Aging of ferrous martensites

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2. CLASSICALTHERMODYNAMICS There are some aspects of the thermodynamics of shear transformations that are particularly attractive for study:°6) (1) Unusual, but predictable, solid solutions and superlattice structures are created. (2) Regardless of how reversible the transformation is, the energy of the transformed state can be determined from simultaneous mechanical and calorimetric measurements. The transformation can be studied either isothermally or adiabatically.°7-~9) The temperature range is conveniently low, below where diffusion becomes appreciable. The energy change for a complete cycle of transformation and its inverse is an indication of stored energy from other processes. If this is small, it is an indication that the transformation is structurally, if not thermodynamically, reversible. (3) The calorimetric measurements give lower bounds on the entropy changes; calorimetric measurements on both the transformation and its reverse give upper and lower bounds to the entropy of the transformed state relative to that of the initial and the reverted state. (4) The dissipation, as seen in hysteresis loops, resulting from these shear transformations can range from very large to very small. If it is rather small, then experimentally determined bounds on the entropy converge to permit an accurate determination of this quantity for the non-equilibrium sheared state. This is an example where entropy changes are rigorously measured along a reversible, but highly non-equilibrium, path. (5) Because of the exact correspondence, the configurational entropy does not change as a result of the shear. Therefore this is an irreversible process for which a major contribution to the entropy change is known, and need not be measured. The entropy change that results from electronic, vibrational, and other contributions can be measured directly. (6) For highly ordered structures, there is the possibility of a third-law entropy measurement for the vibrational, electronic, and other contributions to the absolute entropy of the transformed crystals. (7) Energy and entropy changes can be combined to give free energy changes for use in calculations of stresses, permissible isothermal reaction paths, and equilibria. 2.1. Energy Classical experimental thermodynamic measurements are instructive here, because they give energy changes of the specimen regardless of whether the processes are reversible or not. We can experimentally determine whether untwinning or reversion reestablishes many aspects of the original structure, particularly if there is an energy difference between the original and reverted phase. We shall denote with a prime the thermodynamic properties of a crystal that has been transformed, and with a double prime one that has been returned to the untwinned or austenitic state to distinguish it from the unprimed initial state. If there is exact reversibility in the correspondence, the unprimed and the doubly primed states should be equal, and there would be no stored energy. One can monitor both the heat added (6q) and work done on the system ( - 6 w ) ; for 6w, which is force (load) times distance moved, a large strain formulation would be appropriate for expressing the work in terms of stress, transformation, strain, and fraction transformed. Although awkward in this context, the historical sign convention is followed; heat added and work done by the system are positive, while both the expected heat given off and work done on the system are negative. At any stage in the twinning/untwinning or martensite formation/reversion cycle, the change in internal energy (dE) is then dE = c~q - c~w. JPMS 36/I-4~F

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The heat and work terms can be integrated for whatever part of the process is under consideration to give the change in E. To form the primed state

E'- E =

6q -

6w.

The energy of the twin, pseudotwin, or the martensite is thereby readily determined relative to the initial phase. Similar equations will compare the energies of all three states. If there is a temperature difference between two states, a standard correction can be made to obtain isothermal energy changes, if the heat capacities are known. With E ' and E corrected to the same temperature, the validity of any solution model can be tested. These equations give the actual energies of the transformed state, even if there is significant dissipation; the extra work is dissipated and shows up as a temperature rise or heat given off. For a complete cycle that reestablishes the original structure with no stored energy, i.e. structural reversibility,

E"-E=~fq-~,w = C(T"

-

T)

where C is the heat capacity of the initial state (assumed to be constant), and T" - T is the temperature change resulting from the cycle. The stored energy E s would be

Es is unlikely to be negative, and likely to be small in magnitude compared to the work and heat terms. If the specimen is either kept isothermal or returned to the original temperature, the net work done on the system equals the net heat released less the stored energy

If there is no stored energy, the integrals are negative, and are a measure of the dissipation; net work is done on the system and heat is given off. If the cycle is done adiabatically 6q = 0, and the temperature will rise C(T" - T ) = - ~ 6 w > O. If the system is kept isothermal, this implies that net heat is given off and net work is done on the system,

2.2. Entropy According to the second law of thermodynamics, entropy changes are always greater than or equal to 6q/T. In a cycle in which the system is returned to its original state, and ~[tSq/T] = 0, the cycle is thermodynamically reversible. The entropy difference between any two states in that cycle is then exactly equal to S[¢Sq/T] integrated between those two states. This is true even if the states created during this cycle are not in equilibrium; it means that there were no dissipative process while the system was in these states.

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Even if ~[¢Sq/T] is measured to be a small negative quantity, say - E , we can bound the entropy change between any state along the cycle and the initial state to lie between S[~Sq/T] and S[Sq/T] + E. The entropy S can be bounded in a cycle that returns the system to its original state. Since dS > Sq/T throughout the cycle, the entropy change S" - S is bounded by

S " - S > ~[Sq/T]. lid

If we have structural reversibility, S " = S, the measured ~[6q/T] is the quantity -E, the uncertainty in the measured entropy along the deformation path.

q/T] +E > S ' - S >

q/T].

More generally, when there is stored energy Es, the system is not returned to its original state, the entropy is bounded from below relative to that of the initial state and from above relative to that of the reversed state with unknown entropy that must be estimated. If we assume that the stored energy of the reversed state has not contributed to the entropy, then the bound in the above equation holds as well. This is a reasonable assumption; in these processes, the configurational entropy should be unaffected, the phonon and electronic entropy is not likely to change, and the most likely sources of stored energy are internal stresses and dislocations, which can be assumed to have little contribution to the entropy. If the transformations have not introduced significant disorder, then S, S', and S" can be independently determined by a heat capacity determination from low temperatures to the transformation temperature, provided that the latter is below the tempering temperature. The differences in the electronic, magnetic, and lattice vibration contributions to the specific heat between the transformed and initial states can then be used to find their contributions to (S' - S). The total (S' - S) found from the specific heat should be within the bounds found from the integrated heats of transformation. The shear transformation can be studied with an isothermal calorimeter, but the temperature difference between the heat source or sink at the transformation front and the reservoir must be kept small if c is to be kept small; heat flow is a dissipative process and c is quadratic in the temperature gradients. Alternatively the transformation can be considered to occur adiabatically and the 'system' is confined to the moving interface, and then T can be taken as the temperature at the moving transformation front. °) (In classical thermodynamics, 6q is the heat added to the system and T is the temperature of the reservoir. Here the system would be the interface and the remainder of the specimen the heat reservoir.) For an adiabatic process when there is a temperature change to T', S'(T') - S(T) >_ 0. To convert this to an isothermal estimate we can integrate S'(T') - S(T) > - C In

C/T to obtain

(T'/T).

Because there is no configurational entropy change in any of the shear transformations, this is a measure of the entropy change from all other sources. For deformation twinning and pseudotwinning, these are vibrational and electronic, and have in the past been assumed to be negligible; recently this assumption has been questioned for the vibrational entropy, and tested by low temperature differential scanning calorimetry that compares ordered and disordered samples of the same alloy3 TM For the martensite transformations, the temperature

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To is defined by a zero free energy change for the transformation. With a sizeable contribution to the free energy change from the difference in the energies of the basic structures, this is an explicit recognition that vibrational, electronic, and magnetic states of ordering contribute to the entropy change. 3. DEFORMATIONTWINNINGAND PSEUDOTWINNING Deformation twinning and pseudotwinning of alloys are in some ways simpler than martensite transformations in that non-equilibrium versions of the original phase are created with a different atomic order and a different symmetry. Whereas the martensite transformations can only occur in those alloys in which the free energies of two different structures cross, and then only in the vicinity of the crossing temperature To, in principle twinning can occur over a range of temperature in all alloys, because the free energy of the ideal twin should be equal to that of the initial state. Ordering will affect the free energies of both the initial and the sheared states. For martensites this results in a shift of To that may be compensated by a change in the stress required to induce the martensite transformation. For twinning and pseudotwinning this difference in free energies will require an increase in stress to create the non-equilibrium structure. Because there is no need to consider the differences in stability between two underlying structures (lattice stabilities ~21)) many of the phenomena discussed below are simpler to understand for twinning and pseudotwinning than for martensite transformations. With martensites we have more control, because we can affect the transformation with both stress and temperature changes. Of particular interest for this paper is the study of the two kinds of alloys that give rise to structural changes: (1) Partially ordered or segregated alloys that are neither ideal random solid solutions, nor perfectly ordered nor segregated into two or more such ideal phases. Such alloys will have a wide variety of local configurations that will be changed by the shear transformation. (2) Highly ordered alloys that can be deformed to transform into another crystal structure, i.e. another superlattice of the same underlying disordered structure. The twinning shear deformation destroys all the point group symmetry elements present in the initial crystal except those that happen also to be symmetry elements of the deformation process as well.(22)The remaining symmetry elements form a crystallographic point group that is the intersection of the symmetry groups of the original crystal and the twinning strain. With partial local order, the new local order and the pair correlations in the untempered twin will have a symmetry that is different from that of the original crystal. If one neglects the lowered symmetry of the pair correlations, the twinned structure would appear to be the same. The reduction of point group symmetry to the appropriate subgroup of the initial symmetry is easily determined because there are so few symmetry elements in a deformation. Homogeneous deformations can be represented by strain quadrics. When all three principal strains are different from each other, as they are in a simple twinning shear without shuffles, these shears have orthorhombic (mmm or D2h) symmetry.°°'m If an even-fold axis of the crystal is parallel to a principal strain axis, a 2-fold axis survives in the twin. A mirror perpendicular to a principal strain axis also survives in the twin. Twins therefore have orthorhombic or lower symmetry unless new symmetry elements appear. Any deviations from perfect order, or perfect randomness give rise to distortion that prevent the new symmetries. Twinning shear deformations in superlattice structures have received considerable attention/ 23-28)It is often possible to find deformations (usually with an inhomogeneous part, called shuffles) that will give the product phase with the same structure as the original phase, but this is of interest here only in that one should find methods of loading the crystal to avoid

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(a)

(b)

(c)

FIG. 1. (a) Two adjacent layers of { 112} of a B2 structure; the two species are indicated by different sizes. The first layer is connected by - - ; the long axis is the (111) twinning and pseudotwinning direction. (b) Twinning carries the second layer to the right, over a saddle formed by the atoms in the first layer. (c) Pseudotwinning carries the second layer to the left by half the twinning distance, and no equivalent saddle, but now the arrangement of the two species is different.

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FIG. 2. The C m m m structure can be obtained by a ferrous Bain strain along a cube axis of an fcc that has become a 2-fold axis in an L12 (CuAu) ordering. The same structure is obtained in a {112} pseudotwinning of a B2. The bcc cube basis vectors are the short axis and half of the diagonals of the large face.

these modes. The deformations that recreate some superlattices should be easy to avoid: they are usually larger than those of the disordered versions of the structure. For B2 structures the usually quoted twinning displacements on {112} are twice as large as that found in the usual bcc twinning, and are in what would be antitwinning directions of the disordered bcc structure (Fig. I); in addition there are steric hindrances on the { 112} slip plane for the motion of twinning (partial) dislocations, but not for the pseudotwinning motion that creates an orthorhombic Cmmm superlattice (Fig. 2). The {1 14} has been found for the twinning and {112} for the pseudotwinning of B2 in NiTi329'3°) Even if the twinning and pseudotwinning modes compete, it should be possible to avoid the one that recreates the superlattice and to favor the one that produces the different superlattice with a properly oriented loading system. Similar care must be taken to avoid ordinary slip, which is a dissipative process that may also leave stored energy. In some alloys the change in configurational ordering energy may be so large that the stress required to effect the twinning may not be reached before other modes are achieved. Alloys that undergo ordering transitions have relative low ordering energies; a well-documented study of pseudotwinning exists for such systems315~ The deformation near the order~lisorder transition is likely to be the same as for the disordered structure, and therefore one that does not necessarily restore the superlattice. Major effects of prior heat treatment are found, and many can be understood qualitatively in simple thermodynamic terms. If the pseudotwinning is to create a new superlattice, the required stresses must be large enough to supply the change in energy. Assuming that creating a superlattice structure with wrong bonds ought to take an energy comparable to that of disordering, this energy should be of order RTc per mole, where R is the gas constant and Tc is the equilibrium disordering temperature for the alloy. Assuming T~ to be of order 103 K, a molar volume of 10 -5 m 3, and a shear strain of order 1, the required stress is of order 1 GPa, which is very large. Lower stresses are required for twinning partially ordered structures, ~31)or, if there is a choice of twin modes, twinning into those new superlattices that have structures with lower energies.

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Because ideal twinning recreates the same phase, while the To condition involves two phases, the phase rule indicates that twinning should occur with an extra degree of freedom. Since twinned and untwinned structures have the same free energy, To for twinning occurs at all temperatures and compositions where the phase occurs; it fills the entire phase field instead of being confined to a To curve (or To hypersurface for multicomponent alloys) as it is for martensite transformations. For the pseudotwinning of an alloy that undergoes an order-disorder transition, this To can be considered to fill the phase diagram portion for the disordered phase, terminating at the transition. The stress required to provide the energy change on pseudotwinning can then be thought of in terms of distance from To, much as it is for the martensite start temperature, Md in stress-induced martensite.

4. MARTENSITES As with twinning, our interest is in alloys that give elastic martensite transfolTnations. We distinguish two kinds of alloys: (1) When the original phase, called austenite, has local order but no long range order, a distortion of this local order is inherited by the martensite. The transformation strain, which we will generally call the Bain strain, is not affected except for minor distortions. (2) Austenites with long range order that transform to give superlattice structures in the martensite that differ from superlattices that would normally occur at equilibrium, as revealed by tempering or deduced from alloy theory. Unlike pseudotwinning where the structure is identifiable as a different ordering (of the disordered version) of the original structure, there may be no clearly defined parent structure in some cases. Usually though, the martensite of an ordered austenite is clearly a derivative structure of a higher symmetry structure. The diffuse scattering that has the symmetry of the austenite is distorted in the martensite by approximately the inverse of the transformation strain. For a ferrous martensite transformation the Bain strain has two equal principal axes. Because the infinite rotation axis of the strain is parallel to a 4-fold axis of the austenite, this 4-fold axis, and all mirrors parallel and perpendicular to it, survive to give a martensite with tetragonal (4/mmm or D4h) symmetry. In very special cases, or after tempering, a new symmetry arises, a 'supergroup' of the tetragonal intersection group, and the martensite becomes bcc with cubic (m3m or On) symmetry. In order for this to happen four of the twelve (110) 2-fold axes in the austenite, that according to the correspondence became (111) axes with no special symmetry in the tetragonal virgin martensite, have to become the 3-fold (111) axes in the bcc. Tempering (of a ferrous martensite) reveals that the more symmetric bcc structure is lower in free energy than the tetragonal one, and in a pure perfect single-component material the virgin martensite could be cubic. Many factors prevent the establishment of cubic symmetry in the virgin martensite. In non-random alloys in the absence of diffusion, the local atomic arrangement has been distorted by the strain, and the different chemical arrangements along directions that must become equivalent prevent the creation of the 3-fold axes, leaving the martensite tetragonal. The deviation from ideal cubic symmetry depends on the deviation from randomness in the original austenite. Iron and carbon occupy different Wyckoff positions. Even if the carbon is randomly distributed on its (interstitial) sites, the austenite can be considered an ordering, closely related to the NaC1 structure; iron occupying one

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position, and a mixture of carbon and vacancies occupying the other. In the austenite, there is a one-to-one ratio among the numbers of the two sites; in cubic ferrite the ratio is one-to-three, only one of the three carbon sites per iron site in the bcc corresponds to the carbon site in the fcc. If the carbons are carried to a single site in the martensite, its structure is tetragonal. Whatever distributional configurations the carbon has in the austenite are reproduced in the distribution of carbon on that site in the martensite with the same configurational entropy. Tempering by distributing the carbon among the other two sites will raise the entropy. The homogeneous deformation that is normal bcc twinning does not carry all the interstitial carbon in a steel into the same kinds of octahedral sites; the deformation twinning requires shuffles. The effects of interstitial solutes on the twinnning stresses have been found to be in reasonable agreement with estimates of the free energy changes. ~32) The original reports ~33)that martensites were tetragonal correctly implicated carbon, but as Winchell and Cohen ~34)demonstrated, even in a binary Fe-Ni alloy martensite there is a residual tetragonality due to non-random distribution of substitutional atoms. This distribution has a cubic correlation in the austenite that is distorted by the shear to a tetragonal symmetry, and prevents the martensite from being cubic. The twins that according to theory(35)are part of the martensite structure have been found to be those that do not require shuffles.(36) If the austenite is an ordered derivative of fcc, the martensite will have a distorted superlattice of bcc that is predictable from the Bain strain. For example, an L10 austenite (P4/mmm), transforming with the usual ferrous Bain strain, will give B2 martensite domains, that may be tetragonally distorted, when the unique axis of the Bain strain coincides with the 4-fold axis, and orthorhombic martensite domains where the unique axis is along a 2-fold axis. This orthorhombic martensite should have the same Cmmm structure (Fig. 2) predicted in 1952 by Laves01) for pseudotwinning of B2, and found in the pseudotwinning of FeBe°5) (in terms of bcc coordinates, a = [100]a0, b = [110]a0, c = [110]a0; there are 2 atoms of one species in the a position and 2 atoms of another species in the c position337~) Since several variants are needed in a single martensite plate to approximate the invariant plane strain, martensites from L10 ordered austenites are predicted always to be mixtures of variants with different superlattice structures. This structure was once proposed for the orthorhombic martensite in Au-Cd. (38) In contrast, if the austenite has cubic ordering, such as LI2, and undergoes a ferrous Bain strain, all variants that comprise a martensite plate will have the same structure. The superlattices imposed on the martensite by the Bain strain can be those that would be in the equilibrium martensite, but that should not be the general rule. Thus the martensite transformation should also be looked upon as a way of creating phases or mixtures of phases with non-equilibrium states of order. Ordering of the austenite changes both the energy of the austenite and the martensite. These two changes result in a shift of the To temperature. The lowering of the austenite free energy due to its spontaneous ordering is measurable by standard thermodynamic means. A martensite with non-equilibrium ordering will have a free energy that can be raised or lowered depending on whether the change in order imposed from the austenite brings the martensite closer or further from equilibrium; its energy and the non-configurational parts of the entropy can be measured calorimetrically. If the shifted To and the martensite start temperature Ms are in an experimentally accessible range, there should be a stress-induced martensite above Ms. The required stresses should not be large, they should depend on how far the temperature is from To.

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5. MECHANICALPROPERTIES The unusual elastic behavior of these alloys is understood in terms of their thermodynamics. For small stresses there is an elastic region with stiffness coefficients that are normal for metals. At a certain stress level, the transformation sets in, and large deformations resulting from the transformation occur at almost constant stress resolved on the transformation shear system. On reducing the stress level somewhat, this large deformation is recovered almost elastically as the transformation reverts. In ideal twinning, where the same phase is recreated, work is done on the system both to twin and untwin, but this work is entirely dissipated. In alloys there will be an energy and entropy change that is stored in the twin or pseudotwin, and available to aid in the reverting. In the Fe-Be system Green and Cohen °5) managed to find a wide variety of what they called pseudo-elastic behavior, including spontaneous reverting on unloading. The area in their stress-strain curves, between loading and unloading is a measure of the work lost in dissipative processes and was shown to depend on heat treatment. In the specimen with the least dissipation in their Fig. 5, the dissipation is of order 20 MJ/m 3, about 1/2 of the work done to pseudotwin. This is a large fraction, but there is certainly room for reducing or manipulating the dissipation. Since they did not monitor the heat release or temperature changes, only partial thermodynamic inferences can be made. We can use the thermodynamic relations and solution models to predict the minimum work done on the twinning or pseudotwinning systems required to transform and the maximum work that these systems can do to revert. In so far as heat treatment prior to transformation changes the order of the initial state, and tempering changes the twinned and pseudotwinned states, we can alter these thermodynamic factors, and hope to extend the range of behavior found by Green and Cohen. The hysteresis in a martensite transformation, as measured by a temperature shift between the transformation and reversion, can vary by two orders of magnitude. Most of the small hysteresis systems are non-ferrous and have superlattices. As we have seen, there is the possibility of obtaining different superlattices in the different variants in a single martensite plate; some of these may have a much higher energy and may trigger reversion for the entire plate.

6. STATISTICALMECHANICALSOLUTIONMODELS Statistical mechanics provides an alternate way of calculating thermodynamic properties of materials. This method requires a different data base; the properties could be calculated from a knowledge of the energies and multiplicities of all the ways of arranging the atoms or molecules of the system. But these data are almost never obtainable from experiment on condensed systems. A large number of schemes for approximation have been used. The parameters in such approximate models can come either from fitting thermodynamic data, or, more recently, from 'first principle' approximate quantum mechanical calculations. The important states in a statistical mechanical calculation are the most probable states, those low in energy and/or high in multiplicity (entropy).

6.1. Cluster Models The statistical mechanical prediction of the thermodynamic properties of solid solution phases is a formidable two-part problem; the formulation of the energies of all the states of JPMS 36/I-4~F*

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the entire system, and the calculation of the partition sum from which the probabilities of occurrence of any of the various states of the system and the thermodynamic properties are evaluated. The second part has never been solved rigorously even for the simplest idealization of a solid solution, the three-dimensional Ising model, in which the total energy of any particular state is obtained from summing near-neighbor pairwise interactions on a rigid lattice of two kinds of atoms (or up and down spins). (39) The severity of the problem is most apparent near an order~lisorder transition, where neither perfect order nor perfect randomness (on the lattice or any sublattice) are even remotely valid approximations. The correlation length, the effective distance of correlation due to the interactions, grows to infinity near a higher order phase transition even in the Ising model where the energy is assumed to be limited to near-neighbor interactions. For real alloys the first part of the theoretical problem is compounded by the complexity of the interactions among atoms, which are neither easily nor accurately defined and determined from experimental data. No part of the solid solution is sufficiently isolated for its energy to be determined. In the theoretical modeling of the thermodynamic properties, the energy of any specific static configuration of the entire system is commonly expressed in terms of a sum of the assumed or calculated energies of all the local clusters (clusters can be defined to include pairs that are not near neighbours); a sum of products of effective cluster interactions (ECI), and the number of times each cluster occurs. Recently quantum mechanical calculations have advanced to the point where the ECI can be obtained directly from 'first principle' calculations with sufficient accuracy for modeling of alloy phases. (4°'41) More usually, the ECI are used as fitting parameters in solution models such as the regular solution, or in accurate, but still approximate, statistical mechanical procedures such as the cluster variation method (CVM), (42~) to conform to measured properties. Both the choice of a limited set of ECI and the statistical mechanical modeling introduce approximations, but the fitting procedures results in some cancellation of errors when the models are then used to predict properties. The successes of the thermodynamic predictions obtained from solution models are in part due to the cancelling errors in this circular procedure. The measured properties that are used for fitting come from several kinds of data: (1) Calorimetric measurements yield energy changes of entire systems, but usually only for changes near equilibrium or between equilibrium states. (2) Phase equilibria have been most widely utilized. The location and temperature dependence of phase diagram lines are sensitive to heats, entropies, and partial molar quantities. (3) Structural measurements can give information about local configurations; diffuse scattering observations (intensities in reciprocal or k-space) on an equilibrated solid solution can give the pair correlations to quite distant neighbors (45-47) (and an estimate of multi-atom correlations(484°)). The measured properties are then used in the same theory to predict phase transitions and other equilibrium properties. There is a need for additional kinds of input data. The shear transformations should provide valuable input data, experimental energies of well-characterized structural changes to states that are not part of the usual equilibria. The validity of the assumptions about the statistical mechanical calculations for a given set of ECI can be tested by, for example, computer experiments (e.g. Monte Carlo calculations~51'52)) on a system with exactly the same interactions. But this is a test of only the second part of the problem. As the sophistication of the statistical mechanical procedure increases, the accuracy does also, but all models in use with finite range interactions are

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known to fail in the vicinity of critical points. On the other hand, an experimental test on a real alloy is a test of both parts, but is not as clean a test of just the statistical mechanical assumptions as the computer experiment on a hypothetical alloy. This is because the comparison is also sensitive to errors that arise from assuming a limited set of ECI, and neglecting contributions from static displacement and phonons. The determination of a limited set of ECI by fitting can cancel many of the errors in the statistical mechanics, and result in an apparently good agreement between theory and experiment. These approximate solution models have one great advantage: Because they inherently make assumptions about which limited number of states are accessible to the system, they can deal easily with non-equilibrium metastable and even unstable states. Nonequilibrium states thus present predictive challenges, and provide additional data to test the assumptions in the models. Experimental thermodynamic measurements on alloys explore variations with temperature and composition at equilibrium. But even rapid temperature variations can only create a limited range of non-equilibrium states for exploration, confined to the vicinity of equilibrium states. These states are dominated by low free energy configurations, that is, low energy states and those higher energy states that have many equivalent configurations and thus a high entropy. Changing the temperature alters the relative probabilities of these states, but really unusual high free energy configurations never contribute significantly. Some models are useful as a basis for predicting the paths and kinetics of relaxation toward equilibrium and of order~lisorder processes. But since all the measurements involve states that are never far from equilibrium, the full range of the predictions have not been tested. Hence the range of ordered structures available for study is limited to disordered phases with some local order that are close to equilibrium, and to those ordered phases that are stable or metastable in some part of the phase diagram. Only some of the first principle methods can be used for modeling non-cubic structures, but the ECI, even if determined from cubic structures, can be used for predicting the energies of all states obtained by ordering on the 'lattice'. Sheared states provide energy data that have hitherto not been available for comparison with first principle calculations and the various methods of choosing the ECI. Let us use the example of the pseudotwinning of a B2 structure with A and B atoms by the bcc mode to give the Cmmm structure (Fig. 2). Initially there were only A-B near-neighbor pairs; afterwards there were 1/4 A-A, 1/2 A-B, and 1/4 B-B, the same as a random solution. A model with the simplest ECI, near-neighbor pair interaction only, would predict that the pseudotwinning energy is equal to the energy of disordering. But this pseudotwin is not a random solution. Initially there were only A-A and B-B second-neighbor pairs in equal proportion; afterwards 2/3 are A-B and 1/6 each are A-A and B-B, compared to respectively 1/2 and 1/4 for the random solution. Consider a model with just two ECI; first and second pair-wise interaction. If wI and w2 are the near- and second-neighbor bond ordering energies, the heat of formation per atom of the B2 and the disordered phase from the separated bcc elements would be respectively -4wi and - - 2 W 1 - - 3 / 2 w 2 , and the heat of disordering of B2, 2Wl -- 3/2w2. Standard heat measurements can be fitted to these two ECI with no independent check. The pseudotwinning energy would be 2W 1 - - 2W2, and would give an independent check. First principles calculation on bcc superlattices have been used to give the 4-atom ECI. In the bcc the 4-atom clusters comprise a skew quadrilateral with near-neighbor edges and second-neighbor diagonals. They can also be described as tetragonally distorted tetrahedra.

