is a unit of a twin GB. Given is the coincidence S, the misorientation angle, the GB plane and the tilt axis [82]. Courtesy S. Pennycook.

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Fig. 11. The S13 {510} h001i symmetrical tilt GB in Si [82]. Courtesy S. Pennycook.

367

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Fig. 12. Maximum entropy ®ltered Z-contrast image of a S ˆ 85, 258 {920} (001) symmetric tilt GB in SrTiO3 bicrystal and the SU identi®ed in this GB [82]. Courtesy S. Pennycook.

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369

Fig. 13. Maximum entropy ®ltered Z-contrast image of a 458 [001] tilt GB in SrTiO3 bicrystals and the SU identi®ed in this GB [80,82]. Courtesy S. Pennycook.

latter case they are only on one surface. This extension of the SE-ledges model to asymmetric GBs permitted Lojkowski et al. [73±76] to interpret the rotating spheres experiments in grain boundaries. 2.1.4.2. Structural elements in covalent materials. In covalent materials, SEs take the form of rings of atoms. The structure of the low energy coincidence GBs in silicon [77], silicon carbide and SrTiO3 displays SEs having ®ve-, six- or sevenmembered rings of atoms. The most important feature of SEs in covalent materials is that the co-ordination number 4 is preserved except for the various car-

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Fig. 14. Structural elements for tilt GBs in SrTiO2 [82]. Courtesy S. Pennycook.

bon hybridisations [55,78±81]. This seems to be the basic di€erence between SUs in covalent materials and metals, where this rule does not hold. Fig. 10 shows SEs in a symmetric S9 GB in silicon. The six-membered ring is a unit of a twin GB. The ®ve- and seven-membered rings are in fact a dipole of dislocations with a non-zero total Burgers vector. Fig. 11 shows SEs in a S13 symmetric tilt boundary in Si. The SEs marked 2 and 2', 3 and 3' are dislocations dipoles compensating each other, i.e., they are not structurally necessary and cannot be predicted based on crystallography. This adds a new dimension to the SE model, since the set of SEs cannot be predicted based only on crystallography of the interface. In tilt GBs in the SrTiO3 spinel, Pennycook et al. [55] found similar SUs as observed in SiC or Si. Figs. 12 and 13 show the maximum entropy ®ltered Zcontrast images of GBs in SrTiO3 bicrystals and the corresponding SEs. All the [001] tilt GBs are composed of SEs from the S17 (410), S5 (310) and S5 (210) symmetrical GBs and of the S1 (001) and S1 (011) GBs (Fig. 14). An important feature of these SEs is that atomic rows can be occupied by cations and anions at every second atomic place. The term wrong bond means wrong neighbour compared to the bulk crystal. The spacing of some planes in the GB shown in

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371

Fig. 15. Model of the S13 grain boundary made of steel spheres in an alternating magnetic ®eld. The SUs of that GB are outlined [91].

Fig. 13 is smaller than in the case of bulk. Hence, it is possible for a covalent GB to have negative free volume. Kohyama [84] analysed the lengths of distorted bonds in computer simulated low energy GBs in Si and SiC. The bonds in Si were stretched by several percent and the angles between them distorted up to 238. Since in GBs of covalent materials the co-ordination number is preserved, the idea arises to consider bonds equally important members of the structural units as atoms. This may be a fruitful idea as shown by Lu et al. [85±87] who studied the pressure e€ect on the migration of the amorphous silicon±silicon interface. The observed acceleration of the migration rate could be understood according to the Spaepen±Turnbul [88,89] dangling-bond boundary migration model.

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2.1.4.3. Steel spheres and bubble raft models of grain boundaries. It is interesting to note that SUs were identi®ed during studies of hard sphere or bubble raft models of GBs with rather unrealistic interatomic potentials. For instance, Ishida observed SUs in GBs formed in bubble rafts [90] where the interaction potential has a very simple form: contacting bubbles attract each other, non-contacting do not. On the other hand, Pino Plasencia et al. [91] studied models of GBs produced by introducing steel spheres between two glass plates (Fig. 15). The plates were situated in a variable magnetic ®eld. The induced electric currents in the spheres caused a repulsion of the spheres and the external frame kept them together. Such a model permitted to identify SUs in GBs similar to those observed during computer simulations. The observation of SUs for such variety of interatomic potentials, and in particular for the case of no attraction at all, raises the question whether SUs have a physical meaning from the point of view of energy. Pino Plasencia et al. [91] have also found other characteristic features of GBs, which were known from extensive computer simulations of GBs and TEM investigations: faceting of GBs parallel to closely packed planes, asymmetric GBs, curved GBs. These results strengthen the view that some structural features of interfaces are just governed by the principle of dense packing of atoms, which is the principle of the bubble raft models of crystals and the SUs are topological constructions describing the coordination of atoms in their clusters as in the Bernal model of random close packing of spheres [60]. 2.1.4.4. Structural units and multiplicity of structures. The question of unique description of the GB structure has arised after the computer simulations of Vitek et al. [92], followed by HREM studies [93] of GBs in metals indicating that there are GBs where for constant crystallographic parameters at least two structures have equal energy. It is interesting to ®nd out what are the properties where the two GBs are di€erent from each other. 2.1.4.5. Structural units and structural elements. In the light of the above considerations, it seems that there is no evidence that the topological SUs identi®ed during many computer simulations and HREM observations play the role of SEs. It can be safely stated, however, that they play the role of ``packing units'', i.e., simple ways of arranging atoms in GBs, exactly as the Bernal polyhedra are ways to arrange spheres in a random closely packed structure. On the other hand, it will be shown later (Section 2.6) that the SEs, which were discovered by Bishop and Chalmers (Section 2.1.4.1) and called Structural Unitsledges, play the role of real building blocks for GBs and interfaces. Presumably, SUs-rings found in covalent materials (Section 2.1.4.2) play the same role. In summary, SEs or SUs must be de®ned together with the interactions between them. If the interactions are between atoms, then SUs are just topological descriptions of atoms arrangements.

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Fig. 16. A polytype GB in aSiC. On the left side is the 15R and on the right side the 6H polytype. The arrows indicate the interface. The structure consists of periodically repeating segments of the single crystal and of the S9 GB [83]. Courtesy Y. Ishida.

2.2. Relation between geometry and properties of interfaces 2.2.1. De®nition of interface periodicity What is the interface periodicity? When do interfaces have a periodic structure? Assume that S represents a structural feature of the interface in a small neighbourhood {x} around the point x. The de®nition of interface structure is given in Section 2.1.2. For instance, if the SEs are atoms, S(x) may represent the set of vectors connecting the atoms in the neighbourhood {x}. Including also the phase space, {x} may represent the atomic positions and vibrations. This is a restrictive description of the structure SR. There can also be less restrictive de®nitions: the structural feature S(x) could be the density of matter in a

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neighbourhood {x}, the number of closest neighbours, the set of SEs. We de®ne an interface as periodic if there is a vector L such that: S…x ‡ L† ˆ S…x†:

…37†

When S(x) represents the set of vectors connecting the atoms in the neighbourhood {x}, for each atom with co-ordinates x, there is an interface atom with co-ordinates x + L: x 2 finterface gˆ)x ‡ L 2 finterface g:

…38†

Fig. 16 shows an example of such an interface. An important consequence of the above equation is that the structure of the periodic interface does not change when a rigid body translation vector T = L relative to the other translates one of the crystals. As discussed in Section 2.2.3, the periodicity of the interface depends on the periodicity of the surfaces of the adjoining crystals. The situation is relatively simple when joining two crystalline surfaces that have a common period. However, in the general case, the ratio of the periodicity of both surfaces can be irrational. Here, we will present a simpli®ed approach to irrational interfaces based on simple geometrical considerations. A comprehensive treatment of irrational interfaces is given by Sutton [94]. If the deviation from periodicity is small, we may replace Eq. (37) by a more general equation: S…xL † ˆ f…x†
S…L ‡ x ‡ x…x†† ‡ g…x†,

…39†

where f(x), g(x) and x(x) are small perturbations and xL ˆ x ‡ L: The interface is approximately periodic if: S…xL †1S…x†:

…40†

It follows that whether an interface can be assumed periodic or not is arbitrary and depends on the accuracy of measurements of the characteristic structural features. 2.2.2. Periodicity of global and local properties Let us consider a property F of the interface. In some cases, the given property can be obtained by averaging of the property F(x) over a small surface of the interface: … F…x† ds , …41† Fˆ s s where s is the surface of the interface. An example of such a property is the di€usivity of the low angle boundary assuming that all the di€usion goes along the cores of dislocations. In this case, F(x) is the local density of dislocation cores.

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The total di€usivity results from summation of di€usivity of all cores and depends on the number of cores per unit length of the interface. Another example is the interface energy according to the broken bonds concept, where the energy of the interface is approximately the sum of the energy of all the broken bonds [95]. In this case, S(x) is the number of broken bonds in the neighbourhood {x} and F(x) is the energy of a broken bond. We will de®ne F, obeying Eq. (37), as a property of local character (FL). There are properties that have a meaning only for a suciently large number of atoms in each crystal. We will call them global properties (FG). For instance, such a property is the grain boundary mobility or the susceptibility of the grain boundary to sliding. Global properties would also be there where long range interactions are involved. Such an example is the energy of an interface according to the Franck and Read model [96]. The relevant structural feature S(x) is the density of dislocations per unit length of the interface. For an interface build from just a few dislocations, its energy can be approximated as the sum of the energy of the dislocations. However, the energy of a dense array of dislocations is considerably less than the sum of the energy of the individual dislocations. In that case, the interface energy cannot be expressed as an integral of the energy of each neighbourhood. The di€erence between the above properties is that FL is connected with a position dependent structural feature S(x) and is a function in real space: FL ˆ F…x†,

…42†

whereas FG is a function in the space of RBT vectors: FG ˆ F…T †:

…43†

As a consequence, for a periodic interface, a local property is periodic both in the real space: FL …x ‡ L † ˆ FL …x†

…44†

as well as in the space of translation vectors:

Fig. 17. Schematic representation of an interface periodic both in the real space and in the space of RBT vectors. L is the period and T is the rigid body translation vector.

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FL …T ‡ L† ˆ FL …T †:

…45†

For a global property, FG …x† has no meaning and only Eq. (45) is valid. Fig. 17 illustrates an interface periodic both in the real space and in the space of RBT vectors. It follows from the above that for an interface where some structural feature is periodic, the related local property is periodic both in the real and translations space, whereas the global property is periodic only in the translations space. The question arises where is the boundary between ``global'' and ``local''? A global property concerns a structure composed of many atoms. So, for a given property, the question is what is the smallest relevant group of atoms? To some extent, the limit is set by the minimum physically meaningful length of the translation vector and by the in¯uence of external surfaces or other interfaces. For the case of grain boundary migration, the limit would be the smallest bicrystal where surface e€ects would not in¯uence migration. Essentially, this is the limit that separates the nanocrystals from polycrystals, since nanocrystals can be de®ned as those polycrystals where the fact that the grain size is small in¯uences the properties of interfaces. For an approximately periodic global property: F…T † ˆ f…T0 †
F…L ‡ T0 ‡ x…T0 †† ‡ g…T0 †,

…46†

where the meaning of the perturbation functions f, x, g, is same as that in Eq. (39). However, the interface is approximately periodic if: F…T †1F…T ‡ L†:

…47†

In such a case, the interface is periodic under certain conditions (cf. Section 2.2.3.2). We will consider four cases as far as the relation between periodicity of interfaces and their properties are concerned: . . . .

Periodic interface. Interface with periodic spots. Approximately periodic interface (cf. Eq. (46)). Non-periodic interface.

An interface with periodic spots is de®ned as follows: there is a vector L and a neighbourhood x0 such that for x 2 fx0 g: ÿ
…48† F…fx0 g† ˆ F L…fx0 g† and the neighbourhoods fx0 g and L…fx0 g† do not overlap. Here L…fx0 g† is a neighbourhood obtained by translating by the vector L all the points fx0 g: Eq. (48) means that there are some repeating spots with some identical structural feature; however, between these spots the structure is not repeating in a periodic way. The above situation may correspond to an interface at high temperatures, where periodically situated closely packed spots remain ordered, while the less densely packed neighbourhoods are disordered in a random way (cf. Section

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377

Fig. 18. (a) Symmetrical S13 (310) tilt GB in NiO. Courtesy K. Merkle [97]. The approximately identical segments of the structure are outlined. (b) Schematic representation of the structure with the irregularly repeating periodic domains each of length LD :

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Fig. 18 (continued)

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379

Fig. 19. Energy dependence on translation. (a) Periodic interface, (b) non-periodic interface, (c) realistic case, (d) non-equilibrium case.

2.2.3.1) or to an interface where there are several possible structures of similar energy [92]. Fig. 18a shows an approximately periodic GB in NiO [97]. There are periodic domains where the atomic con®guration approximately repeats every eight planes. Close examination shows some small di€erences in the positions of atoms in each periodically repeating neighbourhood (Fig. 18b), which corresponds to the approximately periodic structure, with periodic spots separating non-periodic sections (cf. Section 2.2.3). From basic considerations, it also follows that a periodic interface may have di€erent properties than a non-periodic one. Consider, for instance, the energy as a function of the RBT vector. In general, there can be three types of dependence of energy on the translation: a periodic function (Fig. 19a), a constant function (Fig. 19b,c) and a non-periodic function (Fig. 19d). The last situation is possible in principle, for instance, if there is a gradient of chemical composition, strain or a curvature along the interface. However, the gradient may induce some relaxation processes. For simpli®cation, let us neglect this non-equilibrium state. So, we are left with the periodic and constant functions. Firstly, it can be seen that the periodic interface has a richer spectrum of properties than the non-periodic one, since the properties are a function of translation. In further considerations, the interfaces where the properties are constant in space:

F…x† ˆ const,

…49†

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F…T † ˆ const,

…50†

will be called non-periodic. It can also be expected that the length of the period is important for the interface property. Assume that, in an experiment, we apply a varying RBT to one of the crystals and measure each time the energy. Since the structure is periodic, the energy will be a periodic function of the RBT vector. Say Dg is the maximum possible di€erence of energy between two interfaces of the given type. Let us further assume that @ 2 g=@ T 2 1const, which means that the low frequency shear modulus of the interface is constant during the RBT operation (Section 2.4.2). In other words, the energy changes smoothly with a change of the RBT. For the above assumptions, the derivative of the energy versus translation vector has an upper limit:

Fig. 20. An interface formed by two 1D crystals. (a) Two separate 1D crystals, (b) 1D non-relaxed interface, (c) 1D relaxed interface.

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@g Dg < @T L

381

…51†

Therefore, for long period interfaces, the energy is a slowly varying function of the RBT vector. The formalism of the SDWs (Section 2.3.6) shows in a strict way that interfaces of short periodicity can be connected with lower energy than long period interfaces. 2.2.3. One-dimensional model of an interface In order to illustrate qualitatively the main ideas related to periodicity of the interfaces, we shall use a simpli®ed one-dimensional model as follows. Let us consider two one-dimensional crystals A and B, each consisting of points (Fig. 20a). The lattice constant of one of the crystals is a and that of the other is b. To simplify further equations, we assume a < b. The interface is created by merging the two sets of points into one set lying on a straight line (Fig. 20b). The length of the interface is X. The co-ordinates of the points before relaxation are x r : Assume that the interface structure can relax to minimise the energy by RBT and local relaxation. The RBT is the only Independent Variable (IV) of the interface. Local interactions (Fig. 20c) shift the points to new positions described by the function u(x): u…x† ˆ x ÿ x r :

…52†

If u…x† ˆ 0, there is no relaxation. Such a situation corresponds to low value of the localisation parameter p ( Section 2.4). Furthermore, we introduce the notion of tolerance x: It is de®ned as a relative displacement of a point around its position that does not in¯uence relevant properties of the interface or is too small to be detected. In particular, if mis®t adjustment leads to strain ®elds, a mis®t less than x causes no strain. We will discuss the situation when a/b is rational or irrational, sti€ bonds within the crystals …u…x†  0† and strong inter-crystalline bonds …u…x†6ˆ0† as well as the ideal crystal …x  0† and the realistic crystal …x6ˆ0). Tolerance is essentially an expression of the shape of the interatomic potential. Assume that moving an atom causes a small energy variation, comparable to the energy of thermal vibrations DE1kT: In such a case, the movement of the atom has no e€ect on the interface energy and properties. Assuming that the property considered is energy, the following expression for x is obtained: xˆ

1 kT , a @ E=@ x

…53†

where @ E=@ xjxˆx 0 is the derivative of the interatomic potential against a coordinate of an atom or an angle between atomic bonds. 2.2.3.1. Rational a/b

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Sti€ crystals: u…x†  0. If a tolerance factor is allowed: x > 0, the period of the interface is not exactly determined. A simple calculation shows that the error limit for the periodicity is: ax a ‡ b : …54† DL ˆ jb ÿ aj 2 When DL > L, any length can be a periodicity. In such a case, the concept of periodicity has no physical meaning. It follows that for large tolerance, the physical meaning of periodicity must be treated with care. As already mentioned, tolerance is increased as high temperatures or when the interatomic potential becomes more ¯at. The stochastic factor. For a deterministic relaxation, a periodic initial structure results in a periodic relaxed structure. The situation is di€erent for a non-deterministic relaxation. Assume that owing to entropy e€ects, there is a spectrum of possible relaxation of neighbourhoods. This agrees with the calculated, by Vitek et al. [92], GB structures, where it was shown that a number of GB structures may have the same energy. To take into account this randomness, we introduce the random function osc(x ) where osc takes values from ÿ1 to 1, and the amplitude of the oscillations, A. The co-ordinate of each point after relaxation is therefore: x 0 ˆ x r ‡ u…x† ‡ osc…x†A:

…55†

To express quantitatively the e€ect of randomness on the periodicity, we introduce the stochastic factor fs : It determines the probability that there is a point at the distance L from the given point. This probability depends on the amplitude of vibrations and the tolerance x: Taking the simplest model: what is the probability to hit a target of area A ‡ xa, using a bullet of size xa one obtains: xa : …56† fp ˆ xa ‡ A In the Einstein approximation (independent vibrations of atoms), A is proportional to the temperature: p A ˆ c T, …57† where c is a constant. It follows from Eq. (56) that the fraction of points in coincidence decreases with increasing amplitude of oscillations: SEffective ˆ fs SA, B

SEffective ˆ

x p SA, B , x‡c T

…58†

…59†

where SEffective is the inverse fraction of coincidence points under the condition

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Fig. 21. Non-periodic interface with periodic domains each translated relative to the other by an irrational translation Dx:

that the probability that some nodes that are at coincidence in the ideal lattice is of ®nite probability. If we treat A as a contribution to the error in determining positions, then at the limit A ‡ 2xa ˆ a=2, the vibrations and error exceed the interatomic spacing and there are no coincidence points. Nevertheless, the periodicity of the interface is not changed. Say, there is a function of the type Dx ˆ X0 sin‰2p…x ‡ A osc…x†=L†Š which describes the interatomic distances for atoms of the opposite crystals. Its Fourier transform describing the interatomic distances retains the Dirac delta form: d…L†, even though there is the stochastic factor osc(x ). The fact that periodicity is not lost is not surprising, since the periodicity depends on the bulk crystals mutual orientation, and this is temperature independent. Another question is however the e€ect of periodicity on the properties, which may become negligible. It follows from the above that with increasing temperature some features of periodic interfaces may become non-periodic. Interfaces with high S values may undergo this transformation at lower temperatures than low S interfaces (Eqs. (56) and (58)). The stochastic factor relates to the thermal displacements of atoms in the interface region. This information is accessible though X-ray di€raction techniques [98]. Fitzismmons et al. [98] have shown experimentally that these displacements are di€erent from those in the bulk. It has been shown that for the S13, 22.68 [001] twist GB in gold, at 298 K, the mean square displacement is 50% higher than that in the bulk. The thermal expansion coecient perpendicular to the GB was three times larger than that in the bulk. 2.2.3.2. Case of irrational a/b Realistic crystal: x > 0. A realistic crystal is ®nite. Therefore, a ®nite strain is necessary to bring to periodicity a crystal with irrational a/b ratio. Furthermore, the periodicity of a ®nite crystal is not determined with in®nite accuracy, since the Fourier transformation of a ®nite periodic function gives a range of values for the frequency. We de®ne periodic domains, as in Fig 18b. Within each domain, the

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Fig. 22. 2D model of a 458 h001i twist GB. The small circles represent atoms of the two crystals. Two nearly identical neighbourhoods for two di€erent tolerances are shown. Neighbourhoods are considered identical if dx < x, where dx is the di€erence in relative positions of atoms. The large grey circles represent approximately identical neighbourhoods for tolerance x=a < 0:03 (dark grey) and x=a < 0:06 (light grey). For x=a < 0:03, the nearly identical neighbourhoods do not form a periodic pattern. For x=a < 0:06 there are nearly periodic domains of various lengths LD, each with two or three nearly identical neighbourhoods.

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385

Fig. 23. A non-periodic grain boundary in iron [83]. Courtesy Y. Ishida.

period L should ful®l the following condition: for each n < nD , there is a point of co-ordinates x such that: jx ÿ nLj:

…60†

The length of the domain is: LD ˆ nD
L:

…61†

The length of the domain is de®ned by the condition that after the border of the domain is reached, the points at the positions x and nD L are at a distance larger than x: At this place, instead of forcing the points into periodic positions, another periodic domain may appear, translated by a distance Dx relative to the ®rst domain (Fig. 21). Fig. 18b shows the situation for the example of the real GB represented in Fig. 18a. The arrangement of atoms in the GB core is copied and the periodic domains are highlighted. The new domain may also have a di€erent periodicity. It is clear that to make the domains periodical, some strain is necessary. It can be described in terms of arrays of interface dislocations. The length of the domains and their existence depends on the tolerance x (Fig. 22).

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Fig. 24. A GB in gold parallel to the {111} plane [100]. Courtesy Y. Ishida.

In terms of the SDW theory (Section 2.3.6), the above description is equivalent to de®ning a series of vectors qi , i ˆ 1, 2, 3, . . ., which lead to a continuous adjustment of the two mismatching lattices. The condition K ÿ t ˆ q1 holds at ®rst relaxation, at the border of the ®rst domain. In®nite interface. An in®nitely long interface with irrational a/b cannot be periodic. This will be shown at the end of the present paragraph. However, it can be constructed from a number of domains where there is some approximate periodicity. The domains are non-coherent with each other; i.e., the functions describing the positions of the points are each at a di€erent phase from the other. The structure of the interface will be determined by the set of translations fDxg between domains (Figs. 18, 21 and 22). For an in®nite interface, this is an in®nite dense set 0 < Dx < ja ÿ bj: Any RBT of one crystal relative to the other leaves this set unchanged. Therefore, the interface structure is invariant relative to the RBT: S…x ‡ T †1const…T †, and:

…62†

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F…T †1const…T †:

387

…63†

Hence, the interface is non-periodic. Non-periodic GBs have been frequently observed in TEM. Fig. 23 shows a GB in Fe with a disordered structure [83], as in the early GB models of amorphous like structure (see recent review [99]). Ichinose and Ishida have shown a number of GBs that display a disordered structure [100]. In many cases, the non-periodic GBs is parallel to special planes. Fig. 24 shows a non-periodic GB parallel to a low index plane in gold [100]. E€ect of the tolerance factor on the properties of interfaces. Consider the periodicity of an irrational interface of the length shorter than the domain length …XRLD ). Such an interface shall behave in an approximately periodic way: S…x†1S…x ‡ L2xa†:

…64†

However, the ``improvement'' of the periodicity caused by the tolerance in determining the atomic positions …ax† is at the expense of the periodicity of the property (F ) in question. In fact, from Eq. (64), it follows that: F…T †1F…T ‡ L2xa†: …65† However: @F @F @T ˆ @ x: @T @x

…66†

Therefore, if the given property is insensitive to atomic displacements in the range of x …@ F=@ x10†, it is also insensitive to small RBTs …@F=@ T10). Hence, for large tolerance factor, if there is a minimum or maximum of F for some T value, it is shallow. In other words, for a high tolerance to atomic positions, the properties of interfaces do not strongly depend on RBT and the periodicity of the properties of the interface is not evident. The same reasoning leads to the conclusion that thermal vibrations lead to a decrease in amplitude of the periodic property function. In fact, it is sucient to replace, in Eq. (66), xa by A ‡ xa; since due to oscillations, the precision of determination of the atoms positions is decreased by the amplitude A. The tolerance factor x in fact re¯ects the properties of the atomic bonding: the shallowness of the interatomic potential. Therefore, the physical importance of periodic interfaces depends on the shallowness of interatomic bonding. Qualitative conclusions considering the 1D interface

.1. Periodicity of a structural feature induces a periodicity of the connected

property. If the property is of local character, then it is periodic both in the real space and in the space of translation vectors. If the property has a global character, it is periodic in the space of the RBT vectors. 2. For crystals with very sti€ intracrystalline bonds (high covalence coecient aC † compared to the intercrystalline ones (tolerance x  0), periodicity exists only for exactly rational ratio of the lattice constants. As shown in Section 2.4.12.1, this result corresponds to the narrow range of disorientation angles and small

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Fig. 25. The NW orientation relationship (OR) in steel and brass. For both the fcc and bcc phase three atomic layers are shown, as indicated in the stacking sequence. Brokman and Ballu [104] interpreted the above OR in terms of approximate coincidence at points ABCD. Reprinted from [104], copyright 1981, with permission from Elsevier Science.

localisation parameter p. 3. For an interface with approximate periodicity, the amplitude of properties as a function of translation is smaller than for a perfectly periodic interface, in agreement with the SDW theory discussed in Section 2.3.6. 4. Stochastic structural variations described by the fs parameter cause a decrease of the amplitude of the property as a function of RBT and leads to the transformation periodic interface to non-periodic interface, which is analogous to the special 4 general GB transformation (cf. Section 2.3.5). 5. Any interface can be regarded as coincidence or near coincidence, if an arbitrary tolerance factor is introduced to discriminate whether atoms are in coincidence positions or not.

2.2.4. The two-dimensional case Consider a 2D interface generated by approaching two 2D arrays of points. The 2D interface has three Independent Variables (IV): two independent RBTs and the twist angle of one crystal relative to the other. Therefore, any property is a hyperplane in a 4D space. The twist angle de®nes the Orientation Relationship (OR) of the interface. 2.2.4.1. Parallel interfaces. Fig. 22 represents schematically the grain boundary obtained by rotating one of the crystals relative to the other by 458 around the h001i axis. The period of the crystal A in the x direction is LA ˆ 1: The period of

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389

p the crystal B along the x-axis is LB ˆ 2=2: The ratio of the above two periods is irrational. Inspection of the Fig. 22 shows clearly two characteristic features of the interface: (a) The parallelism of closely packed rows of the two crystals. (b) Some repeating pattern corresponding to nearly identical arrangement of points. If a tolerance ax in determining the positions of the atoms is allowed, then some of the neighbourhoods can be treated as approximately identical. As seen from Fig. 22, the number of such neighbourhoods depends on the tolerance (Section 2.2.3.2). For narrow tolerance, the distances of nearly identical neighbourhoods are not equal and the interface is non-periodic. For larger tolerance, some domains can be distinguished where the neighbourhoods repeat at an approximately constant period (Fig. 22). We will call shortly an interface with the parallelism of closely packed rows of the two crystals as a parallel interface. The interface can be treated as a coincidence one, but this requires arbitrary assumptions for the permissible tolerance. Therefore, as long as a physical criterion is not present, any parallel interface can be treated as coincidence interface, but this is a purely geometrical description. The parallel interface corresponds to low energy according to the SDW theory, as derived in Section 2.3.6. Symmetrical orientations are likely to correspond to special properties of the interface, for instance, maxima or minima of energy, in the same way as short periodicity. This follows from basic considerations. Assume that the interface is symmetric in such a way that a rotation ‡Dy causes the same change of structure as the rotation ÿDy: As a consequence: F…y ‡ Dy † ˆ F…y ÿ Dy †:

…67†

Hence, F can be written in series as follows: F ˆ F0 ‡ ay 2 ‡


and for Dy ˆ 0, F…y† takes a maximum or minimum. A more extensive treatment of the subject can be found in the paper of Kalonji and Cahn [101]. 2.2.4.2. Parallel or atomic matching orientation relationship?. Consider a ®nite interface between two rectangular di€erent lattices with lattice constants …ai , bi †, i ˆ 1, 2: The situation may correspond to a thin plate precipitate or an epitaxial layer. We assume the irrational ratio of the lattice constants: a1 =a2 , a1 =b2 , b1 =a2 and b1 =b2 : For a very thin precipitate or epitaxial layer, the interface must be coherent. Coherent means that closely packed atomic rows are parallel. What is the selection rule for the OR? (a) Maximise the density of approximate CSL nodes or ``0'' lattice nodes? (b) Bring closely packed directions to parallelism? (c) If parallelism, then what rows will be parallel?

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Fig. 26. The low energy cube±cube OR found by Erb et al. for the Ag/NaCl system. The interface is parallel to the {001} plane [105]. Reprinted from [105], copyright 1982, with permission from Elsevier Science.

The selection rule (a) is based on the assumption of best atomic matching at the interface plane in terms of maximum density of the ``0'' lattice nodes [102,103]. One can interpret the Nishiyama±Wasserman and Kurdjumov±Sachs ORs for fcc and bcc precipitates brass and steel along this line of reasoning [104]. Fig. 25 shows an interpretation of that OR in terms of the CSL idea. It was arbitrarily assumed that at the points ABCD, the distance between atoms from the two crystals is small and the two lattices are in approximate coincidence. Hence, it was proposed that the existence of this near CSL lattice is the reason why the precipitates take the mentioned NW ORs.

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However, assuming an arbitrary tolerance, one can interpret any OR in terms of the CSL model. For arbitrary tolerance, parallelism of low index atomic rows leads to near coincidence relationships, since the chances that for some atoms of the adjoining crystals coincide is greater if the closely packed rows are parallel than not. Fig. 26 shows schematically the structure of the low energy OR for the Ag/ NaCl interface found by Erb et al. [105]. This cube/cube OR (the matrix transforming one lattice into the other is a pure dilatation) displays almost periodic arrangements of atoms if an arbitrary tolerance factor is introduced, in the same way as in Fig. 22. However, Erb and Gleiter concluded that it is the tendency to align parallel the low index rows of atoms which determine the low energy of that interface and not the optimisation of atomic matching. 2.2.4.3. Invariant line concept and in¯uence of strain on the orientation relationship. A selection rule for ORs in precipitate and epitaxial systems is the overall strain energy. Dahmen [106] based on a review of OR in precipitate systems concluded that the selection criteria for various parallel ORs like the NW, Kurdjumow and Sachs (KS), Bain, Greninger and Troiano, Pitsch, etc. can be explained in terms of the invariant line concept. He observed that the various OR systems are obtained by a small twist rotation operation followed by pure strain. Both can be optimised so that there is an invariant line along which neither the precipitate nor the matrix is strained. The invariant line is de®ned as follows. Say M is the matrix transforming one lattice into the other: 

a1 b1





m11 ˆ m21

m12 m22



 a2 , b2

…68†

where mij are elements of the matrix M. The points along the ``invariant line'' ful®l the equation: V ˆ MV,

…69†

where the vector V points along the invariant line. An example of that approach is the work of Luo and Weatherly [107], who have shown that ORs for Cr rich precipitates in a Ni±Cr alloy obey at the same time the invariant line criterion and the ``O'' lattice criterion of Bollman. However, Dully [108] has shown that the presence of the invariant line need not be equivalent to the lowest elastic strain OR. This viewpoint was further strengthened by Kato et al. [109], who calculated explicitly the strain necessary to bring to coherency a thin epitaxial ®lm on a (001) substrate. They have shown that for a number of epitaxial systems, the OR corresponds to a minimum elastic energy, but not necessarily to the invariant line criterion. Furthermore, the OR for precipitates may depend on kinetic factors. For instance, during nucleation of the precipitate, a small atomic cluster cannot know

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Fig. 27. A plane matching interface in AlN. Courtesy Y. Ishida.

that a few atomic planes away there is a next CSL node. Therefore, it just grows along closely packed atomic rows. Therefore, there is a range of competing explanations for the given OR observed in real systems: (a) shortest periodicity, (b) symmetry, (c) parallelism, (d) invariant line or plane, (e) lowest strain, (f) kinetic factors. 2.2.4.4. Periodicity in only one direction. In the 2D case, it is possible that there is a common period only in one direction. Such situation occurs for tilt GBs or in hexagonal lattices [110,111]. Plane matching ORs belong to that category of ORs [112]. The plane matching interface can be regarded as an example of interface periodic in one direction but non-periodic in the perpendicular direction. Fig. 27 shows a plane matching GB in AlN. There is a continuous plane across the interface. However, the planes are twisted relative to each other around an axis perpendicular to these planes and there is no periodicity perpendicular to the plane of the photograph. 2.2.4.5. E€ect of rigid body translation on the grain boundary energy and non-equilibrium interfaces. The presently considered 2D system can be regarded as a system where the twist angle is the only variable and the two RBT vectors adjust to the minimum energy. However, externally applied ®eld stresses or ®eld stresses of dislocations may prevent the system to adjust the RBT value. The strong e€ect of

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Fig. 28. Displacement position map for the S11 {131} symmetrical grain boundary in the bcc lattice. The numbers in horizontal row indicate the RBT measured in lattice constant units. The numbers in vertical row indicate the GB plane displacement. The symbols are explained in the text. According to Paidar [114].

RBT on the GB energy is already known from early computer calculations of Weins et al. [68]. Such a situation can be called ``interfacial non-equilibrium''. In Section 3.4.3, results of Inoko et al. [113] will be presented, who have shown that such strain ®elds can even cause GB melting. The structure of GBs as a function of two independent variables: the RBT vector and the GB plane translation was studied recently in detail by Paidar [114].

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Fig. 29. The stable and metastable structures of the (013) symmetrical tilt GB in the bcc lattice. Squares and circles represent atomic positions on two adjacent (001) atomic planes. The rectangles represent the structural units. The four GBs correspond to di€erent values of RBT parallel to the GB plane. The energy of each GB is: (a) 690 mJ/m2, (b) 900 mJ/m2, (c) 1310 mJ/m2, (d) 2200 mJ/m2. According to [116]. Courtesy J. Erhard.

Fig. 28 shows the map introduced by Paidar, the ``Displacement Position Map'' (DPM) Ð a way of presenting the GB crystallography, where the RBT and GB displacement perpendicular to its plane are the independent parameters. The ®gure represents the DPM of the S11 {131} symmetrical tilt GB in the bcc lattice [114]. The symbols M, G, C, o indicate structures estimated to be at least metastable according to computer simulations and the lines joining them Ð paths of possible phase transformations. The dashed line indicates the unit cell of the DPM, including all possible structures. Fig. 29 shows four structures of the S5 (013) GB in a bcc metal according to the computer simulations of Paidar et al. [115] and Erhard and Trubelik [116]. The structures di€er by the RBT parallel to the GB. The structure of lowest energy is 690 mJ/m2, whereas the structure of highest energy is 2200 mJ/m2. The di€erence in energy of the two GBs is over 300%. Therefore, Paidar and Vitek [117] stated that not only misorientation but also the RBT de®nes the special GBs. It follows from this statement that the RBT may cause the special±general GB transition.

