The effect of capillarity on the chemical potential in interfaces

Materials Science and Engineering A249 (1998) 190 – 196

The effect of capillarity on the chemical potential in interfaces E. Rabkin a,*, Y. Estrin b, W. Gust c a

b

Department of Materials Engineering, TECHNION— Israel Institute of Technology, 32000 Haifa, Israel Department of Mechanical and Materials Engineering, Uni6ersity of Western Australia, Nedlands, WA 6907, Australia c Max-Planck-Institut fu¨r Metallforschung and Institut fu¨r Metallkunde, Seestr. 75, D-70174 Stuttgart, Germany Received 23 September 1997; received in revised form 26 January 1998

Abstract A theory of the effect of curvature on the chemical potential of atoms inside an interface, e.g. a grain boundary, is developed under the assumption that the interface is an ideal source or sink for vacancies. Three simple geometries are considered to illustrate the theory. Corrections to the chemical potential appear already in the first order in curvature. The magnitude of the correction term depends on the elastic properties of the phases forming the interface, its intrinsic structure and the relative size of the grains it separates. For migrating interfaces between dissimilar phases, the correction depends on the diffusivities and the molar volumes of the two phases, rather than on their elastic moduli. The implications of the theory for fine-grained materials are discussed. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Grain boundaries; Interface diffusion; Nanocrystals; Thermodynamics of interfaces

1. Introduction The effect of the curvature K of a solid surface on the chemical potential of the surface atoms is described by the classical relation, derived already by Gibbs [1], or its modifications which take into account the effect of anisotropy. In the simplest case of a fully isotropic solid embedded in an inert liquid, it can be written as Dms = Vm( fK+p)

(1)

where Dms, Vm, f and p are the change of the chemical potential of atoms in the solid due to curvature, the molar volume, the principal components of the surface stress tensor, and the pressure in the liquid, respectively. In many applications, an internal interface in a solid, such as an interphase or grain boundary (GB), is treated as a separate, well-defined phase characterised by a distinct chemical potential. For instance, this approach is taken for a description of such processes as grain-boundary diffusion, in which a GB is treated as a thin homogeneous layer of thickness d where diffusivity is much higher than that in the bulk [2], GB segregation [3] and mechanical behaviour of nanocrystalline materi* Corresponding author. Tel.: + 972 4 8294579; fax: +972 4 8321978; e-mail: [email protected] 0921-5093/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0921-5093(98)00610-8

als [4]. It is obvious that the curvature should influence the thermodynamic properties of an interface. However, the use of the Eq. (1) is not justified in this case because it gives only the chemical potentials of the atoms in the bulk phase beneath the curved interface and does not apply to the atoms inside the interfacial phase. To illustrate this, consider a symmetrical, but slightly curved, GB in an isotropic crystal (Fig. 1). A simple coordinate transformation x%“ x, y% “ −y changes the sign of the GB curvature. However, due to full symmetry of the problem, the GB in the new coordinate system is indistinguishable from the initial one. Therefore, corrections to the chemical potential of

Fig. 1. Curved symmetrical grain boundary. The co-ordinate transformation y% “ −y, while changing the sign of curvature, does not change the physical character of the grain boundary.

E. Rabkin et al. / Materials Science and Engineering A249 (1998) 190–196

191

atoms in the GB itself can only be given by even powers of the curvature, starting from K 2. By contrast, if symmetry is broken, as is the case with an interface between two different phases or an asymmetric GB in an elastically anisotropic crystal, corrections to the interfacial chemical potential proportional to K arise. In this paper, such first-order corrections will be calculated for different simple geometries.