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There are six of them seen along the 4-fold cube axis, with vertical and horizontal also being cube axes of the bcc. A A

A A

A [1]

A

A A

B [2]

A

A B

B [3]

B

A B

A [4]

B

A B

B [5]

B

B B [6]

For each of these clusters there is an ordered structure with only this cluster: [1] or [6] would form elemental bcc, A2 (Fe); [2] or [5] would be DO3 (BiF3, Fe3A1); [3] would give B2 (CsCI, CuZn); and [4] would make B32 (NaTI). In a first principles calculation of a binary alloy system, the energies of these six structures would be calculated from a quantum mechanical model. From the four energies of formation from the elemental clusters [1] and [6], four ECI would be obtained for a statistical mechanical calculation of the system. Even if the system never orders into any or all of the structures, all the ECI are needed, because any state of the system that is not perfectly ordered will contain all clusters. Pseudotwinning of a B2 structure to give Cmmm will convert a structure that had only the third cluster into one that is 2/3 of the fourth and 1/6 of the second and fifth. It is highly unlikely that systems that order to form B2, would also order to form B32, which makes it difficult to test that part of the calculation with an experiment. Pseudotwinning, however, provides an opportunity to create an ordered structure that is 2/3 of these clusters; alternatively the calculation method could be tested with a determination of the Cmmm structure with a ratio of axes that conforms to the observed distortion from the bcc. 6.2 The k-Space Thermodynamics From diffuse X-ray scattering, intensity pair correlations among the atoms can be obtained353) With the use of a mean field theory, pair correlations are in turn related to pair-wise ECI, more specifically their Fourier transform V(k). ~a6) This has given rise to a k-space, or reciprocal space, thermodynamics in which the configurational parts of free energies are directly related to diffuse scattering intensities354'55~Because this method is based on a linearization, it is valid only for weak diffuse scattering and not valid for Bragg peaks. A set of ECI can be calculated from scattering measurements on equilibrated disordered alloys at a single temperature. From such a set the equilibrium diffuse intensity at other temperatures and the configurational contributions to the free energy can be predicted. For Au-Cu the pair-wise ECI were found to be composition dependent, which can also be described in terms of multi-atom ECI. (56) The pair-wise ECI of the Fe-A1 system have also been determined in this way, and then used to calculate the ordering phase diagram (from measurements on a disordered phase), that is, through phase changes where the diffuse intensity becomes Bragg peaks357'5s) There are methods for estimating a 'most probable' multi-atom cluster distribution from pair correlations,t4s) but the statistical assumptions make it doubtful that meaningful ECI information for multi-atom clusters could be obtained this way.(49) The k-space thermodynamics has been suggested(59) for predicting from the diffuse scattering in the original crystal the increase in free energy of a martensite due to the non-equilibrium order created by the shear transformation. A similar relationship should hold for twinning of partially ordered alloys. The change in free energy is directly obtained from an integral of the diffuse intensity changes produced by the shear. The inverse of the twinning shear can be used to calculate the distortion of the diffuse intensity of the original

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crystals into that of the twin. All the relevant data for predicting the free energy change for the shear of a crystal, equilibrated at one temperature and sheared at any other, could come from a single determination of the pair-wise ECI from diffuse scattering data. The equilibrium diffuse scattering in the original crystal is used to evaluate the ECI, while an integration, using these ECI, over the diffuse intensity of the twin gives the contribution to the free energy. Once the ECI are known, this contribution to the twinning energy can be calculated for any state of local order in the original crystal, provided the diffuse intensity is known. The use of k-space thermodynamics for the partially ordered, but not equilibrated, austenite is similar. It requires measurement of the diffuse scattering from a high temperature equilibrated austenite for the ECI, and a measurement of the diffuse scattering from the non-equilibrium austenite to estimate the excess in its free energy. The martensite inherits this diffuse scattering distorted by the Bain strain, but the ECI can only be obtained if the equilibrium diffuse scattering can be measured by k-space methods if there is a high temperature version of the phase of the martensite. Iron alloys are unique in that the diffuse scattering behavior of 6-iron can be used to give insight into the ordering energies of the martensite. The ordered phases can be cataloged by the location of a superlattice diffraction vector, while in partially ordered phases the location of the diffuse intensity characterizes local order346) Diffraction vectors characteristic of ordering are few in number; they correspond to special symmetry positions in reciprocal space, and the kinds of superlattices they are associated with are also few. Symmetry considerations dictate almost all of them, and explain why there are so few. Shear, however, moves these ordering vectors to unusual positions in reciprocal space, and in direct space creates pairings of atoms that are rarely found because of their high energy. These structures should therefore be of great interest to test our understanding of the energetics of ordering. 6.3. The Vibrational Entropy Even though there is no configurational entropy change in these shear transformations, there is a definite entropy contribution from other sources. The To for a martensite transformation is defined as the temperature for which the change in Gibbs free energy vanishes; hence the latent heat divided by the temperature is balanced by the entropy change. For most systems the main component of this entropy change is vibrational; for iron there is a magnetic contribution as well. Since the entropy difference between alloys with the same configurational entropy vanishes at 0 K, the entropy difference can also be obtained from low temperature heat capacity measurements. That there can be differences in vibrational entropy between two phases connected by a martensite transformation is easily understood in terms of the differences in the coordination and bonding; what has been ignored in all of the modeling of alloys is the differences in vibrational entropy between an ordered and disordered phase or between two different ordered versions of the same disordered structure. These too could have quite different bonding. It is not feasible to separate vibrational and configurational entropy contributions from equilibrium entropy measurements. As a result it has been convenient to assume that the vibrational entropy is independent of the degree of order, and thus has no influence on the ordering. Twinning and pseudotwinning could provide entropy comparisons between states with identical configurational entropy on the same lattice, determined with calorimetry during reversible deformation, that could be independently verified, as for martensite, with a low

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temperature differential heat capacity measurement. Fultz and coworkers (2°) recently made a differential heat capacity measurement of two different frozen states of order in the same alloy. These measurements indicate that there is a significant dependence of vibrational entropy on the state of order.

7. TEMPERINGOF SUPERLATTICESTRUCTURES By partial tempering of the sheared structures, the new phase is somewhat stabilized, altering the pseudo-elastic behavior. It should be possible to modify both the transforming and reversing stress. If the sheared state becomes metastable, a reverse stress may be required to trigger reversion. The tempering process is also of interest for the insight it might provide about ordering mechanisms. Order~tisorder transitions involve phases that are closely related by symmetry; the symmetry group of one phase is a subgroup of the other. Even here there have been observations of transitional ordered structures, and claims of persistent ordered states with no metastability, called pseudostable states/6°'6J) The ordering transition creates superlattice diffraction Bragg peaks; disordering removes them. Shearing a superlattice structure by pseudotwinning into another structure moves the superlattice spots to new positions, that rarely are special points in k-space. The superlattices that are created can have symmetries that are neither subgroups nor supergroups of the equilibrium ordered structures. A good example is the Cmmm pseudotwin of the B2 structure in which the superlattice peaks appear at {1/2 1/2 0} referred to the B2 reciprocal lattice. The change in ordering during tempering should lead to a reappearance of spots at the original {100} positions. Linearized theories of continuous ordering kinetics predict that the new superlattice spots lose intensity and that the intensity at the correct position grows, leading to transitional ordered structures with both kinds of order in the same volume elements. Many alternatives are possible: domains of the new order could displace the old, with two phases coexisting during the transition, transitional phases could appear, e.g. complete disordering before ordering, the spots could appear at another special point as a transitional phase, or the spots could move continuously, broadening out before sharpening at the new Brags peak.

7. CONCLUDINGSTATEMENT The thermodynamics of work and heat and that of materials chemistry have developed from a common thread, but have gone their separate ways. Shear transformations may be a good case for bringing them together; to connect measurements on phase transitions, especially ordering, with those on the shear transformations, and to relate materials selection, solution thermodynamics, and heat treatment to mechanical properties. These transformations may also provide useful and hitherto unavailable data for guiding the development of alloy theory.

ACKNOWLEDGEMENTS

I have benefited greatly over the years in my association and many pleasant and most stimulating discussions with Jack Christian. His catholic interests and ability to draw from his wide knowledge to bring relevant insights into new areas has made him unique. In the

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preparation of this paper, I have also benefited greatly from stimulating discussions with Leo Bendersky, Ben Burton, Craig Carter, Didier de Fontaine, Frank Gayle, Ryoichi Kikuchi, and Dan Shechtman. This work was supported by DARPA. REFERENCES 1. S. N. L. CARNOT,Reflections on the Motive Power of Fire, translated by R. H. Thurston (edited by E. Mendoza). Dover, NY (1960). 2. M. J. KLEIN, Physics Today, p. 23 (August, 1974). 3. J. W. Gmas, Collected Works Yale University Press (1948). 4. L. C. BROWN, J. de Phys. 43, C4-629 (1982). 5. M. AHLERS, ProN. Mater. Sci. 30, 135 (1986). 6. J. W. CHRISTIAN,Theory of Transformations of Metals and Alloys, V.1. Pergamon Press, Oxford (1975). 7. B. A. BILBY and A. G. CROCKER, Proc. R. Soe. A 298, 240 (1965). 8. M. BEVlS and A. G. CROCKER, Proc. R. Soc. A 304, 123 (1968). 9. M. BEVlS and A. G. CROCKER, Proc. R. Soc. A 312, 509 (1969). 10. J. W. CAHN,Acta Metall. 25, 1021 (1977). 11. F. LAVES, Naturwissenschaften 39, 546 (1952). 12. F. LAVES, Am. Miner. 50, 511 (1965). 13. F. LAVES, Acta Metall. 14, 58 (1966). 13. R. DOCHERTY, E. EL-KOSHY, H.-D. JENNISSEN,H. KLAPPER, K. J. ROBERTSand T. SCHEFFEN-LAUENROTH,J. appl. Crystall. 21, 406 (1988). 15. M. L. GREEN and M. COHEN, Acta Metall. 27, 1523 (1979). 16. J. ORTIN and A. PLANES, Acla Metall. 37, 1433 (1989). 17. C. RODRIGUEZ and L. C. BROWN, Metall. Trans. I1A, 147 (1980). 18. L. C. BROWN, Set. Metall. 16, 1001 (1982). 19. M. SADE, E. CESARI and E. HORNBOGEN, J. Mater. Sci. Lett. 8, 191 (1989). 20. B. FULTZ, L. ANTHONY and J. K. OKAMOTO, T M S 1991 Annual Meet., p. 81, New Orleans, private communication (1991). 21. L. KAUFMAN, The Stability of Metallic Alloys Pergamon (1969). 22. J. W. CAHN and W. ROSENBERG,Ser. MetaU. 5, 101 (1970); J. W. CAHN, Acta Metall. 25, 721 (1977). 23. D. W. PASHLEY,J. L. ROBERTSON and J. L. STOWELL, Phil. MaR. 19, 83 (1963). 24. D. SHECHTMAN and L. A. JACOBSON, Metall. Trans. 6A, 1325 (1975). 25. D. SHECHTMAN,M. J. BLACKBURN and H. A. LIPSITT, Metall. Trans. 5, 1373 (1974). 26. J. W. CHRISTIAN and D. E. LAUGHLIN, Scr. Metall. 21, 1131 (1987); Aeta Metall. 36, 1617 (1988). 27. M. H. Yoo, C. L. Fu and J. K. LEE, Mater. Res. Soc. Symp. 133, 189 (1989). 28. M. H. Yoo, J. Mater. Res. 4, 50 (1989). 29. E. Goo, T. DUERIG, K. MELTON and R. SINCLAIR, Acla Metall. 33, 1725 (1985). 30. W. J. MOBERLY, J. L. PROFT, W. T. DUERIG and R. SINCLAIR, Aeta MetalL 38, 2601 (1990). 31. R. W. CAHN and J. A. COLL, Acla Metall. 9, 138 (1961). 32. C. U MAGEE, D. W. HOFFMAN and R. G. DAVmS, Phil. MaR. 23, 1531 (1971). 33. A. B. GRENINGER and A. R. TROIANO, Metals Trans. A I M E 185, 590 (1949). 34. P. G. WINCHELL and M. COHEN, Trans. Q. A S M 55, 347 (1962). 35. M. S. WECHSLER, D. S. LIESERMAN and T. A. READ, Trans. A I M E 197, 1503 (1953). 36. M. OKA and C. M. WAYMAN, Trans. A S M 62, 370 (1969). 37. International Tables for Crystallography (edited by T. Hahn), Vol. A. Reidel Publ. Co., Dordrecht (1983). 38. H. K. BIRNaAUM and T. A. READ, Trans. A I M E 218, 381 (1960); ibid. 218, 662 (1960). 39. S. BRUSH, History of the Ising Model, unpublished 1975; Statistical Physics and the Atomic Theory of Matter; From Boyle and Newton to Landau and Onsager, Chapt. VI. Princeton University Press (1983). 40. M. SLUITER,D. DE FONTAINE,X. Q. Guo, R. PODLOUCKYand A. J. FREEMAN,Mater. Res. Soc. Syrup. 133, 3 (1989). 41. M. SLUITER, D. DE FONTAINE, X. Q. Guo, R. PODLOUCKY and A. J. FREEMAN, Phys. Rev. B 42, 10460 (1990). 42. H. ACKERMANN,G. INDEN and R. KIKUCHI, Acta Metall. 37, 1 (1989). 43. J. M. SANCHEZ, F. DUCASTELLE and D. GRATIAS, Physica 128A, 334 (1984). 44. D. DE FONTA1NE, Solid St. Phys. 34, 73 (1979). 45. P. C. CLAPP and S .C. Moss, Phys. Rev. 142, 418 (1966). 46. P. C. CLAPP and S. C. Moss, Phys. Rev. 171, 754 (1968). 47. S. C. Moss and P. C. CLAPP, Phys. Rev. 171, 764 (1968). 48. P. G. GEHLEN and J. B, COHEN, Phys. Rev. 139, 844 (1965). 49. P. C. CLAPP, J. Phys. Chem. Solids 30, 2589 (1969). 50. P. C. CLAPP, Phys. Rev. 114, 255 (1971). 51. K. BINDER, J. L. LEBOWITZ, M. K. PHANI and M. H. KALOS, Acta Metall. 29, 1655 (1981). 52. H. ACKERMANN,S. CRUSIUS and G. INDEN, Acta Metall. 34, 2311 (1986).

166 53. 54. 55. 56. 57. 58. 59. 60. 61.

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J. M. COWLEY, Phys. Rev. 120, 1648 (1960). D. DE For,rrAl~nE, Acta Metall. 23, 533 (1975). A. G. KHACHATURYAN,Theory of Structural Transformations in Solids Wiley, New York (1983). B. E. WARREN, Trans. AIME 233, 1802 (1965). S. V. SE~NOVSKAYA,Phys. St. Sol. (b) 64, 291 (1974). S. V. SEr~NOVSKAYAand D. M. UDIMOV, Phys. St. Sol. (b) 64, 627 (1974). J. W. CArIN Scr. Metall. 11, 81 (1977). R. KIKUCHI,T. Momu and B. FULTZ, Mater. Res. Soc. Syrup. 205, (1991). L. ANTHONYand B. FULTZ, J. Mater. Res. 4, 1132 (1989).

Progress in Materials Science Vol. 36, pp. 167-202, 1992 Printed in Great Britain. All rights reserved.

0079-6425/92 $15.00 © 1992 Pergamon Press plc

IRRATIONAL INTERFACES A. P. Sutton Department of Materials, Oxford University, OXI 3PH, U.K.

1. INTRODUCTION Irrational interfaces are ubiquitous in many polycrystalline materials, but it is only in the last few years that any progress in understanding them has been made. By an irrational interface we mean an interface in which there is at most one direction of periodicity. The structures of rational interfaces are periodic in two non-colinear directions. Computer simulations of interfacial structures are almost always based on periodic boundary conditions parallel to the boundary plane, and sometimes also normal to the boundary plane. Consequently no irrational interface can be modelled by such simulations. Apart from the "lattice matched" heterostructures, virtually all interphase boundaries are irrational, as are all grain boundaries in low symmetry crystal systems. The vast majority of theoretical studies of interfaces has been concerned with coincidence site lattice (CSL) grain boundaries in cubic cyrstals. Since Kronberg and Wilson (L) and Aust and Rutter ~2)the existence of periodicity in an interface has led to the interface being described as "special" in some physical sense such as low energy, or high migration rate in the presence of solute, or low diffusivity. But the obvious limitation of this line of thought is that it is restricted to those interfaces where periodicity can exist. Recently, tables of CSL orientation relations in certain non-cubic crystals have appeared (e.g. Ref. 3), but the existence of the CSLs is based on particular relations between the crystal lattice parameters, which are fulfilled only approximately in real materials. Rather than taking ideas that were developed for grain boundaries in cubic crystals, and attempting to graft them onto interphase boundaries or grain boundaries in non-cubic crystals, this article reviews a completely new approach that has grown out of the work of quasiperiodicity. In this approach CSL grain boundaries in cubic crystals are seen as a rather special and simple case. Although the approach we present is new, some of its conclusions are closely related to earlier work on interracial line defects. (4) Pond's classification °) of line defects separating energetically degenerate regions of interface leads to the conclusion that all such line defects are characterized by the product of two space group operations, one from each crystal lattice. As pointed out by Sutton, (5) an equivalent conclusion was reached independently by Gratias and Thalal (6) by embedding the bicrystal in a higher dimensional crystal. Although the bicrystal may display no symmetry it can be related, through the embedding, to a higher dimensional crystal possessing a unique space group irrespective of the orientation relation between the crystals comprising the three dimensional bicrystal. The bicrystal is recovered from the higher dimensional crystal by a projection operation, during which the symmetry elements of the higher dimensional crystal become pseudosymmetries of the bicrystal. The most powerful insights gained by this formalism are that there is always long range order in an irrational interface and that it displays local isomorphism. We describe what this means and how it is proved mathematically in Section 3. 167

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The formalism described in Section 3 is rigorous and general. However, it is helpful to have a structural model to visualize some of its implications. This is provided in Section 4, using the structural unit model of grain boundary structure. In fact Rivier ~7) and Sutton ~s) applied the ideas of quasiperiodicity to irrational grain boundaries using structural units, independently of the work described in Section 3. The two approaches complement each other rather well. In his thesis work O) Sutton gave an algorithm to derive the structural unit sequences of irrational tilt grain boundaries. He appreciated that the existence of the algorithm, which was exactly the same for rational grain boundaries, meant that irrational boundaries had highly ordered structures, at 0 K. But it was not until the work of Rivier tT) that it was shown that the sequence of structural units could be derived from the strip and projection method ~l°-lz) which is a cornerstone of the theory of quasiperiodicity. This is described in detail in Section 4, together with the "inflation" algorithm. In Section 5 we discuss the physical implications of the presence or absence of periodicity within an interface. °3) The physical significance of periodicity within the boundary plane is that there exists a two dimensional Wigner-Seitz cell, within which all relative translations of one crystal relative to the other may be defined uniquely. This cell was introduced by Vitek et al., ~4) who called it the cell of non-identical displacements (c.n.i.d.). The latter, for which the c.n.i.d, is the Wigner-Seitz cell, is defined in a general manner in Section 5 as the reciprocal lattice of the two dimensional lattice of common layer reciprocal lattice vectors. In general it is a sub-lattice of the corresponding plane of the DSC lattice. The c.n.i.d, may be extremely small, as in the case of a twist boundary on a low index plane, or extremely large, as in the case of a symmetrical tilt boundary on a high index plane. The size of the c.n.i.d, is shown to place a restriction of the amplitude of the periodic variations of the boundary energy with translation, which is known as the y-surfaceY ) It is shown that smaller c.n.i.d.'s sustain smaller energy variations. Many physical processes at interfaces sample the resistance of the interface to a relative translation of the two crystals, e.g. dislocation absorption, migration and sliding. The y-surface describes the ground state energy of the interface at a given translation. In general, processes at interfaces such as the absorption of a crystal lattice dislocation raise the energy of the interface above the y-surface. The equilibration processes are thermally activated, and therefore time is required for the interfacial energy to drop back onto the y-surface. These viscoelastic properties are discussed in Section 5. The final part of the review considers an analytic model for expansions and cleavage energies of the most irrational types of grain boundaries in f.c.c. Lennard-Jonesium. 03) These boundaries are the same as those described by Wolf and Phillpot ~6) as "true high angle grain boundaries". The model consists of regarding the boundary expansion as the only variational parameter with which the boundary energy is minimized. It is shown that as the boundary expansion increases the cleavage energy decreases. The boundary expansion increases as the average spacing of planes parallel to the boundary decreases. The predictions are shown to be in good qualitative agreement with full atomistic relaxation of long period twist boundaries. We conclude with a discussion of the limitations of the analytic model. 2. SOMEDEFINITIONS Following Lubensky ~17)we introduce some definitions that will be useful in the following. A lattice is a set of vectors that is closed under addition or subtraction. This means that if a and b are elements of a lattice then + a + b are as well. A periodic lattice is a lattice with a shortest length vector. A quasiperiodic lattice is a lattice without a shortest length vector.

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A reciprocal lattice is a lattice of wavevectors. A structure is a material with a mass density

p(r). A set of vectors is integrally independent if any linear combination with integral coefficients vanishes only when the coefficients vanish. A lattice is said to be generated by a set of vectors, called a generating set, if every vector in the lattice can be expressed as an integral linear combination of the vectors in the generating set. The rank, r, of the lattice is the smallest number of integrally independent vectors in any generating set. Clearly r must be at least the dimension of the lattice. A basis {k~, k 2 , . . . , kr} is any generating set consisting of exactly r vectors. A translationally ordered structure is any structure for which the Fourier transform of the mass density p (r) is non-zero only for wavevectors that are lattice vectors of some reciprocal lattice. Translationally ordered structures are divided into periodic crystals and quasiperiodic structures, depending on whether the reciprocal lattice is periodic or quasiperiodic. Quasiperiodic structures are further divided into incommensurate crystals and quasicrystals. An incommensurate crystal is a quasiperiodic structure with crystallographic point group symmetry. A quasicrystal is a quasiperiodic structure with a noncrystallographic point group symmetry. In addition to the above definitions we will also need to define certain objects associated with interfaces between crystals. Pond and Bollman ~8~ introduced the notion of colour to distinguish sites of one crystal lattice from sites of the other. Let all lattice sites of one lattice be coloured black and all sites of the other lattice be coloured white. We obtain a dichromatic pattern by interpenetrating the two crystal lattices throughout all space. Following Pond and Vlachavas ~9) we distinguish between the dichromatic pattern and the dichromatic complex. A lattice complex is the set of points obtained by carrying out on a chosen point all the symmetry operations of a crystal's space group, as opposed to the crystal lattice's space group. Each point has an identical environment except possibly for orientation. A dichromatic complex is obtained by colouring the two lattice complexes white and black and interpenetrating them throughout all space. A dichromatic pattern (or dichromatic complex because it has the same translational symmetry) may display zero, one, two or three dimensional translational symmetry. A one, two or three dimensional coincidence site lattice (CSL) exists according to whether the dichromatic pattern displays one, two or three dimensional translational symmetry. The CSL is defined by the intersection of the translational symmetries of the two crystal lattices, which may be the empty set. The DSC lattice is defined by the union of the translational symmetries of the two crystal lattices. Thus the generating set of the DSC lattice comprises a basis of the black lattice (3 vectors) and a basis of the white lattice (another 3 vectors). In the absence of any translational symmetry in the dichromatic pattern the rank of the DSC lattice is 6. If there is 1, 2, or 3 dimensional translational symmetry in the dichromatic pattern the rank of the DSC lattice is 5, 4 and 3 respectively. If the rank of the DSC lattice is 3, as in a CSL misorientation between cubic crystals, the DSC lattice is periodic and there is a shortest length DSC vector. However, as the size of the 3D periodic unit cell of the dichromatic pattern tends to infinity, Grimmer's theorem ~2°~indicates that the shortest length DSC vector tends to zero. Grimmer's theorem states that the DSC lattice formed from the black and white crystal lattices is the reciprocal lattice of the CSL formed from the reciprocal lattices of the black and white crystal lattices. We note that Grimmer's theorem relies on the existence of a 3D CSL. However, it is always possible to define the DSC lattice as the union of the translational symmetries of the black and white crystal lattices, regardless of the translational symmetry of the dichromatic pattern. DSC lattices with rank greater than 3 are quasiperiodic and there

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is at least one direction in which there is not a shortest length DSC vector. As the size of a primitive cell of a dichromatic pattern displaying 3D translational symmetry tends to infinity the distinction between periodic and quasiperiodic DSC lattices vanishes. The DSC lattice determines the translations of one crystal lattice relative to the other at which equivalent dichromatic patterns are created. In a rank 3 DSC lattice the set of translations falls on a 3 dimensional discrete lattice with finite shortest length vectors in 3 non coplanar directions. However the origin of the dichromatic pattern is generally displaced by s, where Isl can always be expressed modulo a CSL vector. Thus [sl can be very large as the size of the unit cell of the CSL tends to infinity. The physical significance of s is that it may give rise to a step on an interface, with height given by s- fi, where fi is the interface normal, separating equivalent regions of the interface. As we shall see in Section 3.4, in a rank 6 DSC lattice any relative translation of the two crystals generates a dichromatic patten that is locally isomorphic to the original pattern, and [s[ may be very large indeed. The physical meaning of this result is that an energetically degenerate structure is obtained. 3. EMBEDDING BICRYSTALS IN HIGHER DIMENSIONAL CRYSTALS

3.1. Introduction One of the central results of the theory of quasiperiodic functions is that they are closely related to periodic functions in a higher dimensional space. (2~'22) In this section we shall characterize this higher dimensional space, following Gratias and Thalal, (6) and point out that its existence was implicit in earlier work by Pond. ") Once contact has been made with the theory of quasiperiodic functions we will be able to take advantage of some of the general results of the theory to make statements about long range order in irrational interfaces. 3.2. The Six Dimensional Crystal Consider a six dimensional crystal lattice in which the first three dimensions are filled with the white crystal lattice and the second three with the black. Let El, E 2 . .. E6 be six basis vectors of the 6-D crystal lattice. Then E l = [Wl, 0], E 2 = [w2, 0], E 3 = [ w 3 , 0 ] , E 4 = [0, bl], E 5 = [0, b2] , E 6 = [0, b3] , where the wi's and b/s are basis vectors of the white and black crystals. At this stage there is no misorientation between the white and black crystal lattices. The misorientation between the crystals will be introduced later. The components of the w/s and b/s are expressed in the same coordinate system and reflect any differences in the lattices. Each basic vector b~ may be expressed as a linear combination of the three basis vectors of the white crystal as the following vector transformation: 3

b, = E Pjiwj,

(1)

j=l

where P is a pure deformation matrix. For a grain boundary P is the identity. An arbitrary lattice vector in the 6-D lattice may be expressed as r (6) = (rw, rb) where rw and rb are lattice vectors of the white and black crystals. Let S(w) and S(b) denote arbitrary space group operations of the white and black crystal lattices. A space group operation of the 6-D lattice is denoted by (S(w), S(b)) and may be represented as follows: (S(w),S(b))=([S[(o~)]

[0] '~ [S(b)]]"

(2)

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The quantities in square brackets are 3 x 3 matrices. When (S(w), S(b)) acts on (rw, rb) it transforms (rw, rb) into ([S(w)]rw, [S(b)]rb) = (D(w)irw + t(w)j, D(b)krb + t(b).) where S(w) = [D(w)ilt(w)j] and S(b) = [D(b)k It(b).]" D(w)i is the i'th point group operation of white crystal lattice and t(w)j is the j ' t h translation operation of the white crystal lattice. If the black and white crystal lattices are identical there is a mirror plane m' in the 6D lattice relating the two 3D subspaces:

(t01

\[1'1 [0]]'

(3)

where the primes on the 3 x 3 identity matrices denote colour reversal. The set of all antisymmetry operations (i.e. colour reversing operations) displayed by the 6D lattice is thus {m'(S(w), S(b)} = {(S(b), S(w))'}. Note the reversal in the order of the operations. The symmetry operations displayed by the 6D lattice comprise G6D= {(S(w),S(b))} U{(S(b),S(w))'}. If the black and white crystal lattices are not identical then G 6D = {(S(w), S(b))}, because rn' is not a symmetry operation of the 6D lattice. The white and black crystal lattices are recovered in their final misoriented state from the 6D lattice by applying the 6 × 3 projection matrix ([1], [Rt]) to the 3D "faces" (rw, 0) and (0, rb) of the 6D crystal lattice. [Rt] is the transpose of the 3 x 3 rotation matrix relating vectors of the black and white crystal lattices through the following vector transformation: b i = ~ PjkRkiWj = ( R t p t ) / j w j = (PR)~wj.