Fig. 30. (a) Twist rotation of two crystals, (b) its representation as a sum of in®nitesimally small rotations and in®nitesimally small rigid body translations by Du ˆ R dy:

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2.2.4.6. Relation between the e€ect of rotation and translation on the energy of the interface. It is interesting to consider the relation between the e€ect of twist and translation on the energy or other properties of the interface. Fig. 30 shows that any twist rotation causes in each small neighbourhood, a translation rdy: It follows that there must be a relation between the e€ect of twist rotation and translation on the energy of the interface. The following statements will be demonstrated. If the energy (or other property) is constant as a function of the twist angle for any position of the origin of co-ordinates, the function g…T† is constant:  2    2 @ g @ g ˆ 0 ˆ) ˆ0 : …70† @T 2 @y 2 Eq. (70) follows from the analysis of the values of the full di€erentials: dg ˆ

@g @g @ 2g @ 2g @ 2g dy ‡ dr ‡ 2 dy 2 ‡ 2 dy dr ‡ 2 dr 2 : @y @r @ y @r @r @y

…71†

dg ˆ

@g @g @ 2g 2 @ 2g @ 2g dx ‡ dy ‡ dx dy ‡ 2 dy 2 : dx ‡ 2 2 @x @y @x @ x @y @y

…72†

If dg  0 in Eqs. (71) and (72), all the derivatives are zero: @g=@x ˆ 0, @g=@ y ˆ 0, @ 2 g=@ x 2 ˆ 0, @ 2 g=@y 2 ˆ 0, @ 2 g=…@ x @y† ˆ 0 and g…T† ˆ const: The above considerations concerned the situation when g…y† ˆ const for a range of y: Can any conclusions be drawn for the case that these functions have maxima or minima? In that respect there is one more relationship: if the two crystals are mutually locked at a RBT vector value corresponding to minimum g…T†, g…y† displays a minimum as well: @ 2g @ 2g < 0ˆ ) < 0: @T 2 @y 2

…73†

Here is the derivation. When the energy is a function of local type, the energy change during rotation is the sum of the contributions from each neighbourhood of surface ds, which undergoes translation and rotation: …… dg
ds, …74† DE ˆ s

Fig. 31. Schematic presentation of a row of atoms of one crystal and of the potential on the surface of another crystal. (a) The rows are situated at maximum energy positions both from the point of view of RBT T and of twist rotation y: (b) The rows of atoms are situated in a minimum energy position both with respect to T and y: This con®guration will corresponds to the lock-in model, for LEORs. (c) Intermediate position. The direction of the energy gradient with respect to y is unde®ned.

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where dg is the local change of energy due to rotation and rigid body translation and s is the area of integration. If RBT corresponds to maximum of g, then any small twist rotation causes a decrease of energy. Fig. 31a illustrates that situation. The orientation shown in Fig. 31 represents a symmetrical orientation of two identical cubic lattices situated in such a way that there is a maximum of g…T†: For any rotation, each neighbourhood moves in the low energy direction. Therefore, for any ds, dg < 0: Hence, DE < 0 in Eq. (74) and the OR corresponds to a maximum of g…y†: @ 2g @ 2g < 0ˆ) 2 < 0, 2 @T @y

…75†

which was to be demonstrated. Fig. 31b represents the situation when all the atoms of one crystal are locked in energy valleys on the surface of the other crystal and vice-versa. Hence, any small rotation would bring about an increase of energy …dg > 0†: Therefore, an energy minimum with respect to RBT corresponds to an energy minimum with respect to y: @ 2g @ 2g > 0ˆ) 2 > 0: 2 @T @y

…76†

Fig. 31c shows the intermediate state. Assume that all the points are on an energy slope with respect to RBT. In that case, the rotation would bring about a decrease in energy of one part of the interface and an increase in energy of the other part of the interface: …… … … …77† dg
ds ÿ dg
ds , DE ˆ s1

s2

where s1 is the surface corresponding to energy increase and s2 to energy decrease, respectively. The division of s into s1 and s2 depends on the RBT vector. Hence, the derivative @g=@ y depends on the actual value of the RBT vector. Therefore, in a non-equilibrium system, the derivative @g=@ y can take a

Table 1 Relation between the e€ect of translation and twist on the interface energy Condition

Result

g ˆ const…T† for a range of T (non-periodic interface) g ˆ const…y† for any centre of rotation and a range of y g(T) takes a minimum g(T) takes a maximum T takes a non-equilibrium value

g ˆ const…y† for any centre of rotation g ˆ const…T† (non-periodic interface) g…y† displays a minimum g…y† displays a maximum @ g=gy is not determined

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Fig. 32. A triple junction where three GBs meet, for each of which there is a strong dependence of energy on the RBT vector. It is clear that it is impossible for each GB to reach the low energy RBT value.

range of values for the same OR. Table 1 illustrates the relation between the sensitivity of the energy to RBT and twist. 2.2.4.7. Triple junctions. In this section, we will discuss the contribution of triple junctions to grain boundary non-equilibrium. As discussed in Section 2.2.4.5, the energy of interfaces strongly depends on the RBT vector. For bicrystals, the RBT vector value can be adjusted so that the GB energy is minimised. However, this is not the case for tricrystals. Figs. 32 and 33 show that three GBs cannot simultaneously adjust the RBT vector to the minimum energy value. Otherwise the junction would be symmetric. Hence, close to the triple point at least one of the GBs will be in the state of non-equilibrium. For long GBs, a strain may permit to adjust the RBT value to the low energy. But for a short GBs like in nanocrystals, this may be impossible. Hence, the triple points may be the reason for the observed high energy of GBs in nanocrystalline materials [118,119]. Fig. 33 shows a HREM photo, made by Tanaka and Kohyama [120], of a triple junction in a S27 grain boundary in b-SiC. The triple junction is asymmetric, although the macroscopic OR for the system is symmetric. The plausible explanation is that the RBT of the three crystals cannot correspond to low energy

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Fig. 33. A HREM photo of two triple junctions in b-SiC. The junctions are between the S9 and S3 GB. The triple junction is asymmetric although the macroscopic OR for the system is symmetric. The plausible explanation is that the RBT of the three crystals do not match. In other words, the SUs of the GBs cannot match at the junction in such a way that the junction is symmetric. Courtesy K. Tanaka [120]. Published with kind permission from the Institute of Physics Publishing Limited.

at the same time. Therefore, one of the crystals is forced into a non-equilibrium structure. 2.2.4.8. Interface faceting. The interface may locally change its inclination. According to the considerations for the 2D case (Section 2.2.4.1), there can be several criteria for ``attractive'' planes: . Short periodicity. . Parallelism of closely packed planes or small interface thickness. . Symmetry.

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Fig. 34. A tilt of one grain can be represented as a sum of ``Coble creep'' events. Material must be removed on the compressed side and added on the elongated side.

Furthermore, there can be non-periodic GB planes in CSL misorientations. This is the case if the normal plane is irrational or its position is described by irrational numbers. However, for any non-periodic plane, there is a close periodic plane, similar as close to a non-periodic orientation there is a periodic one (cf. Section 2.2.4.1). For a given CSL value, there are several periodic planes: these are the planes with indexes ‰hi , ki , li Š such that: q q …78† h12 ‡ ki2 ‡ l 12 ˆ h22 ‡ k22 ‡ l 22 ˆ S, where the index i ˆ 1 or 2 indicates the surface of one of the two adjoining crystals. For these planes, there is a common periodicity of the two adjoining surfaces. Of course, the trivial case ‰h1 , k1 , l1 Š ˆ ‰h2 , k2 , l2 Š corresponds to the most investigated symmetrical tilt GB. It is striking that despite the obvious character of Eq. (78), it is only since about 15 years that the importance of asymmetrical CSL GBs was recognised [8,63,97,100], even though there is much experimental evidence for the low energy of GBs parallel to low index planes. For instance, Ishida and Ichinoise observed that about 30% of GBs in gold thin ®lms were parallel to the {111} plane. The relatively frequent observation is that both GB planes are parallel to compact {111} and {100} fcc lattice planes [121,122]. 2.2.4.9. Interface migration and Coble creep. A ``Maxwell Demon'' sitting on a migrating coincidence GB would see a periodically repeating structure. In mathematical terms, the energy of the interface would be an oscillating function of the migrated distance H. However, if the amplitude of energy oscillations would be

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high, then migration by movement of GB steps would be favourable. For such migration mechanism, most of the interface surface remains all the time in an energy valley. Another possibility is motion of DSC dislocations parallel to the interface. In fact, GB migration mechanism involving steps and dislocations was frequently observed. The e€ect of RBT perpendicular to the GB plane on its structure and energy can be understood by considering an imaginary ``Coble-creep experiment''. In such experiment, a tensile stress is applied to the bicrystal perpendicular to the GB plane. The crystals are displaced; and by di€usion, new material is supplied to or removed from the GB region. For instance, the material between the old and new position of the GB, in Fig. 34a, can be supplied by di€usion. For a periodic, in 3D, bicrystal lattice, the structure of the interface should repeat periodically with time. Alternatively, the creep process would take place by climb of edge DSC dislocations and movement of GB steps. On the contrary, for non-periodic misorientations, neither migration nor creep shall be associated with a periodic change of the structure, energy or other properties. 2.2.4.10. Tilt component of the misorientation of the crystals. The question arises whether the conclusions about the relations f@g=@ y $ @ y=@ Tg hold for the case that the rotation is a pure tilt one, where the rotation axis is parallel to the interface plane. From the previous paragraph, it is seen that during the ``creep experiment'' with non-periodic interface, the energy is constant. However, a tilt of one crystal relative to the other is equivalent in each place to such a creep (Fig. 34). We can carry out a similar reasoning as in Section 2.2.4.6. When the tilt angle a is changed, the separation d of the two crystals is changed in each neighbourhood in the interface by the value Ra, where R is the distance from the tilt axis. On one side of the tilt axis material is removed and on the other side inserted. For a nonperiodic GB, the energy g ˆ const …d† (cf. Section 2.2.4.9); and therefore, it does not depend on the tilt angle. On the other hand, if the energy is constant as a function of tilt angle, it must be constant as a function of d. Furthermore, if g displays a minimum as a function of a, it means that the integral: … Eˆ

s

@g @d @a6ˆ0, @ d @a

…79†

which means that @g=@ d6ˆ0 and the energy is a function of separation of the crystals. If any change of separation brings about an increase of energy, then a minimum with respect to separation is equivalent to a minimum with respect to tilt angle. In the same way, if the energy is a function of separation, it must be a function of tilt angle. There is one exception, however, that the structure is asymmetric and a positive change of energy with increase of d on one side is

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compensated by a negative change of energy with decrease of d on the other side of the tilt axis. This conclusion has a considerable importance for the structure of dislocations in GBs. In fact, a plateau in the g…y† function implies that the low frequency shear modulus of the interface GI is either small or zero (Section 2.4.2). Therefore, the cores of the SGBDs are delocalised (Section 2.4.7). This relation was ®rst given by Gleiter [123]. 2.2.4.11. Qualitative conclusions considering the 2D interface. In addition to the conclusions for the 1D case, the following results have been obtained: 1. There is a relation between periodicity and sensitivity of properties to the misorientation angle. The misorientation angle is ``locked'' in a minimum energy valley if the two crystals are locked with one another in a minimum energy RBT. For non-equilibrium value of the RBT, the function @ g=@ y is not strictly de®ned. 2. The 2D interface may have a di€erent kind of periodicity in two independent directions. 3. It may be impossible for the three GBs to reach equilibrium at triple junctions. 4. Asymmetric periodic interfaces are from the point of view of coincidence criteria of low energy identical to the symmetrical ones

2.2.5. The three-dimensional case The OR of a GB in a crystal in three dimensions is fully described by 12 independent variables. This is seen from the following considerations. Assume that we create an interface in the following way. First, two crystals are each cut along a plane. Each plane is de®ned by three independent variables: . Unit vector of the normal to the plane (2). . Place of the cut (1). This makes 2  3 = 6 independent variables. Subsequently, the crystals are joined and the structure of the interface is permitted to relax. The relaxation will follow a trajectory in a six-dimensional space. This number is explained as follows. Assume the following sequence of relaxation: . Two planes get parallel (2). . Their distance (or interface thickness) equilibrates (1). . The two crystals may twist one relative to the other to reach the equilibrium twist angle (1). . The crystals slide one on the other to reach the equilibrium RBT (2). This makes 12 crystallographic parameters describing the interface. However, in experimental or theoretical considerations, some of them can relax to the minimum energy value and, therefore, are a function of the other ones. Usually, the parallelism of the two planes is taken as obvious. However, close to a

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Fig. 35. Spacing of structural dislocations in twist GBs in gold as a function of misorientation [124,125]. Published with kind permission from Taylor and Francis Ltd.

dislocation core, this need not be true. Further, the spacing of planes, or in other words the interface thickness, is assumed to take an equilibrium value. The RBT is generally assumed to correspond to the low energy or to have no e€ect on energy. Nevertheless, the above assumptions need not to be valid in highly strained interfaces. Another common representation of the OR is in terms of misorientation. Assume that we have the initial orientation in space of the two crystals. Now a rotation around an axis will ®x the relation between the directions in both crystals, a RBT will position the centres of the crystals relative to each other and a cut parallel to one plane will de®ne in each crystal the adjoining surfaces. This leads to consider ®ve independent variables as important: . Misorientation: the angle and axis of rotation (3) and . the inclination of the interface plane (2), whereas . the point of intersection of the normal to the interface by each crystal (2) and . the RBT vector parallel to the interface (2) are assumed to be automatically equilibrated in the same way as the thickness of the interface. The independent variables are also called Degrees Of Freedom (DOF) [95]. However, some care must be taken for this terminology. The system can have,

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Fig. 36. Computer simulation of the structure of a non-relaxed (a) and relaxed (b) structure of a S5 GB. The relaxed structure has a lower symmetry than the relaxed one [126]. Reprinted from [126], copyright 1983, with permission from Elsevier Science.

say, n Degrees Of Freedom (DOF) which is equivalent to the statement that the system has n Independent Variables (IV). However, a DOF is a dimension not a variable. Since the term Independent Variable is well known in physics and mathematics, there in no real need to introduce this new term DOF. A more detailed treatment of interface crystallography can be found in the books of Randle [3] and Sutton and Ballu [1]. 2.3. Special crystallographic orientations Consider the function g…y† that represents the energy of an interface as a function of a crystallographic parameter, say, the misorientation angle. For instance, there are energy minima for some special misorientations. The Low Energy OR will be called LEOR. Consider an interface whose crystallographic parameters do not correspond strictly to the LEOR. For instance, the interface plane is inclined to the lowest energy plane or the misorientation angle does not

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Fig. 37. Plane view of a schematic perfectly periodic S5 GB. The light grey circles represent the crystal below the interface plane and the dark grey circles represent the crystal below the interface plane. The twist angle is 36.88 and the twist axis is [001] Ð perpendicular to the interface plane.

correspond to the lowest energy. The di€erence between the lowest energy angle and the real angle is called disorientation. The dislocations accommodating the misorientation di€erence are called Structural Grain Boundary Dislocations (SGBDs). Fig. 35 shows the results of TEM investigations of the structural dislocations in twist GBs in gold thin ®lms. It is seen that there are special misorientations with a large spacing of dislocations. This is in analogy with the low angle GB model of Franck and Reed [96]. In this case, the SEs are the segments of the perfect crystal and the dislocations cores.

Fig. 38. (a) The bicrystal lattice for the S5 GB. Black and white circles correspond to the two interpenetrating lattices. Two equivalent GB positions are shown as well as the cell of the CSL lattice. A DSC vector is also shown. (b) The e€ect of translation of the white lattice by a DSC vector. The bicrystal lattice is invariant with respect to the translation after shifting the origin of co-ordinates by the vector T. The original GB structure is recovered after GB migration over a distance H.

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2.3.1. CSL and dislocations models of grain boundaries Fig. 36a shows schematically a twist GB obtained by rotating one crystal relative to the other by 36.88 around the h001i axis. Triangles and squares represent the atoms of the crystals A and B, respectively. The boundary plane is parallel to the plane of the drawing (plane-view). We denote the inverse of the fraction of points in coincidence for the crystals A and B, respectively, as SA , SB : In the present case, SA ˆ SB ˆ S ˆ 5: The square represents the smallest periodically repeating unit of the interface. Fig. 36b shows the relaxed structure of the S5 GB [126]. As discussed in Section 2.2.3.1.2, for a deterministic relaxation, the relaxed structure has the same periodicity as the non-relaxed one. In this case, the relaxed structure has a higher symmetry than the non-relaxed one (Section 2.2.4). It was observed experimentally that some cusps on the energy versus misorientation curve correspond to low S GBs. (cf. Sections 2.6 and 3.1.1). The concept was re®ned by Bollman [127], who introduced the ``0'' lattice. The ``0'' lattice points are points invariant with respect to the operation transforming one crystal lattice into the other: Ix…0 † ˆ Ax…0 † ,

…80†

where x…0† are the ``0'' lattice points, A is the misorientation matrix and I is the identity matrix. Bollman assumed that the ``0'' points are points where the ``atomic matching'' of the two crystals is best, since the transformation matrix centred at such points causes no action. The ``0'' lattice points need not be real lattice points. The ``0'' lattice theory permitted to predict the dislocations in interfaces, low energy planes and misorientations. However, all the purely geometrical theories including the CSL model, were insucient to fully describe the real systems [9,74]. Important consequences for the structures and properties of interfaces follow from the concept of the bicrystal lattice, which is the superimposed lattice of the two crystals. Fig. 37 shows the bicrystal lattice of the 36.88 [001] tilt S5 GB. The white and black circles represent nodes of the lattices of each crystal. This resulting pattern is called ``dichromatic pattern''. The GB can be created by removing black circles from above the GB line and white circles from below. If the interface plane crosses the nodes of the CSL lattice, its structure is periodic. Further, if the GB plane is translated upwards or downwards, its structure changes in a periodic way. The vectors joining the nodes of the bicrystal lattice form the DSC (Displacement Shift Complete) lattice. A translation of one of the crystals by a DSC vector does not change the bicrystal lattice. Only the origin of co-ordinates changes position. Fig. 38a shows the smallest DSC vector for the S5 GB. After a DSC vector shift, the initial positions of atoms at the interface plane can be recovered by moving the GB plane. This is shown in Fig. 38b. Therefore, a perfect SGBD must have a Burgers vector which is a DSC lattice vector and will be

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409

p Fig. 39. A DSC dislocation with a Burgers vector 1=2 5 [110] in a S5 symmetrical tilt GB.

usually connected with an interface step (Fig. 39). More details about the geometry of the bicrystal lattices can be found in Refs. [1,128]. It was soon realised that what is important is the planar density of coincidence sites [129]. Fletcher [130] went further this track using the Fourier analysis of the interaction of the two crystals. There is the usual analysis of the energy in terms of the strain energy accumulated in the bulk and the mis®t energy caused by disregistry of the lattices. Both the mis®t energy and the elastic energy can be developed in Fourier series in space with frequencies determined by the reciprocal lattice vectors of both the crystals. Only summation parallel to the interface is taken into account, since the strain ®eld u(z ) decays exponentially with distance from the interface: u…z† ˆ u…z0 † exp… ÿ K
z †,

…81†

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where z is the distance from the interface and K belongs to {K} Ð the set of reciprocal lattice vectors of the crystal [130]. According to the Fletcher's treatment, if the two lattices are incommensurate, the interaction terms of the crystals average to zero. However, if both crystals have a common period, negative terms for the energy appear. The Fletcher approach was further developed by Sutton [94,131], who analysed the interaction of two in®nite planar lattices assuming pair interaction potentials. He obtained the following result for the interaction energy EAB of the two lattices: EAB ˆ

1 X V…K, z† exp…iK
x †, A1 A2 K

…82†

where A1 and A2 are the surfaces of the elementary cells of the two lattices, x is the spatial co-ordinate in the interface plane and VK …z† is the Fourier transform of the interatomic potential: … …83† VK …z† ˆ V…x, z † exp…iK
x † dx: Sutton has shown that only the transform of the interaction potential for a single pair enters in Eq. (84). From Eq. (84), it immediately follows that if the lattices are incommensurate, the only element of the set fKg is K ˆ 0 and EAB  0; and the energy is independent of the relative position of the two lattices. This result agrees with the conclusions obtained from the simple model of the interface in Section 2.2.3.2. The result of the above calculations is that only exact coincidence of the two lattices brings about an energy minimum. This is contrary to experiment. A more general solution of the problem of two mismatching lattices can be obtained in the framework of Static Distortion Wave Ð SDW (Section 2.3.6). 2.3.2. Angular distance between coincidence grain boundaries The set of twist angles for which the interface is periodic is dense in the real space, i.e., for any non-periodic misorientation y0 and small angle dy, there is a periodic orientation within the angular range ‰y0 , y0 ‡ dy]. Periodicity of an interface must in¯uence the properties of non-periodic interfaces with a close misorientation. This can be seen by considering a function S…x, y, y†, which describes the structure of the interface (cf. Section 2). Assume that for a given y, we have the equality S…x, y, y† ˆ S…x ‡ L, y, y†; and from experiment, we know that S…y† is a continuous function of y: Therefore, for a small change of y, the change of S is also small. Hence S…x, y, y ‡ dy†1S…x ‡ L, y, y ‡ dy†: As mentioned in Section 2.2.3.2, short period interfaces may have special properties. Assume that there is an interface with a given twist angle, which has no periodicity or has a long period. As shown in Section 2.2.3.2, under some strain, the interface may become periodic or at least periodic in some domains. However, there is a choice of various periodic interfaces. The question is which

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Fig. 40. The disorientation range as a function of the interface period. Each point of the gird represents a symmetrical tilt GB obtained by a mirror operation where the mirror line is that one that crosses the given point.

interface will be approached? This depends on how high S values retain the special character. The general rule is that close to a low S value, there is no short period grain boundary for a relatively large angular interval. This is shown in Fig. 40. The points on the grid represent schematically symmetric and periodic GBs. The representation is as follows: to generate the GB, one should draw a line from the origin of co-ordinates across a grid point (at distance L from the origin of coordinates). Say, the line is inclined to the horizontal axis by the angle f: If it crosses a point at distance L, it becomes the grain boundary line of a GB with misorientation 2f and periodicity L. The point with co-ordinates …m ˆ 2, n ˆ 1† at distance L1 represents the S5 GB. Note that 5 ˆ m 2 ‡ n 2 : This is a special case of the Ranganathan equation [132]: S ˆ n 2 ‡ Wm 2 ,

…84†

where W ˆ k 2 ‡ l 2 ‡ q 2 , and [k, l, q ] is the rotation axis. The GBs with periodicity closest to the S5 GB are the S1, S5, and S13 and the S5 ones. The S1, S5 and S13 GBs indicated by arrows are in the disorientation range from y1 ˆ 78 (the S13 GB) to 648 (the S1 GB). If the disorientation is limited to y3 ˆ 58, then the period of the GBs of closest disorientation increases. Within the y3 limit are the S29, S261, S137, S29, S289, S65, S233 and the S109 GBs. All of them except the S29 GB have the period close to L2, where L22 ˆ 245: The above example shows that close to each short period GB, there is a range of disorientations where there are no other short period interfaces. In fact, assume p that the point (m, n ) represents a periodic interface. The period is L ˆ m 2 ‡ n 2 :

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Fig. 41. Range of disorientation angles for which special properties of interfaces can be found as a function of S: According to Shvindlerman and Straumal [134]. Reprinted from [134], copyright 1985, with permission from Elsevier Science.

The closest GB is obtained by putting m ˆ m21 or n ˆ n21: For the extreme case: m ˆ m ÿ 1 and n ˆ n ‡ 1, the related change of disorientation is: p 2 : …85† Dy ˆ L2 It is seen that the disorientation to the closest GB of similar periodicity decreases with increasing L. The angle de®ned by Eq. (85) is the ``catching radius'' of short p period interfaces. Since L ˆ S, the Brandon [133] criterion for the ``catching radius'' of coincidence boundaries is obtained: const Dy ˆ p : S

…86†

Brandon assumed const ˆ 158: The above angle can be called ``catching radius'' of CSL GBs, since it predicts which GBs will have SEs from the given CSL GB. The higher is the S value, the smaller is the catching radius. The above tendency is commonly observed, as shown in a literature survey of Shvindlerman and Straumal [134] (Fig. 41).

Fig. 42. The interfacial properties of intercrystalline interfaces as a function of the scaled separation a  based on the universal binding energy relation. The ®gure shows the dependence of strain to separate the crystals, the interface energy and entropy, on the scaled separation, respectively.

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The same rule is observed when GBs are not classi®ed according to CSL but according to the spacing of planes parallel to the GB. This leads to classi®cation of GBs according to the spacing of planes parallel to the interface (Section 2.3.4). There is an inverse correlation between the periodicity of symmetric GBs and the spacing of planes parallel to them [63]. The above reasoning can be extended to incommensurate interfaces. Recently, GoÂmez et al. [135] worked out the concept of points of good ®t between two incommensurate lattices, as points halfway between the nodes of the two interpenetrating lattices, under the condition that the distance between the points does not exceed some limit. They provide arguments that their de®nition of good matching is advantageous over the Bolmann's ``0'' lattice theory. With such de®ned approximate coincidence points, the Brandon's criterion will simply apply separately to each sub-lattice. 2.3.3. Universal equation of state for interfaces The above geometrical criteria for special interfaces must be substantiated with physical arguments. Recently, a general equation for the relation between interatomic spacing and energy in solids was proposed [136], which supports the view of low energy for interfaces with large interplanar spacing and dense packing. It was shown (Fig. 42) that the following equation describes the energy/ interatomic spacing relation for a very wide range of materials: E ˆ E0 exp… ÿ x  †… ÿ 1 ÿ x  ÿ 0:05x  †,

…87†

where E0 is the equilibrium binding energy and x  is a scaling parameter. The above equation expresses the total energy of the system, not the pair-wise interaction energy, and re¯ects the universal character of bonding due to overlapping tails of wavefunctions. It does not apply to ionic bonding between ®lled shells (e.g. alkali halides or van der Waals bonding). Fecht [137] has shown that the same equation applies to interfaces. From the above equation, the pressure required to separate two crystals was obtained: s…a † ˆ

E0 exp… ÿ a †… ÿ 1 ÿ 0:15a ‡ 0:05a †, l

…88†

where l is characteristic screening length and a ˆ d=x  is the interplanar spacing in dimensionless units. It is seen that the strain to separate the two crystals increases with decreasing separation of the two crystals (Fig. 42). Furthermore, it was shown that there is a relation between interplanar spacing and interface vibrational entropy as follows [138]:   d , …89† DS…d † ˆ wcv ln ae where w is the GruÈneisen parameter, cv is the heat capacity at constant volume and ae is the equilibrium separation of the atoms. Interfaces with low interplanar

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415

Fig. 43. The hierarchy of GB planes according to Paidar [140]. The dependence of the interplanar spacing d (in a/2 units) on the misorientation angle y for the [ÿ101] rotation axis for the fcc lattice is shown. Having a given plane, one recognises the planes higher in the hierarchy by climbing up along the plotted lines. For instance, the (373) GB is composed of SEs of higher GBs in the following way: (111) + (010) = (131), (111) + (131) = (121), (121) + (131) = (373). Therefore, the (373) GB consists of two units of the (131) GB and one unit of the (111) or of three (111) units and two (010) units. As proposed in the present paper, the e€ect of temperature on the hierarchy is also indicated. As temperature increases, GBs of low interplanar spacing disappear as steps in the hierarchy. It is proposed that this is the nature of the special±general GB transformation. Reprinted from [140], copyright 1987, with permission from Elsevier Science.

spacing have the lowest vibrational entropy. Since cv 13kT, and w12:5 [139], the following result obtains for the entropy contribution to the interface free energy [137]:   a : …90† DFT …a† ˆ ÿ3kTw
ln ae Therefore, the energy of interfaces of various interplanar spacing have a di€erent temperature dependence. Those with low interplanar spacing are the ones that have the lowest energy at high temperatures. The above result was obtained without referring to any details of the interface structure or type of bonding. This fundamental result is very important for further considerations. 2.3.4. Hierarchy of special grain boundaries according to the spacing of planes parallel to the interface Paidar [140] introduced a classi®cation of interfaces according to spacing of planes parallel to the interface plane (Fig. 43). The idea is that GBs with larger spacing provide SEs for GBs with smaller spacing. To determine the SEs for a given GB, one has to simply climb up the

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Fig. 44. Disappearance of energy cusps at high temperatures. According to Erb and Gleiter [146].

hierarchy following the path along the slopes of the classi®cation pyramids. The ultimate decomposition is into SEs of the (111) and (001) GBs. Paidar's extended this approach to non-periodic asymmetric boundaries, where low index planes are parallel [141]. For a GB that is not of pure tilt type or with a high index tilt axis, the question is not only what is the plane of the GB higher in the hierarchy but also what is its misorientation axis. For that case, Paidar introduced the more general concept of a classi®cation tree [141]. The question is where to stop climbing up in the hierarchy for the given interface? According to Section 2.2.3.1, the longer the periodicity, the smaller the temperature at which the stochastic e€ects may cause GB disorder. Therefore, Fig. 43 indicates the order according to which with increasing temperature, GBs may cease to be a step in the hierarchy. It can be seen that at the highest temperatures only the GBs on (111) and (100) planes are the source of building blocks for GBs. According to Lojkowski et al. [73,74], in Ca and Ag SEs build from segments of (111) and (001) planes remain stable up to 5 K below the melting point. Ordering of interfaces according to the spacing of planes parallel to the interface planes was proposed by Wolf [63]. For the same spacing of planes, interfaces were ordered according to the planar density of coincidence. This is an adjustment of the concepts of Brandon et al. [129], who introduced the criterion of the planar density of coincidence points. According to Sutton and Ballu [9], the planar density of coincidence points in combination with spacing of planes seems to be the only geometrical criterion that predicts to some extent the LEORs. Sutton obtained the same result during an analysis of interaction of two crystals using a pair of interaction potential [131]. The above classi®cation has been supported by various computer simulations, and it was shown that it holds

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417

Fig. 45. The temperature of the transition ``special GB 4 general GB'', as a function of S: According to Straumal and Shvindlerman [134].

even for incommensurate interfaces [142±145]. However, the equation of state for interfaces, derived by Fecht [137,138], seems to provide the more general physical argument for the classi®cation according to spacing of planes. 2.3.5. Special 4 to general grain boundary transformation If an energy cusp disappears (Fig. 44), then according to the above GB models, the structure of the GBs being within the catching radius of the given LEOR must change. Shvindlerman and Straumal [134] analysed a large amount of data on the above transition, which are summarised in Fig. 45. As can be seen, there is a line of phase transformation separating pairs of points …S, T† corresponding to GBs with some special properties, like low energy and mis®t dislocations, from general GBs. In the light of the above considerations, the special 4 general GB transformation need not be a transformation from order to disorder. It may also re¯ect the change of the SEs in the GB. SEs that are segments of GBs with lower interplanar spacing can be replaced by SEs built from segments of GBs with higher interplanar spacing. This corresponds to climbing in the Paidar's hierarchy as shown in Fig. 43. The later formulation seems more general, since it includes both GBs and incommensurate interfaces where the concept of coincidence is arbitrary. Furthermore, it is physically explained in terms of the universal equation of state for interfaces [137] (Eq. (90)). 2.3.6. Static distortion wave analysis of periodic and incommensurate interfaces The concept of Static Distortion Waves (SDW) of Novaco and McTague [147] permitted to obtain considerable results for the theoretical analysis of LEORs of

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thin ®lms on solid bulk substrates. The theory is an extension of the Frenkel and Kontorova model of a dislocation [148]. van der Merwe [149] developed the model in a natural way to describe a thin layer of atoms adsorbed on a substrate, with an array of mis®t dislocations. Markov and Stoyanov further developed the SDW theory [150] and applied it to explain LEORs for epitaxial layers. Fecht [151] has shown that the above concepts are suitable for the interpretation of LEORs for interfaces in bulk materials. We shall present shortly the analysis of Markov and Stoyanov [150] for a monolayer of adsorbed atoms on a rigid substrate and then the extension of this concept to general interfaces [151]. Consider a monolayer of adsorbed gas on a surface of a sti€ crystal. The task is to evaluate the potential energy gain due to interactions between an adatom and a substrate and the energy cost to produce the SDW, which are periodic strain ®eld accommodating the ®lm to the substrate. It is assumed that the substrate is rigid. The energy gain for an adatom at distance z from the substrate is presented in the form: V…r, z † ˆ V0 ‡

X VK …z† exp…ir
K †:

…91†

K

The average energy per atom is: Uˆ

N 1X V…rj ‡ uj †, N jˆ1

…92†

where frj g is the set of lattice vectors of the monolayer and uj is the displacement of the jth atom from its ideal lattice site. Substituting Eq. (91) into Eq. (92), leads to the following average energy per atom of the adsorbed layer: U ˆ V0 ‡

ÿ
ÿ
1 XX VK exp iK
rj exp iK
uj , N j K

…93†

which di€ers from Eq. (82) by the strain wave term exp…iK
uj †: The strain vectors uj are considered small, which permits linearisation of the term exp…iK
uj † with respect to uj : U ˆ V0 ‡


ÿ
ÿ 1 XX VK exp iK
rj 1 ‡ iK
uj : N j K

…94†

The above equation expresses the energy of the system as a function of the displacements, for a given OR between the substrate and monolayer. The OR dependence is included in the scalar products. The strain ®eld can be again represented as a Fourier series: uj ˆ

X ÿ
u…q † exp iq
rj : q

…95†

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419

The strain ®eld contains both transverse and longitudinal modes de®ned by wavevectors qw : Minimisation of the energy leads to the following result for the longitudinal amplitude [151]: uL …q † ˆ

VK …K
qw † : O…K ‡ m†q3

…96†

1/K is the 2D compressibility related to the Lame coecients: K ˆ m ‡ l and the transverse amplitude: uT …qw † ˆ

VK …K  qw † : O
m
q3

…97†

It is seen that the mis®t is accommodated by periodic strain waves. The amplitude of the waves rapidly decays as …qw † ÿ3 with increasing wavelength. This indicates that short wavelengths contribute in a signi®cant way to the energy decrease. According to McTargue and Novaco [152], the transverse term contributes little to the total energy. We show the ®nal solution modi®ed for the case of bulk interfaces, where the energy gain on matching of the two interfaces depends on the ratio of the adhesion energy at the interface and in the bulk (Section 2.4.8). This ratio is expressed by the localisation parameter p introduced in Section 2.4. p c 0 for low ratio of adhesion energy at the interface to the cohesion energy of the bulk crystals. The adjustment of the SDW theory to include the degree of mis®t localisation ij interfaces is given in Section 2.4.8. With this correction, the ®nal solution for the energy is [152]: " XX p2 XXX 2 …K  qw † 2
VK dK, t ÿ V K dK, t‡q  UMIN ˆ V0 ‡ p 2n K t q6ˆ0 m t K # …K
qw † 2 =q4 : ‡ …K ‡ m†

…98†

{t} represents the two-dimensional reciprocal lattice in the monolayer, with basis vectors t1
b1 ˆ t2
b2 ˆ 2p and t2
b1 ˆ t1
b2 ˆ 0, where b1 and b2 are the basic vectors of the non-strained monolayer. The total energy is invariant to rigid body translation. The term V0 represents the general adhesion of the substrate and adatoms. The second term shows that if the monolayer and substrate have a common period: K = t, there is a sharp energy minimum (VK are negative). The third term shows that even for incommensurate lattices, there is an e€ect of orientation on energy. It is included in the scalar and vector products. Markov and Stoyanov [150] and Stoyanov [153] analysed several ORs for epitaxial ®rms and give examples of SDWs energy minimisation for epitaxial ORs, for the common precipitate and epitaxial layers ORs. In a real system, the d-like dependence of the wave amplitude on the di€erence

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Fig. 46. Delocalisation of a mis®t dislocation core for low p values.