2. The model Our analysis is based on the work by Herring [5,6] who showed that the chemical potential of an atom in a GB in thermodynamic equilibrium with the crystal bulk and subjected to a tensile stress Tnn normal to it, is altered by an amount DmGB = − VmTnn

(2)

For Eq. (2) to be valid, the interface should be able to absorb and emit atoms freely, or in other words, it should be a perfect source and sink of vacancies. The theory developed below is applicable only to such interfaces, like large-angle GBs or incoherent interphase boundaries. Eq. (2) has been used for the description of electromigration [7], cavitation at the GBs during creep [8] and GB grooving under stress [9]. In a study by Larche´ and Cahn, the Herring relation for the atoms at the incoherent interface was re-derived using the network model of a solid, in which the total number of atomic sites is conserved [10]. However, as distinct from Ref. [10], we consider the interface as an individual phase and calculate the correction to the chemical potentials for the atoms inside the interface. A natural generalisation of Eq. (2) for the case in which the stresses are different on both sides of an interface can be written as follows: DmGB = − Vm[dT ann +(1 −d)T bnn ] a nn

(3)

b nn

where T and T denote the normal to the interface components of the stress tensors in the respective phases (which are assumed to be positive for the tensile stress in each phase). The phenomenological parameter d introduced in this heuristic equation depends on the details of the stress distribution inside the interface. For the simplest case of a step-wise distribution (Fig. 2), it can be interpreted as the fraction of the atoms in the interface which ‘adhere’ to the a phase. For a homogeneous stress, Eq. (3) transforms into Eq. (2). The mechanical balance of forces at the interface implies [11,12] that Tana +Tbnb −div f = 0

(4) a

b

where f is the surface stress tensor, n and n are the exterior normal to the a and b phase, respectively, and div denotes the surface divergence. According to Eq.

Fig. 2. Schematic illustration of a hypothetical stress distribution across the interphase boundary leading to a generalised Herring formula (Eq. (3)).

(4), a curved interface produces stresses inside the solid, which in turn contribute to the chemical potential of the interface phase according to Eq. (3). Complementing Eqs. (3) and (4) with the condition of displacement continuity at the interface and by Hooke’s law, we can determine corrections to the interface chemical potential. The method can be illustrated for the case of an isotropic solid inclusion embedded in an inert fluid, considered in Section 1. Let us assume a=solid and b= fluid. Then d= 1 (an atom on the surface cannot ‘stick’ to the fluid, which does not dissolve the atoms of the solid) and for the case of an isotropic interface div f= − fnaK

(5) a nn

From Eq. (4) we get T + p= −fK, and with the help of Eq. (3) we arrive at the Gibbs relation, Eq. (1). This example also illustrates the fact that the applied stress (in this case it is the hydrostatic pressure p in the fluid) is simply to be added to the capillary stress induced by the interface curvature. Therefore, in what follows we consider the capillary stress only and apply the method suggested to the calculation of DmGB for simple geometries in elastically isotropic solids (the notation DmGB is used both for a grain boundary proper and for a more general interphase boundary).

2.1. Bicrystal with bounded ends For this geometry the displacement u in both grains is given by u= oa,bLa,b

(6)

where oa,b is the strain in the a and the b phase, respectively, and the meaning of the length La,b is clear from Fig. 3. Assuming purely elastic deformation, i.e. using Hooke’s law, Eqs. (4) and (5) combine to make the following equation: fK= u



Ea Eb + La Lb



(7)

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which, together with Eq. (3), yields DmGB =

(1−d)EbLa −dEaLb Vm fK EbLa +EaLb

(8)

where Ea and Eb are Young’s moduli of the respective phases in the direction normal to the interface. In the case of a fully symmetrical GB in a symmetric bicrystal, d =0.5, La =Lb and Ea =Eb. The correction to the chemical potential of atoms in the GB vanishes in the first order in K, in accordance with the symmetry arguments mentioned in Section 1. A striking feature of Eq. (8) is that the correction to the chemical potential due to curvature can be both positive and negative. However, this does not violate the condition of stability of the interface against shape perturbations. Indeed, the curvature-induced diffusion of atoms along the interface should not necessarily be connected with its lateral displacement. If it is, atoms have to be transmitted from one phase to the other across the interface, i.e. positive work has to be done against the interface stresses fK. Other complications for binary systems will be considered in Section 2.4.