(4)

k

Alternatively, the relationship may be interpreted as a coordinate transformation in which the coordinates of a fixed point, expressed in coordinate frames of the white and black lattices as r w = (x'~,x'~ , x~')and r b = ( X lb, X 2b, X3) b , are related by r ~ = RPr b.

(5)

It follows that the dichromatic pattern may be viewed as a 3D projection of two orthogonal 3D faces of a 6D crystal lattice. The 6D lattice is independent of the relative orientation between the crystal lattices: that is introduced in the projection operation. One of the key points of this analysis is that an equivalent dichromatic pattern is created if a symmetry operation of the 6D crystal lattice is performed prior to the projection. Although none of the symmetries of the black and white crystal lattices may survive the projection operation, and appear in the dichromatic pattern, we see that those symmetries relate equivalent dichromatic patterns. An interface created from one of these dichromatic patterns is therefore related by symmetry to interfaces in equivalent dichromatic patterns. However, the symmetry operations belong to the 6D crystal lattice and not the interface. 3.3. Relationship to Earlier Work on Interfacial Line Defects The conclusion of the last section restates an earlier observation by Pond ~4) that the operations characterizing intefacial line defects separating equivalent regions of an interface are products of two space group operations, one from each crystal lattice. Pond had developed a systematic classification of all such line defects that is based on a fundamental principle of geometry, namely the principle of symmetry compensation: (z3) if symmetry is reduced at one structural level it arises and is preserved at another. In the present context, the lowering of symmetry in forming a dichromatic pattern from two crystal lattices is compensated by the existence of variants of the dichromatic pattern. Moreover, Lagrange's

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theorem says that the compensation is exact: the symmetry elements that are lost in forming the dichromatic pattern relate the variants of the dichromatic pattern. Thus for every operation of the black or white crystal lattice that is absent in the dichromatic pattern there is an equivalent dichromatic pattern. Rather than speaking of symmetry being lost or reduced in forming the dichromatic pattern it is perhaps more appropriate to speak of the symmetry being suppressed. Pond's approach (a) is to construct an interface by bonding two crystal surfaces together. An equivalent interface is created by bonding equivalant crystal surfaces. In the white crystal a new equivalent surface is obtained by applying a space group operation, S(w), of the white crystal, and similarly S(b) generates a new equivalent surface of the black crystal. The new surface of the black crystal is brought into contact with the new surface of the white crystal, by applying the operation S(w)S(b)-t to the new surface of the black crystal. The operation S(w)S(b) -1 characterizes a line defect separating the equivalent regions of interface. If the crystals are nonsymmorphic or nonholosymmetric the dichromatic complex may have a lower symmetry than the dichromatic pattern. In that case there are variant dichromatic complexes (called complex variants) related by the symmetry elements that are present in the dichromatic pattern but absent in the dichromatic complex. The presence of complex variants can give rise to important morphological features associated with interfacial line defects such as the presence of inversion domain boundaries. Pond concluded that the set of all operations relating equivalent regions of an interface is the set {S(w)S(b)}, where S(w) and S(b) are space group operations of the white and black crystal lattices. In Pond's approach it is clear that the composite operation S(w)S(b) characterizing an interfacial line defect represents two operations, one applied to the black crystal lattice and the other to the white. Gratias and Thalal made this explicit by introducing the 6D crystal lattice and representing S(w)S(b) as a space group operation (S(w), S(b)) of the 6D lattice. The two approaches are entirely equivalent. Both approaches reveal the symmetries that lay "hidden" from view in a bicrystal possessing no symmetry itself. The underlying principle uniting both approaches is the principle of symmetry compensation. 3.4. Long Range Order at Irrational Interfaces At this point we can make contact with the fields of quasiperiodicity and incommensurate structures, following Gratias and Thalal. ~6) Consider the electron diffraction pattern of a bicrystal. The incident beam undergoes multiple scattering events in both the black and white crystals and the diffracted amplitudes consist of a convolution of the diffracted amplitudes from the two separate crystals. For example, a reflection, Gb, in the black crystal subsequently enters the white crystal and is elastically scattered again so that the reflection becomes Gb + Gw. The set of reflections is therefore {Gb + Gw}, where Gb and Gw range over all reciprocal lattice vectors of the two crystals. The diffracted amplitude from the bicrystal is therefore described as follows: p(k) = ~ A(Gb+ Gw)5(k - Gb-- Gw)

(6)

Gb,Gw

where A(Gb + Gw) is the amplitude of the reflection Gb + Gw, arising from all the multiple diffraction events. We are ignoring here any extra reflections that may arise from relaxation at the interface. They are irrelevant to the present argument. At a commensurate interface there is a two dimensional repeat cell of finite area in the interface plane. If there is a matching periodicity in only one direction the interface is said

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to be commensurate along that direction and incommensurate in all other directions. If there is no matching periodicity in any direction in the interface plane the interface is described as incommensurate. Commensurate interfaces are also described as rational interfaces, while irrational interfaces are those in which there is at most one commensurate direction. In the case of commensurate grain boundaries with CSL misorientations it is possible for the adjoining crystals to be commensurate with each other. In that case not only is the interface commensurate but there is a matching periodicity normal to the interface. The set of reflections {Gw+ Gb} from the bicrystal may be indexed with at most 6 independent sets of integers, 3 for the white crystal reflections and 3 for the black. This would be the case where there is no direction of matching periodicity in the bicrystal; this is the completely incommensurate case. For a grain boundary with a CSL misorientation all the reflections may be indexed with only 3 integers; this is the completely commensurate case. The intermediate cases require 4 or 5 integers to index all the reflections. A function in 3 dimensional space whose Fourier spectrum requires more than 3 independent sets of integers to index all the reflections is defined as a quasiperiodic function. As we shall see a quasiperiodic function is a projection of a strictly periodic function in a higher dimensional space. The question now is what is the quasiperiodic function in three dimensional space that gives rise to the Fourier spectrum described by eq. (6)? Since the spectrum is a convolution of the diffracted amplitudes from the adjoining crystals the simplest function that will give rise to the set of reflections {Gw+ Gb} is p (r) = Pw(r)pb (r),

(7)

where pw(r) is the mass density distribution function in the white crystal, and similarly for pb(r). Evidently, p (r) is not the mass density distribution of the bicrystal, or even of the dichromatic complex. The mass density of the dichromatic complex is pdc(r) = pw(r) + pb(r). But it is clear that if pw(r) and pb(r) are commensurate with each other along a particular direction then p (r) will be a periodic function in that direction. If r is confined to the plane r. ~ = 0 then p (r) will reflect whatever periodicities exist in the interface created by sectioning the dichromatic complex on the same plane. For example, consider the mass distributions along the x-direction within an interface of normal 0. We may express pw(X) and pb(X) as follows:

~ W.exp(2~inx)

pw(X) =n=

--oo

B, exp(2~inx/2).

pb(X)= ~ n=

(8)

oo

The wavelength of the periodic density distribution in the white crystal has been set equal to l and in the black crystal to 2. We require 2 ~< 1 without loss of generality. The periodicity of the interface in the x-direction is determined by the periodicity of pw(X) + pb(X). If 2 is an irrational number pw(X)+ pb(X) is not periodic. If 2 is rational, say 2 =p/q where p and q are coprime integers, then pw(X) + pb(x) is periodic with a wavelength ofp. Now consider fl(X) = pw(X)pb(X). If 2 is irrational then p(x) is also not periodic, whereas if ~, =p/q then the periodicity of p(x) is also p. Thus the periodicity of pw(X) + pb(X) is the same as the periodicity of pw(X)pb(X). Since x can be any direction in the interface it follows that the periodicity of pw(r) + pb(r) is the same as the periodicity of the function pw(r)pb(r), where r is any vector in the interface plane.

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The Fourier spectrum of pac(r)=pw(r)+pb(r) is a subset of the spectrum of p(r) = pw(r)pb(r). This is seen immediately by expressing pw(r) and pb(r) as Fourier series: pw(r) = ~ W(Gw)exp(iGw" r)

(9)

Gw

pb(r) = ~ B(Gb)exp(iG b • r), c,b

(10)

where W(Gw) and B(Gb) are appropriate structure factors. Whereas the Fourier spectrum of pat(r) comprises the sets {Gw} and {Gb}, the Fourier spectrum of p(r) also includes {Gw + Gb }. Since the periodicities of p(r) and Pdc(r) are the same, p(r) can be used to determine the long range order in the dichromatic pattern, or in an interface if r is confined to lie in a plane. Long range order exists if the autocorrelation function, C(r), otherwise known as the Patterson function, continues to take on finite values as ]r] tends to infinity. It is defined by

C(r) = f p ( r ' ) p ( r + r') dr'.

(11)

We shall see that regardless of whether the interface is periodic or not long range order exists always. This is one of the main insights gained by making contact with the field of quaisperiodicity. Let us use p (r) = pw(r)pb(r), eq. (7), together with eqs (9) and (10), to explore the properties of p(r) further. It has the following Fourier series representation: p(r) = ~ ~ W(Gw)B(Gb)exp(iGw.r + iGb "r).

(12)

Gw Gb

Consider the most incommensurate case, where there is no matching periodicity in any direction. Six independent sets of integers will be required to index all the reflections {Gw+ Gb}. Although P ( 0 is not periodic it is closely related to a periodic function in six dimensional space as may be seen by expressing p(r) as follows: p(r) = R(r, r),

(13)

~(rw, rb) = E ~ W(Gw)B(Gb)exp(iGw"rw + iGb" rb) = p,~(rw)pb(rb). 6w C,b

(14)

where

We recognize the density distribution R(rw, rb) as that of the six-dimensional crystal, which is obviously periodic in six dimensional space. It is clear from eq. (13) that p(r) is the three dimensional density distribution in the hyperplane rb----rw of the six dimensional crystal. On the other hand, we recall that Pd~(r) is the linear superposition of the mass densities on the (rw, 0) and (0, rb) "faces" of the 6D crystal. 3.5. Translation Vectors and Local Isomorphism If p(r) and pdc(r) are not periodic functions they are quasiperiodic functions, which means that one can define "translation vectors" for them. t21'22)z is said to be a translation vector, belonging to e i> 0, of a function f(r) if l f ( r + ~ ) - f ( r ) l ~<8 for all r. Obviously i f f ( r ) is periodic and • is a repeat vector then e = 0. But for a quasiperiodic function ~ must be greater

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than zero and the inequality expresses the approximate nature of the replication of the function. A translation vector belonging to 5 belongs also to any 5' > 5. If ~ is a translation vector belonging to ~, then so is - ~ . If ~t, ~2 are translation vectors belonging respectively to el, 52 then ~1 +--~2 is a translation vector belonging to 5~ -4- 52. Let the set of all translation vectors o f f ( r ) belonging to e be denoted by E{5,f(r)}. It is convenient to rewrite eq. (12) for p(r) as follows: p(r) = ~ A, exp(iG, • r),

(15)

n

where G, is a combination of Gw and Gb, An is the corresponding product of structure factors, and the G,'s are numbered in order of increasing magnitude. In order for • to be a translation vector of p (r) belonging to ~ it must satisfy some number of N of Diophantine inequalities IGn" z - 2nvl < 6 for n = 1, 2 . . . . . N, where v is an integer and 6 is a positive number that is less than 7t. The same set of N diophantine inequalities has to be satisfied by all translation vectors of E{5, p(r)}. The proof (2~'22) of this statement is very instructive about the properties of translation vectors of quasiperiodic functions, and is given in Appendix A. Consider now the autocorrelation function C(r), defined by eq. (11), for the quasiperiodic function p(r). Substituting eq. (15) for p(r) into eq. (11) we obtain: C(r) = )-" IA~I2 exp(iG,, r).

(16)

n

Therefore, C(r) is itself a quasiperiodic function and therefore it repeats, within an accuracy of 5, ad infinitum. We conclude that p(r) displays long range order regardless of whether or not the crystals are commensurate. Since Pdc(r) can also be expressed in the form of eq. (15), where Gn's are either Gw's or Gb's, it follows that the mass density of the dichromatic complex is also a quasiperiodic function with its own set of translation numbers and an autocorrelation function that always displays long range order. Moreover since the long range order displayed by an interface depends on the long range order of the dichromatic complex from which it was sectioned, it follows that

an interface between two crystals always displays long range order. The final point we shall make in this section concerns local isomorphism. Two structures are said to be locally isomorphic if, and only if, given any point P in either structure and any finite distance d, there exists a translation of the other structure such that the structures coincide exactly from P out to at least distance d. Thus every finite region of one structure appears in the other and vice versa. The diffraction patterns and free energies of two locally isomorphic structures are indistinguishable. If the black and white crystals are incommensurate with each other then two dichromatic complexes, which differ only by a relative displacement of the crystal lattices, are locally isomorphic. The relative displacement of the crystal lattices is the "phason" degree of freedom associated with quasiperiodic structures. 07) The significance of this is that two dichromatic complexes that are related by symmetry operations of either crystal are locally isomorphic. The symmetries of the adjoiningcrystals, which are symmetry operations of the 6D crystal, become pseudosymmetry operations after the irrational projection to form an incommensurate dichromatic complex. They are pseudosymmetry operations because after they are applied, unlike a normal symmetry operation, not all of the structure is superimposable on the original structure. Instead an infinite number of points in the two structures are superimposable, but not the whole structures. We shall return to the concept of local isomorphism in Section 4.6.

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4. STRUCTURAL UNITS 4.1. Introduction In the previous section it has been shown that any irrational interface has a quasiperiodic structure. The main consequence of this conclusion is that the interface displays long range order and local isomorphism. In order to visualize what this means we have to consider a model for the structure of an interface. In this section we adopt the structural unit model ~z4~ of high angle grain boundary structure, as formulated by Sutton and Vitek (2~-27)Indeed, the first observation that irrational interfaces are quasiperiodic structures was made by Rivier ~7) using a structural unit model in phyllotaxis. The relationship between Rivier's ideas and earlier work on structural units by Sutton and Vitek (2~-:7) was discussed by Sutton, (8) who developed these ideas further. In this section we will discuss irrational tilt grain boundaries in terms of structural units. The applicability of these ideas to other interfaces is limited by the validity of the structural unit model itself. As discussed by Sutton (28) and Sutton and Balluffi, (29) there are simple, geometrical reasons that restrict the usefulness, though not the validity, of the structural unit model as a predictive description of twist and mixed tilt and twist grain boundary structure.

4.2. The Structural Unit Model of Tilt Grain Boundary Structure According to this model the structure of a longer period boundary can be described as a sequence of structural units of two shorter period boundaries. This statement was first made by Bishop and Chalmers324) For a particular tilt axis and mean boundary plane ~26~consider two short period boundaries that are separated by some misorientation range. These short period boundaries may be "favoured boundaries", ~25) in which case their structural units cannot be broken down into units of other boundaries, or they may consist of combinations of a finite number of structural units of other boundaries. The latter were called "multiple unit reference structures" by Sutton et al. ~3°~To be consistent with the terminology of others °1) our two short period boundaries are called "delimiting" boundaries because they delimit the misorientation range. The choice of these delimiting boundaries is not completely arbitrary since there are selection rules ~27) governing successive delimiting boundaries. The selection rules are based on the requirement that all boundaries between two delimiting boundaries can indeed be decomposed into structural units from the delimiting boundaries. This requirement is logically necessary in the model and the condition for its fulfillment is entirely crystallographic. Computer simulations of tilt boundary structures ~25'z6) indicated that the boundaries in the misorientation, range between two delimiting boundaries are satisfactorily described as sequences of structural units of the delimiting boundaries. The description is approximate in that the structural units of the delimiting boundaries are inevitably distorted by their coexistence in intervening boundaries. However this distortion can always be reduced below any desired limit by choosing delimiting boundaries spanning smaller misorientation ranges. There is, therefore, a balance to be struck between the accuracy of the description in terms of ideal delimiting boundary units and the misorientation range they span. It has been shown ~28)for twist boundaries and mixed tilt and twist boundaries that this balance tends to favour smaller misorientation ranges as the boundary normal becomes higher index. In that case a greater number of delimiting boundaries is required in a given misorientation range. This is the reason for the restricted usefulness of the structural unit model as a predictive description of twist, and mixed tilt and twist boundaries.

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Let the structural units of two delimiting boundaries be called A and B. In principle these units are infinitely long normal to the boundary plane since there is no translational symmetry in that direction. But in practice we truncate the units to some manageable size by noting that beyond some distance, the boundary atoms are effectively in ideal crystal positions. The choice of the shape and size of the structural unit parallel to the boundary plane is arbitrary except it must fill two-dimensional space and its area must be commensurate with one coincidence site in the delimiting boundary plane. Since all intervening boundaries and the delimiting boundaries share the same tilt axis it is convenient to choose a rectangular cell with sides parallel and perpendicular to the tilt axis. This was the choice made in Refs 25-27. The intervening boundaries are then described by an infinity of possible one-dimensional sequences of columns A and B structural units with the columns along the tilt axis. The determination of the structures of the intervening boundaries is thus reduced to the determination of the sequences of A and B structural units perpendicular to the tilt axis. An algorithm to determine these sequences was first presented by Sutton (9) and Sutton and Vitek.(25-27) Let u be the vector perpendicular to the tilt axis bounding the rectangular cell of the A structural unit in the plane of the delimiting boundary. Similarly let v be the vector bounding the rectangular cell of the B structural unit. Then if an intervening boundary is composed of m A units and n B units, where m and n are coprime integers, the sequence of units in the intervening boundary repeats after p = mu + nv. If m and n are finite the intervening boundary has a periodic structure and the boundary plane is rational. Otherwise there is no periodicity perpendicular to the tilt axis and the boundary is irrational. It was noted in ~9'27) that u and v may be thought of as the basis vectors of a two dimensional lattice which was called the "decomposition lattice". The repeat vectors of the structural unit sequences in all intervening boundaries are primitive vectors of this lattice. However, it was thought (9,27) that it was not possible to determine the sequences of the structural units from this lattice alone. Instead, the algorithm presented in (9'27) made use of a pattern that was deduced from computer simulations. We shall describe that algorithm in the next paragraph, but we note here that it is quite possible to determine the sequences of structural units from the decomposition lattice alone using the "strip and projection method". This will be described in the next section. It is interesting to note how close Sutton (9) was to discovering this algorithm, which became so important in the development of quasiperiodicity. But, in the end, it was the work of Rivier (7) that established the link between the strip and projection method and the structural unit sequences of irrational boundaries. The algorithm developed in 0'27) was based on the premise that as the misorientation varies between the delimiting boundaries the boundary structure changes as continuously as possible, with the constraints that the boundaries are composed of A and B units. Suppose the boundary consists of r A units for every s B units, where r and s are coprime integers and r < s. The ratio of A and B units in the boundary is r / s which is a rational fraction if the boundary plane is rational, otherwise it is an irrational number. The boundary lies between two boundaries in which the mixing ratios are 1/p and l(p + 1), where p is the smallest integer such that l/(p + 1) < r / s < l i p . In order for the structure of the boundary in question to be continuous with the A(pB) and A[(p + 1)B] boundaries it must consist of sequences of A(pB) sequences and A[(p + I)B] sequences. Thus the A(pB) and A[(p + 1)B] sequences themselves behave like units, which we may call C and D units respectively. Now we seek the sequence of C and D units which describes the boundary in question. It is readily verified that there will be mC units for every n D units where n = s - r p and m = r(p + 1) - s, and m < r and n < s. We now go through the same operation of finding two fractions 1/q

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and l(q + 1) such that 1/(q + 1) < m/n < 1/q where q is the smallest integer satisfying these inequalities. (If m/m > 1 then we consider n/m.) Continuity of boundary structure then requires that the structure of the boundary in question consists of a sequence of C(qD) and C[(q + 1)D] sequences which become new units E and F say. Eventually the situation will be reached in which the mixing of new units, so as to ensure there are rA units for every sB units, will be integral and the sequence of A and B units is then determined. As an example consider r/s = 13/19. Clearlyp = 1 because 1/2 < 13/19 < 1/1. The " C " unit is thus ABB and the " D " unit is AB. The ratio, m/n, of C to D is readily found to be 6/7. Now 1/2 < 6/7 < 1/1 so the boundary will consist of a sequence of CDD and the CD sequences, which are the E and F units. There is one E unit for every 5F units in the boundary. Thus the unit sequence is E(5F) = CDD[5(CD)] = D[6(CD)] = AB[6(ABBAB)] = ABABBABABBABABBABABBABABBABABBAB.

(17)

This sequence is repeated periodically in the boundary plane perpendicular to the tilt axis. In principle the algorithm we have just described will also provide the sequences of A and B units for irrational boundaries where r/s is an irrational number. Sutton (9) argued that this would be done by approximating the irrational number by a rational number to any required accuracy. Successive rational convergents to the irrational number produce structural unit sequences which converge to the sequence of the irrational boundary. The clear implication of this conclusion, which was fully appreciated in (9) is that the structural unit sequences of irrational boundaries were just as determined as those comprising the rational boundaries, and therefore the irrational boundaries have ordered structures. It should be noted, however, that we have ignored configurational entropy in this analysis, and therefore these conclusions apply only at 0 K. 4.3. Structural Unit Sequences Obtained by the Strip Method The strip method is a widely used method to construct quasiperiodic structures and was originally developed by de Bruijn, °°) Elser °1) and Katz and Duneau. (12)The idea is to project a slice of a higher dimensional lattice (in our case 2-dimensional) onto a lower dimensional plane (in our case a line). The relevant 2-dimensional lattice is the decomposition lattice and the line is the repeat vector of the boundary. If the boundary is rational then the repeat vector of the boundary is finite in size and there is a periodic sequence of structural units. It is convenient to map the decomposition lattice onto a two-dimensional square lattice. This mapping is an affine transformation and does not affect our conclusions. The vectors u and v of Section 4.2 outline the unit square in this lattice, as shown in Fig. 1. The line W, shown in Fig. 1, passes through the point (13, 19) and hence it represents the boundary in which there are 13A and 19B units. Consider now the strip obtained by sliding the unit square of the lattice along this line. Inside the strip there is a unique broken line (shown bold in Fig. 1) which joins all the lattice points falling inside the strip. This line is now projected orthogonally onto W and it is seen that two "tiles", A and B, are produced which are the projections of the horizontal and vertical edges of the unit square. These tiles are the structural units A and B and it is readily confirmed that the periodic sequence of these tiles seen along W in Fig. 1 is identical to the sequence obtained in Section 4.2.