K ÿ t is not realistic in the same way as the coincidence of real lattices can be considered with some tolerance factor. Furthermore, for small crystals, the wave frequencies are determined with some error. If the interatomic potential is ¯at, then although some interference of waves can take place, the energy gain is small. This issue is discussed in Section 2.2.3.2. 2.4. The localisation parameter 2.4.1. The Peierls model of a dislocation Nabarro [154] reviewed the recent developments of the Peierls model of a dislocation. An interesting review of the Peierls model is given in Ref. [156]. The essence of the Peierls model is that the core width of dislocations is the result of the balance of energy. Increasing core width causes an increase of the energy of the stretched bonds in the dislocation core and a decrease of its elastic energy. The Peierls model permits a simple analytical estimate of the core width. A lattice dislocation is represented as an in®nite sum of in®nitesimal dislocations with Burgers vector density du=dx: …1 du dx: …99† bˆ dx ÿ1 The Burgers vector density function for dislocations of two di€erent core widths s is shown in Fig. 46. The core width can be de®ned as the half-width of the du=dx function. The elastic energy of all the in®nitesimal dislocations can be integrated and the result is:

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421

Fig. 47. E€ect of the shear modulus in the core region on the deformation ®eld of a dislocation. A denotes a dislocation in the bulk and B in the Ag/Ni interface, respectively. According to Dergia et al. [155], the shear modulus in the interface is about 70% less than in the bulk. The core width of the interfacial dislocation is two times larger than for a bulk dislocation. See Section 2.5.3.2.

EEL 1a1 Gb 2 ln

  R , s

…100†

where EEL is the elastic strain ®eld per unit length, a1 is a numerical constant depending on the Poisson coecient and type of dislocation, G is the shear modulus of the bulk, and R is the crystal size. It can be seen that the elastic energy decreases if the core width increases. However, an increase in the core width causes an increase in the core energy. Assuming a sinusoidal relation between strain u and force between atoms, Peierls [156] obtained the following result for the core energy: ECORE ˆ a2 Gbs,

…101†

where a2 is a constant. The minimum energy requirement: d…EEL ‡ ECORE † ˆ 0, ds

…102†

leads to the equation for the core width: sˆb

a1 : a2

…103†

However, the above result was obtained assuming that the shear modulus in the core and bulk are equal. s ˆ Ge:

…104†

This need not be true for the case of a dislocation in an interface. The strain

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energy of a dislocation is proportional to the bulk shear modulus G, but the core energy depends on the interface shear modulus GI. For two di€erent GI values, GI1 > GI2 , the core widths are di€erent as well: s2 > s1 : The core width of a mis®t dislocation and the interface shear modulus will be introduced in the next paragraph. 2.4.2. Surface tension, surface stress and interfacial non-equilibrium To discuss the dislocation core width in an interface, it is ®rst necessary to de®ne the interface shear modulus. Cammarata [157] reviewed recently the problems related to interface stress and interface excess free energy. Already Gibbs [158] observed that the surface stress term can be introduced, which describes the reaction of a surface to an applied stress. It is di€erent from the surface energy. If a surface is created in a liquid, the high di€usion coecients permit new atoms to ®ll the newly created surface states. In that case, the surface stress s and surface energy g are equal to the work spent. The amount of reversible work dw performed to create new area dA is: dw ˆ g dA

…105†

However, if stretching the surface of a solid creates the new surface and the number of surface atoms is not altered, the work spent per unit surface area is the surface stress. The surface stress tensor sIij is expressed as (cf. [2]): sIij ˆ gdij ‡

@g , @ uij

…106†

where uij is the strain tensor in the interface. For an isotropic interface, Eq. (106) reduces to [159]: sI ˆ g ‡

@g : @u

…107†

Based on the above equations, we propose a simple de®nition of GB nonequilibrium DgN : DgN ˆ sI ÿ g

…108†

In fact, g is the energy of the interface after adjustment of the position of the atoms to the new stretched surface. If di€usion and viscous relaxation processes are active, then after a ®nite time, the surface stress and surface energy will become equal. There are other important di€erences between sI and g: Besides the fact that sI is a tensor, in general, sI ˆ sI …x†, where x is a co-ordinate in the interface plane. To illustrate this fact, imagine extending an interface with an array of dislocations pertaining to the equilibrium structure. Fig. 48 shows that the interface stress is an oscillating function of co-ordinates, since most of the work to extend the interface

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423

Fig. 48. The di€erence between the surface stress and surface energy during stretching at constant structure a grain boundary containing dislocations. The surface energy is constant and isotropic. The surface stress is a function of position and direction.

is done at dislocation cores. Fig. 48 also shows the relation between interface stress and energy; in that case, the average stress and the energy are equal. It is also clear that in this case, sI is anisotropic. For stretching the interface parallel to the dislocation lines, sI is constant; whereas for stretching perpendicular to the dislocation lines, it is periodic. It has to be pointed out that the above straining of the interface can be caused by external stresses or capillary forces, as discussed by Cammarata [157], or by internal stresses and interface non-equilibrium caused by Trapped Lattice Dislocations (TLDs). This issue is discussed in Section 3.2. 2.4.3. Interface stress at triple junctions We will show here an experimental result that illustrates both the interface stress introduced in Section 2.4.2 and the e€ect of structural units for properties of interfaces. Shvindlerman et al. [160] provided evidence that in certain conditions, the triple junctions may limit the interface mobility. Fig. 49a shows a schematic drawing of the specimens for assessment of the mobility of the triple junctions, and Fig. 49b shows the shape of the triple junction during GB migration. Two high angle GBs intersect a low angle GB. At low temperatures, the mobility of the triple junction controls the rate of the high angle GBs migration. At high temperatures, it is the mobility of the GBs that determines the

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Fig. 49. (a) The tricrystal for the measurements of the mobility of GB triple junctions [160]. (b) The angle at the tip of the tricrystal as a function of temperature. The tricrystal is made of Zn and the tilt GBs are 568h11ÿ20i, 568h1120i and 38. (c) The original pictures. Courtesy L.S. Shvindlerman.

mobility of the triple junction. Fig. 50 shows a plausible explanation for the ``triple junction drag'' when one of the GBs is a low angle GB. At ®rst glance, a low angle GB should not produce this drag e€ect, since it is of low energy compared to the high angle GBs. However, when the length of the low angle GB increases, dislocations must be generated. At this point, it is useful to recall the concept of GB stress (Section 2.4.2). Generation of dislocations causes that the stress necessary to extend the GB is a periodic function of the triple point position

W. Lojkowski, H.-J. Fecht / Progress in Materials Science 45 (2000) 339±568

Fig. 49 (continued)

425

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Fig. 50. Illustration of the generation of dislocations during migration of a triple junction joining a two general GBs and a low angle GB.

(Fig. 48). Therefore, each time a new dislocation is generated in the triple junction, there is an energy barrier to overcome. Hence, the idea of Shvindlerman et al. [160] that triple junction itself is an obstacle for GB migration ®nds support when the di€erence between surface stress and surface energy is taken into account. The above phenomenon can also be regarded as a manifestation of the fact that at the triple junctions SEs of the three GBs have to match and three RBT vectors must adjust. This may lead to strains and microscopic asymmetry of the triple junction, as observed by Tanaka and Kohyama et al. [120] in S3, 9, 27 GBs in bSiC (Fig. 34). Hence, the triple point drag e€ect could be the manifestation of the fact that GBs are build from SEs and that creation of SEs requires application of some GB stress. It is seen that SEs and incompatibility of SEs from three GBs lead to the appearance of periodic surface stress, which in¯uences the triple junction migration rate. It is interesting to note that this triple junction drag disappears at high temperatures.1 2.4.4. Interface shear modulus Analysis of interface stress±strain relationships inevitably leads to de®ning interface elastic constants [157]. To understand what this means, let us imagine a bicrystal with in®nitely sti€ crystals. All the elasticity of this system will be the result of the stress±strain relationships for the interface. The theoretical background for calculations of these constants from ®rst principles are given in Sections 2.3.3, 2.5.3.4 and 2.1.3.6. Cahn and Larche [161] introduced two types of interface stress. Assume that the interface is a thin layer between two crystals and the whole system is stretched 1 The interest in GB triple junctions is rapidly growing. Volume 7 of Interface Science (1999) is completely dedicated to triple junctions.

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427

Fig. 51. Shear modulus of an interface.

elastically. Then, the stress h that builds up in the interface need not be the same as in the crystals. On the other hand, one can imagine one crystal sheared against the other. The resulting stress was called ``interface stress g''. Fig. 51 shows the e€ect of application of shear stress to a bicrystal. It is assumed that the crystals are perfectly sti€ and only the interface displays some elasticity. GI0 is the high frequency elastic modulus of the interface. It describes the stress±strain relation for times shorter than the interface relaxation time (Fig. 51a). However, if there are relaxation mechanisms (for instance, GB creep), the same strain could be obtained at low or zero stress (Fig. 51b). The strain±stress relation after relaxation depends on the low frequency shear modulus GI. The Maxwell model of a viscoelastic solid describes well the above reaction of the interface to external stress (Fig. 52) [162]. For GB di€usion relaxation of own strains, the characteristic relaxation time t depends on …G=dDGB †1=3 (Section 3.2). The interface shear modulus depends on the interatomic potentials in the interface. The term ``interatomic force'' follows from the assumption of a linear relationship between the interatomic forces and the atom's displacement: F ˆ ku,

…109†

where u is the bending of atomic planes and k is the force or spring constant (cf. [2]): Fˆÿ

df ˆ ÿkr: dr

…110†

Consider a ``good matching'' sector of the interface or a coherent interface. The interface shear modulus can be approximated as:

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Fig. 52. Maxwell viscoelastic model of a solid applied to a grain boundary [162].

k GI 1 , d

…111†

where d is the interface thickness. Here, the di€erence between the shear modulus and Young's modulus was neglected, since k re¯ects stretching bonds while GI their bending. However, Eq. (110) introduces a basic complication to the concept of interface shear modulus. In fact, there are interfaces with periodic structure, where the derivative df=dx would be a periodic function of x. Therefore, one may consider the local shear modulus GI and the average shear modulus hGI i: The above considerations concern the isotropic case. In the general case, the elastic constants are tensors expressed by the relation: sij ˆ sijkl ukl ,

…112†

and: sijkl ˆ

@ 2g : @ uij @ ukl

…113†

For the isotropic case, this reduces to: hGI i @ 2g ˆ , d2 @u 2

…114†

where d is the interface width and hGI i is the average value of GI. Therefore, the energy of the interface is a function of strain e_ ˆ u=d: gˆ

e 2 dhGI i ‡ g0 : 2

…115†

The ®rst term is the non-equilibrium, strain energy. For the further considerations, we will assume the shear modulus in Eq. (111) to correspond to the coherent interface or to the good matching sectors. On the

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429

Fig. 53. Random oscillations of the forces opposing TLD core spreading along its line.

other hand, the average shear modulus hGI i (Fig. 51) corresponds to an interface where a fraction of the atoms form perfect bonds between the two crystals. hGI i is misorientation dependent: hGI i ˆ hGI …y†i and: GI rhGI …y †i:

…116†

In summary, the following expressions for the interface shear modulus can be envisaged: (i) hGI0 i Ð average low frequency shear modulus, (ii) hGI i Ð average high frequency shear modulus, (iii) GI0 Ð low frequency shear modulus for good matching sectors and (iv) GI Ð high frequency shear modulus for good matching sectors. Which one is applicable depends on the question discussed. We will consider this question on the example of spreading of Trapped Lattice Dislocations. 2.4.5. Interface shear modulus and the core width of interfacial dislocations Consider a lattice dislocation that was stopped at a grain boundary. We will call it a trapped Lattice Dislocation (TLD). Assume that its line direction is random, and the Burgers vector is parallel to the GB plane. In Section 3.2, it is shown that TLDs in GBs are in a metastable state. At low temperatures, their strain ®elds are localised and the core width is similar to that of bulk dislocations. Upon heating, however, a local Coble creep process takes place (spreading of dislocations [163,164]). Application of the Peierls model to the TLD case [162,165] requires estimating the average GB shear modulus hGI i: Consider a GB with a partially periodic structure (Section 2.2.1). Assume that the GB is parallel to the x±y plane and the dislocation line is parallel to the y-axis. The Peierls expressions

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were modi®ed [165] in such a way that the shear stress which prevents TLD core delocalisation was not constant along the dislocation line (Fig. 53):   Gb 4pu…x† sin osc…y †: …117† sxz …x, y † ˆ 2dc b Here u(x ) is the bending of planes in the direction of the Burgers vector, c is a constant that depends on whether the dislocation is of screw or edge type, and osc( y ) is a function with values in the range 0±1, which represents the randomness of the GB structure. Integrating Eq. (117) along the dislocation line gives: … …118† sxz …x† ˆ s…x† osc…y † dy, L

where L is the interface period and s…x† is the shear stress of the equivalent bulk dislocation. If osc( y ) takes random values, the shear stress decreases for increasing period of integration. Therefore, for a non-periodic interface, the equilibrium core width of the TLD is in®nite. This agrees with the observed TLD core delocalisation at high temperatures. The shear stress of the TLD extends over a relatively large area of the GB, and we have to do with the average shear modulus. At low temperatures, the TLD core width depends on the high frequency shear modulus hGI i, which is approximately equal to that of the bulk. On the other hand, at high temperatures, the TLDs delocalise and the low frequency shear modulus controls the TLD core width. In the case GI ˆ 0, the equilibrium core of the dislocation becomes in®nite and the dislocation disappears. However, mis®t dislocations (MDs) or Structural Grain Boundary Dislocations (SGBDs), contrary to the TLDs, belong to the equilibrium structure of GBs. Their lines are not randomly situated but correspond to low energy interface structure. The degree of their cores localisation depends on the shear modulus in the good matching sectors GI. Since MDs are stable in a wide range of temperatures, both the low and high frequency shear modulus must take non-zero values: GI > 0 and GI0 > 0: It is seen that the degree of mis®t localisation depends on the nature of interfacial bonding and it is expected that for GBs in materials with a strong covalent component to the interfacial bond mis®t is localised [166,171,172]. However, for such materials, the range of disorientation angles for the MD or SGBD description of the interface is small ([171,172] and Section 2.4.12.1). 2.4.6. Relaxation of interface mis®t strains by increasing interface width or its dissociation It follows from Eq. (115) that increasing the interface width leads to a decrease of strain energy at constant strain. In fact, consider a given maximum mis®t um. The strain is e ˆ um =d: It is seen that an increase of the thickness by a factor 2 decreases the strain energy by a factor 4. Splitting of an interface into two

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431

Fig. 54. HREM photo of an extended GB in b-SiC, which was hot isostatically pressed without sintering aids [169]. Courtesy S. Tsurekava.

interfaces may be energetically favourable. This is particularly true for interfaces of high shear modulus, for instance, where bonding has a high covalent component. The above considerations point at the possibility of local interface widening in places where mis®t is particularly high or splitting of the interface if the average mis®t is high. Splitting of interfaces was observed experimentally. Ernst et al. [167] have observed splitting of an asymmetric twin GB in copper into two GBs separated by a layer with bcc structure. Tsurekava et al. observed wide interfaces in SiC (Fig. 54) [168,169]. Splitting of interfaces is frequently observed in silicon, where high S GBs are split into several parallel GBs, each of low S value [170]. Garg et al. [170] have found that S9 {122} GBs dissociated into two coherent S3 GBs and S27 GBs split into a series of 100 nm grains consisting of a coherent S3, a {122} S9 and a {111}/{115} S9 GB. The latter one further dissociated to form 10 nm twin grains consisting of two coherent and one non-coherent S3 GB. Split of GBs is essentially equivalent to e€ective increase of GB width. This permits to take account of larger mis®ts, since the mis®t can be gradually reduced at each GB. Ultimately, the mis®t energy may be suciently high so that in favourable thermodynamic conditions the interface is replaced by two liquid/solid or amorphous phase/solid phase interfaces (Section 3.4).

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Fig. 55. The model taken for calculations of the energy of a mis®t dislocations. L is the periodicity of the interface, u is the bending of atomic planes. d is the local mis®t across the interface, r is the mis®t of atoms across the interface: r ‡ u ˆ d [171,172].

2.4.7. Core width of a dislocation in an interface and the localisation parameter It is possible to derive a numerical parameter which will be called localisation parameter, characterising to what extent sectors of good matching in the interface are energetically favourable. For this purpose, we will use a simpli®ed model of the interface (Fig. 55) [171,172]. The interface is perpendicular to closely packed planes in each crystal. Its structure is periodic with a period L. To reach an analytical solution, we applied the Peierls model (Sections 2.4.1 and 2.4.5). We calculate the interface energy as a sum of the stretched bonds energy and elastic energy stored in the bulk. The stretched bonds energy is calculated based on Eq. (109), where u takes values from 0 up to a maximum value um. Owing to the periodicity of the interface, u is a periodic function. Integrating the energy of the stretched bonds over a period, we get the expression for the stretched bonds energy [171,172]:

W. Lojkowski, H.-J. Fecht / Progress in Materials Science 45 (2000) 339±568

gS ˆ GI b…1 ÿ Z† 2 ,

433

…119†

where Z ˆ um =D is the relative um value. The elastic energy stored in the crystals can be calculated by integrating the work to introduce the surface strain ®eld u(x): gEL ˆ

bGZ 2 : 2

The total energy is:   g ˆ b GI …1 ÿ Z† 2 ‡0:5GZ 2 :

…120†

…121†

Eq. (121) contains two energy terms. The ®rst one represents the interface mis®t and the second one the strain ®eld of the mis®t dislocation, respectively. In order to decrease the strain energy mis®t must be increased. The situation is the same as in the bulk with various stacking fault energies. However, interfaces may di€er from each other as far as the shear modulus is concerned. If the two joined crystals have di€erent shear modulus, the elastic energy for each crystal should be calculated separately and the above equations would need to be adjusted accordingly. However, in the present calculations, we focus our attention on comparing di€erent interfaces rather than precise estimates of mis®t dislocation width. For such a purpose, the present approximation is sucient. We will simply assume that G in Eq. (121) is the smaller of the two. In fact, it is the more elastic crystal that will determine whether MDs are localised or not. The minimum energy condition leads to the following equation for the relative bending of atomic planes: Zˆ

p : p‡2

…122†

We introduce the notation: pˆ

GI : G

…123†

The parameter p is the localisation parameter. Here is the reason: For pˆ)0, Zˆ)0, atomic planes do not bend and the mis®t of the lattices is not localised between the sectors of good atomic matching. For pˆ)1, Zˆ)1 and um ˆ b=2: Hence, all the planes except one are continuous. The mis®t is localised in the narrow cores of a mis®t dislocation. The relation between the p parameter and the interface energy is: gˆ

bGp 2 2bGp ‡ ‡ g0 , 2 2…p ‡ 2† …p ‡ 2† 2

…124†

where the ®rst term is the elastic energy stored in the crystals and the second is

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the dislocations core energy. g0 is the general adhesion between the crystals, which does not depend on the structure of the interface (Section 2.5). The ®nal expression for the interface energy is: gˆ

bGp …4 ‡ p† ‡ g0 : 4 …2 ‡ p† 2

…125†

For small p values, this is: gˆ

bGp ‡ g0 : 8

…126†

2.4.8. Adjustment of the static distortion waves theory to include the localisation parameter The Static Distortion Waves (SDW) theory was developed originally for thin ®lms on rigid substrates where the long-range elastic strains could be neglected. This assumption need not be true for interfaces between rigid crystals. The localisation parameter can be used to modify in a simple way the SDW equation for the system energy, so that the resistance of the system to mis®t localisation is included (Eq. (98)). For that purpose, it is sucient to replace the amplitudes VK by the expression: V K0 ˆ p
VK :

…127†

Here p is the localisation parameter de®ned in Section 2.4. For sti€ and weakly bonded crystals p ˆ 0, the strain waves of each crystal are independent, and the energy of the interface is independent of the matching of rows or OR. In real materials, either the crystals have a ®nite size or the interatomic potentials are shallow to some extent. The shallowness of the potential can be taken into account using the Sutton's equation (83), modi®ed to include the localisation parameter: … …128† VK ˆ p V…x, z † exp…iK
x † dx It is seen that there is a continuity of transition between the perfectly commensurate situation and the incommensurate state. The SDW concept is coherent with the lock-in model for interfaces between crystals [173] where of crucial importance is the interaction of atomic rows and energy valleys on the surfaces of the adjoining crystals. The idea is related to minimisation of energy of interaction of strain waves rather than local atomic matching. Minima of energy are obtained for parallel strain waves of the two crystals. 2.4.9. The collectivity coecient Consider an interface where the bonding energy is temperature independent,

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435

Fig. 56. Schematic presentation of strains along crossing closely packed rows of two crystals. GB plane view. Only closely packed rows are shown.

like a metal (Section 2.5.2). However, thermal vibrations and a high stochastic factor cause a decrease of the importance of interface periodicity (Section 2.2.3.1). The following simple model provides a way to qualitatively explain the e€ect of temperature on special ORs of interfaces. We assume that a fraction w=S of atoms belong to good matching sectors. w is the ``collectivity coecient'', i.e., the number of atoms that are bounded in closely packed groups Ð SEs, which are stable against thermal vibrations (Section 2.1.3.9). In other words, instead of single atoms w-membered groups are in good matching positions. If the energy decreases proportionally to the fraction of atoms in good matching sectors, the depth of the energy valley related to such an interface is: Dg ˆ

DE , wSO 2=3

…129†

where DE is the energy di€erence between an atom in a ``good matching'' sector and in a sector of poor matching of the crystals. We may assume w ˆ 1±2 for twist GBs, since for twist GBs closely packed rows of atoms of the two crystals cross each other (Fig. 56). On the other hand, for say [011] tilt GBs in fcc metals, atoms are organised in closely packed rows and w > 2 (Sections 3.6.1 and 2.6.1). Assume that the energy valley disappears when: DE1kT

…130†

Eqs. (129) and (60) permit to correlate the temperature and the depth of the energy valley which disappears at the given temperature. Table 2 shows the results of calculations for a material with average GB energy 1000 mJ/m2, and atomic volume O ˆ 20 ÿ30 m3. T is the temperature at which Eq. (60) is ful®lled. The data given in Table 2 are based on a very crude model, but nevertheless properly re¯ect the temperatures for the special 4 general GB transformation

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Table 2 Temperature of thermal disordering of the good atomic matching sectors as a function of coincidence S, change of energy in the good matching sectors Dg and collectivity coecient w 1=S

1 1/3 1/5 1/7 1/9 1/11 1/13 1/15 1/17

Dg (mJ/m2)

1085 360 220 155 120 100 80 70 60

T (K) w=1

w=2

w=3

5800 1930 1160 830 640 530 450 390 340

11,600 3870 2320 1660 1290 1050 890 770 680

17,400 5800 3480 2480 1930 1580 1340 1160 1020

[134] and the fact that twist GBs tend to undergo this transformation at lower temperatures than tilt GBs. It follows that at high temperatures, low energy interfaces are build from SEs having the form of compact groups of atoms, like closely packed planes (Section 2.6.1). If to consider the parameter p as a criterion whether good matching or parallelisms are criteria of low energy interfaces, the importance of this parameter depends on temperature. Fig. 57 shows a proposed map of the applicability of the two general criteria.

Fig. 57. Proposed map of the relative importance of atomic matching and parallelism for low energy interfaces.

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437

Fig. 58. The degree of mis®t localisation as a function of the p parameter value.

The high collectivity coecient corresponds to interfaces build from compact SEs-ledges according to the SE models of Bishop and Chalmers [64], extended to asymmetric interfaces by Lojkowski [73,74]. 2.4.10. Relation between localisation parameter and degree of localisation of mis®t dislocations cores The following equation relates the core width of the MDs and the localisation parameter [171]: s ˆp s0

…131†

where s is the MD core width and s0 is the core width of a lattice dislocation. Fig. 46 illustrates the distribution of the Burgers vector density close to the MD's core for two p values: 1 and 0.25. It is seen that for low p values, the MD cores are delocalised. When the interfacial bond energy is comparable to the bonding energy within the crystals and GI 1G, then p11 and the width of MD cores and cores of lattice dislocations are comparable. This situation is illustrated in Fig. 58. An interesting situation results when GI > G: An example is the Nb/NbO interface [174]. The shear modulus of NbO is higher than that of Nb. If the interfacial bond is the Nb±O bond, then GI is the shear modulus of NbO. Therefore, the shear modulus of the interface is higher than that of bulk Nb. The result is that p > 1 and the core width of MDs should be smaller than the core width of lattice dislocations in Nb. Since at small core widths the strain energy increases strongly (Eqs. (100) and (103), Fig. 47), it is energetically favourable that MDs quit the interface plane and move into the soft material (Fig. 58d). This phenomenon was in fact experimentally observed for the Nb/NbO, and was called

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``stand-o€'' [174]. Mader correlated the structure of the mis®t dislocations cores with the strength of the anion±metal bond in metal±oxide interfaces: for weak bond the mis®t is delocalised, for a strong bond, stand-o€ [175]. The detailed elastic model for stand-o€ can be found, for example, in the paper of Kamat et al. [175], but for the purpose of assessment of the interface structure it is sucient to know that a very high p value stand-o€ can be expected under a certain condition. This condition is that the crystal with low shear modulus is under a dilatation stress, i.e., has a smaller lattice constant than the hard crystal. This result can be explained based on image force calculations [177]. In simple terms, if the mis®t dislocation is not in the interface, it is situated in the crystal with low shear modulus, since there the elastic energy is lower compared to the situation when the dislocation is in the harder crystal. The situation is di€erent if the softer crystal is under compression (larger plane spacing than for the hard crystal) or under dilatation (smaller plane spacing than for the hard crystal). In the ®rst case, the Burgers vector of the MD is that of the soft crystal, with small lattice spacing. The only way to avoid a smaller core width than the Burgers vector is by moving the dislocation into the softer crystal. On the other hand, for a soft crystal with large lattice spacing, it is possible that p > 1 and the MD is situated close to the interface. This is because the lattice spacing of the hard crystal, which is smaller than that of the soft crystal, determines the Burgers vector of the MD. 2.4.11. Calculation of the localisation parameter p We will use the following expression for GI: GI ˆ

EAB , O

…132†

where EAB is the bonding energy between two atoms across the interface and O is their average atomic volume. A justi®cation of this equation is that for the bulk, GO is roughly the cohesion energy. More correctly, Eq. (132) is an approximation of the Young's modulus, but since the present calculations have an approximate character, we assume that it expresses the shear modulus. A more re®ned theory should take this di€erence into account. Within the above approximation: pˆ

EAB : OG

…133†

In Section 2.5.2, we will attempt to develop simple rules which shall permit to assess EAB for various materials. 2.4.12. E€ect of the p value on properties of interfaces and the coherent 4 noncoherent transformation Delocalisation of SGBDs or MDs cores is a qualitative change of the interface structure. The most important examples are the transformation of a semicoherent interface into a non-coherent interface at a critical thickness of an epitaxial layer

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439

Fig. 59. The interface transformation connected with mis®t delocalisation.

or precipitate, as well as the special 4 general GB transformation for increasing misorientation. Some reasons for the transformation could be: 1. a change of interfacial bonding induced by thermodynamic conditions (temperature, pressure, chemical activity of elements), 2. a change of dislocations spacing caused by a change of mis®t, for instance, due to di€erent thermal expansion coecients or di€erent compressibility of the two crystals, and 3. misorientation change. Fig. 59 shows schematically the structural transformation when the cores of dislocations overlap. The transformation can be tentatively called ``mis®t delocalisation transformation''. It includes both the special 4 general transformation when the misorientation angle or temperature are increased and the semicoherent 4 non-coherent interface transformation. A schematics of mis®t epitaxial interfaces is given in Ref. [178]. 2.4.12.1. The range of disorientation for localised mis®t dislocations. The spacing of MDs decreases when disorientation increases. When the MDs cores overlap, the transformation of the semicoherent interface into a non-coherent interface takes place (Fig. 59). Let us estimate the disorientation angle for the transformation as a function of the localisation parameter. The density of MDs depends on their Burgers vector and disorientation [179]: Dy ˆ br,

…134†

where r is the dislocations surface density in m/m2 units. The limit on the width of MD cores is:

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Table 3 The maximum disorientation angles for some groups of interfaces The p value

Maximum disorientation

Type of interface

p ˆ 1:0 p ˆ 0:5 p ˆ 0:2 p ˆ 0:1

Dy < 188 Dy < 98 Dy < 48 Dy < 28

Metals, oxides of high cohesion energy Interfaces of medium bonding SiC van der Waals bonding

s<

1 : r

…135†

Combining Eqs. (131), (153) and (154), we obtain: b Dy ˆ p : r0

…136†

We assumed that the minimum core width of a dislocation is equal to three interatomic spacings. The values of Dy for various p values interfaces are given in Table 3. Having in mind the approximate character of present calculations, Table 3 is not intended to give exact values for the disorientation ranges but to provide a qualitative assessment of the properties of di€erent interfaces. Table 3 shows that for metals and ceramics with p value similar to metals, interfaces in a wide range of disorientations can be interpreted according to the dislocation models. However, for such materials like SiC, the range of disorientations is limited. Hence, the applicability of CSL and Structural Units models (cf. Section 2.1.1) is limited to a narrow angular range as well. For van der Waals bonding, the MDs are delocalised for the whole range of misorientations. The results of comparison of the present model with experimental observations are summarised in Section 2.4.13. It follows that the Brandon equation (Eq. (86)) has to be modi®ed to include the susceptibility of the interface to mis®t localisation: p158 Dy ˆ p : S

…137†

For p ˆ 0, there are no mis®t dislocations and for p ˆ 1, the Brandon equation is obtained. 2.4.12.2. The shallowness of the energy minima. Consider a special LEOR with misorientation angle y0 : Introducing a dislocation with Burgers vector b causes a change of misorientation and energy increase. Assume that the only change of the GB structure is related to the dislocation core and strain energy:   @g D s 2 ˆ a1 Gb ln …138† ‡ a2 G I b 2 : @y s s0

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441

The ®rst term on the right side of Eq. (138) is the elastic energy and the second term is the core energy of the dislocation, respectively. a1 > 0 and a2 > 0 are constants and R is the crystal size. At the limit of large delocalisation, s4 R: @g ˆ a2 Gb 2 p10 @y

…139†

On the other hand, for narrow dislocation cores: @g > 0, @y

…140†

and the energy minimum is sharp. In other words, for interfaces where MDs or SGBDs have wide cores, special ORs are connected with shallow energy cusps. This result agrees with calculations of Gleiter [123]. 2.4.12.3. The maximum mis®t for a coherent interface. One of the de®nitions of the mis®t at the interface is as follows: cˆ

2jaA ÿ aB j , aA ‡ aB

…141†

cˆ

D , a0

…142†

or:

where aA and aB are both lattice constants, D is jaA ÿ aB j and a0 is the average lattice constant. The spacing of MDs is: lˆ

b , D

…143†

For suciently high values of p, dislocations will accommodate the mis®t. However, if the dislocations have wide cores, they will overlap and the semicoherent interface will transform into a non-coherent one. Taking into account that s ˆ s0 p (Eq. (131)), the condition for a semicoherent interface is: p>

aA ÿ aB : b

…144†

Assuming b ˆ 0:2 nm, aA ÿ aB ˆ 0:03 nm, the value p > 0.15 is obtained. Therefore, mis®t dislocations of various core widths are expected for a wide range of p values. Eq. (144) is the interface analogy of Eq. (136). For GBs in cubic materials, mis®t depends on misorientation; and for interfaces or GBs in noncubic materials, it depends both on misorientation and lattice mis®t.

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Fig. 60. Modes of mis®t relaxation of an epitaxial ®lm.

2.4.12.4. The transition coherent 4 semicoherent 4 non-coherent precipitates and epitaxial layers. The questions of strains in heteroepitaxial layers and relaxation mechanism were reviewed recently [180]. Here we will bring only the simplest ideas to see how the concept of localisation parameter and strains in heteroepitaxial layers are connected. Fig. 60 illustrates a simple method to estimate the strain energy caused by mis®t as well as the energy relaxation methods. The ®lm elastic energy per unit surface area can be estimated as the energy to strain a ®lm of thickness h and Young's modulus EY: gSTRAIN ˆ c

EY h, 2

…145†

The elastic energy is proportional to the thickness. Decohesion at the interface would lead to a total relaxation of the strain energy. This imposes an upper limit on the thickness of a coherent layer: hMAX <

2gADH : Ec 2

…146†

gADH is the adhesion energy of the interface (Eq. (153)). Nabarro proposed a similar equation for a spherical precipitate, but instead of decohesion he assumed local melting at the interface [181]. Using the adhesion energy instead of the melting energy, his equation transforms into:

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443

Fig. 61. The coherent 4 semicoherent 4 non-coherent interface transformations. Three interface structures and the independent parameters: h Ð thin ®lm thickness, c Ð mis®t of the two lattices, p Ð localisation parameter.

RMAX

  gADH 4G , ˆ 1‡ 3K 2Gc 2

…147†

where RMAX is the maximum diameter of the coherent precipitate and K is its compressibility. Eq. (147) overestimates the radius for loss of coherency for the case of high adhesion energy [182]. It is experimentally observed that strained ®lms tend to grow in the form of hills and valleys (Fig. 60b). In such a way, while the ®lm grows, the strain energy remains low, since the strain ®eld is limited to a layer of thickness h. Mis®t energy is also decreased by introducing a bu€er layer between the two crystals. It causes a decoupling of the two crystals and a decrease of strain in the deposited ®lm. The above considerations concerned the transition of coherent 4 non-coherent interface. Let us now consider the transition: coherent 4 semicoherent interface. Such a transition will take place when the GB structure with dislocations will have less energy than the strained coherent structure. The present treatment is similar to that in Section 2.4.7. We have to compare the strain energy of the coherent ®lm with the energy of a semicoherent ®lm, where the core end strain energy depends on the localisation parameter. The transition will take place when the values of Eqs. (145) and (125) or (126) are equal. For low p values, this is: hˆ

bGp : 8EY c 2

…148†

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Fig. 62. Phase diagram illustrating the regions of stability of each of the above structures. The plane abcd is a plane of constant mis®t. For low p values, the path (1) 4 (3) is the direct transformation from a coherent to a non-coherent interface. At high p values, the sequence is (1) coherent 4 (2) semicoherent 4 (3) non-coherent.