2.2. Cylindrical grain embedded in a cylindrical matrix Following Cahn and Larche´ [12], we define the bgrain in a stress-free state as a cylinder of radius Rb, and the a-grain as a co-axial tube with outer radius Ra and inner radius Rb. After the b-grain is inserted into the tube it contracts under the action of interface stress, the new radius becoming (1−o)Rb (see Fig. 4(a)). A general formula for the displacement field in the cylinder is [13]: b u(r)=ar+ r

(9)

where a and b are constants. An expression for the radial stress can be obtained from Hooke’s law:

Fig. 4. Same as Fig. 3, but for (a) a cylindrical and (b) a spherical interface.

Trr =

Ea Eb − (1+ s)(1− 2s) (1+ s)r 2

(10)

where s is Poisson’s ratio, and the constants a and b can be determined from the conditions Trr (Ra )=0 and u(Rb )= − oRb. The correction to the chemical potential satisfying Eq. (3) now reads (1− d)Eb dEa (R 2a − R 2b ) − 1− 2sb (1+ sa )[R 2a + (1+ 2sa )R 2b ] DmGB = Vm fK Eb Ea (R 2a − R 2b ) + 1− 2sb (1+ sa )[R 2a + (1+ 2sa )R 2b ] (11) If the elastic constants in the two grains are equal and the interface is fully symmetric (d= 0.5), then Eq. (11) reduces to DmGB =

(2+ s)(1−2s)+3sj 2 V fK 2s(1−2s)+ (2− s)j 2 m

(12)

where a new variable, j= Ra /Rb, has been introduced. The correction to the chemical potential is always positive; it decreases from 0.5VmfK at j“1 to DmGB =

3s V fK 2(2− s) m

(13)

at j“ . The former value can be easily understood: if the interface is formed by a cylindrical grain b ‘coated’ with a thin single crystalline film of a, which is almost stress-free, then in the case considered (d=0.5) only half of the interface is stressed, which results in a numerical coefficient of 0.5.

2.3. Spherical grain embedded in a spherical matrix

Fig. 3. Bicrystal with the bounded ends. The two crystals are ‘welded’ together along the curved interface and are allowed to contract/expand under the action of curvature-induced stresses. The resulting displacement u is the same in both grains.

Considerations similar to section can be used for the of radius Rb embedded in with external radius Ra and 4(b)). It follows:

those given in the previous case of a spherical b grain a hollow sphere (a phase) internal radius Rb (see Fig.

E. Rabkin et al. / Materials Science and Engineering A249 (1998) 190–196

(1−d)Eb 2dEa (R 3a −R 3b ) − 1−2sb (1 +sa )R 3a −2(1 −2sa )R 3b DmGB = (14) Eb 2Ea (R 3a −R 3b ) + 1−2sb (1 +sa )R 3a −2(1 −2sa )R 3b Again, in the case of equal elastic constants of the two crystals and a fully symmetric interface, Eq. (14) reduces to DmGB =

(5s −1)j 3 Vm fK 8(2s − 1)+ 6j 3(1 − s)

(15)

which yields 0.5VmfK in the limit of j “1 and DmGB =

5s −1 Vm fK 6(1−s)

(16)

for j“ . It is interesting to note that substitution of the value of s =1/3, which is typical for metals, in Eqs. (13) and (16) yields a chemical potential correction of 0.3VmfK for the case of an infinite cylinder and only about 0.16VmfK for the case of an infinite sphere. This difference demonstrates a pronounced effect of the geometry of the system on the magnitude of the curvature-related correction to the chemical potential.

2.4. Complications with binary systems In the above considerations of interphase boundaries, it was assumed that both kinds of atoms are distributed homogeneously across the interfacial ‘phase’. In this case, the above expressions for DmGB can be rewritten for each individual component, assuming the molar volume Vm is replaced with the partial molar volumes of the respective components. However, if the species are distributed non-homogeneously, the non-uniformity of the distribution has to be taken into account. Let us consider the simplest example of an absolutely rigid interface discussed recently by Sridar et al. [14] in connection with the problem of stability of lamellar and fibre-reinforced composites. In this model, the two adjoining phases are assumed to be insoluble and the interface diffusion coefficients of the two species are taken to be the same. It would be reasonable to suppose that in this case the interface is subdivided in two layers: A atoms stick to the a phase and are in the same stress state as the a phase, while B atoms stick to the b phase. For this case the Herring relation, Eq. (3), can be reformulated in the following way: a,b a,b a,b Dm GB = −Vm T nn