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Different orientations of the line W will result in different sequences of tiles and also different relative sizes of the tiles. This is true also of the sequences and relative sizes of the structural units in boundaries between the delimiting boundaries. It is clear that the relative distortions of the ideal A and B units will vary throughout the misorientation range. In Fig. 1 the distortions are exaggerated by mapping the decomposition lattice onto a square lattice. A typical misorientation difference between two delimiting boundaries is 30° for which the angle between u and v in the decomposition lattice is 15°. Thus the variation in the relative lengths will be less than 4%. Why is the sequence of structural units produced by the strip method the same as that prescribed by the algorithm of Section 4.2? The answer lies in the sequence of steps on a free surface parallel to the line W in Fig. 1. A symmetrical tilt boundary can be constructed by bringing such a free surface into contact with a mirror image of itself which is obtained by reflection in the plane containing W and the tilt axis. Thus the sequence of steps seen on the two free surfaces is the same as the broken line inside the strip of Fig. 1. On bonding the two surfaces together relaxation changes the local misorientation locally at each structural unit. In this way incipient, distorted structural units in the unrelaxed configuration become less distorted through changes in the local misorientation towards the misorientations of the delimiting structural units. In effect, the local changes in misorientation are equivalent to the projection operation of the strip method. Thus, the projection operation is physically realized in the grain boundary relaxation. The greater the distance each lattice site has to be projected the larger the strains that are set up in the adjoining grains. The broken line inside the strip is the path joining (0, 0) to (13, 19) that minimizes the sum of these projection distances. Deviations from this line are effected by changing the width of the strip or making the strip

i~,I!)

I\~ /__Y~' / h,k°l /1\~ / y B

/ ",yko / ~ / ~ '

' rv• /~/A \ / f

sT, B FIG. 1. To illustrate the strip method for obtaining the sequence of units in a boundary composed of 13A and 19B units. The bold broken line lies within the strip obtained by sliding the elementary square S along the line W connecting the origin to (13, 19). When the broken line is projected orthogonally onto W a sequence of short and long units is obtained, which are designated as A and B units.

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wavy and they cost internal energy. In the case of an asymmetrical tilt boundary, where there is a change in the interplanar spacing on either side of the boundary plane, the same argument applies provided the two surface structures are commensurate with each other. 4.4. Inflation Levine and Steinhardt (32) have presented a simple iterative algorithm for generating quasiperiodic sequences that are also self-similar. The simplicity of the algorithm is very appealing. The algorithm is exact for those boundaries in the decomposition lattice in which the ratio of the numbers of B and A units is the sum of a rational and a quadratic irrational. (A quadratic irrational is an irrational number that satisfies a quadratic equation with integer coefficients.) In this section we discuss the application of this algorithm to irrational tilt boundaries. Consider the matrix M:

M=jk

•

Each matrix element is either a positive integer or zero. Let an A unit be represented by the column vector (1, 0) and a B unit by the column vector (0, 1). A self-similar sequence is generated by repeated operation of the matrix M on say the vector (1, 0). The first iterate is the vector (g,j), which means that an A unit has been replaced by gA units a n d j B units. If we had started with the vector (0, 1) the first iterate would have been (h, k), which means that a B unit is replaced by hA units and kB units. Whichever we start with we ensure that the sequence (g,j) or (h, k) satisfies continuity of boundary structure. To do this we may use the algorithms of Section 4.2 or 4.3. In the next iteration we operate with M on (g,j) and produce (g2 + hi, gj + kj) and so on. In successive iterations each A unit is replaced by the appropriate sequence of g A units and j B units, and each B unit is replaced by the appropriate sequence of h A and k B units. These substitution rules describe the process of "inflation", and the resulting sequence displays the property of self-similarity. In the limit of an infinite number of iterations we produce an aperiodic sequence of units that is entirely self-generative. Levine and Steinhardt (32) pointed out certain restrictions on the allowed values of the M U matrix elements. Those restrictions apply here also and they may be stated as follows: the determinant of M must be non-zero and the eigenvalues must be irrational. However, in order to satisfy continuity of boundary structure, in the sense defined in Section 4.2, it is necessary to impose a stronger restriction on the determinant of M, namely it must equal _+ 1, i.e. M must be a unimodular matrix. For example, M = I~21],

(19)

corresponds to the inflation rule A ~ ABB and B --* AAB, and the determinant of M is - 3. The resulting sequences do not satisfy continuity of boundary structure because AA and BB sequences may not coexist in the same boundary. Interchanging A and B units is equivalent to interchanging columns of the matrix M. Therefore, it is unnescessary to consider both det M = 1 and det M = - 1. In the following we assume det M = - 1 and the eigenvalues are irrational. The eigenvalues of M satisfy the equation 22 _ ~2 -- 1 = O, (20)

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where 4 = g + k, which is the trace of M. One of the eigenvalues, 21, is always greater than unity, while the other, 22, lies between 0 and - 1 . 21 is given by (21)

)],, = Z ( 4 ) = [4 4- (4 2 4- 4)1/2]/2.

By finding the eigenvectors of M, and using - 1 < 22 < 0, it is easy to prove that in the limit of an infinite number of operations of M on some initial sequence of units the ratio of the number of B units to the number of A units is given by R = [Z(4) - g]/h =j/[Z(4) - k].

(22)

Probably the most famous example of this algorithm is the generation of the Fibonacci sequence. This sequence corresponds to R = 1/z = z - 1, where z = (1 + x/~)/2 = 1.61803398 . . . . is the golden mean, for which the generating M is

,23) In this case the inflation rules are A ~ AB and B --* A. Thus, starting from AB we obtain the following iterates: AB ABA ABAAB ABAABABA ABAABABAABAAB ABAABABAABAABABAABABA etc.

1/1 1/2 2/3 3/5 5/8 8/13 (24)

On the right of each sequence is the ratio of the number of B units to the number of A units. In the limit of an infinite number of iterations this ratio converges to 1/z and the whole sequence is a unique motif, which is a quasicrystal (Gratias and Cahn°3)). As noted by Gratias and Cahn ~33) quasicrystals are defined as limits of increasingly larger periodic structures in exact analogy with irrational numbers being limits of rational fractions involving increasingly larger numerators and denominators. Any irrational tilt boundary in which the ratio, R, of the number of B units to A units is given by eq. (22) consists of a self-similar quasiperiodic sequence of A and B units. Since the only restrictions on g, h, j and k are that gk - j h = - 1 and that 42+ 4, which equals (g 4-k)24- 4, is not a perfect square, it is clear that there is an infinite number of such irrational tilt boundaries. Nevertheless, this infinite set is only a subset of all possible irrational tilt boundaries between a given pair of delimiting boundaries. We discuss the irrational tilt boundaries in which R is not given by eq. (22) but some other irrational in Appendix B. It is instructive to express Z(4) as a continued fraction. Because Z(4) satisfies eq. (20) it has the following continued fraction expansion 1

z ( 4 ) = 4 + 1 / z ( 4 ) = 4 -~

4+

(25)

1 1 1

4 + - 4+'"

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The continued fraction is periodic (i.e. successive elements repeat) as required by Legendre's theorem for a quadratic irrational, o4) It is this feature of the continued fraction representation of Z(~) which results in the self-similar property of the quasiperiodic sequence. If this continued fraction is substituted into the expression for R in eq. (22) we may obtain successive rational approximants to the infinite quasiperiodic sequence. For example, in the case of the Fibonacci sequence R = 1/[Z(1)- 0] = [ Z ( 1 ) - 1]/1 from which we may obtain rational convergents by truncating the continued fraction of Z(1) at successive levels: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13. . . . . These are the fractions appearing on the fight of the successive iterates to the Fibonacci sequence. In the general case truncating the continued fraction for Z(~) at the m'th level produces Am where Am is given by Am = Pm/Qm = (~Pm- I "~ Pm- 2)/(¢Qra- I "~ Qm- 2)

(26)

with P0=¢,

Pl=¢Z+l

and

Q0=l,

Q,=¢.

(27)

If we start with an A unit successive iterates have mixing ratios o f j / ( A m - k), whereas if we start with a B unit successive iterates have mixing ratios of (Am - g)/h for m = 0, 1, 2, 3. . . . . It can be shown, °4) that successive rational convergents Am to Z(~) are alternatively smaller and larger than Z(~), become more accurate as m increases, and that they are the best possible approximation at any order in that better approximations will have larger denominators than Q,,. The error in the approximant Am is less than 1/(QmQm+~). 4.5. Local Isomorphism Revisited Local isomorphism was introduced at the end of Section 3.5. It is obvious that the inflation algorithm described in Section 4.4 leads to structural unit sequences that display local isomorphism. For a fixed irrational slope in the decomposition lattice any finite sequence of units appears infinitely many times, but not in a periodic manner. Furthermore, in the strip and projection method for generating structural unit sequences the projection line in Fig. 1 was drawn through the origin. But this was an arbitrary choice and other structural unit sequences would have been obtained by choosing a projection line with the same slope but non-zero y-intercepts. But it can be shown (e.g. Ref. 12) that any finite patch of structural units in one irrational boundary appears in all boundaries with the same slope in the decomposition lattice. All structural unit sequences derived from the same slope in the decomposition lattice are locally isomorphic. Thus, the regions of interface on either side of a step in an irrational tilt boundary are also locally isomorphic, regardless of the step height. It is interesting to consider how far apart two copies of a given patch are in a particular irrational boundary. It turns out o2) that for quadratic irrationals the mean distance between two copies is proportional to the size of the patch. 5. A MODEL FOR GRAIN BOUNDARY ENERGY 5.1. Introduction Up to this point we have confined ourselves to purely geometrical aspects of irrational interfaces. In this section we develop a simple analytic model for grain bounary energy that reveals clearly the significance of periodicity within the boundary plane when it is present. There have been several attempts to relate geometrical features of interfaces to their energy,

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and these were reviewed by Sutton and Balluffi. °7) On the whole these attempts have failed to predict reliably low interfacial energies. In Sections 5.2-5.5 we present an analysis that was discussed by Sutton, °3) and which is similar to earlier analyses by Fletcher and Lodge. <3s) Although our model ignores relaxation within the interface the conclusions concerning the significance of periodicity within the boundary plane are independent of relaxation. Relaxation can be treated to some extent following Fletcher and Adamson. °9) The analysis reveals the reasons for the failure of the criterion of planar coincidence site density34°) According to this criterion the higher the density of coincidence sites in the boundary plane the lower the boundary energy. A more restricted form of this criterion was proposed in Ref. 41: for a given boundary plane the higher the planar coincidence site density the lower the boundary energy. In this restricted form of the criterion one considers boundaries that differ only in the angle of twist about the boundary normal. Our analysis explains why this restricted form of the criterion works. The implications of the analysis for viscoelastic properties of interfaces are discussed in Section 5.6. In the absence of periodicity in the interface it turns out that it is much easier to model analytically the boundary cleavage energy than the boundary energy itself. The variation of the boundary cleavage energy with the boundary expansion and the average spacing of planes parallel to the boundary is described in Section 5.7. The model consists of regarding the two bicrystals as perfect rigid crystals that are free to float normal to the boundary plane. Thus all local relaxation is ignored, except an expansion is permitted at the boundary plane. The results of this simple model are compared with full atomistic relaxations of long period, commensurate grain boundaries in Section 5.8. The atomistic simulations confirm the trends predicted by the simple model. The work described in Section 5 is taken from Ref. 13. 5.2. The Energy of Interaction Between Two Parallel, Rigid, Planar Nets Consider an infinite two dimensional planar lattice L x. It is assumed that there is an atom at each lattice site. For simplicity it is also assumed that there are no other atoms in the basis although the results are readily extended to the more general case. We assume that the total energy of the atomic assembly is represented by a sum of pair interactions and that there are no other contributions (such as density dependent energy). Let the set of lattice vectors be denoted {X~}. Consider the potential V l at a position r = (x, z) arising from all atoms in L I. (X,Z)

I I I I I

FIG. 2. To illustrate the construction of the potential at a position (x, z) above a two dimensional lattice L t comprising lattice vectors {Xl}. x is parallel to L I. JPMS 36/1-4--G

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Here x is an arbitrary vector parallel to L t and z is the height above L l (see Fig. 2) and Irl2 = Ixl2 + z 2. Vl may be expressed as follows: Vl(x, z) = ~/)(x xl

- X I, z ) ,

(28)

where v(r)= v(Irl) is the pair potential at a separation Irl. Since the potential Vt(x, z) is periodic parallel to L I it may be expanded in a two dimensional Fourier series: VI(x, z) = ~ I~I(Gt, z)exp(iG I. x),

(29)

GI

where --1 t V (G, z) = ~

1

'

Vl(x, z ) e x p ( - iG I. x) dx,

(30)

A ~ is the area of a primitive unit cell of L l and N~ is the number of unit cells in L l, which is infinite. If a I and b I are primitive, non-parallel, lattice translation vectors in L l, and eI is any vector normal to L l, then a l* = (b I x d)/(a I. b 1 x e l) and b l* = (eI x al)/(a I. b I x e l) are basis vectors of the layer reciprocal lattice comprising the vectors {Gt}. The integral in eq. (30) extends over Lk Substituting eq. (28) into eqs (29) and (30) we find 1

Vl(x, z) = ~ ~ ~(G t, z)exp(iG l" x),

(31)

c GI

where 6(G l, z) is the two dimensional Fourier transform of the pair potential: /7(G t, z ) = .IV(X, z)exp(--iG t. x) dx.

(32)

The integral eq. (32) extends over all two dimensional space. Equation (31) may be used to calculate the interaction energy between two rigid, parallel, two dimensional lattices L l and L II. The lattices do not have to share the same unit cell or the same orientation. Let t denote an arbitrary relative translation of L" relative to L l parallel to the interface and let the lattices be separated by z. The energy of interaction of the two lattices per unit area is given by El-n where

Vt

=

1 / V l t ~ II ~ ~'C A~C X11

V t ( X tl

.-]- t,

z),

(33)

is given by eq. (31). Substituting eq. (31) into eq. (33) we obtain E I-n = - -

1

~VIclA Il A c1

~ ~ exp(iG I. Xn)exp(iG I. t)t~(Gl, z). G1

Xll

(34)

Using the relation exp(iG 1" X n) = NlJ 60,G., Xll

(35)

where 6cl.gH= 1 if G t = G ll and zero otherwise, it is finally deduced that E '-ll =

1

¢

~ g(G, z)exp(iG ~. t),

~7

(36)

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185

where G c is a common reciprocal lattice vector. If the two lattices are incommensurate with each other the only G c entering eq. (36) is G ~= 0 and the interaction energy is independent of the relative translation of the two lattices. At commensurate orientations of L ~and L n the magnitude of the contribution to the interaction energy from each matching G ~ vector is a maximum of I~(G~, z)l and it is modulated by the phase factor exp(iG ~. t). The wavelength of the modulations of the interaction energy as t is varied is determined by the condition that exp(iG c. t) = 1. We see that t may be stated uniquely if it is expressed modulo a reciprocal lattice vector of the (one or two dimensional) lattice of G c vectors. Thus, t may be expressed uniquely within the Wigner-Seitz cell formed from the reciprocal lattice vectors of the lattice of G c vectors. This Wigner-Seitz cell is precisely the same as the "cell of non-identical displacements" or "c.n.i.d." defined in Ref. 14. In general, the lattice that is reciprocal to the lattice of G c vectors is a sub-lattice of the DSC lattice plane parallel to the boundary. 5.3.

The Fourier Components of the Pair Potential

The two dimensional Fourier components of the pair potential are defined by eq. (32) and they appear in the interaction energy E l-n in eq. (36). To gain some insight into their form, and also for use in subsequent sections, we shall take the example of a simple Lennard-Jones potential. For a general two dimensional wavevector q parallel to L ~ and L u the Fourier transform of a pair potential v(r) is given by eq. (32): ~(q, z) = _Iv(x, z ) e x p ( - iq. x) dx

= 2re

yJo(qy)v(~ + z 2) dy,

(37)

where J0 is the zero'th order Bessel function. The Lennard-Jones potential that we shall use is as follows:

This potential has a minimum at r = r0 and the depth of the minimum is e. The parameters and r0 are the natural units of energy and length for this potential. Substituting eq. (38) into eq. (37) we obtain

~" " "erZ{f qr°'~SKs(qz) 2(qr°~2K2(qz)), vtq'z)= --4-tt~~o) ~ \z/ro,]

(39)

where K 5 and K2 are modified Bessel functions. At q = 0 this expression reduces to ~

2

1

v(O'z)=~er°(5(z/ro)'° whereas at large values of

qz its

1

(zlro)')'

(40)

asymptotic form is

, rc~r2o( 1 (qro~S_2(qro'~2"~(Tz'~l/2 6(q,z)=---4- ~-~\z/ro j \z/ro,].]\Eqz ]

exp(--qz).

(41)

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.'= '-'1 i ,.-,1

\

(c)

',

20

800 .

.

MATERIALS

I i

\ 0

IN

'=~1\ ;; "='1 \

80

100

0

(d)

.

IN

......

°o

;

I I

I #

40 q 60

.

SCIENCE

1'o Is

.

z'o i?--3o

-

"~ 0

2' ' 4. . 6 . 8. 10 16 18 20 . . 12. 14 .

q

q

q

(e)

0

1

~ ~" /

2

q 3

4

S

(f)

0.0

~o.

~

-O.26

-O.0O7

O:S

1;0

I:S

2.0

Z.S

~

Fxo. 3. The two dimensional Fourier transform ~(q, z) given by eq. (39) as a function ofq for z / r o = O. 1, 0.3, 0.5, 0.8, 2.0 and 5.0 in (a-f) respectively. In each plot the vertical axis is in units of her2~4 and the horizontal axis is in units of r~q.

3.0

/

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INTERFACES

It is seen in eq. (41) that at large values of qz the Fourier transform g(q, z) is exponentially damped. Figure 3 shows plots of g(q, z) for a range of values of z and q. It is seen in Fig. 3 that the sign of g(q, z) can be positive or negative depending on both z and q. The magnitude of g(q, z) generally decreases as z or q increases, g(q, z) always diverges to plus infinity as z tends to zero regardless of q owing to the dominance of the repulsive contribution in the Lennard-Jones potential at small separations. 5.4. Surface and Grain Boundary Energies in Terms of Layer Interactions Consider the creation of two surfaces of a crystal by ideal cleavage. We imagine that all atomic interactions across a cleavage plane between two adjacent atomic layers are "switched oW'. Both halves of the crystal may then be separated and ,in this imaginary process no relaxation is allowed to take place. Thus we create two ideal surfaces with an energy given by -'

2Es=-

Eee,,

~ t"= -m

(42)

t'~ 1

where Etc is the interaction energy between layers ~' and ~". Layers parallel to the surfaces have been labelled from - oo to + oo and the cleavage plane is between layers - 1 and + 1. There is no layer 0 in order to keep the symmetry in the layer numbering on either side of the cleavage plane. Using eq. (36) it is straightforward to write down an expression for Eet.. 1

Etc = - ~ ~ g(G, (: + I:'1 - 1)d)exp(iG • (tt - tt,)).

(43)

z'Jt c G

The G vectors in eq. (43) are the appropriate layer reciprocal lattice vectors as defined in Section 5.2. It is important to note that the layer reciprocal lattice vectors are not, in general, reciprocal lattice vectors of the three dimensional lattice. Layers : and : ' are separated by (: + I:'l - l)d where d is the interplanar spacing. Ac is the area of a primitive unit cell in the layers, tt is the relative translation of layer ~ parallel to the surface (relative to an arbitrary origin) which is a consequence of the stacking sequence of crystal layers along the surface normal. We see that in order to use eq. (42) to calculate ideal surface energies we require many Fourier components of the potential. The energy of an ideal, unrelaxed grain boundary between crystals I and II may also be expressed in terms of layer interactions:

E~B= ~ #ffi--~

~-'11-I ~-Ic

-

_I/"~'~II-II A.. IU'I-I~ 2 ~tt,

7- .-.t:.

(44)

~'ffil

where, ?/" ---~ i___~ EII-I A ~-~,~(G c, z~l _ z~.)exp(iG ~ . (~l _ t~. + t)), c~c

E~?~,= ~

Gc

~ g(G 1, z~ - z~,)exp(iG !. (t~ - tI,)), e

c G1

1 En-n tt"

=

_ _ ~' ~(G A l I A 11 ~ ', c 11 C G 11

n

'

z~l _

z~!)exp(iG u . (t~l _ ~)).

(45)

E11-I t'~" is the interaction energy between layer ~' in crystal II and layer ~" in crystal I. GI(G n) is a layer reciprocal lattice vector of crystal I (II) in a layer parallel to the grain boundary.

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t~(t] ~) is the relative translation of layer f parallel to the grain boundary as determined by the stacking sequence of planes in crystal I (II). t is a rigid body translation parallel to the boundary of the whole of crystal II relative to crystal I. z :i- z:,t is the separation of layers f and f' in crystal I and is equal to (f + [f'] - 1)d ~ where d I is the spacing of layers in crystal I. Similarly, z II - z~! - (Y + I:'1 - 1)d n. Setting the density of the unrelaxed grain boundary to be the same as that of the perfect crystal then results in z~~- z~, = Yd" + [g'ld ~ - (d ~+ d~)/2. By comparing eq. (42) with eq. (44) it is seen that the unrelaxed grain boundary energy is equal to EGB -_E s +1E

s. +

~

~-H-I

:'=-oo

(46)

t=l

where EI~ is the surface energy of crystal I on a plane parallel to the boundary plane. By rearranging eq. (46) we obtain an expression for the energy of cleaving the unrelaxed grain boundary: Eel ----E,' + E~" - EGS =

--

E

~'g:'plI-I"

(47)

g'=-oo :=l

The advantage of working with the cleavage energy rather than the grain boundary or surface energy is that the layer interactions on the right hand side of eq. (47) involve summations over only common reciprocal lattice vectors. At a symmetrical tilt boundary the sets {G~}, {G n} and {G ~} are identical. They are also identical at those asymmetrical tilt boundaries which display the same interplanar spacing on either side of the boundary. But for those asymmetrical tilt boundaries at which the interplanar spacing changes across the boundary {G I} is a subset of {Gll} (or vice versa) and {G c} then coincides with {G ~} (or {Gn}). At a twist boundary the sets {G t} and {G II} are identical except for the twist misorientation and the set {G~} forms a subset of both {G ~} and {Gn}. It is only at 180 ° twist boundary, which is the symmetrical tilt boundary or twin, that all three sets become identical. A mixed tilt and twist boundary may be generated by first producing an asymmetrical tilt boundary and then rotating one crystal about the boundary normal. (42)In that case {G c} is a subset of the smaller of the sets {G I} or {GII}. 5.5. The Significance o f Periodicity in the Boundary Plane: The y-Surface For a given boundary plane E~s and EIs~ are independent of the misorientation about the boundary normal. In that case variations in the unrelaxed boundary energy are determined by the last term on the right of eq. (46): AEGB =

1

~ ff(G~)exp(iG c" t),

(48)

where, -1

ff(G c) =

~ :'=-~

~] g(G ~, (Yd H + lY'ld' - (d' + dn)/2))exp(iG ~. (t~~- t~,)).

(49)

t'=l

An equation of the same form as eq. (48) was derived by Fletcher and Lodge, °s) but as far as we are aware the expression for ff(G c) in eq. (49) was first obtained by SuttonY 3) Equation (48) indicates that the energy of a commensurate boundary is a periodic function of t, and, as in eq. (36), t may be specified uniquely within the c.n.i.d, of the boundary. The

IRRATIONAL

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energy surface AEcB (t) is known as the "~-surface". ~5) Vitek introduced the y-surface in order to discuss the stabilities of stacking faults, with arbitrary translations t, between otherwise perfect crystals. The set {Gc} contributing to the sum in eq. (48) varies in magnitude and direction as the misorientation about the boundary normal varies. However, note that A IA ~¢i is constant for all boundaries sharing the same boundary plane. In eq. (49) we see that the Fourier coefficients ff(G °) for the unrelaxed boundary depend on the Fourier components of the potential and the stacking sequences of planes on either side of the boundary. Although eq. (48) was derived by Fletcher and Lodge 15 years ago its significance seems to have been largely unnoticed. It states that the unrelaxed boundary energy is a periodic function of the translation vector t, with the periodicity of the reciprocal lattice of the lattice of G ¢ vectors. Provided the boundary retains the same periodicity after relaxation the fully relaxed boundary energy must also be expressible in exactly the same form. On relaxation the only difference is that the Fourier coefficients ff(G c) change to ffr(GC). Although we cannot evaluate the Fourier coefficients ffr(G¢) for an arbitrary boundary it is possible to make some general observations about them that are quite adequate for our purposes. Each Fourier coefficient, ff~(GC), can be expressed as the Fourier coefficient of the potential t~(G¢, s(GC)), where s is some characteristic separation of the two crystals in the relaxed bicrystal that depends on G ¢. Thus, we can write, ~r(G~)exp(iG ~. t) = ~ t~(Gc, s(GC))exp(iG ~. t). Gc

(50)

Gc

s(G c) is determined as some average taken over all layer interactions at q = G ~ in the relaxed bicrystal. The dominant contributions to this average come from those layers adjacent to the boundary, and therefore, we estimate that each s(G c) is about a nearest neighbour spacing. Since I~(G°, s)[ decreases exponentially at large values of [Gels, the most significant terms in the Fourier expansion of the relaxed boundary energy are those with the smallest values of IG°I. Therefore we conclude that the most significant Fourier coefficients, ffr(G¢), of the relaxed boundary energy are those corresponding to the smallest IG~I. Boundaries with smaller primitive G ~ vectors have larger c.n.i.d.'s in real space. Thus, as the c.n.i.d, decreases in size so the variations in the relaxed boundary energy within the c.n.i.d, also decrease. In the limit of very small c.n.i.d.'s the variations of the relaxed boundary energy with translation decrease exponentially. We have just given an argument for the planar coincidence site density criterion applied to those boundaries sharing the same plane. This criterion, which is a restricted form of the planar coincidence site density criterion of Brandon et al., ~4°)was first proposed by Wolf. <4~) It was the only geometrical criterion for low interfacial energy that was found to be successful by Sutton and Balluffi ~37)in their survey of the experimental literature. For a given boundary plane, the higher the planar density of coincidence sites the larger the c.n.i.d., and thus the larger the variations in the boundary energy with translation t. If we wish to compare the energies of boundaries lying on different planes it is helpful to rewrite eqs (46 and 48) as follows: Ii EcB = E~s+ Es + ~

1

~

~kr(0) + ~

1

~

ffr(GC)exp(iG c" t).