It is clearly seen that h decreases with p value. In other words, for delocalised MDs cores, the thin layer becomes non-coherent at small h values. However, for delocalised MDs cores, this is a direct coherent 4 non-coherent transition instead of a coherent 4 semicoherent transition (Fig. 61). Eq. (144) gives the condition for the direct transition. For p ˆ 0:1 and c ˆ 2%, Eq. (148) gives the value h150b, i.e., about 50 atomic layers, which is a reasonable value. In the case of medium or high p values, it is expected that the sequence of transformations will be: coherent 4 semicoherent 4 non-coherent. Furthermore, p de®ned as the ratio of the core widths (Eq. (131)) should decrease as the ®lm thickness increases p ˆ p…h†: The reason is that the elastic energy increases with ®lm thickness, while the mis®t energy does not depend on thickness. The semicoherent 4 non-coherent transition will take place when the cores will overlap. We simply assume that the elastic energy is zero for ®lm thickness equal to one layer of atoms and continuously increases to the value given in the ®rst term of Eq. (124) as the ®lm thickness become larger than the MDs spacing. The strain ®eld of the interface is limited to the distance of the spacing of dislocations. The localisation parameter for one monolayer is p ˆ 1, since the MD core is just a missing row of atoms of width s0 ˆ a0 : For increasing thickness, p(h ) decreases up to its minimum value p. If p…h† < …aA ÿ aB †=b (Eq. (144)), the interface becomes non-coherent. Fig. 62 shows a phase diagram illustrating the regions of stability of each of the above structures. The plane abcd is a plane of constant mis®t. For low p values, the path (1) 4 (3) is the direct transformation from a coherent to a non-coherent interface. At high p values, the sequence is (1) coherent 4 (2) semicoherent 4 (3) non-coherent. Some maps of relaxation structures of interfaces depending on the mis®t and

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Table 4 Correlation between the value of the localisation parameter and properties of interfaces The localisation Interface ( p value) parameter value range

Characteristic features: low energy ORs, degree of loalisation of mis®t dislocations

p>1

Al/Al2O3 (2.1); Nb/Al2O3 (1.5)

Stand-o€; coherence; all features the same as for the 0.35±1 range

0.35±1.1

Al/Al (1.2), Al/GaAs (1.1), NiCr (1.1), Au/Pd (0.9) MgO/CdO (0.9), MgO/MgO (0.8), Au/Au (0.8), Pb/Sn (0.8), Au/Ag (0.7), Cu/Cu (0.7), Ag/Cu (0.7), Cu/Fe (0.7), Ag/Ag (0.6), M/Mo (0.6), Al2O3/ ZrO2 (0.7), Al2O3/Al2O3 (0.7), Fe/Fe (0.5), Si/Si (0.5), MgO/ZrO2 (0.4)

Mis®t dislocations; periodic interfaces have low energy; asymmetric low energy interfaces parallel to vicinal planes observed; adhesion, surface and combined LEORs observed

0.3±0.35

Fe/Al2O3 (0.3), Ni/MgO (0.3), NiO/NiO (0.3), ZrO2/NiO (0.3), MgOAl2O3 (0.3), Ag/Ni (0.35), SiC/SiC (0.3)

Parallelism; partially delocalised mis®t; dislocations; LEORs of ``surface type''

0.02±0.25

Cu/NiO (0.2), Cu/MgO (0.2), Au/MgO (0.1), Al/MgO (<0.1), Ag/MgO (<0.1), Ag/NaCl (0.02), Au/NaCl (0.02) Au/LiF (0.01)

MDs are delocalised; parallelism; random twist angle at high temperatures; all LEORs are ``surface type''

adjoining metals are also given in the book of Howe [2], based on the papers of Stoop and van der Merve [183] and Kato [184]. 2.4.13. The localisation parameter and criteria for low energy interfaces It is impossible to calculate the interface shear modulus for all possible interfaces. However, in many cases, it is possible to estimate the p value knowing which type of bond is active across the interface. In Section 2.5, approximate methods for calculating the bonding energy between crystals and interface shear modulus are given (cf. Sections 2.4.11 and 2.5.2). On the other hand, a discrepancy between the prediction of the model and the experimental observations may indicate that the bonding energy was under- or over-estimated. The proposed methods of calculations are rather inaccurate and do not permit to calculate the exact width of mis®t dislocation cores. However, they are suciently accurate to divide interfaces into four categories shown in Table 4. The table shows the correlation between the value of the p parameter and localisation of mis®t dislocations as well as crystallography of LEORs, based on a compilation of Lojkowski [171,172,185]. Experimental results concerning the structure of interfaces will be discussed in Section 2.5. A compliation of data for the localisation parameter value for a number of interfaces is given in Ref. [185]. It can be seen from Table 4 that interfaces can be divided into four categories. (a) For p > 1, the interface shear modulus is higher than in one of the crystals.

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(b) For p > 0:35, their structure is similar to the structure of GBs in metals, which have been extensively investigated. Coincidence and parallelism to vicinal planes are crucial factors determining LEORs. Mis®t dislocations are localised. (c) For 0:25 < p < 0:35, the main factor for the energy of interfaces is parallelism to low index crystallographic planes, although some coincidence boundaries and mis®t dislocations have been observed. (d) For p < 0:25, the low energy ORs are parallel. In Section 2.5, it will be shown that the p value may depend on thermodynamic conditions: temperature, partial gas pressures in the atmosphere, impurity contents, etc. Table 4 refers to room temperature conditions and thermodynamic stability of both the interface and the bonded crystals. It is interesting to note that the p value for covalent materials (SiC, Si) is low (<0.5). This follows from the fact that the ratio of the shear modulus to the cohesion energy is rather low for such materials compared to, e.g., metals. The low value of the p parameter for covalent materials has signi®cant consequences for the structure of their interfaces (cf. Section 2.5.3.4). 2.4.14. E€ect of localisation parameter on the mode of growth of epitaxial layers In Section 2.4.12, we discussed the role of the p value for the structure of interfaces. In the present section, we will discuss the e€ect of the p parameter on the growth mode and stability of thin ®lms on bulk substrates. Vook gives an extensive review of the structure and growth modes of thin ®lms [186]. A more recent review is given by Mahajan [187]. The relations concerning mis®t, thickness and interface energy based on the van der Merwe theory are also given in a very transparent way in the book of Howe [2]. Bonnet [188] gives a more re®ned approach to the calculation of the mis®t dislocations in limited inhomogeneous media. In a recent review, Markov and Stoyanov [150] review the role of adhesion energy for the mode of growth of epitaxial layers. There are two extreme cases for the growth mode: 3D islands growth mode and monolayer growth mode. The 3Dgrowth mode is favoured when: 1. the interfacial bonding is weaker than the bonding in the deposit itself, 2. lattice mis®t is large, and 3. the substrate surface is parallel to a plane of low density. A dense substrate plane favours monolayer growth. The above observations may lead to introduce a parameter determining the mode of growth, which is similar to the localisation parameter: pe ˆ

EAB , OEY

…149†

where EAB is the bonding energy between adatoms and the substrate, O is the atomic volume of the deposit and EY is the Young's modulus of the deposit. Contrary to the localisation parameter de®ned in Section 2.4, which was a function of the shear modulus, Eq. (160) depends on the Young's modulus. The

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Fig. 63. Illustration of the work of adhesion and factors decreasing adhesion. (a) Bonded interface. (b) Cleaved interface and the dangling bonds. (c) Decrease of the free surface energy and adhesion owing to chemical reactions at the surfaces. (d) Decrease of the free surface energy and adhesion due to segregation of impurities to the surfaces. (e) Decrease of the surface energy and adhesion due to surface reconstruction. (f) Decrease of adhesion due to mis®t.

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reason is that the deposited layer is thin and its strain ®eld can be approximated as pure tension. Markov and Stoyanov [150] introduced the parameter l0 to describe the ratio of the interfacial adhesion and cohesion energy within the ®lm, a parameter of similar form to p: 1=2  ka 2 , …150† l0 ˆ 2cEAB where c is a factor varying from 1/30 for van der Waals bonding to 1/3 for covalent bonding, and l0 is the inverse of the mis®t limit for generation of dislocations. Since the force constant k ˆ EY O, O ˆ a3 , the following relation is obtained between l0 and the localisation parameter given in Eq. (149): pE ˆ c

a , l 02

where c is a numerical constant. The following relation obtains: r a fMS ˆ c : pE

…151†

…152†

The general relationship is that the smaller the localisation parameter, the larger mis®t can be accommodated without dislocation generation. 2.5. Work of adhesion and interface structure It is obvious that the structure of interfaces depends on the energy of interaction of the two crystals. The crucial parameter in this respect is the work of adhesion. It is de®ned as follows: gI ˆ gA ‡ gB ÿ gADH ,

…153†

where gA, B are the surface energies of the crystals A and B, respectively, gI is the interface surface energy, and gADH is the work of adhesion or adhesion energy. Fig. 63 illustrates the concept of work of adhesion. The work per unit surface to separate two crystals along the interface is the work of adhesion. According to the convention of the present paper, if the two surfaces attract each other, the work of adhesion is positive. Low adhesion means the work of adhesion is positive but small. For constant adhesion, the higher the surface energy, the higher the interface energy (Eq. (153)). The energy of the free surfaces depends on the chemical composition of the atmosphere and on surface segregation of impurities as well as surface reconstruction. Therefore, it may depend on the chemical activity of small amounts of impurities (Fig. 63c and d). It can be expected that phase transformations and chemical reactions at the

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surfaces should in¯uence the interface adhesion and structure. An example is wetting of GBs by liquid phases, discussed in Section 3.4. The energy of the free surfaces is roughly proportional to the cohesive energy of the crystals [2,189]. However, a correction has to be taken for surfaces exhibiting substantial decrease of surface energy because of surface reconstruction [2,189] (Fig. 63e). In semiconductors, surface reconstruction may lead to a decrease of energy of the order of 25% [189]. Reconstruction decreases the stability of grain boundaries compared to materials where surface reconstruction does not alter signi®cantly the energy. A strong e€ect of surface reconstruction on the work of adhesion for the Cu/Al2O3 interface was found by ®rst-principle's study by Zhao et al. [29]. Somorjai reviewed recently the e€ect of the pressure of various gases on the surface reconstruction of metals [190]. It was shown that the structure and energy of the surfaces in noble metals such as Pt or Rd strongly depend on the partial pressures of gases such as hydrogen, oxygen, carbon monoxide. It may be concluded that the atmosphere in¯uences the terms gA , gB in Eq. (153), and, therefore, should in¯uence the preferred faceting of the GBs as well as the low energy GB planes. One example of this phenomenon is the well-known diculty in sintering diamond. At high temperatures, the free surfaces of diamond form a graphite-like structure with sp2 hybridisation and three bonds parallel to the surface so that the graphite planes are not strongly bonded to each other. Another problem is the ``internal reconstruction''. Subramanuian et al. [191] have found that in Ni3Al, nickel enrichment of grain boundaries leads to a decrease of energy compared to stoichiometric grain boundaries. The reason is that Ni-enrichment decreases the contribution of Al±Al interactions to the energy. It is known that besides enrichment in nickel, a boron addition improves the mechanical properties of this material. Muller et al. [192] have shown that nickel and boron segregate in GBs and that boron segregation enhances inter-granular cohesion. It was shown that B-rich regions have a bonding similar to that in Ni3Al, whereas the GB sectors with low B content have bonding similar to that in Ni±Ni, which is less strong than in the bulk [193]. These results agree with the decreased mobility of dislocations in B-doped Ni3Al as measured using the dislocation spreading technique [194] referred in Section 3.2.1. Segregation may have two basic consequences: (a) the work of adhesion and interfacial bonding is modi®ed, and (b) the structural units can change. The ®rst point is obvious since the segregated species modify the interface chemistry and the surface energy (Fig. 63d). A recent paper on that e€ect, including calculations of the type of bond modi®cation in Fe due to P and P±Mo couple, is given in Ref. [195]. The subject of interface segregation was reviewed recently in Ref. [196]. The electronics of the e€ect of segregation on interface cohesion is discussed in Refs. [29,50,195,197]. The recent paper of Rittner et al. [198] brings rather pessimistic news about the possibility to extract microscopic

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information on the GB structure and segregation using standard macroscopic models (cf. [196]). The second point follows from the consideration that the variety of polyhedra that can be constructed from di€erent atoms is larger than when only one type of atoms is available. It might be also expected that denser packing is possible when the atoms have di€erent size than when they are all of the same size. This factor is well known for random close packing and is one of the reasons why the glass transition temperature is higher for multi-component metallic alloys than for alloys with one or two components [359]. The above two types of reconstruction make it dicult to predict the interface energy based solely on the theoretical interaction energy between the atoms as, for instance, in the Miedema and den Broeder [189,199], Becker [200] and McDonnald±Eberhard [201] models. It follows that the work of adhesion depends on the atomic matching at the interface, bonding energy and energy of the free surfaces created when the two crystals are separated. 2.5.1. Surface and adhesion determined energy minima Interfaces where due to geometrical constraints the atoms cannot form strong bonds may be expected to have a low adhesion (Fig. 63f). Let us de®ne the bonding energy between atoms in positions of good matching as EAB. There must be some relation between the adhesion energy and bonding energy. We will follow the Miedema and den Broeder [199] approach to the problem, where the adhesion between two solids is split into a chemical and geometrical contribution. The geometrical part takes into account the geometrical mismatch of the crystals. The chemical part is the heat of solution of an atom of one crystal in the matrix of the other crystal. Furthermore, we apply the simple correlation that a cusp in the interface energy corresponds to a maximum in adhesion, and that cusps related to vicinal planes are shallow compared to energy cusps related to good matching and connected with the presence of mis®t dislocations [202]. For simplicity, we assume a linear relationship between the adhesion energy and bonding energy: gADH ˆ a

EAB , O 2=3

…154†

where O is the average atomic volume and a is the ``degree of good matching'' de®ned as the fraction of perfect bonds that would give the same adhesion energy as in the real interface: … EABI dI , …155† aˆ N nEAB where EABI is the energy of the Ith bond and n is the number of bonds. For a perfect coherent interface with small strain, a11: The fraction of good

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451

Fig. 64. Energy of an interface as a function of the crystallographic orientation of the two adjoining planes [171].

bonds depends on the orientation relationship: a ˆ a…y† and therefore gADH ˆ gADH …y†: A maximum of a…y† implies a maximum of adhesion energy (Fig. 65c). Naidich [203] used a similar equation to correlate the bonding energy and adhesive energy: gADH ˆ

nB EAB , nO 2=3

…156†

where n is the co-ordination number and nB is the number of broken bonds. The energy of the two interfaces depends on the orientation of the two planes. In metals, the more open surfaces have a higher energy than the closely packed surfaces. The sequence for fcc metals is: (111) < (001) < (011) and for bcc metals: (011) < (001), respectively [189]. The low energy surface planes are called vicinal planes. The energy di€erences between the (111), (001) and (011) planes in copper at 8278C are up to 3% [2,6]. For a pair of planes, this makes up to 6% variation of the sum gA ‡ gB in Eq. (153). This idea is illustrated in Fig. 64. Such a variation is of the same order of magnitude as the variation of gI as a function of misorientation in typical metals for high S GBs. Hence, the orientation of the interface plane is equally important for the interface energy as the misorientation.

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Fig. 65. Surface and adhesion determined energy cusps [171]. (a) Surface type minimum with one low energy surface (e.g. {random plane}6{111} plane). (b) Surface type minimum with two low energy surfaces (e.g. {001}6{111}). (c) Adhesion type minimum (e.g. coincidence GB). (d) Combined minimum (e.g. S11 {1,1,17}6{111}, cf. Section 2.6.1).

In Fig. 65, the interface energy is regarded as a function of two independent variables: the angles y1 , y2 determine the tilt of each crystal. Consider the possible minima of gI as a function of y1 and y2 : If one of the planes is vicinal, either gA or gB is a minimum. This situation is depicted in Fig. 65a. It is clear that since gA and gADH are approximately constant and gB has a minimum, gI has a minimum. Minima of energy for vicinal surfaces are shallow [2], therefore such a minimum of gI is shallow as well. This situation corresponds to a GB in an fcc metal where one plane is, for example, (111) and the other plane is of random orientation. Fig. 65b shows the situation when both planes are vicinal. Now gA and gB have a minimum, so that the energy minimum for the interface is deeper than in the former case. This situation corresponds, for instance, to the (111)//(001) GB in an fcc metal. Both the above energy minima can be called ``surface determined minima''. Fig. 65c represents the situation when the relative orientation of the crystals corresponds to good atomic matching at a large fraction of the surfaces of the two crystals. This leads to a cusp in the adhesion energy and consequently to a cusp in the total energy. Such a con®guration will be called ``adhesion determined minimum''. Fig. 65d shows the case when both conditions are approximately ful®lled: high adhesion and vicinal planes. In such a case, the deepest energy minimum is expected. This will be called a ``combined minimum''. It is seen that interfaces which can combine the two features: parallelism of vicinal planes and good atomic matching are connected with deepest energy minima. How this can be achieved, since good atomic matching corresponds to periodic interfaces and parallelism to non-periodic? Merkle and Wolf [204] have

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shown that such an interface in gold displays mis®t dislocations, so that combining of atomic matching and low energy plane is possible. This issue will be further discussed in Section 2.6.5. The above considerations show that energy minima for interfaces parallel to low index planes follow from very basic considerations of the work of adhesion as function of interface plane. In summary, in the present section, based on very simple arguments, three types of energy minima for interfaces were explained: Surface type energy minimum Adhesion type energy cusp Combined type energy cusp

an energy minimum of the interface caused by the fact that the free surface of the crystal for the given OR has a minimum. an energy minimum of the interface caused by the fact that there is a good atomic matching at the interface. an energy minimum of the interface caused by the fact that at the same time there is the surface type minimum and adhesion type minimum

2.5.1.1. Cleavage kinetics and interface adhesion. When assessing the adhesion energy and interface bonding, it is necessary to take into account kinetic factors. In fact, let us compare two situations. (a) There is a ®ssure rapidly propagating along the interface. In such case, there is no time for absorption of gases or for segregation. So, the work of adhesion is obtained by comparing the interface energy and the energy of two clean surfaces. (b) The two crystals are separated slowly along the interface. In such case, segregation, adsorption or surface di€usion can lead to a decrease in the energy of the surfaces compared to the clean surfaces. Which of the above cases is relevant for assessment of the interface structure (localisation parameter)? It seems that no matter how fast segregation or adsorption takes places, always a clean surface is created ®rst, even for an in®nitely short time. Therefore, for evaluation of the localisation parameter and structure, one has to take into account the work of adhesion for clean surfaces. 2.5.2. Estimation of interfacial bonding energy and the p value If the p parameter has to play its role for constructing maps of interface properties, there must be ways to estimate the work of adhesion without referring to computer calculations of interface bonds. We present approximate equations permitting to assess the bonding energy and work of adhesion for some groups of interfaces.

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Fig. 66. The relation between work of adhesion, Gibbs free energy of dangling bonds formation and Gibbs free energy of bonding of the bulk [172]. The importance of the broken bonds energy is shown in the example of MgO. (a) MgO formation from O2 gas and Mg metal. The surface terms of the free energy of Mg can be neglected. (b) Illustration of the importance of surface terms.

2.5.2.1. Physical and chemical nature of the bond between the two crystals. The true nature of bonding can be calculated by using various computer codes. However, it will take some more years to calculate all the interfaces. Therefore, we are forced to use some crude simpli®cations. Let us consider the energy of adhesion for a metal (Me) bonded to an ionic crystal (Me 'O). According to the McDonald±Eberhard approximation [201]: gADH ˆ cDGAB ‡ gWaals ,

…157†

where c is a constant, DGAB is the free energy of the Me±O oxide formation and gWaals corresponds to the van der Waals adhesion energy. Naidich proposed an alternative correlation: gADH ˆ f…DZ †,

…158†

where: DZ ˆ DGAB ÿ DGMe 0 O ,

…159†

where DGMe 0 O is the energy of formation of the substrate oxide Me 'O. The above

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equation corresponds to the situation when at the interface the following bonds are created: Me±O±Me 0 :

…160†

For such an interface, the interfacial O±Me ' bond competes for electrons with the bulk Me±O bond. For strong bonding within the substrate, the bond at the interface is expected to be weak. Both the above equations (157) and (158) do not account for the fact, that separation of the two crystals creates broken bonds at the crystal surfaces. The free energy of the MeO and Me'O compounds formation, used in Eqs. (157) and (158), is usually taken from thermodynamic tables and does include the contribution of surfaces, which for chemical reactions, in usual conditions, is small. However, when adhesion at interfaces is considered, surface energies cannot be neglected. The need to include surface terms is illustrated on the example of a grain boundary in MgO in Fig. 66. Fig. 66a shows how the free energy of MgO formation is calculated without taking into account surface terms. The energy of the MgO crystal is compared with the energy of oxygen gas that consist of O2 molecules with saturated bonds, and the energy of the Mg bulk metal. The contribution of the surfaces is negligible. Fig. 66b shows that separation of the two crystals results in generation of broken bonds for each considered atom, which corresponds to separation of the MgO into single oxygen atoms and single Mg atoms rather than into the O2 molecules and Mg bulk metal. The cohesive energy of materials, their surface energy and dangling bonds energy are correlated [2,6]. It follows that the EAB value can be estimated based on the cohesion energies of the two components A and B, and the free energy of the AB compound formation, which represents the chemical term, and the van der Waals energy: EAB ˆ CB

EA ‡ EB ‡ DGAB ‡ EWaals , 2

…161†

where EA and EB are the cohesion energies of the A and B component, respectively. CB is a constant that takes into account the degree of saturation of dangling bonds A and B when the AB bond is formed. EWaals is the van der Waals bonding energy. The contribution of the van der Waals energy to the adhesion energy is: gWaals ˆ

HrC , d

…162†

where H is the polarisability of the atoms, rC is the density of the atomic planes in contact and d is the interplanar spacing [46,203,204] (Section 2.1.3.5). The ®nal expression for the localisation parameter:   CB …EA ‡ EB =2† ‡ DGAB ‡ EWaals : …163† pˆ G

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To understand the factor CB, let us consider a situation when two crystals are brought together but their broken bonds are not satis®ed. If, nevertheless, CB ˆ 1 in Eq. (169), one would get a high bonding energy. Therefore, there must be some correlation between the degree of saturation of the dangling surface bonds and the Gibbs free energy of formation of the interface bond. CB expresses quantitatively up to what extent the dangling bonds are satis®ed in the interface. This coecient can be calculated by ab-initio methods or using simple thermodynamics, since it re¯ects the di€erence of the free energy of separated single atoms of the given compound and the chemical compound itself. We shall discuss the issue for some cases of bonding. Although the following approach lacks the rigidity of present advanced computer calculations, it may permit to assess qualitatively the e€ect of some variables, like temperature, pressure, type of bond on the interfacial adhesion, interface structure and low energy orientations. 2.5.2.2. Assessment of the interfacial bond based on the experimentally measured adhesion energy. It is possible to get an estimate of the interatomic bond energy just by dividing the adhesion energy by the surface. However, this approach may o€er some accuracy only for coherent interfaces, where all bonds are satis®ed. The adhesion energy can be calculated from Eq. (153). Assume that the surface atoms of each crystal form a dense layer and each one contributes to the adhesion proportionally to the number of saturated dangling bonds. The following equation for the bonding energy obtains: EAB ˆ aNa O 2=3 gADH ,

…164†

where a is the fraction of broken bonds, Na is the Avogadro number and O is the atomic volume. For simplicity, we have taken its average value. For large di€erence of the atomic volumes, the above equation must be corrected: the broken bond energy must be calculated for each surface separately:   …165† EAB ˆ Na aA OA2=3 ‡ aB OB2=3 gADH : Therefore: pˆa

Na gADH : GO1=3

…166†

In the above equation, the adhesion energy refers to a contact of two coherent interfaces. 2.5.2.3. van der Waals bonding. For systems such as Ag/LiF or Au/NaCl with (001) surfaces, bonding is determined by the van der Waals interaction. Its energy was experimentally determined by the scratching technique, when a needle scratches the surface of an evaporated ®lm [205]. The adhesion energy is estimated to be of the order of 0.01 J/m2 (see Section 2.1.3.5 and Eq. (162)).

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2.5.3. Grain and phase boundaries in metals and covalent materials 2.5.3.1. Grain boundary case. The bonding energy in a grain boundary in a metal can be approximated as the cohesive energy ECOH. EAB ˆ ECOH :

…167†

The cohesion energy of metals can be assumed constant as a function of temperature. For the case of interfaces between mutually soluble metals or metals forming intermetallic compounds, we assume that the interfacial adhesion depends on the average cohesion energy. The chemical factor is accounted for by adding the Gibbs free energy of the A±B bond formation: EAB ˆ

EA ‡ EB ‡ DGAB : 2

…168†

The bonding energy for Ni3Al, one of the strongest bonded intermetallics, is about 60 kJ/mol. On the other hand, the cohesive energy of Ni is approximately 420 kJ/mol. Therefore, DGAB is 15% of EAB : For comparing phase boundaries in metals that do not form strongly bonded intermetallics, the following approximation is sucient: EAB ˆ

EA ‡ EB : 2

…169†

However, for large DGAB values, the chemical bond may strongly in¯uence the interface structure. Lissowski and Fionova [206] attempted a more re®ned approach, where the e€ect of the electronic structure of metals on the GB structure is explained. DGAB is strongly dependent on thermodynamic factors, like temperature, pressure, chemical activity of impurities, etc., which may lead to phase transformations in GBs. 2.5.3.2. Interfaces between mutually non-miscible metals. Imagine an interface between metals A and B, where the A±A and B±B bonds are much stronger than the A±B bond. What happens at the interface where A±B bonds are forced? A simple assumption is that the average cohesion energy is decreased by the value of repulsive energy between A and B atoms. The Miedema and de Broeder calculations [199] predict, despite the repulsion term, a positive bonding energy between mutually immiscible metals. In his review about wetting of materials, Naidich [203] states a good wetability of Ni by Ag if the surfaces are clean. Howe [2] calculates the energy of a coherent Ni±Au interface using the Becker model [200]. gI ˆ NI n…cA ‡ cB † 2 e,

…170†

where NI is the number of atoms per unit surface in the interface plane, n is the co-ordination number, cA and cB are the equilibrium concentrations of the A and

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B atoms at the interface, and e is the interaction term according to the regular solution model. Howe found the energy 0.107 J/m2 for the Ni±Au coherent interface. Comparing with surface energies of the order of 1 J/m2 for these metals, it is seen that the work of adhesion is high despite immiscibility of the metals. Further models permitting the estimate of the interface energy are the Cahn±Hillard [207] Gradient Energy Model and the Discrete Lattice Plane Model of Lee and Aaronson [208], also discussed in the Howe's book. The above considerations show that the energetic factors in¯uencing grain boundary energy in metals may be temperature dependent when chemical bonds are present, like in intermetallics. The structure of such interfaces may display a strong dependence on thermodynamic factors. Furthermore, the additional repulsive or additive chemical energy of bonding at the interface may in¯uence the structure. It is expected that whenever an additional covalent component or the repulsive factor is present, p is lowered and the importance of parallelism of low index planes is increased compared to coincidence. 2.5.3.3. Ag/Ni interface. An extensively studied case was the Ag/Ni interface [209]. As far as the low energy interfaces are considered, Maurer and Fishmaister studied the Ag/Ni interfaces by means of the rotating spheres technique [210]. All the interfaces found correspond to the parallelism category, like for the weakly bonded systems. Gao et al. [211] calculated the structure of some Ag/Ni interfaces using computer simulation models. Fig. 47 shows that the core width of the mis®t dislocation is a factor of 2 wider than that of the bulk dislocation. A considerable delocalisation of mis®t dislocations seems to be evidenced by HREM observations of Gao and Merkle [212]. All the above information indicates that for model systems such as Ag/Ni, the system displays less enhancement of bonding for good atomic matching at the surface than for typical GBs in metals. Gao and Merkle [212] have shown that interfaces parallel to the {011} plane facet in a saw-tooth manner parallel to the {111} planes. The above results are consistent with the general scheme that at low temperatures, both parallel and near coincidence interfaces are of low energy; whereas at high temperatures; only parallel ORs retain the special character. This scheme is consistent with rather low p value, which is a consequence of low geometrical factor for adhesion (Section 2.5.2), so that only the general attractive cohesion energy between two metals contributes to the bonding. A considerable diculty in predicting interface bonding arises from the fact that condensation of vacancies can reduce the interfacial strains, as calculated by Gumbsh et al. for the Ag/Ni case [213]. 2.5.3.4. Covalent materials. Interfacial bond in covalent materials is to some extent similar to the metallic bond, since it is based on sharing electrons. Therefore, the same equations for EAB apply as for metals (Eq. (168)). The bonds in covalent ma-

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Fig. 67. The 10.608 …S101† GB in Si. (a) Zero-temperature relaxed GB, (b) snapshot at 1500 K. Courtesy P. Keblinski [214]. Reprinted with permission from The American Ceramic Society, Post Oce Box 6136, Westerville, Ohio 43086-6136. Copyright 1997 by The American Ceramic Society. All rights reserved.

terials have a limited stretching and straining tolerance. Therefore, the mis®t factor plays a more important role than in metals (Fig. 63e and Eq. (154)). Another peculiarity of covalent interfaces is the possibility of denser packing than in the bulk. Fig. 13 shows schematically the structure of the 458 asymmetric [001] GB in a SrTiO3 bicrystal. The GB is more densely packed than the bulk crystal. Silicon and diamond. The most studied covalent material is silicon. It is quite striking that most of the studied, by means of TEM, GBs in silicon are of low S type, where S is a multiple of 3. This indicates a high energy of general GBs in that material. Keblinski et al. [214] carried out computer simulations of the energy and structure of twist high angle GBs in silicon. They found that except some low

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angle GBs, the minimum energy structure correspond to an amorphous ®lm about two atomic planes thick. The four-fold co-ordination is preserved. The thermodynamic stability of GB phases that are unstable in the bulk is explained in Fig. 113 (Section 3.4). It was found that the structural dislocations cores in the low angle S101 10.68(100) twist GB are considerably delocalised. This result agrees well with the relatively low p parameter value for silicon (Table 4). Fig. 67 shows a ``snapshot'' of the core structure of SGBDs in the 10.608 …S101† GB depending on the temperature of relaxation. The above study leads to the conclusion that non-special GBs in silicon have a glassy structure. The authors are not aware of HREM studies of general GBs in silicon that might directly con®rm the results of calculations. However, the results of Keblinski et al. [214] seem to be con®rmed by Tsurekava et al. [168], Ikuhara et al. [215] and Kaneko et al. [169] HREM observations of GBs in sintered SiC. They have found that general GBs are about 0.5 nm thick, indicating splitting of the GBs into two interfaces separated by an intermediate layer. The extended GB is not a glassy phase of oxides, since SiC retains a high mechanical strength up to 2070 K. This is con®rmed by EELS analysis, which has shown that the ratio of C to O in the extended layer is 10:1. The wide GBs are not observed for special ORs and vicinal planes [169]. Some coincidence GBs were split into several twin GBs in a similar way as in silicon, which is also a way to increase the e€ective GB width. The results of Ikuhara et al. [215] should explain the sinterability and high strength of SiC despite the expected relatively high GB energies and low value of the p parameter. For covalent materials like silicon, surface reconstruction takes place. In such case, the energy of the surfaces is lower than expected purely based on

Fig. 68. The types of S9 GB in SiC: non-polar (stoichiometric), polar Si terminated and polar C terminated [218,220]. Courtesy M. Kohyama. Published with kind permission from the Institute of Physics Publishing Limited.

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consideration of cohesion, and the stability of high energy grain boundaries may be relatively low with respect to separation of the two crystals (cf. Section 2.5 and Fig. 63). A similar situation takes place in diamond, where the surface layer may take graphite-like form at the sintering temperature. Thus, it cannot be excluded that general GBs in diamond would include an intermediate graphite-like layer. In fact, recently, Keblinski et al. [216] have shown that whereas in Si, the sp3 hybridisation for GB atoms and the four-fold co-ordination is preserved, for GBs in diamond the sp3 hybridisation (as in graphite) is possible so that the interfacial adhesion is relatively weak. Silicon carbide. Kohyama et al. [218,220] calculated the structure and energy of the S9 GBs in SiC using the self-consistent tight-binding method. Particular attention was given to the di€erences between polar and non-polar interfaces. The SiC crystal surfaces can be either neutral, Si or C terminated. Accordingly, the structure and energy of GBs depend on the adjoining planes. Fig. 68 shows three GBs of the same misorientation but of di€erent polarity: stoichiometric Ð non-polar and two types of polar interfaces. For polar interfaces, either C±C or Si±Si wrong bonds are present. The energy of the polar interfaces is smaller than that nonpolar, owing to smaller Madelung sum and higher covalent contribution. Kohyama et al. [217] recalculated the structure and energy of this GB using abinitio calculations and obtained identical results as with the self-consistent tightbinding method. Furthermore, the charge transfer is smaller for polar than for the non-polar interface. The behaviour of SiC is di€erent than expected for ionic solids since the charged states originate from polarisation of bonds and not from charged states of the ions [218,219]. The presence of wrong bonds: Si±Si and C±C is a major contribution to the energy of a S9 GB [220]. As far as the stretching and bending of bonds is concerned, for the symmetrical S9 GB in Si, the deviation of the angle between bonds was in the range ÿ168 to +19.98. The bond length distortion was in the range ÿ1.9 to +1.58. The static charge ¯uctuations were in the range ÿ0.02e to +0.02e. For the same GB in SiC, the deviation of the angle between bonds was in the range ÿ23.1 to +24.18. The bond length distortions were in the range ÿ2.5 to +2.28. The higher bond distortions compared to Si were in part caused by the presence of the wrong Si±Si Table 5 Contributions to the bonding energy in the polar and non-polar symmetrical tilt S9 GB in silicon carbide Energy (eV)

Si

SiC non-polar

SiC polar

EBOND EMADELUNG EOVERLAP ECOVALENT EB/cell gGB (J/m2)

0.274 ÿ0.008 2.345 ÿ0.762 1.849 0.335

ÿ1.785 4.021 7.645 ÿ4.808 5.072 1.425

ÿ1.411 3.682 7.566 ÿ5.321 4.516

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Fig. 69. Uniaxial tensile strain test of the non-polar symmetrical tilt S9 GB in silicon carbide. Courtesy M. Kohyama. (a) Ab-initio calculations using the density-functional calculations of the energy change and (b) stress±strain relationship during the strain test calculated using the same method [224,225].

and C±C bonds, the former 4.4% shorter and the latter 3.8% longer than in bulk Si or C, respectively. The mis®t caused by wrong bond was accommodated by a RBT translation of 0.5% perpendicular to the interface and 1% parallel to the interface. Also, in SiC, the static charge ¯uctuations were relatively high compared to Si: +0.20e for the wrong Si±Si bond and +0.14e for the wrong C±C bond. It is interesting to compare the various contributions to the above GB energy in Si and SiC [220,221]. Table 5 shows the respective values. EBOND is the bonding energy related to charge transfer, EMADELUNG is the contribution of the Coulomb summation of energies, which clearly decreases interface cohesion, in agreement with Wolf's calculations for MgO [222]. These bonds contribute 84% of the Coulomb energy to the interface, and decrease the interfacial adhesion. EOVERLAP is the contribution from repulsion of atoms when the electronic shells become too close, ECOVALENT is the energy gain when a covalent bond is formed, EB is the calculated energy of one cell and, EGB is the GB energy [223]. The Madelung sum along a S9 GB in SiC depends on the atom site and displays strong variations. Furthermore, Kohyama et al. carried out computer simulations of tensile tests on the above interface. Fig. 69 shows the energy dependence on separation and the stress±strain relationship for non-polar symmetrical tilt S9 GB in silicon

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Fig. 70. The bond stretching versus bulk strain curve of individual bonds, a, b, c,. . . in the non-polar symmetrical tilt S9 GB in silicon carbide [225]. (a) The structure of the GB. (b) The stress±strain curves. It is seen that each bond is rapidly stretched and broken if the bond stretching exceed 20%.

carbide. Fig. 70 shows the bulk stress versus individual bond strain curves for some individual bonds [224,225]. It is seen that each bond is rapidly stretched and broken if the bond stretching exceed 20%. The stress±strain curves for individual bonds do not, however, permit to estimate the GB shear modulus GI as a local property, since the local stress is not known. Nevertheless, they permit to estimate the interface average low frequency shear or Young's modulus. The value 519 GPa was obtained, while the Young's modulus of the perfect crystal of SiC along the h111i direction is 610 GPa [226], and the orientation averaged value is 470 GPa [227] for SiC obtained by sublimation. The Young's modulus of the above {122} boundary in SiC has an intermediate value between the maximum and average bulk value. It seems that it should not di€er much from the bulk value for the h112i direction, since the GB is well reconstructed [225]. Polytype grain boundaries. Covalent materials, like SiC frequently display polytypism. This adds a degree of complication to the interface structure. Interfaces can separate various polytypes, so that three categories of interfaces are present: polytype boundaries [83], grain boundaries and phase boundaries. One can envisage also polytype-interfaces when the two covalent materials forming the interface display polytypism. Vermaut et al. [228] analysed recently in detail the dislocation structure of polytype interfaces between SiC and AlN. Fig. 15 shows an example of a periodic structure of a polytype GB a-SiC [229].