(17)

a,b with corrections Dm GB being introduced to the chemical potential for the a and b sublayers. Here V a,b m stands for the respective molar volumes of the components. Following the same analysis as in Section 2.1, one obtains

Ea,bLb,a a,b a,b = Vm fK Dm GB EbLa +EaLb

(18)

193

It follows from Eq. (18) that for the case of full symmetry (Ea = Eb, La = Lb, V am = V bm), DmGB = 0.5VmfK, in accordance with the intuitive assumption that only one half of the total difference of the chemical potential across the interface, VmfK, is available for each component as a driving force for diffusion along the interphase boundary. The analysis presented in Ref. [14] is valid in this case, but the value of the isotropic interface energy f used should be divided by two. However, in the case of any asymmetry of the problem (e.g. different thickness of the matrix and the lamellae of a strengthening phase or different elastic moduli of the lamellae and the matrix), the driving force for diffusion is different for the two species. The resulting divergence of the atomic fluxes along the interphase boundary implies that some kind of relaxation and redistribution of the interface stresses will occur in the bulk phases. As a result of such relaxation, a new distribution of stresses will be established, which makes the atomic fluxes of two species equal. This condition replaces the condition of equal displacements in both phases at the interface employed in previous analysis. Let us consider this problem in more detail. The flux of volume resulting from the interface curvature can be written in the well-known form [14]: Ja,b = −

da,bD a,b s a,b 9s Dm GB RT

(19)

where D a,b stands for the surface diffusion coefficients s in the sublayers of the a and the b phases in the model of an absolutely rigid interface; 9s denotes the gradient taken along the interface and da,b means the thickness of the respective sublayer. By analogy with Eq. (3), we express da,b through the parameter d and the total thickness of the interface: da = dd and db=(1−d)d. The condition of continuity at the moving interface, together with Eqs. (17) and (19), gives the following relation for the normal stress at the interface: dV amD as 9sT ann = (1− d)V bmD bs 9sT bnn

(20)

The situation simplifies considerably if there is some point at the interface at which both T ann and T bnn vanish simultaneously. This is the case if the curved part of the interface is connected continuously with the uncurved one, or if a sinusoidal perturbation of the initially flat interface is considered. Then integration of Eq. (20) can easily be performed along the interface and, together with Eq. (4), this yields the following expressions for the corrections to the chemical potential: Dm aGB = −

(1−d)V amV bmD bs fK dV amD as + (1− d)V bmD bs

(21a)

and Dm bGB =

dV amV bmD am fK dV amD as + (1− d)V bmD bs

(21b)

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E. Rabkin et al. / Materials Science and Engineering A249 (1998) 190–196

These corrections are still proportional to the curvature of the interface, but the amplitude is determined by the ratio of the diffusion coefficients along the interphase boundary rather than by the elastic constants. Any of Eqs. (21a) and (21b) can be substituted into Eq. (19) to get the normal velocity of the interface, 7n, according to the relation 7n = − div Ja,b : d(1− d)V amV bmD as D bs f div(9sK) 7n = RT[dV amD as +(1 −d)V bmD bs ]

(22)