(51)

Gc~O

The first three terms on the right of this equation change when the boundary plane changes. Only the fourth term depends on periodicity in the boundary plane. It is clear from this equation that the unrestricted form of the planar coincidence site criterion ~4°)does not account for all the terms that discriminate the energies of boundaries on differing planes. Even though

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there is some cancellation among the first three terms their sum may easily dominate the one term that is accounted for by the planar coincidence site density criterion, particularly when the c.n.i.d.'s are small. 5.6. Interfaces as Viscoelastic Media So far in this article we have addressed the structures and energies of interfaces in their ground states. But absorption or emission of crystal lattice dislocations, interfacial sliding, migration and plastic deformation, etc., send the interface through a series of transient states before a new equilibrium state is reached. The new equilibrium state is determined by the new constraints that are imposed on the interface, such as the rotational and translational relationships between the adjoining crystals, and the average interface plane normal. The re-equilibration process is time and temperature dependent because it involves atomic displacements of the order of the nearest neighbour spacing. In this section we discuss these viscoelastic aspects of interfaces. To focus the discussion consider a 1° (001) twist grain boundary in aluminium. This boundary consists of an orthogonal grid of 1/2(110) screw dislocations, separated by large patches of elastically strained crystal. There is no periodicity in the boundary because it is not at a CSL mirorientation. The screw dislocations are spaced quasiperiodically. The c.n.i.d. for the boundary has zero area because there are no Go # 0 vectors. Therefore the ),-surface, eq. (48), is fiat, meaning that the ground state energy of the boundary is independent of the relative translation, t, of the two crystals. This seems to imply that there is no energy cost in imposing a new relative translation instantaneously on the bicrystal. But we know this cannot be true because, by imposing some other t, we introduce stacking faults into the regions of elastically strained crystal between the screw dislocations. How is the paradox resolved? The paradox is resolved by recognizing that the ),-surface represents the energy of the gorund state of the boundary at any given value of t. When a new value of t is imposed by simply sliding the boundary instantaneously it is no longer in the ground state. To see this, consider the dichromatic pattern. After the imposition of the new translation a new, locally isomorphic dichromatic pattern is obtained. However, the origin of the dichromatic pattern may be displaced a large distance, compared with a crystal nearest, neighbour spacing. For instance, if the orientation relation were a CSL orientation then the shift of origin of the dichromatic pattern could be up to half a primitive CSL vector. In the case of a twist boundary the shift origin of the dichromatic pattern means that the screw dislocations in the boundary Plane have to relocate. The relocation distance can be as large as half the average dislocation spacing. In other words, after the new rigid body translation is imposed on the twist boundary, all the screw dislocations have to glide in the boundary plane by up to half the average dislocation spacing before the boundary returns to its ground state. Initially the energy of the boundary rises above the ~-surface, and returns to the ),-surface only after the dislocations have rearranged themselves throughout the entire boundary. Irrespective of the magnitude of the imposed translation the dislocation rearrangements involve atomic displacements that are comparable to the nearest neighbour spacing. The dislocation rearrangement is likely to be thermally activated even for screw dislocations in a twist boundary, because of the Peierls barrier to glide and because of the dislocation interactions. Therefore, we expect the equilibration process to be time and temperature dependent. The atomic rearrangements we have described are associated with the phason degrees of freedom of the quasiperiodic twist boundary structure. Atomic rearrangements

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(a)

Co)

I/

I

FIG. 4. The response of an incommensurate bicrystal subjected to a constant shear strain: (a) the instantaneous response, (b) the equilibrium response in which no shear stress is sustained.

associated with phason modes in three dimensional quasicrystals are also very extensive, thermally activated and comparable to the nearest neighbour spacing. °7) The bicrystal containing the 1° twist boundary supports an instantaneous shear stress parallel to the boundary plane. Let a constant shear strain of magnitude s0 be applied to the bicrystal parallel to the boundary plane, as shown in Fig. 4a. The boundary is unable to equilibrate immediately because the relaxation processes require a finite time T. After a long time, compared with ~, the shear strain is relaxed and the bicrystal is as shown in Fig. 4b. The shear stress at time t after the strain is applied is given by = a0e-in

(52)

where ao is the initial shear stress in the bicrystal. Now consider a constant stress experiment. A constant shear stress a 0 is applied at time t = 0 and maintained. The instantaneous elastic shear strain is (7o/M, where M is the shear modulus of the bicrystal. At time t the shear strain is given by e(t) = g0/M(1 + t / Q .

(53)

If the stress is removed at time tl the elastic strain recovers leaving a permanent strain of ~ o h / M ¢ . This is illustrated in Fig. 5. The mechanical behaviour we have just described is that of a Maxwell s o l i d , (43) which may be modelled by a spring and a dashpot in series. Let the elastic modulus of the spring be M and the viscosity of the dashpot be M~. The differential equation relating the stress g and total strain e is then: M'c~ = "c(r + a. (54)

strain

L/

) ,e I

1

)

'e I

)

1

=

time tI Fio. 5. The response o f a Maxwell solid subjected to a constant stress at time t = 0, which is removed at time t, leaving a permanent shear strain e~. JPMS 3 6 / I - 4 ~ *

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We note that this is not the usual anelastic model of a grain boundary that is discussed in connection with internal friction (e.g. Ref. 44). A Maxwell solid is a viscoelastic material. The key difference between these solids is that there is complete recoverability of the response upon release of the applied stress or strain in an anelastic material but not in a viscoelastic material. There are many physical processes in which the time and temperature dependences of the mechanical response of an interface are evident. In the present context perhaps the most relevant is the absorption of a crystal lattice dislocation. We may think of the dislocation as a probe that tests the response of the interface to an instantaneously applied relative translation equal to the Burgers vector. In general the Burgers vector has components that are parallel and perpendicular to the boundary plane. The slope of the y-surface of the boundary into which the dislocation is introduced determines the localization of the density of the Burgers vector parallel to the boundary plane. (45) If the c.n,i.d, of the boundary has zero area then the y-surface is flat and, at equilibrium, the Burgers vector density parallel to the boundary plane is completely delocalized. However the delocalization process may involve long range adjustments of the boundary structure and will certainly be time and temperature dependent, as observed experimentally. (46)We note that although the y-surface of the boundary into which the crystal dislocation is introduced may be fiat, this does not imply that intrinsic dislocations in the boundary are delocalized. To see this consider the 1° (001) twist boundary again. The localization of the intrinsic screw dislocations in this boundary is determined by the y-surface of the (001) plane of the perfect crystal, which is not flat. The localization of intrinsic dislocations is determined by the slope of the y-surface of the boundary structure to which they are referenced. When the boundary does contain localized intrinsic dislocations, and its y-surface is fiat, the delocalization of the parallel Burgers vector density of an absorbed crystal lattice dislocation involves long range rearrangements of the intrinsic dislocation positions. The localization of the density of the Burgers vector normal to the boundary plane is determined by the cohesive forces acting across the boundary plane347) Provided the boundary is stable with respect to cleavage there is always a force tending to constrict this Burgers vector density, irrespective of the y-surface. 5.7. Cleavage Energies of Incommensurate Grain Boundaries In the case of an incommensurate interface the set {G~} reduces to G~= 0. In practice this amounts to saying that the nearest matching G vectors are long and that the Fourier components of the potential to which they correspond are negligible. In that case the unrelaxed boundary energy is independent of the misorientation about the boundary normal and dependent only on the interplanar spacings d ~ and d~: --1 E~omm

o0

1 ~ ~ ~(0, (:d" + - E , + Es q AIcA~:,=_o~:= l __

I

II

I:'ld I - ( d ~ + d " ) / 2 ) ) .

(55)

Thus the cleavage energy of an unrelaxed, incommensurate grain boundary is simply -I

E~?°°mr"=

1 .4 tail =-c'-c

~

~, ~(0, (:d" + I:'ld'- (dX+ d")/2)).

~"= --oo d=

(56)

1

I f eq. (40) is substituted into eq. (56) it is found that almost all unrelaxed incommensurate grain boundaries in fic.c, crystals are unstable with respect to cleavage. Indeed as the interplanar spacings d' and d" tend to zero it is found that the unrelaxed grain boundary

IRRATIONAL

193

INTERFACES

energy tends to plus infinity. This is obvious from the divergence in eq. (40) as z ~ 0 arising from the repulsive part of the pair potential. In reality the boundary relaxes and the cleavage energy becomes positive. In the case of an incommensurate grain boundary the relaxation consists of an expansion normal to the boundary plane as well as individual atomic relaxation in which each atom optimises its local environment. At a commensurate boundary, i.e. where the set {Gc} includes non-zero vectors, the rigid body translation t parallel to the boundary is also an important relaxation mode. This mode is absent at an incommensurate grain boundary because no new boundary structure is created by a rigid body translation: the structures generated by varying the rigid body translation are locally isomorphic. The divergence in eq. (40) as the interplanar spacing tends to zero may be removed by including the boundary expansion, e, in the model. The model is then the same as that used by Wolf (48) when he considered the "random boundary limit". In this model of an incommensurate grain boundary the two adjoining crystals are treated as rigid objects that may be displaced normal to the boundary plane. By minimizing the boundary energy (or, equivalently, maximizing the cleavage energy) with respect to e it has been shown(/s) that positive cleavage energies may be obtained even in the limit of zero interplanar spacing. In addition, the variations of the incommensurate boundary energy and the boundary expansion with the interplanar spacings d I and d n have been derived. The neglect of local atomic relaxation in our model is an approximation, the severity of which can be tested only by comparison with full atomistic relaxation calculations. This comparison is given in the next section. The incommensurate boundary cleavage energy is now expressed as Eic~.

. . . .

A I1 A 1I fl

CZa C ['

~ =

~ f ( O , ( [ d 11+ I:qd I - (d' + dn)/2 + e)).

--o0

(57)

=

Substituting eq. (40) into eq. (57) we obtain the following expression Eic~. . . .

nero4 =

~

,

g(d /r o, dn/ro, e/ro),

(58)

where f~ is the atomic volume and g is a dimensionless function: (59)

g = Sl0 - 5S4, and

"0 00 I[

u'/,o>U".o> ro

l

<6o>

3 _1

To evaluate S. it is convenient to express it as an integral:

(dX/r°) (dn/r°) f : S, =

-(n - ~ .

y(.- l) e x p [ - (e + (d))y/ro] ~ v dy (1 - exp[-(d'y)/ro])(1 - exp[-(dny)/ro])'

(61)

where ( d ) = (d' + dn)/2 is the average interplanar spacing. Maximizing the cleavage energy with respect to e yields the following condition, which is effectively a relationship between the equilibrium expansion and the interplanar spacings d ~ and dn: Su = 2S5.

(62)

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IN MATERIALS SCIENCE

(a) 0.7

(t

-1.5

0.6

-2.0

0-~

-2.$

0.4

-3.0

~m eL 0.3

-3.5

0.1

-4.5 ]

0.0 0.0 0.1 0.2 0.3 0.4 0.$ 0.6 0.7 0.8 0.9 Interplanar spacing

-5.0

~k .

.

.

.

.

.

.

.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Interplanar spacing

FIG. 6. (a) The equilibrium expansion as a function of the interplanar spacing for incommensurate pure twist boundaries, in units of r 0. The solid line is computed using eq. (62). The open circles show the calculated expansions for fully relaxed twist boundaries listed in Table 1. (b) The function g, given by eq. (59), for the same boundaries as in (a); the cleavage energy is proportional to the negative of g.

O n c e eq. (62) is s o l v e d the e q u i l i b r i u m e x p a n s i o n is s u b s t i t u t e d i n t o eq. (59) to o b t a i n the m a x i m u m c l e a v a g e e n e r g y in eq. (58). T h e results are p r e s e n t e d g r a p h i c a l l y in F i g s 6 - 8 . C o n s i d e r first the s i m p l e r case w h e r e d ~ = d n , w h i c h i n c l u d e s p u r e twist b o u n d a r i e s . F i g u r e 6a s h o w s the e q u i l i b r i u m e x p a n s i o n as a f u n c t i o n o f the i n t e r p l a n a r s p a c i n g o n e i t h e r side o f the b o u n d a r y . T h e e q u i l i b r i u m e x p a n s i o n is z e r o o n l y w h e n the i n t e r p l a n a r s p a c i n g is

(a) 0.7'

(b) -1.4 .1.6

0.6

-1.8

-2.0

-2.2

ra 0.4

-2.4

.2.6 0.3

o.o

o'.1 o'.2

" o'.3o'.4

Average lnterplanar spacing

o.s

0.o

0.1

o'.2

o13

o14

Average interplanar spacing

FIG. 7. (a) The equilibrium expansion as a function of the average interplanar spacing (d~ in units o f t 0 as computed using eq. (62). The symmetric case, corresponding to J = 0, is represented by circles and the msot asymmetric case, corresponding to ~ = 1, is represented by squares. Note that for a given ( d ) the symmetric case always has a higher expansion than the asymmetric case. (b) The function g, given by eq. (59), for the same boundaries as in (a); note that the symmetric boundaries (with ~ = 0) always have a lower cleavage energy than the most asymmetric boundaries (with & = 1) for a given value of (d~.

o.s

IRRATIONAL INTERFACES (a)

0.36'

~)

-2.6S"

0.34

.2.70"

0.32'

.2.75" IM

0.30

-2.80"

~,1 0 . ~ '

-2.85"

0.26'

-2.90-

0.24 0.0

0.'1

0.3 d(1)

014

o.s

195

.2.95 0.0

o.l

0.2

0.3

0.4

o.s

d(1)

FIG. 8. (a) The equilibrium expansion (in units of r0) as a function of the interplanar spacing d I (in units of r0) with (d) = 0.5ro, The symmetriccase, corresponding to 6 = 0, is represented by d I = 0.5 and the most asymmetric case, corresponding to 3 = 1, is represented by d I= 0. (b) The function g, given by eq. (59), for the same boundaries as in (a). 0.8794r0 = 0.6402ar, which is greater than the m a x i m u m spacing (0.5774ar) in f.c.c, crystals; thus the model predicts that there is always an expansion at incommensurate twist boundaries in f.c.c, crystals. The model also predicts that as the interplanar spacing tends to zero the expansion tends to 15-1/6r o = 0.4636af and the function g tends to -154/3/24 = - - 1 . 5 4 1 4 . The function g is plotted in Fig. 6b. It is seen that for all interplanar spacings in f.c.c, crystals the cleavage energies of incommensurate twist boundaries increase monotonically as the interplanar spacing increases. Note that to determine the variation of the incommensurate twist boundary energy with interplanar spacing we also require the variation of the surface energy with the interplanar spacing. F o r the more general case where d i s ~ d n it is convenient to work with the average interplanar spacing ( d ) = (d I + dll)/2 and 6 = (d I - d n ) / ( d I + dII). Figure 7a shows the equilibrium expansions for two extreme sets of boundaries as a function of ( d ) . The set represented by circles corresponds to the case of 6 = 0 and ( d ) = d~= d" which is also shown in Fig. 6a. The set represented by squares corresponds to the case of d n = 0, d ~= 2 ( d ) , 6 = 1 (or d ~ = 0, d" = 2 ( d ) , 6 = - 1). It is seen that for a given average interplanar spacing ( d ) the boundary with d [ = d a always has a greater expansion than the boundary with d I = 2 ( d ) , d" = 0. The corresponding curves for the function g are shown in Fig. 7b, where it is seen that the cleavage energy of a boundary with d~= d " = ( d ) is always less than that of the boundary with d ~ = 2 ( d ) , d" = 0. Moreover, the larger the average interplanar spacing the greater the difference in expansions and cleavage energies for the two extreme cases. The intermediate cases where d ~ varies between 0 and ( d ) for a given value of d ~+ d u = 2 ( d ) are shown in Fig. 8 for ( d ) = 0.5r0. Similar curves are obtained for other values of ( d ) . It is seen that the boundary expansion increases monotonically, and the cleavage energy decreases monotonically, as d ~ tends to ( d ) , i.e. as 6 tends to zero. This is an unexpected and interesting result because it predicts that the equilibrium expansion of an incommensurate mixed tilt and twist boundary is less than the expansion of an incommensurate twist boundary with the same value of ( d ) . Furthermore the cleavage energy of the former is predicted always to be greater than that of the latter.

196

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5.8. Comparison with Full Atomistic Relaxations of Twist Boundaries The model we have used in the previous section neglects individual atomic relaxation because the boundary energy is minimized by varying the boundary expansion while otherwise keeping all atoms in their ideal crystal positions. The severity of this approximation is examined in this section where comparison is made with expansions and cleavage energies of twist boundaries that are fully relaxed using the potential of eq. (38). The twist boundaries selected for this study are listed in Table 1. Ideally we would relax incommensurate twist boundaries but this is not compatible with the use of periodic boundary conditions in the boundary plane. For this reason we have selected commensurate twist boundaries with relatively large unit cells in the boundary plane as approximations to incommensurate twist boundaries. Table 1 gives the smallest non-parallel, non-zero G c vectors in the boundary planes. Whereas in our analytical model no cut-off in the potential was introduced it is unavoidable in the computer simulations. The cut-off was set at two f.c.c, lattice parameters, so that each atom in the perfect crystal interacted with 140 neighbours. The energies of the boundaries were minimized using a conjugate gradient energy algorithm. In addition to individual atomic relaxation one grain was free to translate relative to the other both parallel and perpendicular to the boundary plane. This relative translation occurred over several planes adjacent to the geometrical boundary plane but the largest contribution always arose between the first layers of either crystal. The fraction of the total expansion that occurred between the first layers of either grain was 95%, 91%, 94%, 97%, 111%, 116% and 129% for the (111), (100) (110), (113), (112), (115) and (114) boundaries respectively. In Fig. 9a,b we show the displacements of each layer of the (100) and (114) boundaries normal to the boundary plane. The oscillations in the relaxed interplanar spacings of the (114) twist boundary, that are seen in Fig. 9b, were found in all the simulated twist boundaries except those on (111) and (100) (see Fig. 9a). The amplitudes of the oscillations increase as the unrelaxed interplanar spacing decreases, and they decay exponentially into the adjoining crystals. Similar oscillations are seen at the corresponding relaxed free surfaces. The error in assuming that all the boundary expansion occurs between the first layers of either grain tends to increase as the interplanar spacing decreases. Nevertheless, even in the worst case, i.e. the (114) boundary, the error is 29%, while for the (113) boundary the error is only 3%. Table 1 Table I. Parameters for Fully Relaxed, Long Period, Twist Boundaries Boundary Plane

cos 0

(111)

- I/7

(100)

20/29

Smallest G c vectors + 2~/a/ 2/3[-i-, 5, 2[] 2/3[1, 4, 5] [0, 3, 7]

g = d(ro)

e(ro)

EGB(O/r2)

Es(e/r 2)

-5Ec)~2/zcer~

0.7930

0.0647

0.9927

3.4222

- 3.9085

0.6868

0.1589

2.1240

3.5536

-3.3284

0.4856

0.2369

3.4040

3.7535

-2.7406

0.4141

0.2541

3.6630

3.7667

--2.5851

0.2804

0.2718

4.1033

3.7475

--2.2654

0.2643

0.2759

4.1624

3.7677

-2.2529

0.1619

0.2847

4.3067

3.7802

-2.1733

[0,7,~ (110)

- 1/I 7

[2, ~, 3]

13,~,~q (113)

1/10

[0, ~, 2]

1/1i[20, ~, ~;l (112) (115) (114)

1/49 -1/7 1/17

1/3122, ~, ]-0] [-i-,9, 2q 1/311, 19,~ 1/3122, ~, [4, 0, -1]

[I, 17, 4]

0 is the twist angle, G c denotes a common planar reciprocal lattice vector, d is the interplanar spacing, e is the equilibrium expansion, EoB is the grain boundary energy, E s is the energy of a surface parallel to the grain boundary plane and g is the function, given by eq. (59), which is proportional to the negative of the cleavage energy.

IRRATIONAL INTERFACES

197

0.12(

(a)

0"101 I~1 0.081 ~

0.06'

D, eml

'~

0.04"

~ 0,02'

°1

Z

o.~

-20

|

0

|

|

|

!

|

-15

-10

-5

0

5

10

15

20

Layer

(b)

0.25

0.20

I~ 0.15 un

otN

0.10

al

~ 0,O5' 0

Z

41.05 -70

i

i

i

!

!

!

.5o

.ao

-lo

lo

ao

so

70

Layer FIG. 9. Normal displacements (in units of at) of layers parallel to the fully relaxed (a) (100) and (b) (114) twist boundaries. The boundary plane is located between layers - 1 and 0. Note that the majority of the expansion occurs at the boundary plane.

198

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summarizes the simulation results. The boundary expansion was calculated by examining the normal displacement of layers remote from the boundary. These expansions are plotted as circles in Fig. 6a. The ideal cleavage energies were calculated by subtracting the relaxed twist boundary energy from twice the energy of a relaxed surface parallel to the twist boundary. The g functions (eqs (58) and (59)) corresponding to these cleavage energies are plotted on Fig. 6b as circles. It is seen in Fig. 6a that the simulated boundary expansion increases monotonically as the interplanar spacing decreases for the fully relaxed twist boundaries, in agreement with the analytic model for unrelaxed twist boundaries discussed in the previous section. However the model overestimates the expansion as the interplanar spacing tends to zero by about a factor of two. The quantitative agreement between the predictions of the model and the full relaxation for the cleavage energies (Fig. 6b) is better. It is noted that the energy of the unrelaxed (114) twist boundary in Table 1, in which no expansion is allowed, is of the order of 107 e/r~, the exact value depending on the relative translation t parallel to the boundary plane. Allowing the boundary to expand, while not allowing any individual atomic relaxation, not only brings the boundary energy down through 6 orders of magnitude to the correct order but the trends in the boundary expansion to increase and the cleavage energy to decrease as the interplanar spacing decreases are correctly reproduced. This provides strong support for the model of the previous section. Strong support for our model is also provided by the atomistic calculations of Wolf. <49) 5.9. Discussion of the Model for Cleavage and Expansions of Incommensurate

Grain Boundaries The analytic model we have used to derive expansions and cleavage energies treats the boundary expansion as the only variational parameter with which to minimize the boundary energy. The model has been applied to incommensurate grain boundaries, in which the nearest G c vectors are long and give rise to negligible additional interactions. It predicts that the boundary expansion increases to an upper limit as the average, unrelaxed spacing ( d ) of planes parallel to the boundary decreases. The boundary cleavage energy decreases as the boundary expansion increases. The physical reason for the decrease in cleavage energy with increasing expansion was analyzed in detail in Ref. 13. It was shown that the boundary expansion increased with decreasing (d> because of the increasing repulsive interactions between layers adjacent to the boundary, in agreement with Wolf340 However, the main consequence of the increased expansion for the boundary energy was to decrease the attractive interactions across the boundary plane. This is also the main reason why the boundary cleavage energy decreases as the boundary expansion increases. This view is in direct conflict with Wolf and Phillpot, °6) who argued that the boundary energy continued to be dominated by repulsive interactions (which they defined as interactions involving atomic separations of less than the ideal nearest neighbour distance) after the expansion was allowed. The limitations of the model stem from the complete neglect of local relaxation. As ( d ) tends to zero the surfaces of the crystals presented to the boundary plane consist of steps on lower index surfaces. Relaxation changes the local tilt components of the boundary misorientation at and between the steps to introduce structural units of the lower index boundary planes. The result is to localize intrinsic edge disloctions in the high index boundary plane, with extensive patches of the low index boundary plane between successive dislocations. In reality, therefore, the boundary expansion and cleavage energy do not change discontinuously if an infinitesimal tilt component is introduced into a low index pure twist boundary. We note

IRRATIONAL INTERFACES

199

that the edge dislocations will a l w a y s be localized provided the low index twist b o u n d a r y is stable with respect to cleavage, and, therefore, the structural unit model will always apply in this case. But in our analytic model the local relaxations are absent and the model does predict discontinuities in the expansion and cleavage energy as ( d ) changes discontinuously. Hence, the m o d e l m u s t b r e a k d o w n a t s o m e l o w e r l i m i t o f ( d ) . But the fact that the model shows the same variation o f the b o u n d a r y expansion and cleavage energy with ( d ) as the full atomistic relaxations, indicates that this lower limit o f ( d ) is less that o f the (114) twist b o u n d a r y , i.e. 1 / ( 2 x / ~ ).

ACKNOWLEDGEMENTS O n the occasion o f his (formal) retirement, it is a pleasure to record my sincere thanks to Jack Christian, with w h o m I have enjoyed m a n y searching and thoughtful discussions a b o u t interfaces over ten years. Jack first drew m y attention to the importance o f irrational interfaces in our collaboration on interfaces in martensitic transformations. He also introduced me to m a n y fundamental difficulties in dislocation descriptions o f interfaces, with which I a m still wrestling. I have always f o u n d our interactions deeply inspiring and enjoyable, not least because o f Jack's boundless patience and warmth. Section 5.6 o f this review grew out o f discussions with Vasek Vitek and Bob Bulluffi, to w h o m I a m grateful. Finally I wish to thank The R o y a l Society for giving me the o p p o r t u n i t y to think for eight years.