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Fig. 71. Charge density distribution along the h001i axis in each cell of the SiC (001)/Al interface. Courtesy M. Kohyama. (a) C-terminated and (b) Si-terminated [218,219]. Copyright 1998 by the American Physical Society.

2.5.4. Importance of polarity on the example of the SiC/Al interface Fig. 71 shows the charge density distribution in the SiC/Al interface for two terminations of the SiC plane. The adhesive energy of the interface is 6.42 J/m2 for the C-terminated case and 3.74 J/m2 for the Si-terminated case. This indicates that the C±Al bond is almost twice as strong as the Al±Si bond. The latter one has a metallic character, whereas the former one has mixed covalent±ionic character. The di€erence in bonding has an important in¯uence on the electronic properties of the two interfaces. Kohyama et al. [218,230] show that Schottky barrier height is 0.08 eV for the C-terminated case and 0.85 eV for the Siterminated case. This example illustrates the importance of polarity of the crystal surface for interfaces in polar materials. It also shows that care is needed as far as predicting the structure of such an interface when the polarity of the polar crystal is not known. 2.5.5. Oxides and nitrides 2.5.5.1. Oxide±oxide interfaces Work of adhesion for the oxide±oxide interface. For simplicity of the theoretical analysis, the simple oxide Me±O will be considered. The conclusions can be easily generalised to the MexOy case. The standard assumption is that the ``joining layer'' is oxygen, as in the work of Dickey et al. for the NiO/ZrO2 interface [231]. In such case, one would use the Eq. (161) for the EAB term. However, the meaning of the cohesion terms EA and EB is di€erent from that of the metals. The cohesive energy is connected with the Gibbs free energy of oxide formation, which depends on temperature and pressure. The same remarks, which were discussed in Section 2.5.5.1, have to be taken into account. The cohesive energy of a bulk ionic crystal includes the Coulomb interactions with all the ions in the lattice. This is taken

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into account by the Madelung sum. For the NaCl structure, this is 1.74, so that the energy of one bond per atom in a Me±O molecule is approximately 40% less than the cohesive energy per atom in the bulk Me±O crystal. The question arises which value should be taken into account for interfacial bonding: the value of the Gibbs free energy of reaction between bulk materials or the smaller by about 40% value for reaction between two atoms? The values of the Madelung sum provided in standard text books are calculated by integrating Coulomb interactions between an ion in a perfect lattice site and all other ions. However, atoms in interfaces may occupy positions shifted from the perfect lattice sites. The data for the Madelung sum for atoms shifted at the perfect sites are scarce. There are calculations for atoms situated outside the crystal [232]. Wolf [222] discusses the problems with Madelung summations for GBs in MgO. In the Wolf's calculations, the Coulomb forces did not contribute signi®cantly to the cohesion of the (001) GBs in MgO. The variation of the Madelung sum as a function of position in the GB was evidenced by Kohyama et al. [223] (Section 2.5.3.4). Since the Madelung sums for interface ions are not known, we have to assume that it is between 1 (no long-range electrostatic interactions) and 1.74 for NaCl. The uncertainty of the Madelung factor introduces an error up to 40%. Most likely, the Coulomb interactions lead to an increase of the interface energy and decrease of adhesion as compared to a metal of the same cohesive energy. Let us consider the bonds compensation factor CB (Eq. (159)). For the case of the interface atoms that do not form any chemical bond with each other: DGAB ˆ 0: Therefore, the surface energy term will not be compensated and CB ˆ 0: On the other hand, for a high value of DGAB , the surface dangling bonds must become strongly saturated. Therefore, for the strong interaction case CB ˆ 1: It follows that there is a relation between CB and DGAB : Assume that the degree of surface bonds saturation is proportional to the energy of the Me '±O bond formation at the interface. A simple estimation leads to the conclusion that: CB ˆ

2DGAB : EA ‡ EB

…171†

Substituting this CB value in Eq. (161) leads to the following result: EAB ˆ 2DGAB ‡ EWaals :

…172†

Eq. (172) shows a signi®cant di€erence between metals and oxides. In metals, the cohesive terms are relatively insensitive to temperature. In oxides, the interfacial bonding depends strongly on the thermodynamic stability of the oxide. Hence, it may be expected that the interface structure will change considerably as a function of temperature or chemical activity of the components. In contrast, in metals, the interfacial bonding energy remains relatively constant up to the melting point. Structure observations. The structure of stoichiometric GBs and the SEs in materials such as NiO [233] and Al2O3 [234] does not seem to di€er qualitatively from that of GBs in metals. As in the case of metals, the bonding does not have a directional character. Coincidence GBs and structural dislocations are observed

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Fig. 72. Possible low and high-energy metal/ionic crystal interfaces for three di€erent image charge ®elds. (a) Lock-in type of interface, where charges compensate locally. (b) Low energy interface where surface charges are neutralised. (c) High energy metal±ionic crystal system, where the increased adhesion at the interface is compensated by the presence of surface charges and long range electric ®elds.

[234]. Similarly, like in metals, non-symmetric interfaces parallel to low index planes are frequently observed. Merkle reviewed the results of extensive studies of metal/oxide interfaces [233]. A clear tendency for parallelism of the interface plane parallel to the (111) plane was shown, in agreement with image charge considerations (Section 2.1.3.5), concept of Structure Elements-ledges (Section 2.1.2) and the hierarchy of planes of Paidar (Section 2.1.3.9). Merkle et al. also pointed out the importance of kinetic factors when judging low energy interfaces in precipitate systems [235]. Dynamically changing OR was also found by McIntyre et al. during studies of Pt epitaxy on MgO [236]. Pt islands of thickness less than 20 nm aligned in such a way that the azimuthal orientation was (001) or (111), which minimised mis®t. Thicker and continuous ®lms were oriented parallel to the (111) direction, which minimised the surface energy. The transformation occurred by grain boundary migration.

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2.5.5.2. Metal±oxide interface Work of adhesion. For the metal±oxide interface, Eq. (172) takes the form: EAB ˆ EA ‡ DGB ‡ DGAB ,

…173†

where DGB is the bonding energy of the gas molecule divided by the number of atoms in one molecule. The image forces introduce a complication as far as assessment of the adhesion energy between ionic crystals and metals are concerned. Besides the chemical contribution related to charge transfer and of creating free surfaces when interfaces are separated, there are the image forces (Eqs. (19) and (20)). The result of these forces may be that the real adhesion energy is much higher than estimated from the nature of bonding of the crystals. (154)±(172) were derived under a number of simplifying assumptions: isotropic crystals and interface, broken bond energy proportional to the surface energy which in turn is proportional to the cohesion energy, etc. As far as image forces are concerned, they may strongly in¯uence which interface plane is of low energy [48]. Fig. 8 shows a possible faceting of such an interface so that the image forces more e€ectively increase the interfacial adhesion than in the case of a {001} surface. The image forces may have implications for the LEORs sintered to ionic substrates (Section 2.6.2) or precipitates growing in a metal matrix as well as stability of thin metallic ®lms. The subject was analysed in detail by Stoneham and Tasker [48]. As shown in Fig. 72, the same interface may possess three di€erent energies. Fig. 72a shows a lock-in structure (Section 2.6), where the energy is lowered compared to the structure when the interface is exactly parallel to the (001) plane (Fig. 8b). Intrusion of metal in valleys in the surface of the ionic crystals may permit to increase the image forces and adhesion, without changing the overall charge neutrality (Figs. 8a and 72a). The image forces lower the surface energy when the two metal ®lms deposited on the surfaces of di€erent polarity contact each other (Fig. 72b) and cancel the long range electric ®elds. On the other hand, if the two ®lms are not connected (Fig. 72c), the long range electric ®eld is not cancelled, and the system might have a higher energy than in the case shown in (Fig. 72b). One of the basic questions, however, is whether the bond is of the Me±O±Me ' or Me±Me'±O type. Usually, it is assumed that this is the Me±O±Me ' bond. However, recent experimental techniques permitted to identify situations when the interface is bonded according to the Me±Me'±O scheme. Further, the type of bond may depend on thermodynamic conditions and/or segregation. The metal± oxide interface is strongly susceptible to thermodynamic factors in the same way as the oxide±oxide interface. Here we will list some recent experimental results. A recent study of the interfaces Cu/Al2O3 and Ti/Cu/Al2O3 by Dehm et al. [237] clearly indicates the role of gas partial pressures for the bonding at the interface. Considering the growth mechanism of Ti on sapphire, the authors stated that the classical Bauer's model [238] predicts a 3D growth mechanism because

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the surface energy of sapphire is less than the sum of the surface energies of titanium and of the Ti/sapphire interface. The authors applied the Markov and Kaischev thermodynamic correction to the Bauer's criterion for a 2D growth model: gS rgI ‡ gTi ÿ

Dm , O 2=3

…174†

where gS is the surface energy of sapphire, gI is the energy of the Ti/sapphire interface and gTi is the Ti surface energy, O 2=3 is the area of an adatom on the surface plane [237,239,240]. The chemical potential Dm ˆ kT ln w, where w is the supersaturation of the Ti vapours during Ti evaporation at the substrate temperature T. The above thermodynamic term relatively increases the energy of the surface of sapphire and thus increases interfacial adhesion. Structure observations and calculations. Dickey et al. [241] have shown that for Ni/ZrO2 interface formed by reduction of NiO±ZrO bonding occurs between Ni and Zr instead of Ni and O. On the other hand, Zhao et al. [29] observed that in the Al2O3±Cu interface the usually assumed competition between Cu and Al for oxide formation is complicated by the signi®cant Cu±Al metallic/covalent component to that bond. TEM observations of the interfaces in the MgO/Ag, MgO/Au and CdO/Ag systems, con®rm the prediction of relatively delocalised cores of mis®t dislocations. Hoel describes the structure of the MgO/Au interface as parallel bands of delocalised mis®t [242]. Mis®t is delocalised over several interatomic distances in the Ag/CdO interface [174]. In the Ag/MgO interface, the mis®t dislocations seem to be delocalised. The ``stand o€'' distance for mis®t dislocations does not exceed one atomic spacing [243]. Jang et al. [244] were in the position to determine uniquely the type of bonding for MgO precipitates produced by internal oxidation of copper. They stated by means of HREM studies faceting of the surfaces parallel to the {111} planes, which agrees with the idea that charged planes are the low energy ones when the precipitate is embedded in a matrix. Furthermore, they determined the chemical composition of the interface by using a ®eld ion microscope combined with a mass spectrometer. This method permitted to state that the interface bond was of the Cu±O±Mg type and not of the Cu±Mg type. The above orientation of the precipitate facets is typical for precipitates grown in equilibrium conditions. Special techniques may lead to interfaces of di€erent, non-equilibrium shape [245]. The Pd±O or Pt±O bond is expected to be weak so that mis®t dislocations should have delocalised cores, as in the Ag±MgO case [246]. Groen and de Hosson observed recently the structure of interfaces between Pd and ZnO [247]. The interfaces were produced by internal oxidation of a Zn±Pd alloy. It seems that no mis®t dislocations were observed. The preferred surface plane is {111} and {001} on the Pd side. When a material containing the Pd±Al2O3 interface is annealed in an oxygen pressure, the interface structure depends on the weak Pd±O bond [248,249,250]. However, the noble metal can contact the metal of the oxide,

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thus forming a strong bond. Such strong bonding would explain the frequent observations of localised dislocations networks, as in Ref. [251]. Lipkin et al. [252] predict a high work of adhesion for the Pt and Pl±Al2O3 interface due to the metal±aluminium bond active here instead of the van der Waals forces. Zhao et al. [29] have shown that surface reconstruction can lower the work of adhesion by a factor of 3 in the case of Al2O3/Cu interface. On the other hand, Gegner et al. have shown a strong e€ect of oxygen and hydrogen segregation on the structure of the Ag/MgO and Pd/MgO interface structure [253] and therefore on its energy. The interface structure in¯uenced oxygen or hydrogen segregation, which in turn in¯uenced the energy and structure itself. The control parameter was the partial oxygen and hydrogen pressure. Another example of interface bonding engineering is given in the paper of Shieu et al. [254]. The shear strength of the Pt±NiO interface was measured as a function of thermodynamic conditions during bonding of Pt and NiO. For the case of untreated interfaces, Pt atoms are neighbours of the O atoms and bonding was via van der Waals forces. A suitable choice of annealing temperature, time and oxygen partial pressure permitted to produce at the interface either a layer of the intermetallic compound PtNi or a NiPt solid solution. Therefore, the two materials were bounded by a metal/metal interface and a metal/ceramic interface where the principal role was played by the strong Ni±O bond and not by the weak Pt±O bond. The above treatment resulted in a ®ve to ten times higher interface shear strength. As far as vacancy segregation in oxides is concerned, it can introduce charged states which considerably modify the interface potential, as shown, for instance, by Du€y et al. for the NiO/Ag interface [255]. In frequently studied precipitate systems where the oxide grows by internal oxidation, Backhaus-Ricault [256] has shown the complexity of factors in¯uencing the orientation relationships and morphology of the precipitates: chemical reaction at the atomistic level, energy minimisation by minimisation of volume, elastic strains and interface energy, kinetic factors. Therefore, before drawing conclusions for the relationship between the interface energy, structure and OR, all the factors controlling the system energy must be well understood. 2.5.5.3. Interfaces in nitrides. The analysis of interfaces in nitrides leads to the same conclusions and equations as for the case of oxides. The qualitative di€erence is, however, that the N±N bond is strong (4.88 eV/atom). The competition between the N±N bond and Me±N bond causes that the nitrides are unstable at high temperatures, unless annealing is carried out under sucient nitrogen pressure or in an atmosphere of atomic nitrogen with unsaturated bonds [257]. Formation of the N±N bonds is equivalent to strong surface reconstruction illustrated in Fig. 63e, which cancels the cohesive contribution to the interface bond. Furthermore, nitrides may have a higher degree of covalence (Section 2.1.3.6.2) of the bond than oxides. This observation qualitatively explains the observed instability of general GBs in nitrides like Si3N4, where a thin oxynitride ®lm is present

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Fig. 73. The interface between MgIn2O4 spinel and the MgO (001) substrate. Courtesy Hesse et al. [12,13]. (a) The case of rapidly growing thin ®lm. Migration of the interface occurs by glide of mis®t dislocations on (011) planes. Glide is enabled by two components of the Burgers vector, viz a/2 [010] parallel to the interfaces and a/2 [001] perpendicular to the interface. A consequence of the perpendicular component is the slight tilt of the MgIn2O4 lattice. (b) The case of slowly growing thick ®lm. In this case, the perpendicular component of the Burgers vector is missing and migration must occur by dislocation climb. The MgIn2O4 lattice is not tilted in this case. Published with kind permission of Taylor and Francis Ltd.

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[258,259]. Studies of interfaces in an internally nitrided Ni±Ti alloy have shown that the common criteria for ORs: plane matching, O-lattice, CSL, each explains some orientation, but no single criterion explains all the ORs [260]. One of the most recently studied interfaces is the GaN/Al2O3 interface [261,262]. The high density of mis®t dislocations is the main problem for the development of the semiconductor blue laser [263]. 2.5.6. E€ect of interface kinetics on its structure Hesse et al. [11,12] studied recently the reactive interfaces for the MgIn2O4 spinel on a MgO substrate. They have found that the interface structure depends on its kinetics. Fig. 73 provides an example how thermodynamic non-equilibrium in¯uences the interface dislocational structure [12]. The HREM image shows mis®t dislocations that accommodate the di€erence of the lattice constants of the two phases. Hesse et al. [11] have shown that this interface has a di€erent structure for two di€erent thicknesses of the growing spinel and di€erent growth rates. In Fig. 73a, the Burgers vectors of the mis®t dislocations are parallel to the interface. The MgO layer grows slowly by a parabolic law while the mis®t dislocations climb. On the other hand, in Fig. 73b, the Burgers vectors of the mis®t dislocations are inclined to the interface plane. The MgO ®lm grows with a combination of dislocations glide and climb so that the rate of the ®lm growth is faster and the low linear growth obeys. Hence, the dislocation structure of the interface is di€erent for the two growth regimes. For the interfaces studied mis®t, dislocations cores are the place where the actual reaction leading to new phase growth takes place. In Ref. [11], the example of the formation of MgAl2O4 spinel by reaction of MgO and sapphire shows MD that retard the reaction which is proceeding faster along coherent sections of the interface than the rate of MD dislocations climb. It follows from the above that MD dislocations may play a crucial role for the interface kinetics in reaction interfaces. Furthermore, the interface structure depends on the kinetic conditions. In the given example, its structure adjusted to increase the reaction rate. 2.6. Sintering of spheres experiments In the present section, we will report some results of experimental investigations of LEORs using the ``sintering of spheres technique''. This technique permits to determine the set of low energy interfaces by sintering single crystalline spheres sintered to single crystalline substrates. Shiroko€ et al. [264] reviewed the evolution of this technique, originally introduced by BaÈro and Gleiter [265]. They have shown that the technique perfectly predicts the low energy epitaxial orientations. The technique is as follows. A large number of spheres of random OR are put on a single crystal surface and sintered. During sintering, the spheres rotate to decrease the energy of the interface at the neck of the sphere (Fig. 74). The low energy ORs correspond either to parallel planes (surface type cusps) or periodic interfaces (adhesion type cusps, Section 2.5.1) (Fig. 75). One can imagine

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Fig. 74. Single crystalline silver spheres sintered to a single crystalline silver plate. (a) Initial state. (b) After sintering 500 h at 5 K below the melting point.

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Fig. 75. Schematic presentation of the rotation of a sintered sphere. An OR with parallel planes and two equivalent coincidence ORs.

this process as movement of the spheres towards energy valleys in the 2D space, where the independent variables are the two angles determining the OR of a sphere. Afterwards, the OR can be determined from the texture plot using calculation techniques [73,74]. However, during the interpretation of the results, special attention must be paid to two factors. First, there is a stochastic factor in the texture plot, both due to the statistical nature of X-ray scattering and number of spheres gathered by each energy valley [266]. Disregard of this factor may lead to completely unsubstantiated conclusions [266], as in the work of Maurer [267], who claimed to identify ORs which in fact were represented by less X-ray maxima than for a completely stochastic texture. On the other hand, this statistical approach permits to identify all the energy valleys in the 2D OR space. Second, the interfaces at the sphere neck need not be exactly parallel to the single crystal surface. 2.6.1. Sintering metal spheres to metal substrates Fig. 75a shows silver spheres on a silver plate before sintering. Fig. 75b shows the same plate after sintering. Using this technique, Lojkowski et al. [73,74] determined the low energy GBs formed after sintering of silver and copper plates

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at the temperature 5 K below the melting point. All the GBs found were of [011] tilt type, with the GB plane of the sphere of the type: {n, n, m }, where m > n, n ˆ 0 or 1. In other words, the pole of the plane lies between the [001] and [111] directions. Furthermore, the {111}6{001} asymmetrical GB was identi®ed as well. The above result is in excellent agreement with the Paidar's hierarchy [140] of GBs (Fig. 43). The decomposition of the identi®ed GBs into symmetrical and

Fig. 76. In GB plane or ``plane view'' presentation of the structure of low energy GBs at the necks of the sintered spheres (Fig. 75). Circles with points represent atoms crossed in the centre by the drawing plane. Small circles represent atoms situated deeper below the drawing plane [73,74].

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Fig. 77. (a) The low energy {111}6{100} GB. (b) The initial increase of energy one grain is tilted and steps-dislocations are introduced. Every atom on the ledge has two more broken bonds than on a surface atom. ``Locking'' of every nth row of atoms on the surface of the upper grain into valleys in the surface of the lower grain gains three new bonds, so that an edge atom has more neighbours than a surface atom [73,74].

asymmetrical SEs-ledges was demonstrated. All the identi®ed GBs can be regarded as built from SEs from (111) or (001) symmetrical GBs or from (111)6(001) asymmetrical GBs. In fact, the interpretation of the above results was based on the idea of predominant faceting parallel to {111} planes. This assumption agrees with HREM observations of the interfaces in the Ag/Ni system by Gao and Merkle [211,212]. Fig. 76 shows schematically the proposed structure of the low energy GBs found in the above experiment. To visualise stepped surfaces, atoms were represented as spheres cut by the plane of the drawing. A small circle means that the edge of the sphere just touches the plane and the centre of the atom is below the plane. Fig. 76a shows the (111) surface of the plate. The S3 twin GB is obtained by putting the (111) plane of the sphere (Fig. 76b) on the plate. Putting the (ÿ1, ÿ1, 5) plane of the sphere (Fig. 76c) on the (111) plane of the plate also results in the S3 GB. It is clearly seen that this representation corresponds well to the Structural Units-ledge model of Bishop and Chalmers [64]. For the case of the (1, 1, 5) plane (Fig. 76d), the S9 GB is obtained. Fig. 76e and f show the (1, 1, 19) and (ÿ1, ÿ1, 19) sphere planes which correspond to the S11 and S33 GBs. Finally, Fig. 76f shows the (001) plane of the sphere which when combined with the (111) plane of the substrate gives the only non-

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Fig. 78. Comparison of the S11 {1, 1, 19}6{111} and the {13, 13, 5}6{111} GB. The former is connected with one of the most important cusps, the latter collected no spheres [73,74].

coincidence GB observed. The interface from Fig. 76f is a low energy GB of ``surface type'' (cf. Fig. 65), according to the terminology of Section 2.5.1. Fig. 76 shows that for the structure of low energy, GBs results from a compromise in an attempt to achieve two goals at the same time: a period as short as possible and long segments of {001} and {111} planes. They all are examples of LEORs of ``combined surface and adhesion type'': cf. Fig. 65, Section 2.5.1. How such a compromise is possible is explained in Fig. 77 [74]. In the following qualitative considerations, we will assume that the GB relaxation

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introduces a small energy change to the energy resulting from the contact of two non-relaxed surfaces. Fig. 77a shows the (001)6(111) GB, corresponding to a ``surface type'' energy minimum. Fig. 77b shows that tilting the (111) planes is equivalent to introducing an array of steps. For each atom on the ledge of the step, there are two broken bonds. Initially, this causes an increase of energy since steps are introduced. This increase takes place for each tilt direction, so the (001)6(111) GB clearly corresponds to a minimum. Every atom on the edge of the step has two broken bonds [74]. However, if the condition 1 2 ‡ 1 2 ‡ n 2 ˆ m 2 , where n, m are integers, is ful®lled, the ledges of the steps can ``lock'' in valleys on the surface of the (001) plane, thus forming three additional bonds. In this way, the ledges of the steps do not contribute to energy. Fig. 77c shows the S11 GB created in the above-described way. It combines the two features that lead to a decrease of energy: periodically locked atomic rows and low energy structure between the locked rows, similar to that of the (111)6(001) GB. As a ®nal step, the structure of the interface is permitted to relax. Thus, this GB corresponds to a ``combined energy minimum'', cf. Fig. 65. At the same time, this type of GB may facet into symmetrical (111) and (001) GBs according to the hierarchy of Paidar (Fig. 43). The above GB corresponds to a particularly important energy minimum, since it was represented by second after the twin number of spheres during the rotating sphere experiments [73]. Fig. 77d represents the S11 GB where the segments of the GB between the locked rows are not segments of the (111) and (001) planes. This GB was not found during the rotating sphere experiments. Fig. 78 shows the two S11 GBs on a projection along the [011] tilt axis. The above picture is consistent with the hierarchy of GBs according to the interplanar spacing introduced by Paidar [140,141]. It seems that close to the melting point, only GBs parallel to the {111} and {001} planes are represented in the hierarchy. In other terms, the special±general GB transformation [134] is in reality a climb in Paidar's hierarchy. All GBs where the spacing of planes parallel to the GB are less than for the {111} and {001} planes disappeared as hierarchy steps. Hence, we propose the following model of GB transformations with increasing temperature. At low temperatures, the structure of GBs is in¯uenced by GBs on the bottom of the Paidar's hierarchy (Fig. 43). As the temperature increases, the GBs of small interplanar spacing disappear as hierarchy steps, and the GBs higher in hierarchy take the role of sources of SEs. At the melting point, only the S ˆ 1h011i f001g, S ˆ 1, 3 symmetrical tilt h011i f111g and the non-coincidence asymmetrical tilt h011i {111}6{001} GBs remain as hierarchy steps. The stability of these GBs can be explained in terms of stability of the structure formed from segments of closely packed atomic planes and rows. In the fcc lattice these are only the {111} and {001} planes and h011i rows. A further consequence is that the GBs even at 5 K below the melting point (sintering temperature) remain structured.

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Fig. 79. The LEORs found during sintering of single crystal spheres from noble metals to ionic substrates [146,173]. All types of parallel ORs, which were found, can be schematically plotted as symmetric arrangements of triangles, squares and parallelepipeds. The particular ORs found during a sintering experiment depend on the system, temperature and substrate orientation [172].

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2.6.2. Sintering metal spheres to ionic substrates Fecht et al. [76,173] and Shiroko€ et al. [264,268] carried out a systematic study of the LEORs for spheres made of various noble metals sintered to ionic crystal substrates. The method they used permitted to sinter in one experiment up to 107 spheres, so that the obtained results are particularly statistically meaningful. The variables in the experiment were: (a) Lattice mismatch: comparison of the Au/NaCl system and the Au/KCl system. (b) Crystalline structure of the substrate: comparison of cubic and hexagonal substrates, various substrate orientations. (c) Chemical e€ects: comparison of gold and silver. (d) Temperature. Consider the LEORs for the Au/LiF system. This system is characterised by a very small mis®t: 1.3% and a relatively large number of LEORs. Here they are listed, ordered according to the relative number of cached spheres. 1. 2. 3. 4.

(001)Au6(001)LiF (every one locked row): (1ÿ11)Au6(001)LiF (every seventh locked row). (2ÿ21)Au6(001)LiF (every third locked row). (1ÿ12)Au6(001)LiF (every fourth locked row).

For all the LEORs, the [110] directions of the NaCl substrate and Au are parallel. Identical LEORs have been found for the Au/MgO system. The above result can be regarded from several points of view. 1. Number of locked atomic rows [173]. The above con®gurations ensure that inducing a 1.2% strain, every one, seventh, third and fourth atomic row of gold is locked in an energy valley on the surface of the NaCl substrate (Fig. 31b). However, it is seen that the energy does not scale exactly with the number of locked rows. 2. Parallelism of closely packed planes Ð Paidar's hierarchy (Fig. 43). According to this hierarchy the parallelism of the (131) and (313) planes of Au should be observed, which is not the case. 3. Faceting of the interfaces and parallelism of low index directions and planes (Figs. 8 and 72). All the interfaces observed by Fecht et al. [76,173] can be schematically presented in the form shown in Fig. 79. The low energy orientations are represented by means of geometrical ®gures: the square stands for the {001} plane, the triangle for the {111} plane, the parallelepiped to the {011} plane. The above planes are aligned in such a way that low index directions are parallel. By such alignment closely packed rows of atoms of one crystal that happen to extend along valleys on the surface of the other crystal are able to ``lock'' and thus decrease the distance separating the crystals. Therefore, the van der Waals and image forces bonding the crystals are strengthened [151]. On the contrary, if such rows are not parallel, strains can be induced at their crossing (Fig. 56).

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Fig. 80. The pole-®gure for Ag spheres sintered to a LiF substrate with plane macroscopically parallel to the (011) plane. The points represent intensity maxima for X-rays scattered form the {111} planes of the Ag spheres [75].

A striking result is that in the case of the Ag/NaCl interface nearly all spheres gathered in the cube/cube OR, while in the case of the Au/NaCl interface all the above LEORs were represented. The di€erence in mis®t is only 0.2%. This result may be connected with di€erent electronic structure of the two metals and energetics of the image forces in the two cases. Despite identical lattice constants, an atom of gold is almost two times heavier than that of silver and has 79 electrons compared to the 47 of silver. There is also evidence of faceting of the substrate according to the idea of maximum image forces (Fig. 72). Fig. 80 shows the pole ®gure of Au spheres sintered to the LiF substrate of surface macroscopically parallel to the (110) plane. The points represent intensity maxima for X-rays scattered from the {111} planes of the Ag spheres. It is seen that the {111} planes of the spheres are parallel to the (111) and (111) planes of the substrate. Furthermore, the {111} planes of the

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Fig. 81. Sections of the {111} texture for Au spheres sintered to a sapphire substrate. At 5508C, the (111) plane of the spheres is parallel to the (0001) plane of the substrate, and the twist angle is well de®ned. At 6508C, the (111) plane of the spheres is parallel to the (0001) plane of the substrate but the twist angle is random. At 9008C, there is no preferred orientation any more [75].

spheres are parallel to the (100) and (010) planes of the substrate. To interpret the above ®gure, it was assumed that the substrate is facetted parallel to {001} planes and the spheres adjust the orientation to produce a {111}Ag6{001}LiF interface. It can be seen that various triangle/square orientations (Fig. 79) are present. Besides that there is a band of points showing that a number of Ag spheres is aligned, so that their {111} planes are parallel to the two {001} planes of the substrate, but the twist angle perpendicular to the {111} plane is random. This may indicate that the thermal vibration energy is comparable to the energy gain when closely packed rows of atoms are parallel. Temperature e€ects depended on the interface studied. For the Au/MgO interface, special LEORs were observed for temperatures up to 9008C. For the

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Fig. 82. LEED di€raction patterns from thin ®lm evaporated on substrates and annealed. Gold on silicon and CaF2. (a) Room temperature, Si substrate. (b) 413 K, Si substrate. (c) 643 K, CaF2 substrate. The calculated and measured pattern is shown. At higher temperature, the di€raction spots are di€use. The ORs can be interpreted either in terms of parallelism or approximate coincidence [270].

Cu/MgO and Au/MgO case, LEORs were observed at 5508C, but not any more at 9008C. An increase of temperature causes ``de-locking'' of the atomic rows and only the planes remain ®xed [269] (Fig. 81). This takes place in two steps: ®rst the atomic rows become unlocked and the twist angle becomes random for a given interface plane. In the second step the interface plane becomes random as well. In the case of the Au/MgO case, the ®rst LEOR to disappear is the one with the lowest fraction of locked rows. Fig. 82 shows LEED (Low Energy Electron Di€raction) from gold ®lms evaporated on silicon and CaF2 [270]. The gold ®lm formed small islands on the surface of the substrate. In the case of gold on silicon, the gold particles aligned according to approximate coincidence relationship. This corresponds to the relatively high value of the p parameter for that system: p10:5: On the other hand, gold particles on the CaF2 substrate are aligned according to the parallelism criterion, and furthermore, the twist angle is to a high degree random. This corresponds to the low p value: p ˆ 0:1:

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It seems from the above that the most important factor for the energy of the interfaces studied is to maximise the fraction of closely packed atomic rows that are locked in the valleys on the surface of the ionic crystal. The driving force is presumably the image interaction described by Stoneham and Tasker, where the energy of interaction is inversely proportional to the distance between the metal crystal and ionic substrates. Locking of rows permits to decrease the above distance. Locking can be achieved in three ways: (a) by tilting the spheres at constant tilt axis until some fraction of the spheres is locked at relatively small strain, (b) by orienting the substrate and the metal parallel, so that the chance that a closely packed row of atoms is parallel to an energy valley is increased, and (c) by faceting of the substrate so that the conditions (a) and (b) are best ful®lled. Which of the above ways is realised, depends on temperature, mis®t and energy gain caused by lock-in. It has to be stated that the poor theoretical understanding of the interfacial bonding in the above systems makes it dicult to attribute a localisation parameter value. The uncertainty comes from the fact that the contribution of the image forces, its relative role for di€erent systems and the e€ect of lock-in on its enhancement is not known. Nevertheless, it seems that most important is the parallelism criterion, which is typical for interfaces of low p value. 2.6.3. Experiments with single rotating sphere Chan and Ballu [271] investigated in situ by means of TEM the process of sintering of silver spheres, 100 nm in diameter, to a silver substrate. They found that the spheres rotated into several coincidence ORs and one 458(001) twist OR. According to the terminology of the present paper, the last is the 458cube/cube OR. The above result is in agreement with the rotating spheres experiments of Lojkowski et al. [73], where during sintering of copper and silver spheres to substrates both coincidence and parallel ORs were found. 2.6.4. ``Smoke'' experiments An experimental technique that is similar to the rotating spheres experiments is the ``smoke'' technique where particles of one substance are obtained by burning metal in air and collecting them on a substrate. Recently, Gao et al. [272] applied this technique developed by Chaudahari and Matthews to study interfaces between MgO and a Ni single crystal. The above technique is particularly suited to study interfaces for ®xed interface plane on both sides. Assuming some arbitrary values for the tolerance parameter (Section 2.2.3.2), Gao et al. identi®ed a number of LEORs as interfaces with a high planar density of coincidence points [272]. Such an approach is justi®ed in terms of the SDW theory (Section 2.3.6), since near coincidence means that there are vectors of the inverse lattice of the

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Fig. 83. Geometrical factors contributing to low energy of interfaces. A combination of two or three criteria leads to particularly low energy interfaces.

two crystals close to each other and thus static distortion waves minimising the energy. 2.6.5. Summary of the sintering of spheres experiments It is clearly seen from the sintering of spheres experiments and other quoted results that the LEORs depend on the speci®c system and thermodynamic conditions. Nevertheless, three major geometrical factors can be distinguished: (a) parallelism of closely packed atomic rows, for fcc materials primarily the [011] rows, (b) parallelism of low closely packed planes, and (c) coincidence. The criterion of parallelism of closely packed planes explains the low energy of h011i tilt GBs in fcc metals and the LEORs for noble metals on ionic crystal substrates at low temperatures. It can be rationalised in terms of the SDW theory (Section 2.3.6). Parallelism of low index compact planes of the two crystals is an energylowering factor according to various GB models and energy calculations. We may just mention the computer calculations [63,95,142,144,145], the hierarchy of GBs

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Fig. 84. Geometry of a silica precipitate in a GB in a copper bicrystal. According to Mori et al. [273].

according to Paidar [140] and the general explanation in terms of work of adhesion given in Section 2.5. Coincidence as an energy-lowering factor is well documented. The experiments with sintering of spheres show that the deepest energy minima correspond to a combination of the above factors. This is shown schematically in Fig. 83. 3. Grain boundary kinetics In this part of the paper, we will discuss some recent results concerning interface phase diagrams and phase transformations. In the interpretation of the results, the parameters and concepts developed in Section 2 will be used. Furthermore, the possibility of the size e€ect on the interfaces physics will be stressed, in the example of some new results concerning nanocrystalline materials. 3.1. Grain boundary phase transformations 3.1.1. Grain boundary energy in bicrystals As an introduction to the GB phase transformations, we will present some recent precise measurements of the energy of GBs, in which the GB intersects a liquid/solid interface or a crystal±amorphous material interface. During such experiments, it is usually assumed that the energy of the interface between the crystal and the amorphous or liquid state is isotropic. Fig. 84 shows the idea of the measurement [273]. The amplitude of energy variations with misorientation is less pronounced for [011] twist than that for [011] tilt GBs [273]. This result is expected since for twist GBs the collectivity coecient is low (Section 2.4.8). Fig. 85 shows the importance of small changes of oxygen partial pressure for the energy of Cu±SiO2 interfaces. This result con®rms the strong dependence of properties and presumably structure of interfaces in oxides on the thermodynamic

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Fig. 85. E€ect of misorientation angle on the GB energy in copper bicrystals for [011] tilt GBs. The two curves correspond to two partial oxygen pressures. The principle of measurement is shown in Fig. 84. According to Miura et al. [273].