where R and T are the gas constant and the absolute temperature, respectively. One can see from Ref. Eq. (22) that the analysis presented in Eq. (14) is still valid if the value of the interphase boundary diffusion coefficient is properly renormalised. It is interesting to note that, provided that d is not too close to zero or unity, the migration rate of the interphase boundary is determined by the diffusivity of the slower of the two components. Expressions such as Eq. (22) for bulk interdiffusion in a binary system were first derived by Bokshtein and Shvindlerman [15]. They showed that the difference in the diffusion rate of the two species can be compensated for not only by the plastic flow of the lattice, as considered by Darken [16], but also by the pressure gradient in the diffusion zone. In a more detailed analysis of Stephenson [17], it was demonstrated that the stresses in the diffusion zone can exist only at the beginning of the interdiffusion process or in small samples in which the plastic relaxation of stress is difficult. However, in our case the source of stresses is the curved interface, and these stresses cannot relax as long as the interface is curved. Hence, the expression given by Eq. (22), which in the terminology of Stephenson [17] corresponds to the Nernst– Planck regime, is the only possible one for the effective diffusivity at the interphase boundary. Recently, an expression for the effective diffusion coefficient along the interface similar to our Eq. (22) has been obtained during the study of the interface stability in the electric field [18,19]. The transition from the initial stressed state, which is described by the ratio of the elastic constants of the two phases (cf. Eq. (18)), to the new one, determined by the diffusivities (cf. Eqs. (21a) and (21b)), occurs through an adjustment of the elastic strain. This kind of relaxation was neglected in Eq. (17) because of its small contribution to the total flow of matter with respect to plastic flow. However, in the case under consideration, the role of elastic stress and strain is decisive, because diffusion along the interphase boundary is driven by the gradient of the interface-induced stress, and not by the gradient in composition, as in Eq. (17). The change in elastic deformation, too small to contribute to the interface velocity, changes the driving force for the diffusion along the interphase boundary consider-

ably. The normal velocity of the interface during the transition from the initial ‘elastic’ to the ‘diffusion’ regime is given by the following relation: 7n = − div Ja,b +

(ua,b (t

(23)

It should be noted that the transient time is very short because of the smallness of the elastic displacements (see below). Variation with time of the elastic displacements ua,b, different in the two phases, compensates for the initial difference in the diffusion fluxes during this transient regime. As a result, the elastic displacements in the ‘diffusion’ regime are different in the two phases: if they were separated along the interface, the two pieces would no longer match along the interface in the stress-free state. One can make a rough estimation of the duration t of the transient regime by substituting the time derivative in Eq. (23) with the averaged velocity and assuming the initial shape of the interface to be given by a sinusoidal function with a small amplitude and a wavelength l. The elastic distortion from such an interface spreads over distances of the order of l in the bulk phases [20], so one can write (u/(t : lfK/Et. Taking into account Eq. (19), we arrive at the result t:

l 3RT 4p VmEdDs 2

(24)

For the parameter values typical for experiments in which the evolution of the interface morphology is governed by the diffusion along the interphase boundary (see, for example, Ref. [21]: Ds =10 − 11 m2 s − 1, Vm = 10 − 5 m3 mol − 1, l= 10 mm, E=100 GPa, d= 1 nm and RT=5000 J mol − 1), we arrive at t: 10 s. This time is too small in comparison with the usual duration of an experiment of this type (at least some hours), so that detailed numerical investigation of the transient regime is not of great practical importance. In some circumstances, a non-zero integration constant should appear upon integration of Eq. (20). For example, this is the case for an isolated inclusion embedded in a solid matrix. There is a non-zero compressive stress in the inclusion acting everywhere at the interface, which gives rise to a non-zero integration constant containing the elastic moduli of the contacting phases. The expressions for the chemical potentials at the moving interface would then be given by a mixture of expressions of the type represented by Eqs. (18), (21a) and (21b), and the velocity of the interface motion will be determined by both the elastic moduli of the adjacent phases and the interphase boundary diffusion coefficients. A detailed analysis of this situation is beyond the scope of the present paper, however.





E. Rabkin et al. / Materials Science and Engineering A249 (1998) 190–196

DGB = D6 exp

3. Discussion It has been demonstrated that the correction to the chemical potential of atoms in a curved interface can be calculated from a known distribution of interface stresses with the help of the Herring formula, Eq. (3). However, the situation in a real solid differs significantly from the simplest cases considered here. A detailed calculation of interface stresses inside the solid is hardly possible, and only an averaged stress can be calculated, as recently done by Weissmu¨ller and Cahn [22]. They derived an expression for the pressure DP inside a polycrystal stemming from GBs: DP =

2gŽ f  D

(25)

where Ž f , g and D are a suitable average of the surface stress tensor, a geometric constant and the averaged grain size, respectively. It follows from our analysis that the chemical potential of atoms inside the GBs changes in this case by: DmGB :r