REFERENCES L. KRONaERGand F. H. WILSON,Trans. Am. Inst. Min. Metall. Engrs 185, 50 (1949). K. T. AUSTand J. W. RUTTER, Trans. Am. Min. Metall. Engrs 215, 820 (1959). H. GRIMMER,J. Phys. (Paris) 51, C1-155 (1990). R. C. POND, Dislocations and the Properties of Real Materials, p. 71, Inst, of Metals, London (1985). A. P. SUTTON,Phase Transitions 16/17, 563 (1989). D. GRAT1ASand A. THALAL,Phil. Mag. Lett. 57, 63 (1988). N. RIVIER,J. Phys. (Paris)47, C3-299 (1986). A. P. SUTTON,Acta Metall, 36, 1291 (1988). A. P. SUTTON,PhD Thesis, University of Pennsylvania (1981). N. G. DE BRUIJN, Nederl. Akad. Wetensch. Proc. A 43, 39 (1981). V. ELSER, Phys. Rev. B. 32, 4892 (1985). A. KATZ and M. DUNEAU,J. Phys. (Paris)47, 181 (1986). A. P. SUTTON,Phil. Mag. A 63, 793 (1991). V. VITEK, A. P. SUTTON,D. A. SMITHand R. C. POND, Grain Boundary Structure and Kinetics (edited by R. W. BULLUFFI),p. 115, American Society for Metals, Metals Park, Ohio (1980). V. VITEK,Phil. Mag. 18, 773 (1968), D. WOLFand S. PHILLPOT,Mat. Sci. Eng. AI07, 3 0989). T. C. LUBENSKY,Introduction to Quasicrystals (edited by MARKOV. JARIC),p. 199, Academic Press, New York (1988). R. C. PONDand W. BOLLMANN,Phil. Trans. R. Soc. Lond. 292, 449 (1979). R. C. PONDand D. S, VLACHAVAS,Proc. R. Soc. Lond. A386, 95 (1983). H. GglMMER,Scr. Metall. 8, 1221 (1974). H. BOHR,Acta Math. 45, 29 (1924); ibid. 46, 101 (1925); ibid. 47, 237 (1926). A. S. BESlCOVlTCH,Almost Periodic Functions, Cambridge University Press, London (1932). A. V. SHUaNIKOVand V. A. KOPTSIK,Symmetry in Science and Art, p. 348, Plenum Press, London (1974). G. H. BISHOPand B. CHALMERS,Scr. Metall. 2, 133 (1968). A. P. SUTTONand V. VITEK,Phil. Trans. R. Soc. 309, 1 (1983). A. P. SUTTONand V. VlXEK,Phil. Trans. R. Soc. 309, 37 (1983). A. P. SUTTONand V. VITEK,Phil. Trans. R. Soc. 309, 55 (1983). A. P. SUTTON,Phil. Mag. Lett. 59, 53 (1989). A. P. SUTTONand R. W. BALLUFFI,Phil. Mag. Lett. 61, 91 (1990). A. P. SUTTON,R. W. BALLUFF1and V. VI~EK, Scr. Metall. 15, 989 (1981).

I. M.

2. 3. 4. 5. 6. 7. 8. 9. 10. I1. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

200 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

PROGRESS

IN

MATERIALS

SCIENCE

P. D. BRISTOWE and R. W. BALLUFFI, J. Phys. (Paris)46, C4-155 (1985). D. LEXqNEand P. J. STEINHARDT, Phys. Rev. B 34, 596 (1986). D. GRATIAS and J. W. CAHN, Scr. Metall. 20, 1193 (1986). A. YA. KHINTCHINE, Continued Fractions (translated by P. WYNN), Noordhoff, Groningen, The Netherlands (1963). I. C. PERCIVAL, Physica 6D, 67 (1982). N. RlVmR, R. OCELLL J. PANTALONI and A. LlSSOWSK1,J. Phys. (Paris) 45, 49 (1984). A. P. SUTTON and R. W. BALLUFFI, Acta Metall. 35, 2177 (1987). N. H. FLETCHER and K. W. LODGE, Epitaxial Growth, Part B (edited by J. W. MATTHEWS), p. 530, Academic Press, New York (1975). N. H. FLETCHER and P. L. ADAMSON, Phil. Mag. 14, 99 (1966). D. G. BRANDON, B. RALPH, S. RANGANATHANand M. S. WALD, Acta Metall. 12, 813 (1964). D. WOLF, J. Phys. (Paris)46, C4-197 (1985). A, P. SUTTON, Mat. Res. Syrup. Proc. 122, 81 (1988). A, S. NOWlCK and B. S. BERItY, Anelastic Relaxation in Crystalline Solids, Academic Press, New York (1972). H. GLEITER and B. CHALMERS, Prog. Mater. Sci. 16, 219 0972). V. VITEK, L. LEJCEK and D. K. BOWEN, Interatomic Potentials and Simulation of Lattice Defects (edited by P. C. GEHLEN, J. R. BEELER and R. I. JAFFEE), p. 493, Plenum, New York (1972). R. Z. VALIEV, V. YU. GERTSMAN and O. A. KAIBYSHEV,Phys. Stat. Sol. (a) 97, 11 (1986). J. H. VAN DER MERWE, Proc. Phys. Soc. (Lond.) A 63, 616 (1950). D. WOLF, Acta Metall. 37, 1983 (1989). D. WOLF, Phil. Mag. A 63, 1117 (1991).

APPENDIX A:

TRANSLATION VECTORS OF A QUASIPERIODIC FUNCTION

In order to determine the translation vectors of p(r) belonging to e we first note that the series representation in eq. (15) is convergent. This follows from the Parseval identity: the mean value of Ip(r)l 2= ~ IAnl2, (A1) n=0 and the fact that the left hand side is finite. Therefore we can approximate p(r) by a finite expansion s(r): N s(r) = ~ H, exp(iG,, r), (A2) n=0 where N and H n are undetermined. For reasons that will be clear shortly we require I p ( r ) - s(r)[ < e/3 for all r. An obvious means of ensuring that this inequality is satisfied is to set H, = A, and to increase N until it is satisfied. The point is that it must be satisfiable for some finite value of N provided e > 0. Having found a suitable s(r) let C = l n , l+ln21+ ' + l n N I . Now, ]s(r + ~) - s(r)] = n~0 Hn exp(iGn • r)(exp(iG n • ~) - 1

(A3)

and again we reqmre Is(r + ~) - s(r)[ < e/3 for all r. To achieve this we require that lexp(iG~ • *) - 11 = 6

(A4)

for n = 1, 2 , . . . , N. Then exp(iG~ • ¢) - 1 = 3 exp(itp,) and [s(r + ~)

- -

s(r)J = n~0 H~ exp(iG,, r)exp(kpn) 3 ~< 3~.__Ia.I

= ~C.

(A5)

Therefore for ]s(r + ~) - s(r)[ < e/3 we must require 3 < e/3C in eq. (A4). Since we have required Ip(r) - s(r)l < e/3 for all r we may now write

Ip(r+~)-p(r)[<~lp(r+*)-s(r+z)l+[s(r+*)-s(r)l+ls(r)-p(r)l
(A6)

Thus ~ is a translation vector of p (r) belonging to e. To find z we have to solve N diophantine inequalities (eq. (A4)): lexp(iGn" ~) - 11 < e/3C,

n = 1, 2. . . . . N,

(A7)

i.e. IGn" T - 2nkl < 2 sin-I(e/6C) where k is an integer. N is determined by the following condition: __N r) p(r) - ~_~ H~ exp(iGn • < e/3.

(A8)

As e is reduced N increases in order to satisfy eq. (A8), and the increased number o f diophantine inequalities to be satisfied in eq. (A7) results in fewer translation vectors. For any choice o f e greater than zero the set o f translation vectors forms a relatively dense set. (21,22) This means that there is a cubic lattice, with a lattice constant L, such that

IRRATIONAL

INTERFACES

201

every cell contains the head of at least one translation vector. As e is reduced L increases in size because there are fewer translation vectors. We see that although p (r) is not a periodic function nevertheless it does repeat, to an accuracy specified by e, at least once somewhere inside every cell of a cubic lattice. As the accuracy parameter e is reduced so the cubic lattice becomes coarser and the repetitions less frequent. It should be appreciated that not only is the value of the function p(r0) repeated (to within e) at ro + * but the value of the function p(r0 + ~r) is repeated (to within e) at r0 + • + fir, where ~r is any vector. Therefore a Maxwell demon sitting at r 0 could not distinguish the environment around him (or her) from that at r 0 + z to within an accuracy of e.

APPENDIX B: THE NON-QUADRATIC IRRATIONAL TILT BOUNDARIES In this Appendix we consider the generation of approximate self-similar sequences for irrational tilt boundaries in which R is not given by eq. (22). The approach we take is to approximate R to any required accuracy by a periodic continued fraction. This amounts to an interpolation between two rational convergents of R and the generation of a self-similar arrangement of the finite sequences of structural units comprising those two rational convergents. Thus, these boundaries are described by a self-similar arrangement not of the basic A and B units but of finite sequences of A and B units. This is exactly analogous to periodic boundaries being either favoured boundaries or multiple unit reference structures. Consider the infinite continued fraction representation of R: 1

R =q0+

1

q~ +

q2 + . . . . . This is a unique representation of R. The rational convergents to this fraction, Am, are given by the generalization of eq. (26):

A,,,=Pm/Qm=(q,,,P.,

i+P.,-z)/(q,.Q,,,

,+Q.,,-z),

(Bl)

Ql=qt-

(B2)

where P0=q0,

Pl=q0ql+l

and

Q0=l,

We may attach a "tail", t, to the rational convergent Am. Let the resulting number be called Xm(t): 1 x ~ ( t ) = qo +

ql + " " .....~__

(B3) 1

1 q,~+t It can be shown that Xm(t ) is the following interpolation between the rational convergents Am and Am_ ~: Xm(t)=(tP,~ + P~,-t)/(tQ,. + Q,, 1).

(B4)

In the decomposition lattice the rational convergents A,. and A m_ ~ lie on either side of the irrational slope R. The line with slope X,.(t) is given by the vector addition of t times the vector (Q,., P,.) and the vector (Qm - ~, P~ - t). The maximum error in X,.(t) is 1/QmQm_t. If t is the periodic continued fraction Z(~) = ~ + 1/Z(~) this will have the effect of generating a self-similar arrangement of the two finite sequences of A and B units corresponding to (Q,~ _ t, P,. - ~) and (Qm, Pro). This amounts to replacing the exact tail of the infinite continued fraction of R beyond qm by the periodic tail Z(~). If the periodic tail is Z(1) = ~ the number obtained is called a "noble" number °5) and Rivier et al. 136~have discussed the importance of such numbers in phyllotaxis. But, the optimum choice of ~ is the value which makes the error in Xm[Z(¢)] smallest. Consider ~ = qm+l. This choice results in the following expressions for X,.(t):

X,,(t) = [Z(q m+~)P,,, +, + em]/[Z(qm +1 )Qm +1 + Qm].

(B5)

Thus, Xm(t ) is now an interpolation between A m and Am+~ and therefore the maximum error in Xm(t) is reduced to 1/QmQm+~ with this choice of ¢. For this choice of ¢ the generating matrix M is uniquely determined by the condition that the mixing ratio o f (Qm, Pro) sequences to (Qm - I, Pm - I) (or, equivalently (Qm + t, Pm+ i ) to (Q,,, Pm) sequences) is Z(¢). The matrix M is given by

Progress in Materials Science Vol. 36, pp. 203-224, 1992 Printed in Great Britain. All rights reserved.

0079-6425/92 $15.00 © 1992 Pergamon Press pie

SHAPE MEMORY A N D RELATED PHENOMENA C. M. Wayman Department of Materials Science and Engineering, University of Illinois, Urbana, Illinois 61801, U.S.A.

1. INTRODUCTION More than half the papers presented at ICOMAT-89 (International Conference on Martensitic Transformations, Sydney, Australia, July, 1989) were concerned with various aspects of the shape memory effect (SME), a relatively new phenomenon associated with a martensitic transformation. Accordingly, a material undergoes a displacive, shear-like martensitic transformation when it is cooled below a certain temperature, Ms. The transformation is completed when a lower temperature, Mr, is reached, where the material is said to be in the martensitic state. When this martensite is deformed (below Mr) it undergoes a strain which is completely recoverable upon heating. The shape recovery (memory) begins at a temperature, As, and is completed at a higher temperature, Af. The Ms, Mr, As and Af temperatures depend upon the particular alloy system, and recovery strains typically range from 2-10%. The martensite in shape memory alloys (SMA's) may also be isothermally induced above the Ms temperature by the application of stress, known as stress-induced martensite (SIM). This martensite disappears (reverses) when the applied stress is removed, resulting in a mechanical type (as opposed to thermal type) shape memory. The formation of SIM and its reversion gives rise to superlastic behavior. The two-way SME combines aspects of the 'one way' SME and SIM formation. Following a variety of thermomechanical treatments a specimen will exhibit a spontaneous shape change upon cooling between Ms and Mr because of microstresses inbuilt during processing. The microstresses effectively function ('program') as the external stress in SIM formation, but since the internal stresses are not released, the martensite will only reverse upon heating, at which time the one way SME operates. The shape memory effect was initially reported in 1951 by Chang and Read who studied a Au-Cd alloy3I)* Since then many other alloy systems have been found to exhibit SME behavior. A brief history of the shape memory effect can be found in Ref. (2), and an excellent review of the development of SMA's has been provided by Miyazaki and Otsuka. (3) The more practical and engineering aspects of SMA's, which are not covered here, will be found in the book, Engineering Aspects of Shape Memory Alloys, proceedings of an international workshop/conference held in 1988.(4) Additional information on the SME and SMA's can be found in various conference proceedings°-14) and monographs, os-ig) Table 1 presents a brief glossary of shape memory and related phenomena. Apparently recognizing that the rather exotic Au--Cd alloy of Chang and Read °) had little engineering significance, interest in the shape memory remained largely dormant until 1963, when similar *'Immediatelyafter a specimenhas been transformedto the orthorhombicphase, it is very soft and takes a permanent set. The permanentset is recoverable,however,as the specimenis transformedback to the cubic phase.' 203

204

P R O G R E S S IN M A T E R I A L S S C I E N C E

Table 1. Glossary of Shape Memory Terminology(2) Terminology Shape memory effect

Two-way SME (TWSM)

Superelasticity (SE)

Rubberlike behavior

Pseudoelasticity (PE)

Definition A material first undergoes a martensitic transformation. After deformation in the martensitic condition, the apparently permanent strain is recovered when the specimen is heated to cause the reverse martensitic transformation. Upon cooling, the specimen does not return to its deformed shape. A material normally exhibiting the SME is thermomechanically processed, after which, upon cooling through the martensite formation regime, it undergoes a spontaneous change in shape. Upon heating the inverse shape change occurs via the SME mechanism. When SME alloys are deformed isothermally in the temperature regime a little above that where martensite normally forms during cooling, a stress-induced martensite (SIM) is formed. This martensite disappears (reverses) when the stress is released, giving rise to a superelastic stress-strain loop with some stress hysteresis. This property applies to the parent phase undergoing a stress-induced martensitic transformation. Some SME alloys show rubberlike flexibility. When bars are bent, they spontaneously unbend upon release of the stress. In order to obtain this behavior, the martensite after its initial formation usually must be aged for a period of time; unaged martensites show typical SME behavior. The rubberlike behavior, unlike superelasticity (also a rubberlike manifestation) is a characteristic of the martensite phase--not the parent phase. This is a more generic term which encompasses both superelastic and rubberlike behavior. As such, it is less descriptive. Using the terms superelastic and rubberlike behavior is more specific and tends to avoid ambiguity by emphasizing parent phase and martensite properties, respectively. Both are isothermal phenomena.

behavior was found in a nominally stoichiometric Ni-Ti alloy (NITINOL), (2°) a somewhat more realistic candidate for commercial exploitation. Other alloys followed as well, and a year later Russian workers (2~) reported the existence of SME behavior in a Cu-A1-Ni alloy, and it soon followed (22) that the SME in this alloy was closely associated with the occurrence of a thermoelastic martensitic transformation. Most shape memory alloys (SMA's) studied to date involved a martensitic transformation from an ordered parent phase (hence an ordered martensite) and in 1972 it was proposed that SMA's were ordered, and exhibited a crystallographically reversible thermoelastic martensitic transformation. (23) The order criterion is not universally applicable, 24 as seen from Table 233)

2. CRYSTALLOGRAPHY AND SELF-ACCOMMODATION

The martensite formed in SMA's is usually self-accommodating. Figure 1 is an optical micrograph showing the surface relief due to martensite formation in a Cu--40%Zn alloy; the specimen was prepolished to a fiat condition prior to the formation of martensite at a lower temperature. The invariant plane strain surface tilt for each martensite plate can be seen. It is further noted that the observed microstructure consists of pairs of martensite 'wedges', the aggregate of which gives the appearance of an overall diamond-like morphology, each diamond consisting of four crystallographic variants of martensite. Figure 2 is a similar surface relief micrograph for the same alloy, but in this case a few reference scratches were intentionally prescribed on the flat surface before the martensite was formed. It is to be noted in particular that although each long scratch is locally deviated ('sheared') across the width of the individual plates of martensite (the invariant plane strain) the scratch AB as a whole is on the average undeviated throughout the entire field of view. That is, if a given plate

SHAPE MEMORY AND RELATED PHENOMENA

205

Table 2a. Non-Ferrous Alloys Exhibiting Perfect Shape Memory Effect and PseudoelasticityC3)

Alloy Ag--Cd Au-Cd Cu-Zn Cu-Zn-X (X = Si, Sn, Al, Ga) Cu-AI-Ni

Composition (at.%) 44--49 Cd 46.5-50 Cd 38.5-41.5Zn A few at. %

Ni-AI Ti-Ni

28-29 A1 3--4.5 Ni ~ 15 Sn 23-28 Au 45-47 Zn 36-38 AI 49-51 Ni

In-T1 In-Cd Mn--Cu

18-23 TI 4-5 Cd 5-35 Cu

Cu-Sn Cu-Au-Zu

Structure change B2-2H B2-2H B2-9R,rhombohedral M9R B2 (DO3)-9R, M9R (18R, M18R) DO3-2H

Temperature hysteresis (K) ~ 15 ~ 15 ~ 10 ~10 ~ 35

DO3-2H,18R Heusler-18R

-~6

B2-3R B2-monoclinic B2-rhombohedral FCC-FCT FCC-FCT FCC-FCT

~ 10 20~ 100 1~ 2 ~4 ~3 --

Ordering Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Ordered Disordered Disordered Disordered

Table 2b. Ferrous Alloys Exhibiting Perfect or Nearly Perfect Shape Memory Effect¢3)

Alloy Composition Fe-Pt ~25 at. % Pt Fe-Pd ~ 30 at. % Pd Fe-Ni-Co-Ti 33% Ni, 10% Co, 4% Ti (wt%) Fe-Ni-C 31% Ni, 0.4% C (wt%) Fe-Mn-Si ~30% Mn, ~5% Si(wt%) Fe--Cr-Ni-Mn-Si~Co ~ 10% Cr, < 10% Ni, < 15% Mn, <7% Si, < 15% Co (wt%)

Structure change L12-ordered BCT FCC-FCT FCC-BCT FCC-BCT FCC-HCP FCC-HCP

Temperature hysteresis(K) Small Small Small Large Large Large

Ordering Ordered Disordered Disordered Disordered Disordered Disordered

produces an 'up' displacement, its adjacent neighbor undergoes a 'down' displacement to annul the first, and the two plates self-accommodate each other. This self-accommodation effect has been investigated in detail ~25) for Ag~Cd, C u - A I - N i , Ni-AI, Cu-Zn, Cu-Zn-A1 and C u - Z n - G a alloys in which cases the parent/martensite structures are B2/2H, D03/2H, B2/3R, B2/9R, D03/18R and D03/18R , respectively. It has been found that the formation of martensite in shape memory alloys having 2H, 3R, 9R and 18R periodic stacking order structures occurs by the localized formation of four self-accommodating variants in a plate group. The 24 variants of martensite consist of six plate groups, each of which consists of four habit plane variants which are near to and symmetrically clustered about {011} poles of the parent phase. Optical and electron microscopy and computer calculations using the phenomenological crystallographic theory have shown that c o m m o n features of the mutual orientation of the martensite lattices in a plate group and their orientation with respect to the parent phase exist, even though these martensites are both internally twinned and internally faulted. The characteristic four-variant habit plane grouping in a self-accommodating diamond-shape morphology with respect to an {011 } plane proves to be identical for all alloys studied, and therefore a c o m m o n type of behavior during the shape memory deformation and recovery process may be expected. As a case in point, Table 3 lists the calculated shape deformation (invariant plane strain) matrices for martensite formed in a C u - Z n alloy. Results A, B, C and D are for each of the four constituent martensite variants which comprise a diamond morphology, as shown

206

PROGRESS IN MATERIALS SCIENCE

FIG. 1. Surfacetilting caused by the formation of martensite plates in a Cu 39.0%Zn alloy cooled to -40°C after quenching from 850°C to room temperature to retain the parent phase.

schematically in Fig. 3. The matrix representing the average of the A - B - C - D plate group, designated Avg, is essentially the unit matrix with matrix elements aii ~ 1.0 and a~O.

3. KINETICS:THERMOELASTICAND NON-THERMOELASTICMARTENSITICTRANSFORMATIONS In steels, when martensite is heated it undergoes tempering and eventual decomposition into equilibrium phases, ferrite and carbide. However, in carbonless ferrous alloys that form martensite an interesting behavior is found, as reported by Kessler and Pitsch (26)who studied an Fe-33%Ni alloy. Usually, once a martensite plate reaches a certain size its growth ceases because its interface becomes immobile. Such martensite plates do not exhibit 'backwards' movement when a specimen is heated. Instead, the parent phase (austenite) is nucleated as platelets within the martensite plates. Since several variants of the parent are usually nucleated within a single martensite plate, the martensite plate as a whole does not revert to its original austenite orientation. There are wide differences in kinetics and transformation behavior upon comparing different materials undergoing a martensitic transformation. Some alloys exhibit thermoelastic behavior, a phenomenon first reported by Kurdjumov (27) who noted the complete reversibility of martensitic transformations in copper alloys: ' . . . transformation having martensitic kinetics on cooling and heating'. It is instructive to examine the comparison given by Kaufman and Cohen (28)for thermoelastic A u - C d martensite and non-thermoelastic Fe-Ni

SHAPE MEMORY AND RELATED PHENOMENA

207

FIG. 2. Optical micrograph showing surface relief due to the formation of martensite in a C u - Z n shape memory alloy. The overall macroscopic displacement of the scratch running from upper left to lower right is nil, although on a local scale the individual plates of martensite displace the scratch differently.

martensite. This is shown in Fig. 4 where electrical resistance vs temperature plots are given for the two alloys. A substantial hysteresis difference, defined by (At-Ms) exists, comparing the two transformations. The thermoelastic transformation (Au-Cd) is characterized by a small hysteresis while the non-thermoelastic transformation (Fe-Ni) features a large transformation hysteresis. Other differences are found in both the forward (P---, M) and reverse (M ~ P) transformations. In non-thermoelastic transformations, during cooling, a martensite Table 3. Calculated Shape Deformation Matrices for C u - Z n Alloy~25~ A

1.003 0.016 0.018

0.016 1.081 0.095

-0.015 - 0.079 0.907

B

1.003 - 0.018 -0.016

0.015 0.907 -0.079

-0.016 0.095 1.081

C

1.003 0.018 0.016

-0.015 0.907 - 0.079

0.016 0.095 1.081

D

1.003 - 0.016 -0.018

-0.016 1.081 0.095

0.015 -0.079 0.907

1.003 0 0

0 0.994 0.008

0 0.008 0.994

Average

208

P R O G R E S S IN M A T E R I A L S S C I E N C E

)

_(TI T) IT)

0oo)

~.~ (b)

(o) FIG. 3. Crystallography of martensite plate group containing variants A, B, C and D. (25)

plate usually springs full size into existence, and then its interface becomes disarrayed and sessile. In thermoelastic martensites, transformation proceeds by the continuous growth of plates and the nucleation of new plates upon cooling. If the cooling process is stopped growth nucleation ceases, but if it is resumed growth and nucleation continue until the plates impinge with an obstacle such as a grain boundary or another plate. When the specimen is heated, the reverse transformation occurs by the 'backwards' movement of the martensite/parent interface. The martensite plates revert completely to the parent phase and to the original lattice orientation, i.e. complete crystallographic reversibility.