conditions. In Section 2.5.5, the e€ect of the oxide formation energy, including the e€ect of partial gas pressures, on the structure of interfaces was discussed. Otsuki et al. [274] measured the dihedral angle at the intersection of symmetrical GBs in aluminium and the Sn±20at%Zn melt at 2408C, i.e., approximately 0.6 Tm, where Tm is the melting point of aluminium. The technique discussed here is one of the most sensitive techniques for GB energy measurements. Presumably, the reason is the high di€usion coecient in the liquid phase, which leads to quick equilibration of the wetting angle even at low temperatures. Fig. 86 shows a large number of energy minima compared to Fig. 85. This is a manifestation of the low temperature of measurements and high sensitivity of the technique based on wetting with a liquid. In the experiments of Miura et al. [273] the annealing temperature at which the dihedral angle equilibrated was about 0.97Tm. (Fig. 85). The stochastic factor discussed in Section 2.2.3.1 leads to a disappearance of a large fraction of energy cusps at high temperatures. According to Fig. 86, in the case of the [001] misorientation axis the twist GBs

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Fig. 86. The wetting angle at the intersection of GBs in aluminium and a melt of composition Sn± 20at%Zn. Full bullets: [100] twist GBs. Open circles: [100] tilt GBs [274]. Reproduced with permission from J de Physique, Colloque C5, 1988, 49 (Suppl. au no. 10).

have a lower energy than the tilt GBs. This may indicate that for the h001i rows of atoms, which are not so closely packed as the h011i rows, the collectivity coecient is lower than in the case of h011i GBs. As a consequence, the ``lock-in'' factor (Sections 2.1.4.1, 2.2.4.2 and 2.6.5) is important only for [011] GBs where the collectivity coecient has a high value and closely packed rows of atoms interact with each other. The relatively small depth of energy cusps for twist GBs is clearly seen from computer calculations of [001] twist GBs by Vitek [275]. 3.1.2. Relation between grain boundary energy and free volume The study of the pressure e€ect on the GB energy may permit one to assess the GB free volume. Lojkowski and Otsuki [276] attempted to determine the free volume of GBs in aluminium bicrystals using the experimental methods developed by Otsuki et al. [274]. The e€ect of pressure on the wetting angle for aluminium

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Fig. 87. The e€ect of pressure and misorientation on the wetting angle for grain boundaries in aluminium bicrystals wetted by the Sn±20wt%Zn alloy.

bicrystals wetted by an Sn±Sb alloy was studied as a function of misorientation (Fig. 87). The misorientation angle varied from 08 to 168. The grain boundaries were of [100] symmetrical tilt type. It was assumed that the liquid/solid interface is isotropic as far as free volume and the di€erences in pressure e€ect on the wetting angle between GBs of di€erent misorientations will re¯ect the di€erences of their free volumes. However, it was found that up to 1.2 GPa there is no pressure e€ect on the wetting angle in the whole misorientation range. The above results can be rationalised as follows. The pressure P e€ect on the wetting angle …y† is expressed as the derivative [40,42]:   @j @ gGBO ‡ PnGB ˆ arccos , …175† 2gSLO ‡ 2PnSL @P @P where gGBO and gSLO are the surface tensions of the grain boundary and solid/ liquid interface at zero pressure, respectively, and nGB and nSL are the free volumes of these interfaces, respectively. The experiment has shown that the wetting angle does not depend on pressure for the whole misorientation range. The condition that the wetting angle is constant, as a function of pressure and misorientation angle y: @j=@ P ˆ const…y† ˆ 0, leads to the following expression: gGBO g ˆ SLO , nGB nSL

…176†

which means that the di€erence in energy of the liquid/solid and solid/solid

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Fig. 88. (a) The arrangement for the measurement of the ratio of the energy of two GBs. The general GB moves at steady rate. For steady state migration, the dihedral angle is determined by the ratio of GB energies. (b) The arrangement for measurement of GB migration rate at constant driving force. According to Maximova et al. [277]. Reprinted from [277], copyright 1998, with permission from Elsevier Science.

interfaces results from the di€erence of their free volumes, irrespective of the structure. It follows that it is not the topological structure but the free volume which is the major factor contributing to both the GB and the liquid/solid interface energy. In this way we reach the simple equation for the energy of a GB or L/S interface: g ˆ C
n,

…177†

3.1.3. Energy of grain boundaries in tricrystals and grain boundary phase transformations Maximova et al. [277] measured the dihedral angle y at the intersection of two GBs using the tricrystals as shown in Fig. 88a. The angle y depends on the ratio of the surface tension of the two GBs Ð the one described as ``special GB'' and the other as ``general GB''. The special GB was a symmetrical tilt [001] GB with misorientation angle close to the S17 GB. It was found that the ratio …s1 =s2 † displayed a singularity when crossing the

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Fig. 89. Map of the regions of existence of the GB structure with SEs from the S1 and S17 GBs in tin. The variables are the misorientation and temperature. The open circles show the phase transformation line as measured by means of the tricrystals (Fig. 88a). Reprinted from [277], copyright 1998, with permission from Elsevier Science.

phase transition line in Fig. 89 at constant temperature or at constant misorientation. In a further series of experiments, the GB migration rate for a range of misorientations close to the symmetrical tilt S17 was measured using the technique shown in Fig. 88b. The phase transition line measured by both methods was identical. Assume that the entropy of both GBs is di€erent (Fig. 90). At the points of crossing of the two GB energies, a phase transition will occur and the lowest energy structure will be assumed by the special GB. Maximova et al. [278] classi®ed this transformation as the special 4 general GB transformation. However, taking into account the classi®cation of Paidar (Section 2.3.4), it seems that the transformation that really takes place is a ``climb in hierarchy'',

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Fig. 90. Schematic presentation of the temperature dependence of the energy of two GBs and of the phase transformation (change of position in hierarchy).

i.e., change of GBs which are a source of SEs. The S17 GB ceases to be the reference GB and is replaced by the S1 GB parallel to the (111) and (110) planes. 3.1.4. In¯uence of a small additional disorientation component on GB energy The above experimental techniques were used to study the e€ect of ``subsidiary misorientation components'' on the GB energy. Fig. 91 shows two ``subsidiary'' angles describing a small deviation from an exact symmetrical tilt OR shown in Fig. 91a. Fig. 91b shows a small twist rotation yK , whereas Fig. 91c a small tilt yK perpendicular to the [001] tilt axis of the symmetrical tilt GB [278]. Fig. 92 shows the e€ect of these angles on the GB energy, in comparison to the misorientation angle for ®xed tilt axis. The e€ect of the additional disorientation on the energy is smaller than the e€ect of variation of misorientation at constant axis. If to regard the LEOR as a valley in the misorientation space, we see that, the slope around the energy minimum depends on the direction. Perhaps the slope depends on the Burgers vectors of the SGBDs. 3.1.5. In¯uence of a small additional disorientation component on GB di€usion Budke et al. [279] measured the e€ect of the misorientation on Ni and Au di€usion in copper bicrystals. The particular feature of their experiments was that the misorientation angle varied in very ®ne steps of about 0.18 around the S5 36.98 (310) [001] tilt GB. The method of preparation of bicrystals with such precisely de®ned misorientation and ®rst results of di€usivity measurements are

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Fig. 91. The additional small misorientation changes in a bicrystal [278]. (a) Initial symmetric [001] tilt GB. (b) An additional twist yk (c) An additional tilt yH around an axis perpendicular to the initial tilt axis. Reprinted from [278], copyright 1989, with permission from Elsevier Science.

given in ref. [280]. The di€usion coecient does not show a monotonic decrease when the misorientation approaches the exact S5 GB, as expected. The di€usivity increases gradually when the disorientation increases. At approximately 1.58 disorientation, the di€usivity drops by half an order of magnitude. The minimum of the di€usion coecient was found at about 0.58 of the coincidence misorientation (Fig. 93). At a misorientation of 368 and at the exact S5 36.98 misorientation a maximum of the di€usion coecients was found. The most striking result is that the di€usivity varies by 1 order of magnitude for a variation of misorientation angle of 18. Such accurate determinations of the misorientations have not been carried out yet. They imply that determination of activation energy for GB di€usion close to energy cusps is meaningful only if the same bicrystal is

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Fig. 92. E€ect of additional (subsidiary) small disorientation angles on the S17 GB energy in tin in comparison to the change of misorientation. Df is the change of misorientation. Reprinted from [278] copyright 1985, with permission from Elsevier Science.

used at various temperatures. Otherwise, the misorientation is not de®ned with sucient precision, and it is not clear what is the misorientation and what the temperature e€ect. In the temperature range 600±800 K the activation energy for di€usion strongly depends on misorientation and shows the opposite behaviour to the di€usion coecient (Fig. 94). However, at temperatures above 800 K the activation energy does not depend on misorientation (horizontal broken line in Fig. 94). Therefore,

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Fig. 93. The di€usivity as function of misorientation for Cu bicrystals close to the S5 [279]. Reproduced with permission from Budke E, Herzig Chr, Proko®ev S, Shvindlerman LS, In: Bokstein B, Balandina N, editors. Proc. Int. Workshop on Grain Boundary Di€usion and Grain Boundary SegregationÐDIBOS'97, Moscow, May 1997. Switzerland: Scitec Publications, 1998. p. 21.

from the point of view of the structural features that determine the di€usion rate, the GB studied undergoes a phase transition at about 800 K: from a structure dependent on misorientation to one which does not depend on misorientation. Such a transformation may indicate that at 8008C the cores of the SGBDs do not play any more a crucial role for GB di€usion. Budke et al. [279] interpreted the results as follows. The di€usivity is at minimum for the exact S5 GB, because there are no structural dislocations. With increasing disorientation, the density as well as di€usivity of SGBDs cores increases. The di€usivity decreases again abruptly when the cores of the SGBDs overlap. The disorientation at which di€usivity drops corresponds to the structure transformation, from one related to the S5 GB to that related to the S1 GB. The authors called it special 4 general GB transformation, and according to Paidar's classi®cation this would be a climbing in the hierarchy transformation.

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Fig. 94. Activation energy for GB di€usion in symmetrical tilt copper bicrystals with misorientation close to S5 [279]. Reproduced with permission from Budke E, Herzig Chr, Proko®ev S, Shvindlerman LS, In: Bokstein B, Balandina N, editors. Proc. Int. Workshop on Grain Boundary Di€usion and Grain Boundary SegregationÐDIBOS'97, Moscow, May 1997. Switzerland: Scitec Publications, 1998. p. 21.

However, since the bicrystals misorientation is varied and measured with high precision, one has to consider not only the tilt component for the ®xed tilt axis along which the bicrystals were grown, but also the involuntary generated twist components. VystaveÏl et al. [281] studied, by TEM, the bicrystals prepared by Proko®ev [279] and observed networks of dislocations which must in¯uence GB di€usion. The network of dislocations is connected with a twist component of the misorientation, with the twist axis perpendicular to the interface plane. VystaveÏl et al. [281] pointed out at the importance of networks of SGBDs for the di€usivity of nearly special GBs. In fact, assuming a special GB having a closely packed structure, GB di€usion takes place mostly along the SGBDs cores. Further, the Burgers vectors of the SGBDs and their core structure depend on the disorientation matrix. Therefore, in the same way as the GB energy change depends on the direction in the hyperspace of crystallographic parameters (Section

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Fig. 95. In¯uence of alloying with gold on the structure of a small angle [001] twist GB in iron. Courtesy Sickafus and Sass [284]. Reprinted from [284], copyright 1987, with permission from Elsevier Science.

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497

3.1.4) the di€usivity depends on the type and density of SGBDs accommodating the disorientation. It seems, therefore, that SGBDs play a crucial role for the di€usivity close to special ORs. If the S5 GB is a step in the hierarchy of GBs according to the hierarchy proposed by Paidar [140], then the GBs to the left of the S5 GB in Fig. 93 will be constructed from other SEs than the GBs on the right side. Hence, the asymmetry of the di€usivity is not surprising. 3.1.6. In¯uence of impurities on the special 4 general GB transformation In the above reported di€usion studies [279] a strong e€ect of impurities on the misorientation and temperature at which the special±general transformation takes place was observed. There were several papers indicating the e€ect of impurities on the interface structure and energy. For the sake of the present paper, it is interesting to note papers indicating directly or indirectly the e€ect of impurities on the delocalisation of SGBDs cores. Maksimova et al. [282] studied, using the above mentioned tricrystals technique, the e€ect of the Na impurity on the energy of GBs close to the S17 GB in tin. They found that the sodium addition changes the temperature of the special to general GB transformation. They interpreted the result as an e€ect of sodium induced widening of cores of SGBDs [282]. They came to that conclusion based on the theoretical treatment of Gleiter, who has shown that disappearance of an energy cusp and widening of SGBDs cores are parallel processes [123]. The e€ect of impurities on the stability of GB dislocations was directly observed during experiments where Cr and Co were implanted into thin ®lms of gold [283]. Greenberg et al. [283] interpreted their results in the same way as Maximova et al. [282]. Sicka®uss and Sass investigated the changes of the dislocation structure in iron with an addition of 0.17% gold [284]. Fig. 95 shows the e€ect of a small addition of gold on the dislocation structure of a tilt GB in iron. The changes of dislocation structure under small impurity additions seem to be well established, since they were also observed in Fe±Sb alloys and in high angle GBs [285]. The above results introduce the new complexity to the GB phase diagrams, where the misorientation and impurity content are variables [284]. In the light of such results questions about ``scienti®c economy'' arise, since the thoroughly investigated structures of GBs in pure metals represent a very small fraction of possible interfaces in real materials. 3.1.7. Segregation e€ects on GB kinetics Recent work on GB di€usion and migration has shown that the GB di€usivity is extremely sensitive to the segregation of even smallest amount of impurities. Surholt and Herzig [286] have shown that about 1 ppm sulphur in polycrystalline copper can cause an increase of the activation energy for di€usion by about 10% (Fig. 96). They proposed that Cu±S clusters are strongly bonded and contribute to the lowering of the GB energy. The above result shows again

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Fig. 96. Arrhenius diagram for GB di€usion of the 64 Cu tracer along GBs in Cu polycrystals of di€erent purities. A and A' Ð 99.9998% purity. B: 99.999% purity. The material B contains 1 ppm more impurities than the material A. The activation energy for A is 84.7522.75 kJ/mol. The activation energy for B is 72.4721.46 kJ/mol. According to Surholt and Herzig [286].

the importance of using the GB di€usivity as a simple index of the GB structure (cf. Section 3.2.2). Fig. 97 shows the e€ect of misorientation on the activation energy for GB migration in Al bicrystals. The activation energy depends on the misorientation angle for Al of 99.9992 wt% purity and special ORs are observed. However, for the material of 99.9995 wt% purity no misorientation e€ect is observed, providing the GB misorientation is in the high angle region. On the other hand, for a material of low purity, again no special ORs are observed. Kasen [288] has shown already in 1983 that GBs in aluminium and copper

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499

Fig. 97. Activation energy for GB migration in Al bicrystals for three degrees of purity. According to Molodov et al. [287].

Fig. 98. A series of subsequent dark ®eld TEM images of spreading of trapped lattice dislocations in a grain boundary in 4 N aluminium during in situ straining of an aluminium thin foil at room temperature [293].

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Fig. 99. Spreading and splitting of TLDs in a random grain boundary in 5 N aluminium. The annealing was carried out for 1000 s at each of the following temperatures: a) 250 K, b) 320 K, c) 350 K, d) 380 K [291].

undergo structural transformations during grain growth and the reason of the transformation is picking up impurities from the matrix. The impurity level in the matrix was as small as 1 ppm of Fe or Cr in Al or 0.7 ppm of Fe in Cu. The issue was also discussed by Menyhand and Uray [292]. The in¯uence of changing impurity content in very pure materials on GB migration is discussed in a recent paper of Shvindlerman et al. [289]. It follows that extremely small amounts of strongly segregating impurities may in¯uence the GB kinetics and phase diagrams. 3.2. Di€usion and stresses in grain boundaries 3.2.1. Relaxation of dislocation strain ®elds in grain boundaries Spreading of GB dislocations is a process observed during in-situ annealing of thin metal foils: TLDs disappear by an apparent spreading of the dislocation core [290] (see Fig. 98). Fig. 99 shows an example of spreading of a TLD at a grain boundary in aluminium [291]. The TLDs are also frequently called EGBDs (Extrinsic Grain Boundary Dislocations). The problems related to the spreading of

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Fig. 100. Schematics of the di€usion ¯uxes leading to relaxation of a TLD strain energy in a GB [297].

dislocations were recently reviewed by Priester [234] and it is shown that the micro-mechanism of TLD spreading in general GBs is still a question of discussion. Here we will present a treatment of the TLD spreading based on the continuous model of dislocation [164,293]. The advantage of the approach is that it gives elegant analytical solutions, precise enough to illustrate the basic physics of the process, and the equations obtained have a more general application than just to the TLD spreading case. Since the TLDs dissociation and spreading are physically equivalent, the two terms in the present text will be used to describe the same process. The crucial assumption here is that spreading does not cause an

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increase of the GB energy due to structural rearrangements. In the case of continuous spreading, this is because an amorphous GB is assumed. For the dissociation model, it is assumed that the TLD dissociates into small dislocations with DSC Burgers vectors. Movement of such dislocations does not change the GB structure providing the dislocations glide and climb is connected with GB migration (Section 2.2.4). Within the continuous model, spreading is equivalent to an increase of the core width as in Fig. 58. The derivative du=dx is the local Burgers vector density. The di€usion ¯ux j and the strain ®eld u…x, t† are related by the equation: @u @j …x, t† ˆ Od …x, t†: @t @x

…178†

On the other hand, the di€usion ¯ux depends on the gradient of chemical potential which is m ˆ cOds…x† where c is a constant within the range 0.5±1.0. jˆ

aOd @ s…x, t † : kT @x

…179†

Fig. 100 shows schematically the situation. The stress ®eld is build up by summation of the contribution from each dislocation with in®nitesimally burgers vector du: @ s…x 0 † ˆ

G @u : 2p…1 ÿ n † x ÿ x 0

…180†

Combining Eqs. (178)±(180), the equation for the displacement function u(x ) as obtained. @u GOdD @ 2 s …x, t †, …x, t † ˆ @t kT @ x 2

…181†

the ®nal equation is: @u GOdD @ 2 …x, t† ˆ F @x kT @x 2

"…

‡1 ÿ1

# @u dx 0 sE , ‡ @ x 0 x ÿ x 0 FG

…182†

where sE is the external stress, F ˆ c=‰…1 ÿ n†2pŠ: The above equation is the general equation for di€usion in a strained GB. It describes the kinetics of development of the GB strain ®eld u…x, t† under an external stress ®eld sE , taking into account the elasticity of the system, represented by the bulk shear modulus G, and the GB relaxation kinetics. Eqs. (183) and (184) show that the rate constant for relaxation of self-stresses include the ratio of the GB di€usion coecient and the bulk shear modulus with exponent 1/3. Its particular application is in the solution of the problem of a Peierls dislocation, which is unstable because the self-stress of the dislocation induces a di€usion ¯ux which, in turn, leads to stress relaxation. The core width is de®ned as half width of the Burgers vector density function

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503

Fig. 101. Solution of Eq. (180) Ð evolution with time of the displacement ®eld u(x) during TLD core spreading [162].

(Fig. 58). The TLD core width as a function of time is given by: r 3 t s ˆ 0:743 , t

…183†

where the relaxation time t is: tˆ

kT 2p…1 ÿ n † , GOdDGB c

…184†

where n is the Poisson ratio and c is a constant of the order of 0.5±1. Fig. 101 shows the solution of Eq. (182) as a function of time, assuming the border condition u…x, 0† ˆ b=2 arctg…x=b†, exactly as for the Peierls dislocation, and no external stress. For the above considerations, the discussion whether the mechanism of TLD spreading is described by a continuous model as in Fig. 101 or as splitting into discrete dislocations is irrelevant. The change of the energy of the system, the strain ®eld and the e€ective core width of the dislocation will be described approximately by an equation of the form of Eq. (183), where the GB di€usion coecient enters under the cubic root: s1…dD† ÿ1=3 : The exception is when the spreading process can take place by pure dislocation glide. However, since GB dislocations sliding may lead to GB migration (Section 2.2.4), this is a very special geometrical situation. The elastic energy of the dislocation depends on its core width:   R  p : …185† EEL ˆ cGb 2 ln s0 ‡ 0:74 3 t=t This is a very slow function of time (Fig. 102). The TLD image disappears for s equal to three extinction distances, which in most materials is of the order of 100 nm. However, an increase of s by two orders of magnitude in¯uences the energy only by a factor 4. In other words, even though no dislocations are visible any more by means of TEM a large fraction of their strain ®elds persist.

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Fig. 102. The rate of decay of the elastic energy of a TLD [162].

Consider a situation when a large number of TLDs dissociates. The dissociation products repell each other and their distance increases until triple points are reached and dislocation pile-ups are generated. Application of the above formalism leads to the conclusion that for sliding SGBDs large stress concentrations at GB triple joints will be generated. 3.2.2. Distribution function for GB di€usivity determined based on spreading experiments Whatever mechanism is active, the spreading rate depends on GB di€usion. This observation led Lojkowski and Grabski to propose the spreading rate as a probe of the GB structure and properties [164,291]. They have shown that the measurements by means of the spreading technique are useful to determine GB di€usivity and energy. Since the TLD core becomes invisible in the TEM at about three extinction distances, putting sM = 100 nm in Eq. (183) transforms it into an equation relating the spreading time, GB di€usion coecient and temperature tˆ

3s3M : DGdO

…186†

The technique permitted to measure the di€usivity of large numbers of individual GBs as a function of thermal history of the material and/or doping with impurities. The method was developed to measure distribution functions of GB di€usivity in polycrystals [294±296]. An important consequence of the above measurements was that it was stated that di€usivity of GBs in polycrystals vary

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505

Fig. 103. The range of values for the activation energy and pre-exponential factor for Zn di€usion and GB self-di€usion in aluminium [297].

over three orders of magnitude. It was also shown that the activation energy for GB di€usion in Al varies by a factor 5 [297] and depends on the thermal history of the material [296]. In usual tracer di€usion experiments [286] the specimens are annealed at high temperatures to reach a stable grain structure. On the other hand, the spreading method and investigations of the time dependence of the di€usion pro®les [297] permits to assess the di€usion coecients in realistic materials. Fig. 103 shows the distribution of GB di€usion parameters in aluminium measured by the spreading method and conventional methods. The activation energy varies by a factor 5 while the pre-exponential factor by eight orders of magnitude. The linear dependence of the activation energy on pre-exponential factor is called ``compensation relationship'', and was extensively discussed by Bokstein et al. in Ref. [298]. In Section 3.2.3 we will provide a short discussion of the compensation relationship. At this point however, we want to note that the fact that all the di€usion data are close to one line on the compensation diagram is an indication that there is one physical mechanism for di€usion, with di€usion parameters varying from grain boundary to grain boundary. Di€erences in D0 and Q values obtained in various laboratories may re¯ect the fact that in the samples investigated di€erent types of grain boundaries prevailed. In Fig. 103 the line for self-di€usion, which was measured using the spreading method, is shifted parallel to the line of zinc di€usion. This shift can be interpreted in terms of segregation factor or some systematic error in Eq. (183).

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Fig. 104. The e€ect of thermal treatment on the distribution function for the activation energy for grain boundary di€usion in aluminium. An aluminium polycrystal with ®ne grain size of 4 mm is compared with a polycrystal with 22 mm grain size. x stands for the fraction of GBs with activation energy higher than given in the graph [295,296].

In another study, it was shown that the range of values for GB self-di€usion coecient in austenitic steel extends over four orders of magnitude [294]. The distribution of the GB di€usivity can be in¯uenced by thermal treatment of the material [295,296] (Fig. 104). The above wide range of GB di€usivity, in addition to the fact that no reliable information about the correlation between crystallography and properties of GBs is available, has led Grabski to propose GB di€usivity itself as a GB structure parameter [299]. This approach is particularly useful to characterise the whole GB ensemble in the polycrystals in terms of GB di€usivity distribution. This contrasts with the Watanabe [300] concept of GB character distribution, where ``character'' means the whole GB crystallography. Swiatnicki et al. [301] have recently shown the usefulness of the spreading method in obtaining detailed information about GB di€usion and in¯uence of segregation on di€usion. The thermal stability of TLDs (the authors in reference use the term EGBDs) in a near S11 GBs in Ni bicrystals was studied as a function of GB plane, direction of the dislocation line and segregated impurities. It was found that in the presence of segregated sulphur, TLDs are more stable on far from coincidence GBs than on the CSL GB. The direction of the dislocation lines is also important because of the GB di€usion anisotropy. Crystallography strongly in¯uences the spreading rate because segregation is strongly OR dependent. Swiatnicki et al. [302] raised the question how to correlate the distribution

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507

function for GB di€usion and the average GB di€usion coecient in GBs. They distinguished two cases (Fig. 105). (a) the di€usion depth is much smaller than the grain size. Then the average di€usion coecient is the geometrical average of the GB di€usion coecients; s X 2 , DGBi …187† hDGB i ˆ i

(b) the di€usion depth is much larger than the grain size and the GB with low di€usion coecients determine the average matter ¯ux rate. In this case, the average GB di€usion coecient is the harmonic mean: X ÿ1 DGBi : …188† hDGB i ÿ1 ˆ i

In both equations hDGB i is the average di€usion coecient and DGBi is the di€usion coecient in the ith GB. Swiatnicki et al. [304] have shown that the average GB di€usion coecient strongly depends on the method of calculating it [298,302±304]. 3.2.3. Compensation relationship and the activation vector 3.2.3.1. The compensation relationship. The linear relationship shown in Fig. 103 is frequently observed in di€usion studies. Its formal expression is: ln…sdD0 † ˆ A 0 Q ‡ B 0 :

…189†

In the present case A 0 ˆ 1:93  10 ÿ4 20:07  10 ÿ4 Jÿ1 and B 0 ˆ ÿ41:722:1: A ' can be expressed as A 0 ˆ 1=RTC , where TC is the compensation temperature, at which all GBs have the same di€usion coecient. Since ln…D0 †0S  =k, where S is the entropy of the activated complex in the absolute reaction rate theory, S  ˆ AQ ‡ B,

…190†

where A and B are constants. Following Lojkowski et al. [297] we will assume that in the GBs the activation entropy and energy depends linearly on the number of atoms in the activated cluster, Q ˆ anc ‡ b S ˆ cnc ‡ d

…191†

where nc , collectivity coecient, is the number of atoms in the activated complex. In this way, the points along the line in Fig. 103 represent boundaries with

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continuously varying number of atoms in the activated complex. The compensation relationship can be interpreted in the framework of the heterophase ¯uctuation theory, as discussed in Refs. [298,305]. It is assumed that an activated complex during the elementary di€usion event takes the structure of the nearest phase on the phase diagram. Gottstein and Shvindlerman [298] assume that this is a ®rst order transition. A reasonable agreement with the linear regression line in Fig. 103 is obtained assuming nc in the range 1±3 on the left-hand side and nc in the range 11±14 on the right-hand side of the diagram. Indeed, the melting heat per atom is qm ˆ 0:11 eV and the melting entropy is sm ˆ 1:4k: The minimum value of the activation energy corresponds to 0.3 eV/atom, i.e, about three activated atoms. The respective value of the activation entropy corresponds to nc ˆ 1: nc in the range 1±3 may correspond to interstitial di€usion mechanisms. For the right-hand side of the linear regression line, S corresponds to 20k so that nc is 14 and Q to 1.2 eV, which corresponds to melting a cluster of nc ˆ 11 atoms. Both nc values are close to each other. In this treatment the con®guration entropy was neglected, since for 10 atoms this is k ln…10† ˆ 2:3 and is small as compared to the measured value of 20. It follows, that the regression line in Fig. 103 represents GBs where the number of atoms in the activated cluster increases from about 1 to 3 for low activation energies to 10±14 for high activation energies. Eq. (186) can be resolved into a linear relation between Q and S: S ˆ

Q : TC

…192†

According to the data of Fig. 103, TC is about 300 K below the melting point. 3.2.3.2. The activation vector. It can be observed that Fig. 103 represents a map of activated clusters for grain boundary di€usion. Each activated cluster is represented by an activation vector A with co-ordinates (S, Q). Using this terminology, the line corresponding to GB self-di€usion is shifted compared to the line for Zn di€usion by a vector …Ss , Qs ), where Qs < 15 kJ/mol and Ss < 2k: Assuming that Eq. (186) contains no systematic errors causing the parallel shift of the two lines, the di€erence of the two lines can be attributed to Zn segregation to GBs. Therefore, …Ss , Qs † is the segregation vector. The concept of activation vector and segregation vector can be further extended to include pressure or another ®eld. For the case of a system with independent variables, pressure and temperature, the activation vector takes the form:
ÿ …193† A ˆ Q , S  , V  , where Q, S and V are the activation energy, entropy and volume, respectively. As far as the segregation vector is concerned, the notation is: Ss ˆ …Qs , Ss , Vs †,

…194†

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509

where Ss is the segregation vector, and Qs, Ss, Vs are the segregation energy, entropy and volume, respectively. The limit of small concentrations is assumed. The relationships between the components of the activation vector can be explained in the simplest terms based on the ¯uctuations theory of Landau and Lifchitz [306]. The material is divided into small subsystems, undergoing energy, entropy, temperature and pressure ¯uctuations around an average value. If it is so, then heat absorbed by an activated complex is the activation energy Q. The related change of entropy is Q =TK , where TK is the temperature of the activated complex. The temperature is such to permit atom rearrangements. The linear character of the compensation relationship may be regarded as an indication that all the activated clusters leading to GB di€usion have approximately the same temperature, lying 300 K below the melting point. The above treatment can be extended to include the pressure e€ects. Indeed, assume that the activated cluster is characterised by a local pressure PC , which we will call as compensation pressure. Accordingly, the activation free energy G will increase with activation volume proportionally to the compensation pressure: G  ˆ V  PC :

…195†

In Section 3.6.2 we will show that indeed such relationship is experimentally observed. We have shown that for GBs with a variety of structures the activation vector points along de®nite lines whose inclination is characterised by the ``compensation temperature'' and ``compensation pressure''. These parameters might be the characteristics of the activated complex. The actual values of the di€usion coecients may vary a few orders of magnitude as a function of the GB structure. Therefore, it is rather the space of positions of the Activation Vector than the actual GB di€usion coecient, that is a characteristics of GB di€usion. It must be pointed out, however, that the activation vector discussed here is di€erent from the activation volume tensor introduced by Aziz et al. [85±87]. The activation volume tensor re¯ects the anisotropy of the interface and di€erence of the pressure e€ect on di€usion along di€erent directions. 3.2.3.3. Grain boundary non-equilibrium caused by dislocations Lattice dislocations in GBs induce locally strain ®elds, which may induce nonequilibrium values of the RBT vector and a high increase of the GB energy (Section 2.2.4.5). This e€ect is dicult to measure experimentally and assess theoretically. Most of the theoretical e€orts in relation to the e€ect of TLDs on GB energy are connected with the elastic strain energy of TLDs, which has to be added to the GB energy if TLDs are bounded to the GB. Lattice dislocations carry mis®t and strain energy. Once they are trapped by GBs, they contribute to the GB energy. Grabski and Korski [307] proposed that TLDs cause GB non-equilibrium. The question was analysed in detail by Grabski [308]. Besides the additional GB energy connected with the TLDs strain ®elds, there is a possible e€ect of TLDs on the GB structure, and ®nally, the products of

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Fig. 105. Two methods of averaging the GB di€usion coecient based on a known distribution function for the GB di€usivity: (a) for the case of grain size larger than the di€usion depth and (b) for the case of di€usion depth larger than the grain size. The thick lines indicate the di€using tracer. Following Swiatnicki et al. [301].

TLD dissociation may pile up at GB triple junctions leading to grains incompatibility strains. All these e€ects may in¯uence mechanical properties of polycrystals, grain growth, superplasticity, etc. Since the TLDs spreading is so sensitive to GB di€usivity (Section 3.2.1), it is possible to investigate GB nonequilibrium using the spreading rate as a probe. Pumphrey and Gleiter [309] have shown that TLDs spreading temperature on strongly curved GBs seems to be faster than the average, what points at a higher interface energy than in the equilibrium state. In a series of experiments Grabski et al. studied GB non-equilibrium induced by spreading of TLDs introduced into GBs by controlled deformation [310]. A stainless steel specimen was deformed, annealed to spread the TLDs cores, deformed again and the TLDs spreading rate measured. Fig. 98b shows the e€ect of annealing temperature on the spreading temperature for TLDs. It is seen that there is a minimum corresponding to annealing at temperatures just above the spreading temperature. Therefore, for relatively low annealing temperatures, despite the TLDs cores cannot be seen any more, the GB di€usivity is increased, in agreement with theoretical estimations shown in Section 3.2.1. The above

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Fig. 106. (a) Dependence of spreading temperature on the annealing temperature of pre-strained samples. The material was stainless steel. (b) Dependence of yield stress of samples after 1% prestraining and subsequent annealing on the temperature of annealing [310].

observation has led to the concept of two stages of GB non-equilibrium [311±313]: (1) when the cores of TLDs are localised, and (2) when the cores are delocalised but the interface is not fully equilibrated. TLDs spreading in¯uences the average strain at GBs, which is important for the mechanical properties of polycrystals (Fig. 106a). The situation is schematically illustrated in Fig. 107. The second stage is related to the fact that although the TLDs cores are wide, the long-range strain ®eld of the dislocations may still persist [162,307]. Nazarov et al. [313] have shown a

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Fig. 107. The subsequent stages of equilibration of a GB that absorbed TLDs, and the path of the GB in the energy±misorientation diagram.

particular feature of the strain ®elds of GBs with non-relaxed TLDs: since the statistically stored TLDs form a non-periodic pattern, their strain ®eld is not decaying exponentially, as is the case for SGBDs, but decays according to the 1/z law, where z is the distance from the interface. The long-range strain energy of dislocations is relatively insensitive to the core width, which enters in the logarithmic term log…R=s† (cf. Section 2.4.1). As the core width approaches the grain size, the dislocation strain ®eld gradually transforms in the incompatibility strain between grains that have mutually unadjusted shapes. Fig. 108 illustrates that idea. In that respect, the interface dislocations can be divided into two groups in the same way as Ashby [314] divided the bulk dislocations: statistically stored and geometrically necessary grain boundary dislocations [162,315,316]. Therefore, the ®rst stage of GB equilibration is the annihilation of the statistically stored GB Dislocations (SSGBDs) [162], and the second stage is the annihilation of the geometrically stored GB dislocations (GSGBDS) (Fig. 108). The latter stage is expected to involve GB sliding and migration. In terms of the Maxwell model of the interface (Section 2.4.5), the ®rst stage of equilibration is relaxation of stress via short-range rearrangements of atoms and the second stage to relaxation involving long-range rearrangements (Fig. 52).

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Fig. 108. Equivalence of the TLD spreading and the formation of incompatibility strains at triple junctions. The TLD spreading products are the Ashby's geometrically necessary and statistically stored dislocations bounded to interfaces [314]. (a) Tension of the polycrystal. (b) Anisotropy of deformation of the grains. (c) The mismatch of the grains is adapted by means of dislocations. (d) Geometrically necessary dislocations at interfaces.