Vm f D

(26)

where r is a constant which can be either positi6e or negati6e, depending on the local surroundings of a given GB. For example, for a spherical grain inside an infinite matrix, the value of this parameter is given by r = (5s −1)/6(1 − s) (cf. Eq. (16)). To simplify further discussion, we substitute the principal components of the surface stress tensor f with the isotropic GB energy gGB and employ the well-known relation [23]: mv = mGB −vgGB

(27)

where mv and v are the chemical potential of the atoms in the bulk phase and the molar area of the atoms in the GB phase, respectively. A curved GB causes changes in both mGB and mv, because the bulk phases are stressed (strictly speaking, the chemical potential in a non-hydrostatically stressed solid is not defined). From Eqs. (26) and (27), one can estimate the upper limit of the variation of GB energy due to curvature, assuming that mv stays constant: d DgGB :r gGB D

(28)

where d is the thickness of the GB phase. Eq. (28) represents a generalisation of Tolman’s equation describing the dependence of the surface tension of liquids on curvature [23]. In nanocrystalline materials, in which the value of d/D can be as large as 0.1, the change of the GB energy associated with the curvature (i.e. its increase or decrease) can be substantial. A semi-empirical correlation between the GB energy and GB diffusion coefficient [24],

2a gGB kT 2

195

(29)

can be used to evaluate the effect of grain size on the GB diffusivity. Here DGB, Dv and a are the GB and bulk diffusivities, and the lattice parameter, respectively; k is the Boltzmann constant. Estimations on the basis of Eqs. (28) and (29) show that a 10% increase of the GB energy due to GB curvature in a nanocrystalline material can increase the GB diffusion coefficient by more than two orders of magnitude at room temperature. This can enhance the rate of diffusion creep and may even lead to a change of plasticity mechanism from dislocation to a diffusion-controlled one. This would explain the observed anomaly in the grain size dependence of the yield strength, consisting in a deviation from the Hall–Petch relation in the range of sub-micron (nanometer) grain sizes [25,26].

4. Conclusions From the results of the present study, the following conclusions can be drawn. 1. A method for calculating the corrections to the chemical potential of atoms inside a curved interface was suggested. It was assumed that the interface is a perfect source and sink of vacancies, and the generalised expression of Herring was combined with the expression for the interface stresses. 2. The method was applied to three simple geometries: a bicrystalline rod with fixed ends, a cylindrical grain in a cylindrical matrix and a spherical grain in a spherical matrix. In all cases, the correction to the chemical potential inside the interface was shown to be proportional to the curvature, and also to depend on details of the intrinsic structure of the interface, on the elastic constants and on the relative size of the adjoining grains. The correction may be either positive or negative. 3. It was shown that interphase boundary migration is accompanied by a redistribution of the interface stresses in the adjacent phases. As a result of this redistribution, the chemical potential inside the interface is determined by the interphase boundary diffusivities and atomic volumes of the components. The conventional expressions for the evolution of the interface shape can be used, but the interphase boundary diffusivity should be appropriately renormalised. The time needed for transition from the regime in which the chemical potentials of atoms in the boundary are determined by the elastic moduli to the dynamic regime, in which the chemical potentials are determined by the diffusivities, has been estimated. For typical experimental conditions, the transition from one regime to another takes only a few seconds.

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E. Rabkin et al. / Materials Science and Engineering A249 (1998) 190–196

4. It was shown that for the case when interfaces in a fine-grained (e.g. sub-micron) material are curved, their energy would be changed significantly, by up to 10%, for instance. This may lead to a dramatic increase of the GB diffusivity (in the grains whose interface energy is increased) and enhance the diffusion-controlled contribution to plastic deformation appreciably. This effect may be of interest in the context of deformation mechanisms in nanocrystalline materials.

Acknowledgements E.R. wishes to acknowledge support from the University of Western Australia through a Gledden Senior Visiting Fellowship. Helpful discussions with Professor Yves Brechet are gratefully appreciated.

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