]

1.00

Jy-

=_3ooc 0

0.7-=

/ i

0

I

{J

s2.s:47.s /

g o.50 --74'C L , /

70:30

n-

0.2~ ¢

"

i

0

I

-100

I

0

I

I

I

100 200 300 Temperoture

I

l

400

5OO

oc

FIG. 4. Electrical resistance changes during cooling and heating Fe-Ni and Au-Cd alloys, showing the hysteresis of the martensitic transformation on cooling and the reverse transformation on heating, for non-thermoelastic and thermoelastic transformation, respectively. (2s)

SHAPE MEMORY AND RELATED PHENOMENA

209

4. ORDERING,THERMOELASTICBEHAVIORAND CRYSTALLOGRAPHICREVERSIBILITY Thermoelastic martensitic transformations are usually found in ordered alloys, where an ordered parent phase is transformed into an ordered martensite. In such a case, ordering promotes crystallographic reversibility because the reverse transformation path in ordered alloys is unique, in contrast to multiple paths in disordered alloys;(29) that is, disordering is not permitted because the parent phase would have a higher energy state. Ordering also promotes a higher flow stress in the parent phase, (3°'3]) making it less likely for the martensite/parent interface to become damaged (sessile) as the plate grows. In alloys where the parent phase can be either ordered or disordered, no observable changes in crystallography, i.e. the habit plane, occur as the parent phase becomes ordered, although the kinetics change completely from non-thermoelastic to thermoelastic behavior. °l) In ordered alloys, martensitic plates revert as a unit to the original parent phase orientation. (23) In a number of cases where an ordered parent transforms to an ordered martensite the arrangement of atoms, though ordered, exhibits a lower symmetry in the martensite phase following the lattice deformation (Bain strain). Considering that another lattice deformation must operate during the inverse transformation, it follows that the number of reverse lattice correspondences is more limited, compared to the cooling transformation, if it is required that the original ordered atomic arrangement in the parent is rederived. Accordingly the 'incorrect' reverse correspondences would produce a different ordered arrangement, compared to the original parent. The above is made clear with reference to the D 0 3 ~ 2H (orthorhombic) transformation in a Cu-A1-Ni alloy,(23) where it has been shown that there is only one martensite-parent correspondence which maintains the original order/32) This is demonstrated in Fig. 5, which shows the atomic arrangement on the (001) plane in the Cu-AI-Ni martensite, and three possible lattice correspondences for the reverse transformation (large and small circles represent atoms on the 1st and 2nd layers of the (001) plane, respectively, and full and open circles designate Cu/Ni and A1 atoms, respectively). Three possible reverse correspondences are indicated by the rectangles. If the rectangles elongate and contract along X-X' and Y-Y', respectively (with accompanying shuffles as XI

o. •

o. o

?----

~ / b

\

~.

o-A

Ix

•

,-I

•

•

~

•

•

•

B•' o

qlD o

o

o ." "o • 0

•

•

•

•

o •

0 &l

• 0

• o

•

•

• 0

•

o •

Cu or

•

o

0

•

•

Ni

FIG. 5. Three possible two-dimensionallattice correspondences for the martensite/parent reverse transformationin Cu-A1-Ni martensite/TM

210

P R O G R E S S IN M A T E R I A L S S C I E N C E

indicated by arrows) the bcc matrix can be obtained, the (001) plane changing to the (101) plane of the matrix. If the ordered arrangement of atoms is disregarded, the bcc matrix and the (101) planes obtained by the three inverse lattice deformations are equivalent and cannot be distinguished from each other. However, because of the ordered arrangement, the three correspondences lead to different matrix lattices and (101) planes. That is, correspondence (A) results in the (101) plane as shown in Fig. 6(a) and the other correspondences (B) and (C) result in the (101) plane shown in Fig. 6(b). Figures 6(a) and (b) are clearly different in atomic arrangement but Fig. 6(a) is the atomic arrangement of the (101) plane of the D03-type matrix. Moreover, the nearest A1 atoms are 3rd nearest neighbors in Fig. 6(a), while they are 2nd nearest neighbors in Fig. 6(b). The latter configuration would have a higher energy than the former. Therefore only correspondence (A) is likely to operate for the reverse transformation, and result in the same matrix lattice orientation. Accordingly, the ordered martensite is 'obliged' to revert to the original parent orientation, and crystallographic reversibility is indeed the case. 5. RUBBERLIKE BEHAVIOR

Rubberlike behavior was first observed in 1932 by Olander who studied a Au-Cd alloy. (33~ This phenomenon was later confirmed by Chang and Read °) studying an Au-47.Sat.%Cd alloy and Burkhart and Read (34) who studied an In-20.7at.%T1 alloy. Both of these alloys show shape memory behavior. Since, rubberlike behavior has been found in several other SMA alloy systems: Cu-AI-Ni, (as) Cu-Au-Zn (36) and C u - Z n . (37) Considering the Au-Cd alloy, it is interesting to note that the rubberlike behavior is found only after the martensite is aged at room temperature (M, = 60°C) for at least a few hours. Freshly transformed specimens exhibit shape memory behavior and bent rods will not spring back to their original shape. Chang and Read (11 attributed the rubberlike behavior to favorably oriented 'regions' growing at the expense of others, based on microscopic observations, but did not explain the driving force for the springback. These regions are bounded by twin boundaries which move reversibly. Rubberlike behavior also constitutes a mechanical type of shape memory as does the process of SIM formation, and, in fact, these two phenomena cannot be distinguished on the basis of stress-strain curves alone. But rubberlike behavior is characteristic of a fully martensitic structure whereas superelastic behavior is associated with formation of martensite

•

•

•

•

•

•

•

•

•

•

•

O

•

O

•

•

•

•

O

o

•

•

•

•

•

0

o

•

•

•

o

•

o

•

o

•

•

•

•

•

•

•

•

•

•

•

•

0

o

•

•

O

•

O

•

o

•

•

•

•

•

•

•

•

•

•

•

•

•

O

(a)

(b)

FIG. 6. Correspondence A in Fig. 5 results in the parent (101) plane shown in (a), while correspondences B and C result in (b). Only correspondenc~ A produces the correct configuration3TM

S H A P E MEMORY A N D R E L A T E D P H E N O M E N A

211

from the parent phase under stress. These two types of behavior collectively fall in the category of 'pseudoelasticity', but one should use care in the interest of precision. Broadly speaking, Olander's report of rubberlike behavior in 1932 was the first indication of the existence of a shape memory effect. Birnbaum and Read °8) suggested that the interaction between twinning dislocations and order faults inherited from the parent phase could explain rubberlike behavior, but there is no evidence for this. Later, Lieberman e t a / . (39) physically separated the atomic twinning displacements from the attendant twinning shuffles and suggested that the shuffles were time dependent (requiring aging) although the twinning shear occurs immediately. With reference to the similar rubberlike behavior in an In-Tl alloy, Basinski and Christian (4°) pointed out the difficulty of distinguishing between two possibilities: either the twin interfaces move back continuously from their new positions to their original positions, or nuclei of the original orientation left behind during loading grow into macroscopic regions upon unloading which amalgamate to give the original twin-configuration. This problem arises in the Au-Cd case as well; in fact, the mechanism proposed by Birnbaum and Read ~3~)corresponds closely with the second alternative above, while the one proposed by Lieberman e t a / . (39)corresponds to the first. It is quite clear that rubberlike behavior has been a long standing enigma. The most recent work to clarify the situation was undertaken by Otsuka and coworkers<4~)who carried out X-ray diffraction experiments using high intensity synchrotron radiation, capable of detecting subtle aging effects. Intensity changes expected from Lieberman's time dependent shuffling, were not obtained and in fact no change at all in the integrated intensity was found during aging. Thus rubberlike behavior remains a puzzle: why does a freshly transformed specimen exhibit SME and then after aging switch to rubberlike behavior? Tadaki et al. (42) have suggested replacing the terminology rubberlike behavior with twinning pseudoelasticity. 6. CRYSTALLOGRAPHY AND INTERFACES IN SELF-AcCOMMODATING PLATE GROUPS

The first systematic investigation of interrelations in plate groups was conducted by Schroeder <43)who studied morphologies associated with the formation and deformation of martensite and the shape memory recovery process in single crystals of Cu-Zn alloys. On cooling, four self-accommodating variants of martensite with a {2 11 12} habit plane form throughout extensive regions of the parent fl' phase in a diamond-like morphology consisting of the four variants placed back-to-back. For a particular case, the four variants (~ 11 12), (~ 12 11), (2 12 1 l) and (2 11 12), designated as A, B, C and D for brevity, are symmetrically clustered about the (011)8, plane of the parent. Crystallographic analysis, making extensive use of polarized light and the phenomenological theory, showed that a number of inter-variant twin relationships exist: A and C (and B and D) are twin related with respect to (011)8, [(114) martensite]; A and D (and B and C) are twin related about (100)8. [(2 0 10) martensite]; and A and B (and C and D) are (0T1)8, related. During deformation the plate groups of four variants are converted into a single preferred variant by mechanical twinning processes in which B--,A [(0T1) mode], and C - , A [(110) mode]. Such single crystal regions of martensite reverse to the parent phase by the nucleation of only a single variant which produces a single crystal of the parent. During reversal, the recovered strain is the inverse of the strain of the A variant. From the above, it is clear that intervariant conversions involve twinning in the martensite and the relative displacement of boundaries. The atomic and dislocation aspects of the

212

P R O G R E S S IN M A T E R I A L S S C I E N C E

movement of these boundaries has been considered in detail by Christian (44)who stressed that these intervariant boundaries can both grow and be nucleated at stresses below those at which dislocation deformation can begin. Christian has noted that displacement of an interface between two crystalline regions which differ in orientation or structure is conservative if all the atoms in the first crystal swept by the boundary are incorporated into the new region of the second crystal; it is nonconservative if atoms have to be added to or removed from the swept region. An interface which moves conservatively is glissile in the same sense as this term is used of dislocation motion; that is, there is no net flux of vacancies or other point defects, the individual atom displacements relative to their neighbors are less than an interatomic distance, and any activation energy is appreciably smaller than that for self diffusion. An extensive transmission electron microscopy investigation of the crystallography and structure of martensite intervariant boundaries was conducted by Adachi et alJ 4s) Detailed crystallographic analysis was undertaken on the various combinations of martensite plate variants at the boundaries formed between different self-accommodating plate groups in 18R Cu-Zn-A1 martensites. Interplate-group combinations of variants do not originate as self-accommodating pairs; rather, they come together in the growth process after nucleating separately at different nucleation sites, or may originate as a consequence of local accommodation of the martensitic shape strain. Crystallographically, 16 unique combinations were deduced out of the 276 possible combinations which may form between the 24 plate variants. Any one of these combinations is found to be capable of combining in a twin or pseudo-twin relation, owing to certain similarities between near-parallel planes designated as 'convertible' planes. Transmission electron microscopy verifies the validity of the crystallographic analysis as well as revealing a number of intervariant transition morphologies, including 'bridge', 'twin-simulation', and 'boundaryless' types. Together with data from thermal cycling experiments, microscopic observations indicate that many of the interplate-group boundaries are not particularly high in defect content, while others are, depending on the nature of the boundary and its formation mechanism. 7. MECHANISM OF THE SHAPE MEMORY EFFECT In view of the foregoing discussion, the shape memory process can be succinctly described as follows. Upon cooling a single crystal of the parent, typically 24 variants of martensite form. They form in self-accommodating groups of four variants in a diamond-like morphology. As the martensite phase is deformed some variants grow at the expense of others, and eventually only one variant persists. At this point the specimen surface is featureless, showing no relief effects. The surviving variant is that whose shape strain direction is most parallel to the tensile axis; thus permitting maximum elongation of the specimen. When this resultant single crystal of martensite is heated between As and Af the original specimen shape and parent single crystal are regenerated. Because of crystallographic restrictions and the necessity to maintain ordering, the single crystal of martensite has only one way to undergo the reverse transformation. In other words, there are numerous variants of the Bain strain during the forward transformation but only one during reversal, It is to be noted that the 'one-way' memory just described is a one time only occurrence, but can be revived by reforming the martensite, deforming it, etc. Completely recoverable strains of over ten percent have been observed. The shape memory process is schematically described in Fig. 7. The conclusions reached above have, for the most part, been derived from studies of the copper-based SMA's which feature thermoelastic behavior, a self-accommodating

SHAPE MEMORY AND RELATED

213

PHENOMENA

shaped martensite morphology, twin-related martensite variants in plate groups, etc. The same modus operandi applies to the more widely popular Ni-Ti SMA's. Comparing Ni-Ti SMA's to their Cu-based counterparts, the only real difference is that the self-accommodation consists of a triangular grouping of three martensite variants clustered about one of the three {001 }B2 poles in the parent for the former (46)and a diamond grouping of four variants with respect to the six {011} poles in the parent in the latter case. 8. THE REWRSE SHAPEMEMORYEFFECT(RSME) Pops reported on some unusual shape memory behavior in a Cu-Zn-Si alloy in 1975.(47) His alloy was 'overdeformed' (10% strain) at a temperature below or slightly above the Ms temperature, and when heated to above the Af temperature the SME shape recovery was found to be incomplete. But upon further heating the specimen changed shape towards the direction of the formerly applied stress. Unlike the usual SME, this supplemental shape change was found to be time dependent, consistent with the occurrence of the bainitic transformation in the Cu-Zn-Si alloy. Pops termed this behavior a 'reverse shape memory effect'. Similar behavior has more recently been found by Takezawa and Sato (48)who studied a Cu-Zn-A1 SMA. They observed elongations up to 5.5% when heating under stress, which were both time and temperature dependent. The shape change occurred only after an incubation period consistent with the bainite reaction. Some of their data is shown in Fig. 8.

24 Variants of

Mortensite

Single Crystal of

I

Parent Phase

I"

L

L

ll-

l

Self-accommodating Mortensitic Transformation

on Cooling ; No Macroscopic Shape

Change

Reverse Transformation on Heating ; Recovery of the Original

Deformation by Variant Coalescence on Stressing

[below Mf)

Shope

II Single Crystal of Martensite

I

~ I

S i n g l e ~ I

I

,

_1_

_1

No Shape Change on Unloading FiG. 7. Schematic illustration of the various processes involved in the shape m e m o r y effect.

214

P R O G R E S S IN M A T E R I A L S S C I E N C E

o" = 125 M

~

9

5

K

"-3

g

_7 I _./" - ] . / 0

2

4

I

I

6

8

I I0

I 12

Heating time/ks FIG. 8. Effect of treatment temperature on the elongation achieved from the bainitic reaction in a

Cu-Zn-A1 alloy subjected to a tensile stress of 125 MPa above the M~ temperature. 148)

9. T h E ALL ROUND SHAPE MEMORY EFFECT

(ARSME)

This behavior was first noted by Nishida and Honma (49)who studied aged Ni-rich Ni-Ti alloys where a Till Ni~4 precipitate forms during aging. It is necessary that the aging be done under constraint, for example by bending a thin strip of the alloy to conform in a circular manner to the inside diameter of a piece of pipe while being aged. When the strip is unconstrained and cooled after aging, it straightens flat, and with further cooling the strip curves in the opposite direction; that is, the original shape becomes inverted. The original shape is then recovered by heating. The lenticular Till Ni~4 precipitates formed during aging are coherent, thus forming stress fields, in this case tensile along <11 l> directions of the matrix. However, the precipitates form selectively, parallel to the neutral bend axis above it and perpendicular to it below/5°) Honma ~5°) suggests that the martensite forms only 'convenient' variants on cooling to eliminate the internal strain, particularly the strain perpendicular to the precipitates. Thus, preferred martensite variants are stress-induced. Accordingly, the ARSME is explained by the anisotropic precipitation of Tili Nil4 (with respect to the neutral axis) and the subsequent biasing of the martensite transformation by the precipitates, as shown in Fig. 9 where the compression side (inside) of the strip expands parallel to the neutral axis and the tensile side (outside) of the strip contracts, both of these eventually turning the initial (as-aged) strip upside down upon cooling it. 10. STRESS-INDUCEDMARTENSITE(SIM) Most materials capable of undergoing a martensitic transformation will form martensite above the M, temperature if a stress is applied. This was predicted by Scheil(St) in 1932 who noted that the shear stress required to activate the transformation decreases with decreasing temperature (being zero at Ms) whereas the shear stress required for slip in the parent phase increases with decreasing temperature. Thus at temperatures near M, applied stresses should induce plastic deformation by the martensite mode rather than by slip. The critical stress to induce martensite increases linearly with temperature, and there is another temperature, Ma, above which martensite cannot be formed by stress; instead, the parent phase undergoes

SHAPE MEMORY AND RELATED

tension-side

('

PHENOMENA

~

215

~

~

--

tension-side il

compresslon-side

!

)

compression-slde

(

I transf°rmati°n

T direction of deformation FiG. 9. Mechanism o f the All R o u n d Shape M e m o r y Effect. (s°)

plastic deformation. It is to be noted that when stress-induced martensite forms the stress is equivalent to a change in chemical free energy, the usual driving force for the martensitic transformation. In some alloys it is observed that the critical stress to form martensite increases linearly with an increase in temperaiure from M, up to a temperature M~ above which martensite forms after plastic deformation of the parent phase. (m Between Ms and M~ the stress is below the yield strength of the parent and is thus an elastic stress. Accordingly, the nucleation is stress-assisted and existing' nucleating sites are simply aided mechanically. At M~° the stress surpasses the parent yield strength, and because of its plastic deformation new nucleating sites are introduced. Thus, a distinction between elastic stress-induced martensit¢ and plastic strain-induced martensite can b¢ made. In thermoelastic shape memory alloys the stress-strain characteristics corresponding to SIM formation are more or less unique. Figure 10 is a stress-strain curve for a single crystal specimen of a Cu-39.Swt%Zn shape memory alloy deformed in tension at about 50°C above its Ms temperature. (53)Yielding at an essentially constant stress (upper plateau) corresponds to the formation of stress-induced martensite (SIM) from the parent. At about nine percent strain the specimen becomes fully martensitic. When the stress is released the strain follows the lower plateau and fully recovers as the SIM reverts to the parent. This behavior corresponds to a mechanical (as opposed to a thermal) shape memory. A stress-strain relationship such as that shown in Fig. 10 is frequently referred to as a superelastic stress-strain loop. The stress necessary to induce the martensit¢ decreases with decreasing temperature and falls to zero at Ms. When the upper and lower plateau stresses are plotted vs temperature, Fig. 11, the linear behavior is verified, and ind~d the stress to induce martensite drops to zero at Ms. Figure 12, is a photograph taken during the tensile deformation of the same Cu-Zn alloy above Ms .(53)When the upper plateau stress is reached parallel SIM variants are formed as shown in Fig. 12(a). The habit plane variant chosen by the SIM is that whose shape strain JPMS 36/bab-H

216

P R O G R E S S IN MATERIALS SCIENCE Cu-39.8 % Zn Ms = - 125"C 125

-

77"C

:~ 75 50 25

06

15

Parent. ~ _ S I M_/././././././~~

I00

f

Parent -4-- S I M ~

5 I

I

I

I

2

4

6

8

i

I0

0

Strain (%) FIG. 10. Stress-strain curve for a Cu-Zn shape memory alloy loaded above the M s temperature and then unloaded. Stress-induced martensite is formed during loading, which disappears upon unloading/43)

produces maximum elongation in the direction of the tensile axis. Further deformation results in the nucleation of numerous parallel variants, Fig. 12(b). The parallel variants eventually coalesce at the end of the plateau, leaving the entire gage length a single crystal of SIM. Upon unloading (lower plateau, Fig. 10), the inverse process occurs, whereby plates of the parent phase nucleate, coalesce, etc., restoring the single crystal parent in its original orientation. In a number of SMA's consecutive double superelastic stress-strain loops are observed as shown in Fig. 13. The first is due to SIM, as described above, and the second results from the conversion of the SIM to a second martensite (different crystal structure) also by an SIM process. In these same alloy systems, the 'first' martensite can be thermally formed by cooling, deformed into single crystal by variant coalescence, and finally converted to the 'second' martensite by an SIM process. In both of the above cases, martensite-to-martensite transformations are involved.

Cu-39.8% 7n

150

20 16

Z

IO0 12~

lit 1--

5O

(3 "I-130 - 1 2 0

-I00

-70

TEMPERATURE (°C) F]o. 11. Plotting the plateau stresses such as shown in Fig. 10 as a function of temperature gives a linear plot which obeys the Clausiusq21apeyron relationship/37)

SHAPE

MEMORY

AND

RELATED

PHENOMENA

217

FIG. 12. Gage length portion of a Cu-Zn single crystal showing the formation of only one variant (orientation) of stress-induced martensite during tensile loading above the Ms temperatures. (371

~" 120 100 z

~ ~ m

Cu-39.8% -Tn -88oc #arent~ 9R (SI M)

,~,_. 9 R ~ _

6o 4o 20 0 0

I

I

i

2

4

6

I

I

8 I0 Strain (%)

I

I

12

14

FIG. 13. Double superelastic loops resulting from two successive stress-induced martensitic transformations. (43~

218

PROGRESS IN MATERIALS SCIENCE

11. THE TWO WAY SHAPEMEMORY(TWSM) AND 'TRAINING' Over 15 years ago Wassilewski (54)described the two way shape memory as follows: 'under suitable prior deformation conditions in either the martensitic or parent structures a "reversible" expansion/contraction may be developed to accompany the forward transformation; and a change of equal magnitude but opposite in sense will then be present in the reverse transformation'. Evidently, there are two methods of conditioning or 'training' to achieve this behavior, which involve: (55) (1) SME cycling: Cooling a specimen below Mr, deforming it to produce the preferred or 'surviving' variant as mentioned above, and then heating it to above the Af temperature. This is the conventional shape memory treatment, and in such a case the memory is perfect provided the deformation does not exceed some limiting value, say 7-8%. (2) SIM cycling: Deformation of a specimen above the Ms temperature to produce stress-induced martensite (SIM) and then reversal by release of the load. Provided that a certain strain is not exceeded complete superelastic loops in the stress-strain curves are obtained, and the original specimen dimensions are recovered completely when the applied stress is released. The TWSM was observed after both SME and SIM cycling, and the results are termed SME training and SIM training. In both cases, the TWSM was observed to result from the preferential formation (and reversal) of a 'trained' variant of martensite. The training of a specimen to form a preferred variant on cooling to Mf can occur either by the prior deformation of thermal martensite or by superelastic stress cycling above Ms. If a specimen forms SIM with a shape strain which is unopposed by the tensile jig or grip ends, the superelastic training results in regions consisting of very large martensite plates. After nucleation these plates grow and coalesce, resulting in large regions of single variants which continue across the width of the specimen, as shown in Fig. 12. With additional training (SIM cycling) the single variant regions become more predominant in the gage length, which leads to a single crystal of the trained variant being formed at Ms. A specimen can also be trained by cooling to Mr under no stress, and then deforming it. In the as-cooled condition several self-accommodating plate groups are usually present, and preferred variants grow at the expense of others under deformation. After heating to above Ar for reversal (the SME), when the specimen is recooled, a dominance of the preferred variants is observed. These correspond to the variants which grew during the previous SME cycle. The effectiveness of SME training is less than that of SIM training. This is because presence of several plate groups in the former case results in smaller strains being obtained on deformation, and hence a smaller TWSM. The renucleation of several plate groups following an SME cycle results in a rather small 'averaged' strain offset since each plate group will produce an elongation in its respective direction, and only one of the six possible plate groups will contribute the maximum elongation. Some 'retraining' experiments were also carried out, (55)and one of these is now described. After a specimen had undergone several cycles of SME training and was then cooled to Mr, several plate groups were observed. Each had a predominance of one or two variants. Some specimen regions showed a large single variant to form near Ms because of the previous training. When the specimen was heated above Af and then cooled and loaded above M s, it was observed that the SIM variant was not of the same plate group as the large SME trained variant. Unloading the specimen caused reversal of the SIM. Then, on cooling to Ms the newly trained variant which formed was that of the SIM. The SME variant which had previously dominated the region in the form of large plates was now scarcely present below Mf except

SHAPE MEMORY AND RELATED PHENOMENA

219

for small regions between the newly trained variant, in which the SME plate groups acted as 'fill-in' martensite. Additional superelastic cycling above Ms resulted in more pronounced training of the SIM variant and a complete wiping out of the SME training. Thus, a specimen previously trained by SME cycling can easily be retrained, and an even larger strain offset can be realized. Figure 14 shows an optical micrograph of a Cu-Zn specimen which underwent one SME training cycle and was then cooled to Mr and photographed. The A-D variants with nearly the same shape strains are dominant (a). Further loading shows that the D variant is the preferred variant (b) after a strain of 4%. Note the large number of {2 0 10} transverse twins. It is generally held that both SIM and SME cycling result in built-in microstresses from defects which progressively accumulate during the thermo-mechanical treatment involved. These microstresses program the sample so that during cooling from Ms to Mf eventually only one variant ofmartensite forms--hence the 'forward' shape change. The reverse shape change results from the usual SME and crystallographic reversibility. Figure 15 is a transmission electron micrograph of a Cu-1 5at.%Sn alloy after one cycle. 56The regular dislocation arrays so generated presumably bias the next cooling cycle between Ms and Mr. 12. CRYSTALSTRUCTUREOF SHAPEMEMORYALLOYS The most popular SMA's, Ni-Ti, Cu-Zn-Al and Cu-Ni-AI are ordered and the martensite is characterized by a long period stacking order structure (LPSO). Some SMA's are disordered (Table 2) including the most recent class of ferrous alloys. Time and space do not permit a detailed discussion here, but extensive descriptions are found in Refs 3 and 57. Thus far, no mention has been made of the well-known R(rhombohedral)-phase transformation which occurs in a number of Ti-Ni type SMA's. This transformation is a 'weak' martensitic transformation which normally precedes the B19' martensitic transformation, although the latter may be suppressed by ternary alloying. Relative to the B19' martensite, the lattice strains to produce the R-phase are quite small, so for all practical purposes, even if the R-phase precedes the B19' martensite, the martensitic transformation is essentially from the B2 parent to the monoclinic B 19' martensite. The R-phase possesses all of the characteristics of thermoelastic martensites such as self-accommodation, shape memory, ability to be stress-induced, etc. (Ss) 13. SHAPEMEMORYIN DISORDEREDALLOYS As shown in Table 2 some SMA's are not ordered. Excluding the more recently discovered Fe-based SMA's, which will be discussed shortly, the most notable of these is the fcc ---,fct martensitic transformation in In-T1 alloys as first studied by Burkhart and Read. (34) In this case the transformation strains are very small, and in fact so small that it has been argued that the In-T1 transformation does not constitute a stringent test of the phenomenological theory of martensite crystallography. Since the transformation strains are so small it appears that the martensite atoms take the inverse path to the parent phase because this is easier than alternative paths to different orientations. Hence, crystallographic reversibility is obtained. The same argument applies to other 'small strain' fcc --, fct transformations in the disordered In-Cd and Mn-Cu systems,o) Ferrous alloys have recently received much attention because of their, even if not perfect, shape memory behavior. The state of the art here has been reviewed by Maki ~59)from which Table 4 has been taken. The SME in ferrous alloys is associated with both c~' 'thin plate'

220

P R O G R E S S IN M A T E R I A L S S C I E N C E

FIG. 14. Optical micrograph showing martensite in a C u ~ n shape memory alloy; (a) was taken after the initial cooling to below Mr, (b) shows the result of constraining the structure in (a) during heating to above .,If, removing the constraint, and cooling to below Mf again. This treatment results in a dominance of variant D. ~43)

SHAPE MEMORY AND RELATED PHENOMENA

221

FIG. 15. Transmission electron micrograph of Cu-15at.%Sn alloy showing regular dislocation arrays generated after one P ~ M ~ P cycle356~

(non-lenticular) bct martensite and e (banded) hcp martensite in a variety of alloys (Table 4), where the SME results basically from the reverse transformation of a stress-induced martensite. The fcc --, hcp e martensite is not thermoelastic and the transformation hysteresis is large compared to ordered alloys. However, the hysteresis for the fcc ~ bct ,t' martensite varies considerably, from 50-500 K. Nevertheless complete shape memory is found, although the recoverable strain (2-3%) is much smaller than in ordered thermoelastic alloys. The reverse transformation in ferrous alloys occur by the backwards movement of the martensite/parent interface. It is thus necessary that the martensite/parent interface remains glissile after transformation has occurred. This is accomplished by alloy design to ensure that the yield strength of the parent matrix is as high as possible. The effect is comparable to ordering in Fe-Pt alloys which raises the matrix elastic limit. °l) In the fcc ~ hcp case the (111)/(0001) interface is, of course, perfectly coherent. The shape recovery in this case is believed to be due to the preferential multiplication of a single variant of Shockley partial dislocations upon transformation, which allows the accumulation of a stress field, which in turn drives the partials backwards in an inverse manner during the reverse transformation, restoring the original parent orientation.°) Clearly, shape memory in ferrous alloys is still in its infancy, relatively speaking. But ferrous alloys are intriguing because they are inexpensive, not to mention that the theoretical shear for the fcc ~ hcp transformation implies a possible recoverable shape of some 20 percent.