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Nazarov et al. [313] and Lojkowski [162] have shown that the elastic energy of non-equilibrium GB dislocations can increase the GB energy twice. The above theoretical considerations or indirect experimental evidence has been con®rmed during recent observations of Inoko et al. [113] of GB pre-melting caused by high dislocation densities at grain boundaries. These results are discussed in Section 3.4.3. 3.2.3.4. Relation between grain boundary non-equilibrium and mechanical properties of materials Since GBs act as dislocation sinks and sources, and their ability to absorb or generate dislocations is a function of temperature. The rate of dislocation annihilation at GBs is given by the equation [164,308,317]:   dr t ˆ S
j 1 ÿ exp , …196† dt r where j is the dislocation ¯ux, r is their density, and S is the speci®c surface of GBs. Therefore, the mechanical properties of materials must change when the spreading temperature is approached. In fact, the change of mechanical properties of materials at the spreading temperature was experimentally observed [317±320]. The kinetics of TLDs spreading may also in¯uence strongly the superplastic properties of materials. Lartigue et al. have shown that segregation of yttria to alumina leads to stabilisation of TLDs cores, which in turn decreases the creep rate of the material [321]. 3.2.3.5. Generation of dislocations at grain boundaries The mechanism of dislocations generation at GBs is far from being understood [322]. One possibility is a process opposite to TLDs spreading, i.e., ``pressing together'' SGBDs [322]. However, this process requires very high local strains, which can be generated only on some stress concentration sites like hard precipitates. Strain localisation at interfaces and building up of dislocations pile ups which may be emitted from triple junctions is a well known phenomenon for materials with low stacking fault energy and was observed during in-situ TEM experiments [323]. Kurzydlowski et al. [324,325] calculated, using the Finite Elements Method, the necessary conditions for sucient stress concentration factors. The same result can be obtained analytically using Eq. (183) [293]. The process opposite to spreading cannot cause localisation of the TLD core unless a stress concentration factor is present or some other process takes place, like condensation of vacancies. It must be pointed out, however, that generation of dislocations in materials with low stacking fault energy via stress concentration at steps is a plausible process, since shear is localised in shear bands. However, in materials with a high stacking fault energy homogeneous generation of dislocations from GBs was

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Fig. 109. Generation of dislocations at general GBs in an Al±Mg alloy without any visible stress concentration factor [326]. The presence of precipitate free zones prevents the dislocations to be generated in the bulk and reach GBs from the grain interior. So the dislocations seen clearly emerge as half loops from GBs. (a) 0.5% deformation. (b) 2% deformation.

observed [326±328]. Fig. 109 shows dislocation loops emitted from GBs in sections where no strain concentration objects are present. Gleiter [329] proposed an alternative dislocation generation mechanism. Assume a GB migrates by the step mechanism, as shown in Fig. 110, and two steps are

Fig. 110. The mechanism of dislocation generation on migrating GBs. According to Gleiter [329].

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Fig. 111. (a) Schematic representation of the di€usion ¯uxes and direction of motion of interfaces during reaction±di€usion leading to the Cd21Ni5 phase growth during annealing of the Cd/Ni di€usion couple. (b) Schematic representation of the Cd concentration gradients for di€usion controlled and kinetically controlled growth of the intermetallic phase.

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moving towards each other. If the newly generated layers are displaced relatively to each other by a non-lattice RBT vector, a dislocation will be generated at the place the two steps meet. A strain ®eld may cause the generated dislocation to move into the bulk or to climb if the vacnacy di€usion coecients are suciently high. Nevertheless, despite its crucial importance, the mechanism of dislocation generation is not well understood. Fig. 107 shows that no special stress concentration sites are required and it seems unlikely that GB migration has taken place in the alloy studied at room temperature. 3.3. Reactive di€usion at an interface In this section, we will show an example of reaction di€usion, i.e., a di€usion process leading to the growth of a new phase. The case presented is interesting because it can be shown that the phase growth rate depends more on the reaction at the interface than on di€usion. The reaction rate, in turn, must depend on the interface structure. A microstructural picture of such a situation was already shown in Section 2.5.6 for the case of the spinel growth studied by Hesse et al. [11]. Paritskaya and Bogdanov [330] studied in detail the reaction controlled case on the example of the formation of the Cd21Ni5 intermetallic compound during interdi€usion in a di€usion couple Cd±Ni. Evidence for the interface role for the kinetics follows from investigations of di€usion ¯uxes. Consider a reaction between two elements A and B to form the AxBy compound. To form the compound the two elements must di€use across the new phase. The growth rate of the compound depends on the di€usion coecients of the elements A and B as well as on the reaction rate at the interfaces. In this system there is an extended linear stage of the dependence of the thickness of the growing phase on time, l0t, followed by the parabolic stage l0t1=2 : Fig. 111a illustrates schematically the growth mechanism of the intermetallic compound. In the case of the Ni/Cd di€usion couple, the Ni di€usion coecient is negligibly small. Hence, the only place where the new phase can grow is the Ni/ Cd21Ni5 interface. As Cd atoms reach this interface and react to form the Cd21Ni5 compound, the interface moves towards Ni. On the other hand, the Cd/Cd21Ni5 interface moves to the left because Cd di€uses towards the Ni/Cd21Ni5 interface. It is clear that the Cd/Cd21Ni5 interface moves to the left approximately four times faster than the Ni/Cd21Ni5 to the right. Since there is a net ¯ux of Cd atoms to the right, there must be an equivalent ¯ux of vacancies to the left. The obtained Cd di€usion coecient in the intermetallic phase is relatively high (in the range 10ÿ6 cm2/s at T10:4±0:6Tm ). According to Paritskaya and Bogdanov, this is a consequence of nonstoichiometry of the Cd21Ni5 phase, which results from high Cd concentration. The above phase exists in a wide range of non-stoichiometric compositions so that the Cd concentration can be considerably di€erent from the stoichiometric value of 81 at%. A consequence of the high Cd di€usivity is that the kinetic barrier at the Ni/Cd21Ni5 interface plays the crucial role for the new phase growth rate.

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Fig. 111b shows schematically the Cd concentration gradient within the intermetallic layer. The Cd/Cd21Ni5 interface is considered a perfect Cd source and c ' the Cd concentration at the Cd/Cd21Ni5 corresponds to the solubility limit c1. If the Ni/Cd21Ni5 was a perfect Cd sink or vacancy source, c2 should correspond to the stoichiometric value c2 ˆ 81 at%. If di€usion was the limiting factor, the measured c0 value should be smaller than 81 at%. It was experimentally shown that c 00 > 81 at%, which speaks for a di€usion barrier. To compare the di€usion and kinetic characteristics of the process one can use  The term b is the reaction the bl product and the chemical di€usion coecient D: rate at the interface, which is de®ned as the ratio of the concentration di€erence Dc across the interface and interface velocity v: bˆ

v : Dc

…197†

The chemical di€usion coecient, D is DA cB ‡ DB cA : Since DNi ˆ 0: D ˆ D…1 ÿ c1 †,

…198†

where D is the Cd di€usion coecient and c1 is the Cd concentration. Three cases for the growth kinetics were envisaged by [330]: (a) The growth rate of the intermetallic phase is limited by di€usion of Cd across the Cd21Ni5 layer: D  bl: In that case, growth takes place in the parabolic growth regime. The real concentration C0 at the Ni/Cd21Ni5 interface is lower than c2. (b) The limiting factor is the kinetics of the reaction 21Cd + 5Ni 4 Cd21Ni5: D  bl: In that case growth takes place in the linear growth regime. As a consequence, c 00 > c2 and Cd accumulates at the Ni/Cd21Ni5 interface. This situation usually corresponds to thin layers. (c) Mixed situation, where both factors in¯uence the growth rate. This corresponds to the system studied. Paritskaya et al. [330] derived the following equation for the mixed case:   dt 1 1 c2 ‡ : ˆ 2  bl 2k c D … 1 ÿ c2 † dl

…199†

The factor k represents the expression: kˆ

1 1 ‡ , c 00 1 ÿ c1

…200†

where b and c0 are related by: bˆ

v : c 00 ÿ c2

…201†

Eq. (199) permits to determine both the kinetic constant b and the interdi€usion

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coecient D in a system with mixed kinetics. Graphically, the above quantities can be determined from the dt=dl 2 vs. l ÿ1 plot. The slope of the line is 1=b and from its intersection with the vertical axis the chemical di€usion coecient can be calculated. The following results for the temperature dependence of b and D were obtained [330]:   70:9 kJ=mol , …202† D ˆ 20:0 exp ÿ RT  60:1 kJ=mol : b ˆ 9:0 exp ÿ RT 

…203†

The low activation energy and high pre-exponential factor for Cd di€usion in the non-stoichiometric Cd21Ni5 phase correspond well with the idea of quick di€usion along the saturated with vacancies Ni sublattice. The low activation energy of 70.9 kJ/mol would correspond to the jump energy of a Cd atom into a vacancy site in the Ni sublattice. The activation energy of 60.1 kJ/mol corresponds to the event of incorporation of Cd atoms in the Ni/Cd21Ni5 interface. The above energy may represent the activation energy for vacancy generation at that interface. The reason is that the vacancy ¯ux controls the Cd atoms ¯ux. The pre-exponential factor in Eq. (203) represents, therefore, the number of sites in the interface where the reaction can take place. According to Paritskaya et al. [331] the pre-exponential factor b0 can be expressed as: b0 ˆ nao,

…204†

where n is the frequency of oscillations, a is the lattice constant and o is the dimensionless density of possible reaction sites. It is clear that the density o and the e€ectiveness of the interface as reaction barrier depends on the structure of the interface. It seems that the above idea is an extension of the case of interfaces of the Grabski [332] model of GB migration. According to that model the GB migration takes place by transfer of atoms from one grain to the other at special ``holes'' in the GB, and the migration rate depends on the frequency of atomic jumps across the holes. In the case of reactive interfaces, the ``holes'' are places where the reaction takes place. According to the above models, segregation can change the situation from a di€usion-controlled process to a reaction-controlled process, as shown in [333]. 3.4. Grain boundary pre-melting, pre-wetting and wetting 3.4.1. Basic thermodynamics The grain boundary (GB) wetting is a complex phenomenon involving, thermodynamics, kinetics and structure of the GB, solid/liquid (S/L) interfaces

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Fig. 112. The scheme of measurements of Zn concentration during studies of the wetting and prewetting transition in Fe±Si bicrystals wetted by liquid zinc [43]. There are three GB regions: the low Zn concentration region, with GB structure like in the bulk material; the high Zn concentration region, where the GB is in the pre-wetted state; and the thin liquid ®lm of Zn rich melt. cGBO is the prewetting transition concentration of Zn or Zn solubility limit in the GB. (a) Wetted GB, with pre-wetted zone. (b) Partially wetted GB, with no pre-wetting zone. The Zn concentration pro®les are shown in Fig. 119.

and various di€usion paths [334]. The understanding of GB wetting phenomena is still rather rudimentary [335]. Consider a GB in contact with a liquid or amorphous phase. Under equilibrium conditions the contact angle y at the intersection of the GB and the S/L interface (Fig. 112) is de®ned by the equation: cos

j g ˆ GB , 2gSL 2

…205†

where gGB and gSL are the GB and S/L interface energy per unit area, respectively. GB torque terms have been neglected. The transition from y > 0 to y ˆ 0 is called the wetting transition (Fig. 112a). The anisotropy of the S/L interface and GB energy is neglected. It is clear that the wetting transition means that j ˆ 0: Fig. 113 shows the basic thermodynamics of the pre-melted or pre-wetted GB [336,337]. There are two possible structures: the regular GB and the pre-wetted GB, where a liquid ®lm separates two L/S interfaces. The two L/S interfaces repel each other. The work of Kayser et al. [338] has shown that the opposite may be true only for special crystallographic orientations (Fig. 114). On the other hand, since the liquid ®lm is not in thermodynamic equilibrium, its energy g increases as its thickness l increases. Since the presence of the ®lm changes the entropy and free volume of the system, the energy g is also a function of pressure and temperature. If the energy of two S/L interfaces and of the liquid ®lm is less than

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Fig. 113. Energetics of two L/S interfaces separated by a thin liquid ®lm and of the GB pre-wetting transition. Fig. 113a shows the situation when the thermodynamic parameters are far from those corresponding to equilibrium of the bulk liquid phase. Fig. 113b shows the case where the free energy di€erence between the bulk liquid and the solid is small enough to form a thin wetting layer, but not small enough to melt the bulk. Fig. 113c shows the case where the liquid spreads from the GB and the crystal melts.

the energy of the GB, the liquid ®lm is stable. When the system approaches the conditions of equilibrium of the liquid, the inclination of the line which corresponds to the contribution of the liquid to the energy decreases and the width of the liquid ®lm increases to in®nity (Fig. 113). Desre [340] analysed recently the above problem in an elegant way. When the GB is completely wetted, the condition obeys: gGB ÿ 2gSL ˆ gS ,

…206†

where gS > 0 is called spreading coecient and plays the role of driving force for

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Fig. 114. Di€erent coalescence behaviour of solid particles in the melt, depending on whether the OR is special (coincidence). According to Takajo [339].

penetration of liquid or amorphous phases along the GB. The thickness of the equilibrium amorphous ®lm was calculated as:   ÿe exp , …207† gGB ˆ 2gSL ‡ dDG ‡ g…a† S x where d is the interface thickness, DG is the free energy di€erence between the is the amorphous or liquid ®lm and the same phase being crystalline. g…a† S spreading coecient, where the index (a) corresponds to either the amorphous ®lm or the pre-wetting state and x is the correlation length of the disordered state. The thickness of the liquid or amorphous ®lm is: d ˆ x ln

g…a† S , xDG…x†

…208†

where x is the molar fraction of the wetting phase in the amorphous or liquid ®lm. For wetting of GBs in the Fe±Si±Zn system, which will be presented here in detail, the correlation length may take particularly large values, in the range of above 1 nm. The reason is the plausible formation of Fe±Zn associates [41,341]. Fig. 115 schematically shows how the presence of Fe±Zn associates may cause density ¯uctuations, which contribute to the S/L interface thickness. Eq. (206) shows that the wetting pressure or temperature shall depend on the energy, excess entropy and excess volume of each of the interfaces. Let us assume that gGB and gSL are linear functions of temperature or pressure:   j gGBO ÿ sGB T ‡ PvGB , …209† ˆ arccos 2gSLO ÿ 2sSL T ‡ 2PvSL 2

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Fig. 115. Schematic diagram, showing the distance of interaction of two solid/liquid (S/L) interfaces. According to the ``melting iceberg'' model of the S/L interface, segments of the solid detach and di€use away from the solid, until they eventually dissolve in the liquid. The interaction of two S/L interfaces starts at the distance where the solid particles dissolve. Assuming the structure of the solid particles to be that of associates in the liquid, the thickness of the pre-wetting layer would be equivalent to the multiple of several associates [40,341].

Fig. 116. The pre-melting transition in a (100) S29 twist GB in silicon. Below TG the interface has a glassy structure. With increasing temperature, its free volume decreases and reaches the value corresponding to molten silicon at its melting point.

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where gGBO and gSLO are the energies of the GB and S/L interface at zero pressure and room temperature, respectively, sGB and sSL are the excess entropies, vGB and vSL are the excess volumes of the interfaces, and P the is pressure [334,342]. Eq. (209) shows that wetting transitions occurs both with the change of temperature and pressure. It can also take place with a change of chemical composition. The in®nite widening of the interface when approaching the phase transition is in agreement with the Cahn and Hillard Gradient Energy Model for calculation of the interface energy [343]. For the case of wetting by the own melt, the liquid and the solid can coexist only in a temperature gradient. It was proposed that a thin ®lm along GBs coexists with the solid phase at temperatures below the melting point. The subject was revived recently by Phillipot et al. [99]. The idea follows from calculations of Hillard and Cahn [344] and Miller and Chadwick [345] that in noble metals the energy of two S/L interfaces is lower than the energy of a GB at the melting point. In the particular case of GBs in pure materials, transformation at T < TM of the GB into a sandwich structure with two S/L interfaces separated by a thin layer of undercooled liquid is called GB pre-melting (Fig. 118). Keblinski et al. have shown using computer simulations that the pre-melting transition takes place in silicon GBs, as a representative of covalent materials. It was shown that there is a reversible continuous transition from the glassy state to the liquid state as temperature is increased [346]. Fig. 116 shows the decrease of the GB free volume from the value characteristic to the amorphous state at low temperatures to that of the liquid state at the silicon melting point. 3.4.2. Grain boundary pre-melting and pre-wetting Already in the ®rst paper on GB wetting, published in Acta Metallurgica, it was pointed out that GB wetting is connected with GB segregation [347]. From the investigations of the systems: Ni wetted by Bi [348], Fe±Si alloys wetted by Zn [38] and Al wetted by Ga [349,350], a consistent picture emerges. It was shown that the low-melting point metal di€uses along GBs and transforms their structure. The transformed GBs are characterised by high di€usion coecients. This transformation was called pre-wetting transition [38]. The concentration of the di€using metal, which causes the GB pre-wetting transition was called ``GB solubility limit'' [43]. Fig. 117 shows schematically the di€erence between premelting and pre-wetting and the idea of the GB solidus line. However, although the wetting experiments provide evidence of the pre-wetting transition, there is no substantial experimental evidence [95,99] for the pre-melting transition. 3.4.3. TEM studies of pre-melting in thin foils of strained aluminium and copper The in-situ transmission electron microscopy observations of melting of aluminium, by Hsieh and Ballu [351] and Siu-Wai Chan et al. [352], of SGBDs at GBs are not consistent with the pre-melting theory in very pure metals. Contrary results have been obtained by Inoko et al. [113]. They studied by means of TEM melting of non-equilibrium GBs in thin copper and aluminium foils. The di€erence to the Hsieh and Ballu experiment was that the foils were

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Fig. 117. The di€erence between pre-wetting and pre-melting GB phase transformation. The GB solidus line is also shown.

Fig. 118. Strain induced grain boundary melting in copper. Courtesy F. Inoko [113].

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prepared from material strained by 40%. Furthermore, Inoko et al. [113] controlled the thickness of the oxide ®lm on the surfaces of the foils. Inoko et al. found that GB melting takes place at a temperature as low as 0.5Tm (Fig. 118). The melting temperature depends on the type of dislocations interacting with the GB. For screw dislocations, the lowering of the melting point is greater than that for edge dislocations. The new phenomenon was called Strain Induced Grain Boundary Pre-Melting (SIGBPM) [113]. The importance of strain for GB melting was also demonstrated by the in¯uence of the oxide ®lm thickness. If the thin foil was oxidised for long time, the decrease of the melting point was suppressed. The authors attributed this fact to the high strain imposed on the foil by the oxide ®lm, which prevented pre-melting. In non-strained material, GB pre-melting took place only in the region where tangles of dislocations were present. The above result con®rms that interaction of GBs with dislocations may cause a considerable increase of GB energy for constant misorientation (since a change of misorientation in the thin foil is impossible) as proposed Grabski and Korski [307,310]. Melting of the material brings about delocalisation of all dislocation cores and relaxation of strain energy. The paper of Inoko et al. [113] is a ®rst direct evidence of GB pre-melting.

Fig. 119. The e€ect of pressure on the penetration pro®les for Zn di€usion along the tilt axis of a 438 [001] symmetrical GB in an Fe±6at%Si alloy at 9058C. CZn is measured by EPMA Zn concentration close to the GB. The real concentration is expected to be higher owing to GB segregation. CGBO is the GB solubility limit. At this point, the GB di€usion coecient decreases by two orders of magnitude [43]. The scheme of the experiment is shown in Fig. 112.

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Fig. 120. The three-dimensional phase diagram for the GB wetting transition of a 458h001i GB in the Fe±6at%Si alloy [43].

Recently, Mabuchi et al. [353] suggested that GB superplasticity of high strain rate matrix-reinforced metallic materials is related to local melting, where the role of the liquid phase at interfaces is of the so-called ``accommodation helper''. 3.4.4. The pre-wetting transition in grain boundaries in Fe±Si bicrystals Wetting of GBs in bicrystals made of Fe±Si alloys by a Zn-rich melt was extensively investigated [38,40±43]. Fig. 119 shows the measured EPMA (electron probe micro-analysis) concentration of Zn in the bulk close to the 438 [001] symmetrical tilt GB in Fe±6at%Si, as a function of distance from the bulk liquid phase. The pro®les were measured after annealing carried out at various pressures. cZn is the maximum value of the Zn concentration during a line scan perpendicular to the GB. For cZn > cGBO, the slope of the concentration of Zn is small indicating fast GB di€usion. In conformity with the previous work of Bishop et al. [348], this region was considered as having a transformed structure

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Fig. 121. The pressure e€ect on the energy of GB in the pre-wetted state and in the regular state, for two GBs of di€erent free volumes and initial free energy.

owing to the high Zn concentration. Hence, cGBO was called the solubility limit of Zn in the GBs and the small-slope region corresponds to GB pre-wetting. Detailed investigations of cGBO as a function of the Si content, GB misorientation, temperature and pressure permitted to construct the ®rst GB phase diagrams for the pre-wetting transition [43]. Fig. 120 shows the GB solidus surface in the 3Dphase diagram of the Fe±6at%Si bicrystal wetted by liquid zinc. The independent variables are pressure and temperature. Such diagrams were also plotted with the misorientation angle as variable [43]. As far as the physical nature of the prewetting transition is concerned, it was assumed that at high cZn values a thin Znrich liquid ®lm is present at the GBs which would be unstable outside the GB for the same composition and temperature. In other words, GB pre-melting is a preliminary stage before GB wetting by the bulk melt. Fig. 6 shows that the solubility limit of Zn in the GBs is the mirror re¯ection of its solubility in the bulk [38,41]. To explain the pressure induced dewetting transition it was assumed that the energy of the liquid ®lm penetrating along the GBs during the pre-wetting transition is increasing faster with increasing pressure than the non-wetted GB energy (Fig. 121). The present results are also consistent with recent results concerning the pressure e€ect on GB segregation of oxide phases in ceramics [354]. Fig. 122 shows the e€ect of pressure on the wetting angle for the three GBs in the system studied. It is interesting to note that the near S5 coincidence GB displays a higher transition pressure than the others. Let us consider what that means. At low pressure, a liquid ®lm separates the two grains. High dewetting pressure means that the excess volume of the near S5 GB is small compared to the other two GBs studied. It follows that for this periodic orientation of the two

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Fig. 122. Pressure e€ect on the wetting angle for symmetrical h001i GBs in Fe±at%Si alloy [342]. The calculated curves correspond to the best ®t of Eq. (209). Published with kind permission from Kluwer Academic Publishers.

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crystals, there must be some interaction of the crystals across the liquid phase. It seems that even in the pre-wetted state this near S5 GB is less disordered than the other GB studied. 3.4.5. Solidi®cation of grain boundaries with increasing temperature The peritectic temperature in Fig. 6 is singular in two aspects: at that temperature, pressure has the strongest e€ect on the dewetting transition and at the same time the GB solidus limit for Zn is at minimum. This is the result of competition of two processes. One is the formation of Zn±Fe clusters or associates at low temperature. Such clusters are expected to have a similar structure as the G phase. The presence of such clusters prevents GB pre-wetting. However, since the G phase is unstable above the peritectic temperature, the number of such clusters decreases as the temperature is increased. At the same time, the fraction of free Zn atoms in the GB increases. Clusters of free Zn atoms may contribute to the GB pre-wetting transition. So, replacement of G clusters by free Zn clusters leads to GB prewetting. However, as temperature is further increased, Zn desegregates from the GB and the number of Zn-rich liquid like clusters in the GB decreases. Therefore, the higher the temperature, the more solid-like the GB structure. In conclusion, it is expected that GBs should show a higher tendency towards pre-melting at intermediate temperatures, where low-melting point impurities are still segregated on them, than at high temperatures, where desegregation takes place. This idea was expressed as solidi®cation of GBs with increasing temperature [41]. According to this idea, for a material containing low melting point impurities segregating to GBs, three regions of GB behaviour as a function of temperature are expected: (a) Solid GB at temperatures well below the peritectic limit. (b) Structure susceptible to wetting or pre-melting transition, at intermediate temperatures. (c) Solidi®cation of the GB with increasing temperature due to de-segregation of impurities. It is worth to analyse the concept of solubility limit from the point of view of segregation. Usually it is accepted that segregation of an element is enhanced if its solubility limit is low (cf. [196]). However, let us examine this statement as a function of temperature. For most systems, the solidus line has a negative inclination. Therefore, the solubility limit decreases with increasing temperature. However, since the temperature increases, impurities desegregate from GBs. Therefore, taking as the reference the GB, the solubility of the segregating element in the bulk increases with increasing temperature. Hence, we see that the solidus line is not a solubility limit. 3.4.6. Grain boundary wetting and segregation in the Cu±Bi system Another type of GB behaviour was observed during investigations of bismuth

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Fig. 123. The bulk and GB solidus line for the Cu±Bi system. Courtesy Chang [355,356].

segregation in copper. Fig. 123 shows schematically the bulk and GB phase diagram for the Cu±Bi system according to Chang et al. [355,356]. The thick solid line represents the bulk solidus line and the thin line, the GB solidus line. On the left side of the thin line, the GB has a solid-type structure with segregated Bi. According to Chang et al. [355,356] in the region between the thin and solid lines Bi forms liquid ®lm that is stable only along the GBs (Fig. 123). The above conclusions were drawn based on Auger studies of Cu polycrystals with precisely controlled Bi content in the bulk in the range 25±100 ppm. Chang et al. [356] noted that the GB phase diagram, and the bulk phase diagram have a similar shape (Fig. 123). It can be seen that the bulk line can be obtained form the GB line by multiplying the Bi concentration by a factor 1.6. The above similarity was interpreted as follows. The potential energy of a Bi atom is lowered close to the GB plane. This leads to an increase of Bi concentration close to the GB. Hence, the ®rst copper layer is Bi enriched compared to the bulk and display melting even though no melt is in equilibrium as a bulk phase. The situation above may be illustrated schematically as in Fig. 124. The GB solidus line was calculated analytically, based on the thermodynamic properties of the

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Fig. 124. Schematic presentation of the chemical potential gradient and Bi concentration gradient in the vicinity of a grain boundary. It is clear that in the GB vicinity phase transformations characteristic the bulk will take place at bulk concentrations smaller than that in the bulk. (a) Schematical representation of the potential energy of a Bi atom in the vicinity of the interface and within it, (b) the corresponding Bi concentration relative to the bulk value.

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Fig. 125. The GB wetting transition in the Al±Ga polycrystal. The Ga-rich wetting liquid is shown in a dark phase extending along GBs and triple points. T1, T2, T3 are the wetting transition temperatures for various GBs. For the temperature less than T1, the liquid phase is present only at GB triple points. For the temperature T2 between T1 and T3, a fraction of the GBs is wetted. With increasing temperature, the fraction of wetted GBs increases. For the temperature above T3, all the GBs are wetted. According to Straumal et al. [350].

liquid phase in the Cu±Bi system. The transitions abrupt/smooth concentration pro®le of the segregating element at GBs has been analysed in terms of chemical energy by Rabkin [357]. He connected such changes with phase transformations of ®rst and second order at GBs. The above result can be visualised assuming a smooth and not step-wise change of the Bi chemical potential when approaching the interface. 3.4.7. Grain boundary wetting and grain growth in the Ga±Al system Another strongly segregating system was studied by Straumal et al. [350]. They investigated grain growth in aluminium containing a small amount of gallium. It was found that at high temperatures some GBs accelerated their migration rate leading to anomalous grain growth. The fraction of such GBs increased with temperature. This acceleration was attributed to the formation of a liquid Ga ®lm at some GBs. In another work, it was found that small Ga additions enhance GB sliding [358]. Fig. 125 shows schematically the increase of the fraction of wetted

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Fig. 126. Destabilisation of the crystalline lattice by strains arising from mis®t of atomic size.

GBs with temperature. The composition of the alloy corresponds to the coexistence of the liquid and solid phase at the temperatures T1, T2 and T3. At temperatures below T1, the liquid phase gathers at triple corners, but is not able to wet the GBs. At temperatures T2, where T1 < T2 < T3 , some GBs are wetted and some not. At temperatures above T3, all GBs are wetted by the liquid phase. 3.4.8. Grain boundary wetting by amorphous phase and solid state amorphisation Solid State Amorphisation Reactions (SSAR) occur during di€usion in di€usion couples, mechanical deformation, irradiation, application of high pressure etc. (see the review by Johnson [359]). Di€usion induced amorphisation is possible only if the adjoining metals are polycrystalline. This shows that GBs play a crucial role for the nucleation of the amorphous phase [370]. Besides the production of amorphous alloys by mechanical alloying (cf. [374,377]) an alternative method is by repeated cold rolling and folding [360] or shock deformation [361]. Recently, Sagel et al. found that amorphous metallic alloys Zr68Al5Ni9Cu18 can be produced by cold rolling as well [362]. A stack of Zr, Al, Ni, Cu foils corresponding to the composition Zr68Al5Ni9Cu18 was repeatedly rolled and folded up to 100 times. The X-ray di€ractograms of the amorphous phase obtained in this way were identical to those obtained by mechanical attrition and quenching of the melt [363,364]. In the present paragraph, we will discuss the role of interfaces in the above solid state amorphisation process. In this respect two major factors have to be considered: the possibility of di€usion along interfaces at room temperature and the role of interfaces as sites for heterogeneous nucleation of the amorphous phase. Let us ®rst discuss the thermodynamics of amorphisation. As shown in Section 3.4 a liquid or amorphous ®lm can be stable at interfaces although the bulk liquid or amorphous phase is thermodynamically unstable. In that respect it may be useful to recall the Egami criterion for amorphisation of materials by mixing

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535

Fig. 127. Schematic free energy diagram representing the conditions for amorphisation of a Zr±Ni±Al± Cu alloy.

atoms of di€erent sizes [365]. According to that concept, when atoms of di€erent size di€use into the crystalline lattice, strains resulting from atomic size mismatch arise. There is a concentration when the strains reach a critical level and the lattice becomes unstable. The idea is illustrated in Fig. 126. The mismatch of atomic sizes causes forces on atoms, which shifts them out of lowest energy positions for the ideal lattice, in the same way as thermal vibrations during melting. However, since the temperature is low, instead of melting, amorphisation takes place. The above concept agrees with the observed, by Ettl and Samwer [366], mechanical instability of some solid solutions near the crystal-to-glass transformation, associated with a decrease of the Debaye temperature. The above concept is discussed under a di€erent perspective by Phillipot et al. [99] who consider volume expansion leading to instability of the lattice. The volume expansion may be due to di€usion of small atoms like hydrogen. It can be also a consequence of lattice mis®t at grain boundaries. Finally, it is the expression of the negative pressure in the GB region as in the thermodynamic treatment of Fecht (Section 2.3.3) [137]. Therefore, GBs are nucleation sites for amorphisation of materials, as discussed in Section 3.4. In interfaces, in addition to the strain caused by mismatch of atomic sizes,

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mis®t strain caused by misorientation is present. Hence, it may be expected that the concentration of atoms that may cause destabilising of the lattice in the atomic layers adjacent to the GB is lower than that in the case of bulk. This is one more of the factors why interfaces may serve as nucleation sites for the amorphous phase during processes leading to solid state amorphisation. Desre recently discussed the thermodynamics of nucleation of the amorphous phase at interfaces [367]. Fig. 127 shows schematically the free energy diagram for the system Zr±Al±Ni± Cu, which was recently extensively studied. The basic condition for amorphisation, the large heat of mixing of the components [368±370], is ful®lled. In Fig. 127, it is re¯ected by the di€erence of the free energy line of the mixture of the components and the intermetallic phase. The second condition is the existence of the metastable amorphous phase. The conditions for the existence of such a phase were discussed by Johnson [368,370] and by Samver et al. [369]. The third condition is of kinetic nature: the existence of a speci®c temperature range where the di€usivity of the components is too small for the formation of the intermetallic compounds but suciently high for mixing of the components. At room temperature, for di€usion times of the order of 104 s, the di€usion distance of Zr in the bulk is negligible, and for the small Ni, Cu and Al is less than 2 nm. However, the GB di€usion range extrapolated to room temperature is of the order of 50 nm. It is seen that at room temperature the kinetic conditions exist for solid state amorphisation of the above Zr68Al5Ni9Cu18 alloy, providing enough GB di€usion paths exist. Amorphisation caused by rapid GB di€usion is a well-known phenomenon, observed for instance by Samver et al. [371] for the case of fast hydrogen di€usion along GBs. However, in this case for the ®rst time full amorphisation of the alloy was achieved purely by the repeated rolling/folding

Fig. 128. The map of existence of the amorphous and crystalline phase, in the co-ordinates of concentration of the destabilising element and grain size.

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537

Fig. 129. The sequence of events leading to amorphisation during mechanical alloying or cold rolling of a Zr±Ni±Cu±Al alloy. Courtesy Sagel [372]. (a) GB di€usion induced amorphisation on a thin layer. (b) Amorphisation of a next layer due to mechanical mixing.

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technique. Fig. 127 illustrates the proposed path leading to amorphisation of the alloy in question. The high heat of mixing of the components drives GB di€usion that causes locally concentration changes. The increase of concentration of the small atoms Ð fast di€users approach the system to the region of stability of the amorphous phase. This kinetic barrier is overcome by proper mechanical deformation of the material. Either mechanical attrition or repeated cold rolling and folding, introduce such a density of GBs that the material is nanocrystalline. In addition, there is a mechanical mixing of the components. Fig. 127 shows that introducing a high density of interfaces, as in a nanocrystalline material, further destabilises the crystalline phase compared to the amorphous phase, so that the concentration of the destabilising elements leading to amorphisation is decreased [367]. Fig. 128 shows the expected regions of stability of the crystalline phase and crystalline or nanocrystalline phase, as a function of grain size and concentration of Ni or other fast di€users. The same ®gure shows the path of the system evolution for a mechanically alloyed or repeatedly cold-rolled material as a function of time. It was shown experimentally, that these processes lead to a decrease of the grain size [118]. As is seen prolonged mechanical alloying or repeated rolling leads to amorphisation of the material. For small grain size amorphisation occurs at relatively low concentration. Fig. 129 shows schematically the presumed mechanism of material amorphisation owing to di€usion of the smaller atoms along GBs and along the amorphous material/crystalline material interface. 3.5. Nanocrystalline materials Nanocrystalline materials have attracted considerable scienti®c interest because of their unusual physical properties (for a review see Ref. [373]). Such materials are characterised by their small crystallite size, which is in the range of several nanometres. As such, they are inherently di€erent from glasses (ordering on a scale of <2 nm) and conventional polycrystals (grain size of >1 mm). The grains are separated by high-angle grain or interphase boundaries. 3.5.1. Production of nanocrystalline materials by mechanical alloying Cyclic mechanical deformation at high strain rates by mechanical attrition or mechanical alloying leads to large quantities of nanostructured powder particles which can be compacted to bulk samples. In the process of mechanical attrition, powder particles are subjected to severe mechanical deformation from collisions with steel or tungsten carbide balls and are repeatedly deformed, cold-welded and fractured. During the continuous severe plastic deformation at high strain rates (0103±104 sÿ1), a re®nement of the microstructure of the powder particles to nanometre scales has been observed. As a result, a wide range of metals, alloys, intermetallics, ceramics and composites can be prepared in a nanocrystalline, amorphous or quasicrystalline state [374]. Deformation processes within the powder samples during high energy ball

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539

Fig. 130. The average grain size and microstrains as determined from X-ray line broadening as function of milling time for iron powder.

milling are important for fundamental studies of extreme mechanical deformation and the development of nanostructured states of matter. The e€ects of work hardening, material transfer and erosion during wear situations result in microstructures of wear surfaces comparable to those observed during mechanical attrition [375]. In particular, during sliding wear, large plastic strains and straingradients are created near the surface [376]. The elementary processes leading to the grain size re®nement include generally three stages [377]: (i) Initially, the deformation is localised in shear bands consisting of an array of dislocations with high density. (ii) At a certain strain level, these dislocations annihilate and recombine to small angle grain boundaries separating the individual grains. The subgrains formed via this route are already in the nanometre size range with diameters often between 20 and 30 nm. During further attrition, the sample volume exhibiting small grains extends throughout the entire specimen. (iii) The orientations of the single-crystalline grains with respect to their neighbouring grains become completely random. We will illustrate the microstructure evolution during mechanical attrition on the example of iron powder. The microstructural changes resulting from mechanical attrition can be followed by X-ray di€raction methods averaged over the sample volume. The X-ray di€raction patterns exhibit an increasing broadening of the crystalline peaks as a function of milling time. The peak broadening is caused by the decrease of grain size as well as due to internal strain e€ects. The grain size and the microstrain as function of milling time are obtained from the integral peak widths assuming Gaussian peak shapes (Fig. 130) [378]. In the ®rst stage it is

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Table 6 Structural and thermodynamic properties of metal and intermetallic powder particles after 24 h ball milling, including the melting temperature Tm, the average grain size R, the stored enthalpy DH, and the excess heat capacity Dcp [138] Material

Structure

Tm (K)

Fe Cr Nb W Co Zr Hf Ru Al Cu Ni Pd Rh Ir NiTi CuEr SiRu AlRu

bcc bcc bcc bcc hcp hcp hcp hcp fcc fcc fcc fcc fcc fcc cscl cscl cscl cscl

1809 2148 2741 3683 1768 2125 2495 2773 933 1356 1726 1825 2239 2727 1583 1753 2073 2300

R (nm)

DH (% of DHf )

Dcp (%)

8 9 9 9 (14) 13 13 13 22 20 12 7 7 6 5 12 7 8

20 25 8 3 6 20 9 30 43 39 25 26 18 11 25 31 39 18

5 10 5 6 3 6 3 15

2 2 10 13

observed that mechanical attrition leads to a fast decrease of the average grain size to 40±50 nm. Further re®nement to less than 20 nm occurs after extended milling. In addition, the average atomic level strain exhibits an increase to about 0.7%. Let us consider the possible deformation mechanisms leading to the above behaviour. Steady state deformation is observed when the dislocation multiplication rate is balanced by the annihilation rate. This situation corresponds to the transition of stage (i) to (ii)/(iii) as described above. A natural limit to the dislocation densities which can be achieved by plastic deformation is typically less 1016 mÿ1 for edge dislocations. In this stage, further deformation occurs probably via slip of grain boundaries in a way similar to superplastic behaviour [379]. 3.5.2. Thermal properties of nanocrystalline metals Decreasing the grain size of a material to the nanometre range leads to a drastic increase of the number of grain boundaries reaching typical densities of 1019 interfaces per cm3. The fraction of atoms located in the grain boundaries scales with the reciprocal grain size. The thermodynamic properties of nanostructured materials produced by mechanical attrition can be described on the basis of a free volume model for grain boundaries [138]. During heating in the DSC (Di€erential Scanning Calorimeter), a broad exothermic reaction is observed for all of the samples.

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541

Fig. 131. Dependence of stationary grain size and microstrain on annealing temperature [119].

Integrating the exothermal signals gives the energy release DH during heating. Table 6 lists DH for a range of materials and other characteristic values, such as the average grain size and excess speci®c heat after 24 h of mechanical attrition. The stored enthalpy reaches values up to 30±40% of the heat of fusion DH: For the compound phases the stored energies ranged from 5 to 10 kJ/mol, i.e., between 18 and 39% of the heat of fusion for grain sizes between 5 and 12 nm. The ®nal energies stored during mechanical attrition largely exceed those resulting from conventional cold working of metals and alloys. The energy determined can reach values typical for crystallisation enthalpies of metallic glasses corresponding to about 40% DHf : During conventional deformation, the excess energy is rarely found to exceed 1±2 kJ/mol. A simple estimate demonstrates that these energy levels cannot be achieved by the incorporation of dislocations and point defects introduced during conventional processing. The maximum dislocation densities that can be reached in heavily deformed metals correspond to energy of less than 1 kJ/mol. Therefore, it is assumed that the major energy contribution is stored in the form of grain boundaries, and strains within the nanocrystalline grains, which are induced through grain boundary stresses. 3.5.3. Phase stability Because of the cold work stored in the powder particles, nanocrystalline materials produced by mechanical alloying materials are far removed from their equilibrium con®guration. As a consequence, during annealing at elevated temperatures, relaxation and grain growth processes will occur leading to a concomitant increase of the grain size. This behaviour will be illustrated with the example of iron [119]. X-ray di€raction of powder samples annealed for 80 min at each temperature revealed

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the evolution of grain size and strain as function of annealing temperature as shown in Fig. 131. The microstrain is decreasing rapidly below 2008C while the grain size remains nearly constant. Grain growth starts to become signi®cant at about more than 3008C. As such, two regimes with and without grain growth can be distinguished. The enthalpy release can be clearly assigned to the existence of grain boundaries. The reduction of the microstrains is probably caused by grain boundary relaxation. Based on elastic theory it is estimated that this contribution to the overall energy is less than about 5%. By simple geometric considerations [380,381] the speci®c grain boundary excess enthalpy is estimated to be about 2.1 J/m2. This would correspond to a value for non-equilibrium non-relaxed grain boundaries, whereas after relaxation, the grain boundary energy is reduced to 1.5 J/m2. Rabkin et al. estimated a possible increase of the chemical potential of interface atoms due to the interfaces curvature to be of the order of 10% [382]. Values resulting from computer simulations suggest excess enthalpies between 1.2 and 1.8 J/m2 [383]. Therefore, we conclude that grain boundary energies after mechanical alloying are characterised by increased values of about 25% due to their unrelaxed atomic structure. Due to the small crystal size, it is unlikely that dislocations exist either in GBs or in the crystals. Therefore, the barrier to GB relaxation must be connected with incompatibility at triple junctions, as discussed in Section 2.2.4.7. 3.5.4. Instability of nanocrystalline materials with respect to melting or amorphisation Lu and Fecht [384] discuss the relative stability of a nanocrystalline material and the melted or amorphous phase of the same chemical composition. In general terms, the above treatment deals with the question of excess free energy and volume introduced into the nanocrystal due to the high density of high-angle grain boundaries. If the thermodynamic properties of the nanocrystals are not changed, the di€erence between the free energy of a nanocrystal and single crystal is: DG NC ˆ x in DG IN ,

…210†

where DG In is the Gibbs free energy di€erence of the interfaces in respect to the single crystal and x in is the molar fraction of interfaces. Lu and Fecht show that with decreasing grain size the free energy of a nanocrystal approaches the free energy of the amorphous or liquid phase and the melting temperature of the solid is depressed. There is a critical grain size R where the solid becomes unstable in respect to melting. The respective melting is a second-order isentropic and isenthalpic transition. On the other hand, in a stable ®ne-grained nanocrystal the free volume of the interfaces must be small, similar to that of amorphous materials. There are some contradictions as far as the energy of interfaces in nanocrystals is concerned. According to [380,381], relatively high interfacial energies are observed, whereas Refs. [385±387] quote low interfacial energies in

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543

Fig. 132. Fracture surface of a sintered nanocrystalline SiC. Courtesy S. Gierlotka.

Ni±P, TiO2 and Pd, respectively. Nevertheless, the last results are consistent with densely packed interfaces in ®ne-grained nanocrystals. Another way of looking at the amorphisation process of nanocrystalline materials during mechanical alloying is that the nanocrystals became unstable when their thickness is less than two interfaces thickness. The interface thickness is understood as the range over which the bulk material is in¯uenced by the proximity of the interface: strain and free volume ¯uctuations etc. 3.5.5. Mechanical properties of nanocrystals As a further consequence of the grain size reduction, a drastic change in the mechanical properties has been observed. The hardness was 9.3 GPa for d about 16 nm vs. 1.3 GPa for annealed powder. The Young's modulus showed a decrease by 10±20% in comparison with the polycrystal. Therefore, it is suggested that the mechanical properties of nanophase materials prepared by mechanical attrition after extended periods of milling are not being controlled by the plasticity of the crystal due to dislocation movement anymore but rather by the cohesion of the nanocrystalline material across its grain boundaries. From the considerable increase of hardness and the principal changes of the deformation mechanisms, improved mechanical properties can be expected as attractive features for the

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design of advanced materials. This seems to be particularly important for such materials like diamond, which cannot be sintered without additives. The mechanical properties of nanocrystalline materials are reviewed in the recent paper of Gryaznov and Trusov [388]. 3.5.6. Reactive sintering of nanocrystalline SiC±diamond composites under pressure Recently Palosz et al. [389,390] produced compacts of SiC, diamond and diamond±SiC composites. In the latter case the interfacial adhesion between diamond particles was ensured by SiC. In the usual diamond sintering technology metals like Co or Ni are added to facilitate sintering. Sintering of microcrystalline diamond without metal additives is dicult, because at high temperatures and low pressures the actual pressure in the pores can be lower than that outside the agglomerate and the surfaces of diamond crystals reconstruct to acquire a graphite structure. However, presence of metals limits applications of diamond compacts because, at ambient pressure, metals catalyse transformation of diamond into graphite already at 8008C. Therefore, sintering of diamond without metal additives is an attractive option. Practical use of nanocrystalline materials for fabrication of ceramics (including diamond compacts), requires solving several technological problems like a relatively large porosity caused by strong agglomeration of powder particles and high level of internal strains present in the dense compacts after sintering under extreme temperature and pressure conditions [391]. Gierlotka et al. [390] produced compacts of composites SiC±diamond by in®ltration of Si into nanocrystalline diamond powders in a toroid-type press under the pressure of 7.7 GPa at 13008C. The in®ltrated powder contained 18 mol% SiC. The density of the SiC±diamond composite was 3.35 g cmÿ3, and is close to 100% theoretical density (3.55 for pure diamond, 3.2 g cmÿ3 for SiC). The Vickers hardness of the best compacts was 50 2 5 GPa. Liquid silicon ®lls immediately all pores of the diamond polycrystal which leads to hydrostatic pressure conditions in the whole volume and prevents its graphitization at higher temperatures. 3.5.7. Grain boundaries harder than diamond The interface compressibility is an important parameter in a material with a large fraction of atoms in interfaces. Although it is known that the thermal expansion coecient of GBs is larger than in the bulk [392], and it is ®ve times higher perpendicular to the interface than parallel to it [393], there were no data concerning GB compressibility. Palosz et al. [394] studied by means of X-ray di€raction techniques the compressibility of nanocrystalline SiC under pressure up to 40 GPa. They developed a special technique for analysing the X-ray di€raction spectra and assess separately the compressibility of grain boundaries and interior of the nanocrystals. Fig. 133 shows the changes of the lattice constants of the GB material and the crystallites. The line labelled ``Lorentzian'' corresponds to the GBs, while the line labelled ``Gaussian'' corresponds to the crystallites,

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545

Fig. 133. Change of lattice constant of grain boundaries (line denoted as ``Lorentzian'' and crystallites interior (line denoted as ``Gausian'') in nanocystalline SiC. For reference, the line for single crystalline SiC, diamond and gold are given. According to Palosz et al. [394].

respectively. The ``diamond'' and ``gold'' lines are shown for reference purposes. It is seen that the compressibility of the crystallites and diamond are similar over the full pressure range. However, the compressibility of the GB phases changes dramatically with pressure. Below 10 GPa the GB phase is considerably more compressive than the bulk. At about 10 GPa, the GB phase compressibility is similar to the bulk. At higher pressures, the GB phase becomes harder than diamond. Palosz et al. interpreted the above result as follows: high pressure causes a shear

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at the interfaces, which breaks the bonds and permit denser packing of atoms than in the bulk. The above interpretation agrees well with the concept of Lu et al. [402] and Spaepen and Turnbul [88], who proposed that in some covalent interfaces breaking of bonds leads to denser packing of atoms than in the bulk. This observation explains the acceleration of the migration rate of the amorphous/ crystal interface under high pressure, leading to negative activation volume for crystallisation. It has led Lu et al. [402] to predict the ``pressure catastrophe,'' when at a very high pressure there is no energy barrier for bonds breaking. The pressure range where the hardness of the GB phase increases in the Palosz et al. experiments [394] matches well with the pressure predicted by Lu et al. [402] for the pressure catastrophe. Finally, the possibility of material harder than diamond was already postulated by Bunk et al. [395], who compressed to high pressures fullerenes. Their argument was that at the contact of two fullerenes the packing of matter is denser than in bulk diamond. It follows from the above that although the hypothesis has to be veri®ed, it is possible that highly compressed GBs in nanocrystalline SiC and possibly diamond, form a phase which is denser and harder than diamond. 3.6. High pressure investigation of grain boundaries In the present paragraph, we present some selected recent results connected with the study of the kinetics of interfaces under pressure. Since the free volume of interfaces depends on their orientation relationship, the e€ect of pressure on their energy must be a function of misorientation as well [396]. Furthermore, study of the pressure e€ect on GB kinetics may permit to understand the mechanisms of atomic motion in the same way as pressure experiments help in understanding the di€usion mechanisms in solids. 3.6.1. Grain boundary migration in aluminium bicrystals Indirect information about the GB structure can be obtained by analysing the results of investigations of GB migration under pressure. Fig. 134 shows the in¯uence of pressure on GB migration rate in Al bicrystals [287]. The inset shows the idea of the experiment: the GB migrates under constant driving force. Three series of samples were investigated: with symmetrical tilt h001i, h011i, h111i grain boundaries. It is clearly seen that the pressure has the strongest in¯uence on the migration rate of the h001i GB. The pressure e€ect can be expressed quantitatively in terms of the activation volume: V  1 ÿ RT

@ ln M , @P

…211†

where R is the gas constant, M is the GB mobility and P is pressure. The pressure e€ect on the pre-exponential constants was neglected, that may cause an error of less than 10% in the V value. When a thermally activated process causes a

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547

Fig. 134. Pressure e€ect on GB migration in Al bicrystals [287].

positive change of the volume of the crystal under pressure, the enthalpy of activation increases by a factor PV. Therefore, the probability of such an event decreases. For processes which depend on the concentration of vacancies like selfdi€usion in many materials, the V value is of the order of 0.8O. Table 7 shows the results obtained. The activation volume for GB migration for the h011i was in the range of 1.2±3:63O, where O is the atomic volume. For the h111i and h001i the activation volume is ¯at about 1:2O: Molodov et al. [287] Table 7 Activation energy and volume for GB migration in Al bicrystals Bicrystals' misorientation

Activation

Pre-exp. factor

Axis

Angle

S

Energy (kJ/mol)

Volume (V/O)

log(A0)

h100i h100i h111i h111i h110i h110i h110i h110i

36.920.4 31.820.4 37.120.4 32.021.0 38.520.5 36.021.0 32.021.0 30.021.0

5

113.727.9 196.0211.3 161.429.2 194.4245.1 181.4232.2 222.4213.4 274.6256.2 291.4258.9

1.2020.06 1.1920.06 1.1020.06 1.1920.06 1.6520.08 2.2220.10 3.2421,7 3.6320.18

4.4 10 8.3 10.8 7.7 10.3 13.5 14.1

7 9

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concluded that for the h011i GBs a group mechanism of GB migration must be active, contrary to the h001i and h111i tilt GBs where a single atom migration mechanism is active. The special behaviour of the h011i GBs can be understood if one recalls that for these GBs closely packed rows of atoms parallel to the h011i tilt axis are parallel in each grain. This leads to more dense packing of atoms than in the other GBs. It was proposed [397] that to disrupt the continuity of the closely packed atomic rows and permit their jump across the GB, multiple vacancy condensation is necessary, which explains the high activation volume (Fig. 135). It follows from the above experiments that the h011i tilt GBs are most likely constructed from closely packed rows of atoms that are tightly bonded along h011i lines. The group migration mechanism agrees well with the presumed high collectivity coecient (Section 2.4.8) for these GBs. Ichinoise and Ichida directly observed the group migration mechanism for a twin h011i GBs during in-situ observations of the GB migration in incoherent twin GBs in copper [83]. The above high values of activation volume contrast with relatively low values of the activation volume for grain growth in aluminium [399]. However, in the case of GB migration in bicrystals a crucial issue is the material purity. The GB migrating along the bicrystal is able to sweep all the impurities in the material. Hence, it is dicult to estimate at what purity level the GBs can be considered as pure. Kassen studied grain growth in very pure materials and stated that migrating GBs pick up the impurities and consequently change the structure [288].

Fig. 135. Group GB migration mechanism for atoms organised in closely packed rows, with the participation of multiple vacancies.

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549

Fig. 136. Compensation diagram for the relation between activation energy and activation pressure during GB migration in Al bicrystals [398].

Therefore, it is dicult to compare results obtained for bicrystals with results for grain growth where the speci®c surface of GBs is high. 3.6.2. The compensation pressure Fig. 136 shows the relationship between the activation volume and activation energy for GB migration for the experiment illustrated in Fig. 134. It is seen that for the h111i and h001i GBs, there is no dependence of activation volume on energy. On the other hand, for the h011i GBs, there is a clear relationship. The inclination of the line corresponding to the h011i GBs corresponds to a compensation pressure (as de®ned in Section 3.2.3) 5.5 GPa. In other terms, according to the interpretation of compensation pressure and temperature given in Section 3.2.3, for the h011i GBs inside the activated cluster there is a pressure of the order of 5.5 GPa. Having in mind the proposed group mechanism of migration for such GBs, this value seems reasonable. In fact, such is the range of pressures close to a core of a dislocation. Jumping of a whole row of atoms as in Fig. 135 might generate pressures similar to those in a dislocation core. 3.6.3. Pressure e€ect on grain growth in polycrystalline aluminium Lojkowski et al. [399] studied the activation volume for grain growth in aluminium as a function of temperature. Fig. 137 shows that for Al of 99.99% purity, the activation volume is 0:7O for T < 580 K but 0:3O for T > 620 K. At the same time, for Al of 99.999% purity, the activation volume for grain growth

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is 0:3O although the temperature is 560 K. This indicates that in Al4N at approximately 6008C a change of GB migration mechanism occurred, from a mechanism characteristic of a material with some impurity content to that of a pure material. This change is re¯ected by the fact that at high temperatures the activation volume is close to that of vacancy formation and vacancy di€usion, whereas at high temperatures it is closer to values characteristics for interstitial di€usion. The above result is consistent with the idea that at low temperatures or high impurity content, the GBs are supposed to take a closely packed structure, which does not permit GB migration unless vacancies are present. On the contrary, for high temperatures or low impurity content, the GBs take an opened structure, permitting migration by the ``shu‚ing of atoms'' mechanism of atoms movement formulated by Lisowski and Gleiter [400]. The concept is to divide the volume of the crystal into Wigner cells, which are just polyhedra. In two dimensions the polyhedra in the perfect crystal have six faces, whereas close to defects seven or ®ve faces. Combination of such polyhedra corresponds to dislocations (a dipole of ®ve or seven faced polyhedra), vacancies (quadruple of polyhedra) and grain boundaries. Each face corresponds to a contact between atoms. It is seen that interface migration is rather a phenomenon of migration of the contacts between atoms that atoms themselves, like during vacancy di€usion.

Fig. 137. Activation volume for grain growth in Al as a function of temperature [399].

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551

Fig. 138. Grain boundary migration mechanism where small displacements of atoms (shu‚ing) causes migration of the interatomic bonds. Based on the ``shu‚ing'' mechanism of GB migration proposed by Lissowski and Gleiter [400].

Fig. 138 shows how small rearrangements of atoms change the contacts. In Fig. 138b, atoms A and D are neighbours, but B and D are not. In Fig. 138c, atoms C and B become neighbours while the atoms A and D are not any more. Such small rearrangements lead to migration of the ``broken bond'' and thus of the interface. In other terms, it seems that the GB with segregated impurities could become a closely packed structure, which can migrate only by jumps of atoms across the interface with participation of vacancies. On the other hand, a GB with desegregated impurities may acquire a more open structure where atom's rearrangement is possible just by their small movements, which cause migration of atoms bonds (Fig. 139). This concept was developed independently for covalent materials as discussed in the next paragraph. 3.6.4. Pressure e€ect on the migration rate of the amorphous phase/crystalline phase in covalent solids and the high pressure catastrophe Lu et al. studied the pressure e€ect on the crystallisation of the amorphous

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Fig. 139. Change of GB migration mechanism when impurities desegregate and the GB takes a more open structure [399].

phases of Si, Ge, [86±88]. For the case of Si [86] and Ge [401] the Solid Phase Epitaxial Growth (SPEG) phenomenon was studied, where an amorphous ®lm on the surface of a single crystal transforms in a single crystal via amorphous/ crystalline phase interface migration. It was found that pressure accelerates these phenomena, i.e. the activation volume is negative. Its value for Si was ÿ0.28OSi whereas for Ge it was ÿ0.46OGe. The results were explained in terms of the Speapen and Turnbull mechanism [88,89] in which a single bond breaks to form a pair of dangling bonds, each of them migrating along the interface reconstructing the random network into the crystalline network. The model predicts a negative activation volume, i.e., that the activated complex formed during interface migration has a locally higher density of matter than the bulk. It is assumed that during the thermal ¯uctuation interatomic bonds can `jump' forming locally ®vefold co-ordination with denser packing of matter than in the bulk. The above interpretation follows from the idea that in covalent materials bonds are elements of the structure equally to atoms (Section 2.1.2). Therefore, they can migrate, jump, etc, leading to interface migration or reconstruction or even amorphisation, as observed in computer simulations of GBs in silicon and diamond by Keblinski et al. [214,216] and by HREM in SiC by Tsurekava et al. [168] (Fig. 54).

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It follows from the above that formation of dangling bonds in the interfaces studied is connected with a decrease in volume. This has led Lu et al. [401,402] to consider the question of high pressure catastrophe, when at suciently high pressure the energy barrier for interface bonds breaking vanishes. They estimated this pressure to be in the range 3±6 GPa for germanium and 6±12 GPa for silicon. They considered as a possible manifestation of this phenomenon the transition of Si and Ge to the metal state at high pressures [403]. 4. Summary In the present paper, we presented a number of parameters, which are a useful tool for classi®cation of interfaces, interpretation of the experimental results, and selecting interfaces for speci®c applications. These parameters permit to construct ``maps'' of structure transformations as a function of various variables as well as order interfaces into categories according to properties and structures. These parameters provide sometimes a very crude description of interfaces. However, there is a need for a ``bridge'' between the enormous amount of precise information provided by detailed atomistic description of interfaces, and their properties. The structure of interfaces depends on a large number of parameters. For instance the structure of a grain boundary in a mono-atomic crystal depends on up to 12 independent variables (Section 2.2.5). Here also the non-equilibrium structures connected with local bending of atomic planes are considered (Section 2.2). Interaction with lattice defects may cause that the structure of interfaces is non-homogeneous (Section 3.2.1). The number of crystallographic variables increases further in multi-component systems, as shown on the example of polarity and polytypism of GBs in silicon carbide (Section 2.5). The attempts to relate the GB structure and crystallography bring about the perspective of indexing each GB by a large fraction of the above variables. Owing to the complexity of such a correlation, GBs are persistently indexed by means one parameter S Ð coincidence Ð despite the extensive criticism as far as this simpli®cation is concerned. The principal reason of the criticism is that the interface plane is equally important for the GB energy as the misorientation (Section 2.6), but the S value does not include this information. Another problem with the coincidence based theories is that one has to assume an arbitrary tolerance coecient which de®nes the di€erence of lattice nodes positions that are still coinciding (Section 2.2.3). For narrow tolerance, only few GBs are in coincidence. For a large tolerance factor, many GBs are in coincidence, but the importance of coincidence for the energy is small (Section 2.2.3.2). The tolerance factor is useful for describing coincidence in incommensurate lattices, but the same limitations as for grain boundaries are valid. A general approach to the energy of incommensurate interfaces is based on the Static Distortion Wave theory, where ORs corresponding to minimisation of strain ®elds in incommensurate interfaces are strictly calculated (Section 2.3.6).

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In addition to the interface crystallography, there are the usual thermodynamic parameters: temperature, pressure, and chemical composition (Section 2.5). The later factor is particularly important, since segregation e€ects may cause drastic changes of the GB structure even for very small bulk concentrations (Sections 3.1.6 and 3.1.7). Furthermore, the interface structure may depend on kinetics (Section 2.5.6). In nanocrystalline materials, the size e€ects may appear (Section 3.5), leading to such unexpected results as grain boundaries less compressible than the bulk (Section 3.5.6). Finally, the structure and properties of interfaces may depend on the properties of the free surfaces in the given thermodynamic conditions (Section 2.5.1). There is a very simple explanation of the role of interface plane for its energy. We consider two energy lowering factors as independent: one is the presence of good matching sectors, which increase adhesion, and the other is that separation of the crystals requires more work if the resulting surfaces after separation have a low energy. Based on these simple arguments, the energy minima as a function of OR were divided into three groups: surface type energy minimum Ð caused by the fact that the free surface of one or both crystals for the given OR has a minimum. adhesion type energy cusp Ð caused by the fact that there is a good atomic matching at the interface. combined type energy cusp Ð caused by the fact that at the same time there is the surface type minimum and adhesion type minimum. To simplify the description of the structure of interfaces the idea of higher rank structures called Structural Units (SUs) is used. We show (Section 2.1.4.5) that there is a considerable di€erence between the concept of SUs and of Structural Elements (SEs). Structural Units represent a topological description of interfaces. Structure Elements are blocks of atoms interacting with each other (Section 2.1.4). The terms are not exclusive, i.e., SUs can be SEs as well. However, it seems that more predictive power is provided by the idea of SEs Ð segments of close packed planes and rows (Section 2.14.1). The above description of ``Structure'' assumes that the crucial interactions in the interface are strain ®elds and interatomic interactions. In reality what is the ``Structure'' depends on the interaction that is important in the interface (Section 2.1). Therefore, what is a SE depends in fact on the crucial interaction from the point of view of the application of the given interface (Section 2.1.3). Due to the diculty with ®nding a correlation between the interface crystallography and properties, some kinetic parameters were used as indexes of their structure (Section 3.2.2). For instance, it was assumed that GBs of equal di€usivity have an equivalent structure. This approach was fruitful results in analysing distribution functions for activation energies and volumes in GBs in polycrystals. Some other important kinetic parameters are: the characteristic relaxation time for interfacial stress t (Section 3.2.1), and the density of sites for interfacial reactions o (Section 3.3). As far as interface kinetics is concerned, we would like to draw attention to the

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activation vector A ˆ …S  , Q , V  †, as a characteristics the thermally activated complexes. For GBs with a variety of structures the activation vector was shown to belong to de®nite lines in the space de®ned by …S  , Q , V  † variables. Their inclination depends on the ``compensation temperature'' and ``compensation pressure''. These parameters are therefore a common characteristic for all the activated complexes in GBs in the given material. The actual values of the di€usion coecients may vary for some orders of magnitude as a function of the GB structure. In the present paper, special attention was given to the ``special c general'' GB transformation (Section 2.3.5). Usually it is understood as a transformation where at high temperature the S value becomes irrelevant for the GB properties and the structure becomes disordered. We have shown that more insight into this transformation is provided by the idea that the type of SEs is a function of temperature. At low temperatures a large set of interfaces with low interplanar spacing can serve as SE source. With increasing temperature, these SEs are replaced by segments of interfaces with high interplanar spacing. At the highest temperatures, in the fcc lattice, the only source of SEs are interfaces able to facet parallel to the {111} and {100} planes or parallel to both. The above approach is substantiated by the observation that in silver GBs retain some order even at temperatures 5 K below the melting point (Section 2.6.5). The relation between interplanar spacing and interface energy follows from fundamental relationships Ð the universal equation of state for interfaces (Section 2.3.3). In the case of metals a manifestation of this correlation is the proportionality of GB free volume and energy (Section 3.1.2). The low energy of parallel interfaces, both commensurate and incommensurate, follows also from the calculations of their elastic strain ®elds using the Static Wave Distortion Theory (Section 2.3.6). Additional insight on the e€ect of high temperature on the structure of interfaces is provided by the collectivity coecient w, which is the number of atoms that interact as a group with other atoms (Sections 2.6 and 3.6.1). Such a compact group of atoms should be less susceptible to thermal disordering than an arrangement of weakly bonded atoms. Furthermore, the e€ect of temperature on the interface periodicity might be also described in terms of fs Ð the stochastic factor Ð which tells what fraction of atoms is due to stochastic vibrations shifted out of their equilibrium positions (Section 2.2.3.1). In fact, since many interfaces have a periodic structure, it may be expected that they have periodic spots of disorder separated by ordered segments, which is a kind of approximate periodicity (Section 2.2.1). Such structures might be particularly important for the properties of pre-wetted GBs (Section 3.4.4). Investigations of GB wetting resulted in the development of GB pre-wetting phase diagrams with independent variables: temperature, pressure, misorientation angle, and chemical composition. An interesting result is that in alloys GB may become more ordered with increasing temperature, which is manifested as ``solidi®cation of GBs with increasing temperature'' due to desegregation of impurities (Sections 3.4.5 and 3.4.6). The prewetting and wetting phase transitions

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play an important role for di€usion and mechanic alloying induced amorphisation of materials (Section 3.4.8) Transformations of the type special c general GB are similar to the transformation coherent c non coherent interface. The latter depend on the value of interface mis®t c and localisation parameter p These parameters permit to construct ``maps'' of GB and interface structure (Section 2.4). Low p values indicate that the energy gain for extending the sectors of good atomic matching in interfaces is small and mis®t or structural dislocations are delocalised. In other words, lowering of the energy of the system is achieved not by introducing mis®t dislocations and extending the regions of good atomic matching, but by aligning in a parallel way close packed segments of the interface. The above simple concept permits to estimate the critical thickness for the transformation coherent c semicoherent c non-coherent interface, maximum disorientation for special properties of near coincidence GBs, and plot structure maps as a function of c, thickness of the epitaxial layer or precipitate and p value. The localisation parameter value is important for the thermal stability of the interface structure. For low p values, when temperature is increased, ®rstly the Low Energy Orientation Relationships characterised only be periodicity disappear, then LEORs corresponding to parallelism of atomic rows and ``lock-in'' ORs, and ®nally only ORs corresponding to parallelism of vicinal planes remain (Section 2.6). The localisation parameter p is the ratio of interface shear modulus to the bulk shear modulus. It can be estimated analytically (Section 2.4) with due care since there is a strong e€ect of thermodynamic and kinetic conditions on the interface structure and bonding. To calculate the localisation parameter, it was necessary to know the interface shear modulus. However, since GB properties may vary from point to point, it is necessary to understand what modulus is important and di€erentiate local and global GB properties (Section 2.2.2). The later ones are meaningful only for the GB as a whole, while local properties may concern a small neighbourhood and can be averaged over the whole GB. In a periodic GB the local properties are periodic function of spatial co-ordinates and rigid body translation vector, whereas the global properties are a periodic function of the RBT vector only. This factor introduces a considerable degree of complication when the interface shear modulus is concerned. Taking in addition into account kinetics of strain relaxation, there is a need to utilise the following concepts (Section 2.4.4): hGI0 i Ð average low frequency shear modulus hGI i Ð average high frequency shear modulus GI0 Ð low frequency shear modulus for good matching sectors GI Ð high frequency shear modulus for good matching sectors The above parameters concern an isotropic interface. The interface shear modulus is connected with the concept of interface stress, which is di€erent than the interface surface tension. Contrary to surface tension, the surface stress need not to be constant along the interface. The di€erence between interface stress and surface tension can be regarded as a manifestation of

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interface nonequilibrium (Section 2.4.2). The ratio of relaxation of the strain related nonequilibrium depends on the characteristic relaxation time t, which depends on the ratio of the bulk shear modulus and interface di€usivity (Section 3.2). The p parameter value correlates with aC Ð the degree of covalence Ð which is the ratio of the covalent bonding energy to the total bond energy (Section 2.1.3.6). It is a function of thermodynamic variables as partial gas pressures, chemical composition, segregation (Section 2.5). On the other hand, it was shown that since the p value is sensitive to the free energy of the interfacial bond, in the case of ceramics and similar compounds variation of pressure and temperature should lead to much more pronounced structure changes than in the case of metals. Furthermore, since interface adhesion depends on the free energies of the free surfaces, the structure of interfaces depends on CB Ð broken bonds compensation factor. It is a measure of the di€erence of the free energy of single atoms of the given compound and the chemical compound itself. It expresses quantitatively up to what extend the dangling bonds are satis®ed in the interface. The ®nal part of the present paper was written at the same time when new results appeared concerning GBs in covalent materials: Si, SiC diamond. It was shown that interfaces might have an amorphous glass structure of high strength, be more densely packed and harder than the bulk. One way of looking at these results is that for covalent materials dangling bonds are structure elements that have to be arranged in the GB in the same way as atoms. It appears that breaking the bonds o€er locally the possibility for denser packing of atoms than in the bulk. As a ®nal remark, we would like to draw again attention to the fact that one of the crucial parameters for interfaces, the interface mis®t, is a relative value. Although mis®t is well de®ned in geometrical terms, its role for the structure and properties of interfaces depends on the physical conditions. Besides the shape of the interatomic potential at low temperatures, it depends on temperature, pressure, partial pressures of impurities in the surrounding atmosphere, impurity content, misorientation, thickness of the crystals, and ®nally on the crystallographic variables. A consequence of this is that such geometrical parameters as periodicity and coincidence have a relative value for predicting properties. The relative importance of the various geometrical parameters depends mainly on the localisation parameter p, which characterises to what extend the interface or grain boundary mis®t is localised in dislocation cores. Acknowledgements The authors are very grateful to H. Gleiter, M.W. Grabski and W.L. Johnson for the discussions and stimulation which were at the root of the present paper. This paper owes very much to extensive discussions with W. Mader, B. Straumal, E. Rabkin, M. Kohyama, A. Sagel and L.S. Shvindlerman. G. Erdelyi, D. Hesse, F. Inoko, Y. Ishida, P. Keblinski, M. Kohyama, K. Merkle, B. Palosz, S.

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Pennycook, A. Sagel, S. Sass, A. Slawska-Waniewska, K. Tanaka and S. Tsurekava provided us with photographs, manuscripts and comments, which were very helpful for preparation of the present work. S. Gorczyca, W. Gust and T. Massalski encouraged us to carry through this work to the end. Sadly, the late Professors Y. Ishida and P. Haasen, who were at the beginning of the concept of the present paper, are no longer with us. Words of gratitude have to be addressed to our families, who were so patient. WL acknowledges the support of the Polish State Committee for Scienti®c Research. Both authors are particularly grateful for the generous ®nancial support of the Deutsche Forschungs Gemeinschaft (G.W. Leibniz program). Copyright acknowledgements. The authors are grateful to the following editorial oces for permissions to publish ®gures. Journal of Physics: Condensed Matter, Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK; Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands; Elsevier Science Ltd., The Boulevard Langford Lane, Kindlington, Oxford OX5 1GB, UK; Taylor & Francis Ltd., Rankine Road, Basingstoke, Hampshire RG24 8PR, UK.; The American Physical Society, 1 Research Road, Box 9000, Ridge, NY 11961-9000, USA; Materials Research Society, 506 Keystone Drive, Warrendale, PA 15086, USA; Trans Tech and Scitec Publications Ltd., Brandrain 6, 8707-Uetikon-ZuÈrich, CHSwitzerland; Editorial Oce, Philosophical Transactions A, The Royal Society 6 Carlton House Terrace London SW1Y 5AG, UK.; The American Ceramic Society, P.O. Box 6136, Westerville OH 43086-6136, USA. We are grateful for permissions to publish ®gures from M. Ashby, G. Erdelyi, J. Erhard, H. Gleiter, D. Hesse, F. Inoko, Y. Ishida, M. Kohyama, K. Merkle, A. Ohtsuki, V. Paidar, B. Palosz, S. Pennycook, S. Sass, S. Sagel, L.S. Shvindlerman, A. Slawska-Waniewska, B. Straumal, K. Tanaka, S. Tsurekava and P. Wynblatt.

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