Crystal structure o f martensite BCC or BCT (~' martensite)

Table 4. Ferrous Alloys Exhibiting Complete or Nearly Complete Shape Memory Effect~59) Morphology o f Ms As Alloy Fe-Pt (ordered ?) Fe-Ni-Co-Ti (ausaged ?)

Composition "~ 25 at.% Pt 23% Ni-10% Co-10% Ti 33% Ni-10% C o - 4 % Ti

martensite thin plate ('oct) -thin plate

(K)

(K)

131

--

173 146 193

At

ArM,

(K)

(K)

148

17

243 122

2 443 219

2 270 73

343

508

315

Coco 31% Ni-10% C o - 3 % Ti

thin plate

0 0 r~ r~

Oct) HCP (e martensite)

Fe-Ni--C (ausformed 3') Fe-Mn-Si Fe-Cr-Ni-Mn-Si

FCT

Fe-Pd Fe--Pt

31% NiA).4% C

thin plate

<77

--

2400

>320

(thin plate)

2-,300

2410

--

--

(thin plate) (thin plate)

=320 =293

2390 2343

=450 --.573

= 130 2280

(thin plate)

=260

2370

<573

<310

(thin plate) (thin plate)

179 --

Coco 30% M n - l % Si (single crystal) (28-33)% Mn--(4-6)% Si 9% C r - 5 % Ni-14% M n - 6 % Si 13% C r - 6 % N i - 8 % M n - 6 % Si-12% Co 8% C r - 5 % Ni-20% M n - 5 % Si 12% C r - 5 % Ni-16% M n - 5 % Si 2 3 0 at.% Pd 2 25 at.% Pt

---

183 300

r.

4 --

r-,

r~ Z

S H A P E MEMORY A N D R E L A T E D P H E N O M E N A

223

ACKNOWLEDGEMENTS

I am indebted to many former students and colleagues who have travelled with me along the shape memory trail during the past 20 years. I have learned much from all of them. I have also learned much from Prof. Christian. He has been a good teacher, and this is the reason that I have taken a somewhat pedagogical approach in preparing this article. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

L. C. CHANG and T. A. READ, Trans. A1ME 191, 47 (1951). C. M. WAYMANand J. D. HAPa~ISO~,JOM 16 (September 1989). S. MIYAZAKIand K. OTSUKA,1SIJ Int. 29, 353 (1989). Engineering Aspects of Shape Memory Alloys (edited by T. Duerig, K. Melton, D. Stocckel and C. M. Wayman). Butterworth, London (1990). Proe. Int. Conf. on Martensitic Transformations (ICOMAT-79), MIT, Cambridge, MA (1979). Proc. Int. Conf. on Solid-Solid Phase Transformations, Metall. Soc. AIME, Warrendale, PA 0982). Proc. lnt Cont. on Martensitic Transformations (ICOMAT-82), J. Phys. Colloq. 43, Colloque C-4 (1982). Proc. Int. Conf. on Martensitic Transformations (ICOMAT-86), Japan Inst. Metals, Sendal (1987). Proc. Int. Conf. on Martensitic Transformations (ICOMAT-89), Sydney (1990); Martensitic Transformations (edited by B. C. Muddle) Trans. Tech. Publications, Brookfield, VT (1990). Proc. M R S Int. Meeting on Advanced Materials, Mater. Res. Soc. 9, Pittsburgh, PA (1989). Proc. Int. Syrup. on Shape Memory Alloys, China Academic Publishers, Beijing (1986). Proc. 1st Jpn Int. S A M P E Syrup. and Exhibition, Soc. for the Advancement of Mater. and Process Eng. 0989). Phase Transformations '87, The Institute of Metals, London (1988). Proc. The Science and Technology of Shape Memory Alloys, Barcelona, CIDA, Palma de MaUerca, Spain (1989). Shape Memory Effects in Alloys (edited by J. Perkins), Plenum Press, New York (1975). Shape Memory Alloys (edited by H. Funakubo), Gordon and Breach Sci. Publ., New York (1987). Y. SUZUKI, Story of Shape Memory Alloys, Nikkan Kogyo, Tokyo, in Japanese (1988). Y. SUZUKI, Practical Applications of Shape Memory Alloys, Kogyochosakai, Tokyo, in Japanese (1987). Shape Memory Alloys and their Applications (edited by Y. Murakami), Nikkan Kogyo Shimbun-sha, Tokyo, in Japanese, (1987). W. J. BUEHLER,J. V. GmFSl¢}I and K. C. WEmEY, J. appl. Phys. 34, 1467 (1963). I. A. A~BUZOVAand L. G. KHA~DROS, Phys. Met, Metalloved. 17, 390 (1964). K. OTSUKAand K. SmMlZU, Scr. Metall. 4, 469 (1970). C. M. WAYMANand K. SrnM~zu, Met. Sci. J. 6, 175 0972). C. M. WAYMANand T. W. DUERIG, Engineering Aspects of Shape Memory Alloys, p. 13, Butterworth, London (1990). T. SABURIand C. M. WAYMAN,Acta Metall. 27, 979 (1979). H. K~SSI.ERand W. PITSCrl, Acta Metall. 13, 871 (1965). G. V. KURDJUMOV,J. Met. 449 (July, 1959). L. KAtYFMASand M. CoI-IErq, Prog. Met. Phys. 7, 165 (1958). K. OrSUKA and K. SmMIZU, Scr. Metall. 11, 757 (1977). M. J. M~ClNKOWSKI and D. E. CAMPnELL, Ordered Alloys, p. 331, Claytor's PUbl., Baton Rouge, LA (1970). D. P. DtrNr~ and C. M. WA~'}aAIq,Metall. Trans. 4, 137-147 (1973). K. OTSU~, PhD Thesis, Osaka University, Osaka, Japan (1971). A. OLANDES, Z. Kristall. 83A, 145 (1932). M. W. BURrd~ASTand T. A. READ, Trans. AIME 189, 47 (1951). H. SAKAMOTO,K. O r s u ~ and K. Srmalzu, Scr. Metall. 11, 607 (1977). S. MIURA, S. M~d~DAand N. NAr~NISI~I, Phil. Mag. 30, 565 (1974). T. A. SCHROEDERand C. M. WAYraArq,Acta Metall. 27, 405 (1979). H. K. BIRNBAUMand T. A. REED, Trans. AIME 218, 662 (1960). D.S. LIEn~MAN, M. A. SCHM~RHNGand R. S. KARZ, Shape Memory Effects in Alloys, p. 203, Plenum, New York (1975). Z. S. BAKINSKIand J. W. CHRISTIAN,Acta Metall. 2, 101 (1954). T. OH~A, K. OTSU~ and S. SASA[~, Martensitic Transformations, p. 317, Tans. Teeh. Publications, Brookfield, VT (1990). T. T~DAK~,K. OTSUK~ and K. S~I~ZU, Annu. Rev. Mater. Sci. 18, 25 (1988). T. A. SCHROEDER,PhD Thesis, University of Illinois (1976). J. W. Cm~ISTIA~, 1981 Inst. of Metals Lecture, Met. Soc. A1ME, Metall. Trans. 13A, 509 (1982). K. ADACH~,J. I~RK~SS and C. M. WAYMAN,Acta Metall. 36, 1343 (1988). S. M1YAZAKI,K. O'rSUKA and C. M. WAY,N, Proc. M R S Int. Syrup. on Advanced Materials, Vol. 9, p. 93, Materials Research Society, Pittsburgh, PA (1989).

JPMS "~6/I-4- H*

224 47. 48. 49. 50. 51. 52. 53. 54. 55. 56.

PROGRESS

IN MATERIALS

SCIENCE

H. POPS, Shape Memory Effects in Alloys, p. 525, Gordon and Breach Sci. Publ. New York (1975). K. TAKEZAWAand S. SATO, Trans. JIM 29, 894 (1988). M. NtSmDA and T. HONMA, Proc. Int. Conf. on Martensitic Transformations (ICOMAT-82), p. 225 (1982). T. HONMA, Proc. lnt. Conf. on Martensitic Transformations (ICOMAT-86), p. 709, Sendai (1987) E. SCHIEL, Z. Inorg. Chem. 207, 21 (1932). G. B. OLSON and M. COHEN, J. Less Common Met. 28, 107 (1972). T. A. SCHROEDEgand C. M. WAVMAN,Acta Metall. 27, 405 (1979). R. J. WASSILEWSKI,Shape Memory Effects in Alloys, p. 245, Plenum Press, New York (1975). T. A. SCHROEDERand C. M. WAVMAN,Scr. Metall. 11, 225 (1977). N. KUWANO,K. OKI and C. M. WAYMAN,Proc. Int. Conf. on Martensitic Transformations (ICOMA T-86), p. 957, Sendai (1987). 57. L. DELAEY,M. CHANDRASEKARAN,M. ANDRADEand J. VAN HUMBEECK,Proc. Int. Conf. on Solid-Solid Phase Transformations, p. 1429, Warrendale, PA (1982). 58. S. MIYAZAKI,K. OTSUKAand C. M. WAYMAN,Proc. M R S Int. Meeting on Advanced Materials, p. 93, Pittsburgh, PA (1989). 59. T. MAKI, Proc. Int. Conf. on Martensitic Transformations (ICOMAT-89), p. 157, Sydney (1990).

Progress in Materials Science Vol. 36, pp. 225-272, 1992 Printed in Great Britain. All rights reserved.

0079-6425/92$15.00 © 1992PergamonPress plc

A G I N G OF FERROUS MARTENSITES Keith A. Taylor* and Morris Cohen t *Research Department, Bethlehem Steel Corporation, Bethlehem, PA 18016, U.S.A. tDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

DEDICATION

We are most grateful for the opportunity to prepare this paper as a special way of expressing our admiration and appreciation for the productive career and lasting contributions of Professor J.W. Christian. The subject we have selected here for review, although rather narrow in a sense, constitutes a realm of phase transformations that highlights the dynamic interplay among structure, thermodynamics, kinetics, properties, and practical implications--the very spectrum of teaching and research epitomized by the life's work of Professor Christian.

1. INTRODUCTION This review paper focuses on the structure/property changes at play in carbon-containing ferrous martensites prior to the precipitation of carbides that mark the well-known first stage of tempering. All the relevant pre-precipitation phenomena are encompassed herewith by the term aging--in distinction to the subsequent tempering processes which have been elucidated in much more detail. Due to the relatively high mobility of interstitial carbon atoms in ferrous alloys near room temperature, inadvertent aging often sets in before the very first measurements can be conducted, thus creating an indeterminate starting state for analyzing the aging kinetics and underlying mechanisms. This problem invariably arises with Fe-C martensites because they start to form from the parent austenitic phase during cooling to room temperature and so undergo some aging (perhaps even autotempering) before reaching room temperature. This also happens to be the case with F e N martensites. The foregoing dilemma was first recognized and overcome by Winchell (') who devised a series of Fe-Ni-C alloys with subzero Ms temperatures to produce virgin ferrous martensites and to observe their aging without prior structural modification that might otherwise arise from the entree of carbon diffusion. In the present paper, particular emphasis is given to virgin martensites as the starting state for aging in order to access the very beginning of the processes that unfold on aging. Experimentally, then, the martensites are formed and held at comparatively low temperatures where carbon diffusion is effectively nil, and then the aging is carried out below, at, or somewhat above room temperature. In many instances, this type Keywords: aging; axial ratio; carbon clustering; carbon ordering; carbon segregation; conditional spinodal; ferrous martensites; interstitial ordering; iron--carbon (Fe-C) martensites; iron-nitrogen (FEN) martensites; mierocracking; modulated structure; pre-precipitation phenomena; spinodal decomposition; spinodal hardening; tetragonality; {011} twinning. 225

226

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of aging can be stopped intermittently for measurements at a lower reference temperature, typically that of liquid nitrogen. As the following sections will show, numerous techniques have been applied over the years to investigate the complex changes that occur during the aging of ferrous martensites. Systematic measurements of electrical resistivityt~-12) and hardness, 0'4'13) as well as dilatometry(14'15) and calorimetry,°6-18) have been carried out to monitor the stages and kinetics of aging, while the associated structural changes have been examined by X-ray ~19'2°) and neutron t21)diffraction, transmission electron microscopy (TEM), °°'11'22-3°)atom-probe field-ion microscopy (APFIM), ° 1.3J-34)and Mrssbauer spectroscopy.° ~ ) The latter technique provides a sensitive measure of the changes in the energy required to induce a specific nuclear transition within an iron atom, and this in turn is affected by interstitial atoms in the iron atom's vicinity. Hence, careful analysis of the Mrssbauer spectra can furnish valuable information on the distribution of carbon atoms in martensite during aging. Studies of F e N martensites are particularly helpful in this connection, and will be discussed where appropriate. It should be noted that much of the literature on martensite aging is based on steels and Fe42 alloys having Ms temperatures above the ambient, and hence the observed aging does not start with virgin martensite. Reference to some of these results will be included here to disclose certain features of the aging process.

2. CARBON-ATOM SITE OCCUPANCY AND MARTENSITIC SUBSTRUCTURE

2.1. Carbon-Atom Distribution in Austenite To define the structural changes that take place in ferrous martensites during aging, the initial distribution of carbon atoms within the virgin martensite must be established. This information can be obtained from an understanding of the distribution of carbon atoms within the parent austenite, considering the diffusionless nature of the displacive austenite-tomartensite transformation. Investigations utilizing Mrssbauer-effect spectroscopy (MES) (35'36) and neutron diffractiont2~)have shown that carbon atoms in FCC austenite occupy octahedral interstitial sites (OIS). Experimental results have not indicated any significant tendency for carbon atoms to occupy tetrahedral interstitial sites (TIS) within the austenite, and recent molecular dynamics calculations~42)have confirmed that OIS occupation is energetically more favorable than TIS occupation. Each OIS in austenite is crystallographically equivalent, possessing the same cubic symmetry as the host-lattice sites. There are four OIS per unit cell (Fig. 1(a)), each coordinated by six iron atoms located at the vertices of a regular octahedron (hence the name 'octahedral' site). Some investigators have carried their interpretation of Mrssbauer spectra a step further regarding the distribution of carbon atoms among the OIS of Fe-C austenite. 05'36,3s) Two components are generally found in the spectra: one peak is attributed to iron atoms having no near-neighbor carbon atoms and another split peak is assigned to iron atoms having one near-neighbor carbon atom. A peak which could be attributed to iron atoms having two near-neighbor carbon atoms has not been reported (even though a random distribution of carbon atoms should produce such a peak since, for the alloys studied, 8-10% of the iron atoms would then have two near-neighbor carbon atoms). These findings have been taken as evidence for a non-random distribution of carbon atoms in austenite, wherein the relative intensities of the two kinds of Mrssbauer peaks are consistent with a repulsive carbon--carbon interaction that tends to disperse the carbon atoms. A repulsive carbon--carbon interaction in austenite has also been suggested on thermodynamic groundsJ43)

227

A G I N G OF F E R R O U S M A R T E N S I T E S

1' i .........

i 1--7

i

• FCC lattice site

• FCC lettioe site

C) octahedral interstitial site

0 tetrahedral Interstitial site

a

........

i~1 -,

• BCC lattice site

• BCC lattice site

0 octahedrsl Interstitial site

0 tetrehedral Interstitial site

b FIG. 1. Schematic representation of (a) an FCC lattice and (b) a BCC lattice showing octahedral and tetrahedral interstitial sites.

2.2. Carbon-Atom Distribution in Virgin Martensite 2.2.1. Site occupancy M6ssbauer spectra from virgin and freshly formed martensites, although difficult to interpret because there are always several spectral components present, are also largely consistent with octahedral interstitial-site occupation by carbon atoms (4°'41) (Fig. l(b)).

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However, Fujita and coworkers (44.45)and Lysak and coworkers (46-4s) conclude that some carbon atoms occupy TIS in virgin martensite (Fig. l(b)), which they propose is the result of martensite transformation mechanisms that feature an intermediate hexagonal configuration. (45'46)However, the M6ssbauer spectral component attributed to carbon atoms in TIS was found to disappear upon aging at temperatures near room temperature, (45) and this was taken to reflect the migration of carbon atoms from less stable TIS to more stable OIS. Finally, pair potential ¢49) and molecular dynamics calculations (42) have also indicated that octahedral site occupancy is energetically more favorable than tetrahedral site occupancy. While these experimental and theoretical results have shown that OIS are the preferred sites for carbon atoms in martensite, the actual spatial distribution of the carbon atoms in virgin martensite remains controversial. Some M6ssbauer spectra from freshly quenched martensites have been interpreted in terms of a random distribution of carbon atoms, while some investigators have suggested that the carbon atoms are non-randomly dispersed, a configuration presumably inherited from the austenite. 2.2.2. Carbon-atom ordering The martensitic FCC---,BCC transformation in ferrous alloys is a diffusionless, displacive structural change for which a lattice correspondence between the parent austenite and product martensite phases can be defined35°) Figure 2(a) shows two unit cells of an FCC lattice, within which the alternative body-centered tetragonal (BCT) unit cell is also outlined. The BCT unit cell, Fig. 2(b), can be transformed to a BCC cell, Fig. 2(c), by appropriate contraction along [001]r and expansion along [li0]r and [ll0]r. Such a lattice deformation was originally proposed by Bain (5~) and is now known as the 'Bain distortion'. Insights into the distribution of carbon atoms in martensite can be gained from examination of the Bain correspondence. This correspondence has the property that the OIS in austenite are transformed to OIS in the BCC lattice. However, the Bain distortion reduces the symmetry of the OIS from cubic (point group m3m) in the case of FCC to tetragonal (point group 4/mmm) in the case of BCC. Furthermore, additional OIS arise as a result of the distortion; there are six such sites per unit cell (three per lattice site as compared with one per lattice site for FCC), each characterized by a shorter interatomic distance along one of the unit cell axes. Hence, there are three sets of OIS (Fig. 3) which together form three interpenetrating BCC sublattices. Each sublattice is designated as either Oa, Oh, or Oc, depending on whether the near-neighbor BCC lattice points lie along [100], [010], or [001], respectively. Further examination of the Bain correspondence discloses that all FCC OIS are transformed to BCC Oc sites. Hence, on the basis of this correspondence, one would expect carbon atoms in virgin martensite to occupy a single octahedral interstitial sublattice. Interstitial site occupancy has been investigated by Entin et al. (2j) using neutron diffraction on freshly quenched Fe-8 wt % Ni-l.5 wt % C martensite enriched with 57Fe and 62Ni isotopes to increase the contribution from carbon atoms to the total diffracted intensities (57Fe has a small positive scattering amplitude, while 62Ni exhibits a rather large negative scattering amplitude). The relative intensities of their neutron peaks were consistent with full OIS occupancy (i.e. they found no evidence for TIS occupation by carbon atoms) with about 80% of the carbon atoms occupying one OIS sublattice and the remaining 20% occupying the other two sublattices.

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A G I N G OF F E R R O U S M A R T E N S I T E S

[001] ~

,~

) .O,.lO] ~

[li0 -,dl~

[110] a

L--'vJ

b

~(

c

FIG. 2. Schematic representation of the Bain distortion. Two FCC unit cells are shown in (a), as well as the alternative BCT cell (dashed lines). The Bain distortion produces a contraction along [001] 7, and slight expansions along [110] 7 and [110] 7 which transform the BCT cell (b) to a BCC cell (c). In contrast, the symmetry of the octahedral interstitial sites is transformed from cubic to tetragonal ('c-oriented' octahedral interstitial sites are shown as open circles in (b) and (c)).

Consideration is now given to the lattice distortion produced by carbon atoms in octahedral interstitial sites and the resulting effect on the overall (macroscopic) symmetry of the host lattice. Following Khachaturyan, (s2) a tensor, u~, can be defined which describes the macroscopic lattice distortion associated with unit concentration of solute. Some of the elements of this tensor are identically zero in cases where solute atoms occupy sites of high symmetry. For example, an interstitial in the Oc sublattice produces a tetragonal distortion

230

PROGRESS IN MATERIALS SCIENCE ~ [001]

lattice site 0 s site 0 b site (~

0 c site

FIG. 3. Schematicrepresentation of a BCC lattice showing a-, b-, and c-oriented octahedral interstitial sites (denoted O~, Ob, and Oo, respectively).

which is more severe along [001] than along [100] and [010],* requiring the lattice distortion tensor to take on the form: Ull 0 0 uo(Oo) =

0

U,

0

0

0

U33

(1)

Similar tensors apply for interstitials in the Ca and Ob sublattices. In line with K u r d j u m o v and Khachaturyan, ~53) a long-range order parameter, ~/, can be defined which characterizes the relative extent to which carbon atoms occupy a single interstitial sublattiee. Let c represent the overall carbon concentration, given by nc/3NFe, where nc is the total number of carbon atoms and NFe is the total number of iron atoms. This composition parameter is appropriate since, as shown above, the total number of OIS in a *Locally, the displacements of neighboring iron atoms produced by a carbon atom in an OIS are such that the iron atoms form a nearly regular octahedron.

A G I N G OF F E R R O U S M A R T E N S I T E S

231

BCC lattice is equal to three times the number of lattice points. Since the total number of sites on each octahedral interstitial sublattice is equal to the total number of iron atoms, the probability that a given site on interstitial sublattice i is occupied, ni, can be defined by the following relations: na = c(1 - - q ) nb= C(1 --r/)

(2)

nc = c(1 + 2q). The order parameter r/is therefore unity when all carbon atoms are in a single sublattice (na = nb = 0, and n c = 3c); and r/equals zero when all three sublattices are equally populated (na =nb = n¢ = c). Note that this treatment of carbon ordering assumes that the O, and Ob sublattices are equally populated. A treatment of the more general case in which n a ~ nb :~ nc would require the introduction of an additional order parameter. The lattice constants are derivable from eqs 1 and 2 and are given by (for small c): a = b = a0[1 + u.(2 + ~/)c + u33(1 -- n)C]

(3a)

c = a0[l + 2Ull(1 - rl)c + u33(1 + 2q)c]

(3b)

and c / a ~ 1 + 3(u33 - Ull)?/c,

(3c)

where a0 is the lattice constant of pure BCC Fe. Note that the axial ratio (c/a) varies linearly with both the carbon content (c) and the long-range order parameter (q). The macroscopic tetragonality of F e ~ martensites and its variation with carbon content are well established and provide convincing evidence of the preferential occupation of a single octahedral sublattice by carbon atoms. The axial ratios for freshly quenched Fe-C martensites have been measured by a number of investigators, (54'55)and the variation in c/a with carbon content can be expressed as: c/a = 1 + 2.88c

(4a)

c/a = 1 + 0.045 (wt % C).

(4b)

or

2.3. Thermodynamics of Carbon-Atom Ordering Surprisingly, the carbon ordering (in the sense that only one of the three OIS sublattices is populated) is stable (relative to the more distributed occupany among the three sublattices) inasmuch as the macroscopic tetragonality of martensite persists for long times at room temperature, where carbon diffusivity is appreciable. The thermodynamics of carbon ordering has been treated by Zener, (56'57)Ren and Wang, (Ss) and Khachaturyan. (5~) Zener was the first to highlight this carbon ordering, and tetragonal martensites are often referred to as being 'Zener ordered'. Zener and Ren-Wang treated the ordering phenomenologically, while Khachaturyan took a more fundamental approach based upon the elastic interactions between carbon atoms in BCC Fe. These models are formulated within the framework of linear elasticity theory, and they define composition-dependent critical temperatures for carbon ordering. For room temperature, the Zener and Ren-Wang models place the critical carbon content at about 0.25 wt % C, in contrast to about 0.03 wt % C for the Khachaturyan

232

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model. Figure 4 shows the variation in critical composition with temperature as given by the Zener and Khachaturyan models. It is worth noting that a recent TEM investigation of martensite aging in Fe-Ni-C alloys has found diffraction evidence of Zener ordering in an Fe-33Ni-O.03C martensite at room temperature, (H) which is in agreement with the lower critical compositions of the Khachaturyan model. The Khachaturyan equilibrium long-range order parameter varies with temperature and composition according to the following relationship (for small c): (52) kT 22(0)c

-3t/ In[(1+ 2~/)/(1 - ~/)]'

(5)

in which k is the Boltzmann constant and 22(0) is an energy determined by the elastic constants of the host lattice, the severity of the tetragonal distortion produced by the interstitial carbon atom, and a short-range 'contact' repulsion between neighboring interstitials (for BCC iron, Khachaturyan (52) has calculated a value of -50.1 eV for )[2(0)). The variation of ~/with dimensionless temperature (kT/[22(O)[c) is plotted in Fig. 5. At the critical ordering temperature, r/ increases discontinuously from zero to about 0.5, indicating that Zener ordering is a first-order transition. Attempts to detect an interstitial order-disorder transition experimentally have been unsuccessful. When ordered martensites are warmed in an effort to measure the order-disorder temperature, other structural changes intervene, such as the development of compositional modulations, to be described later, or the precipitation of metastable iron

300

r~ o

600

I

I I

t

200

Khaohaturyan

500

o

Zener

Q Q-

E

400

100

m ® Q.

E _o

o

8oo

o

200 -100

100 0

0.1

0.2

0.3

0.4

0.5

Carbon Content, wt pct FIG. 4. Critical temperature for Zener ordering vs carbon content according to models developed by

Zener(~) and Khachaturyan.(52)

AGING

OF

FERROUS

233

MARTENSITES

1 0.9 0.8 0.7 0

E

0.8

m D.

0.8

13 I.

0

0.4 0.3

/

0.2 0.1 0 0

I

I

I

0.1

0.2

0.8

'

I

0.4

0.5

kT/l~t(O)lc FIo. 5. Temperature dependence of the Khachaturyan ~52)long-range order parameter, ~/. See text for symbols.

carbides. Thus, it appears that the Zener-ordered Fe-C solid-solution is not sufficiently stable relative to the other states to permit an appropriate measurement of its disordering. 2.4. Martensitic Morphology and Substructure It is well established that interstitial carbon atoms interact with dislocations in BCC Fe; hence, the martensitic substructure may be expected to exert an influence on the overall aging process. Two distinct martensitic morphologies have been observed in Fe-C alloys. In lowand medium-carbon steels (containing less than about 0.6 wt % C) with high Ms temperatures, the martensitic units take on the shape of laths, separated by low-angle boundaries and grouped into sheaves or packets359) The laths possess a substructure consisting of dislocation cells with estimated dislocation densities of 10]4--10 ~6 m/m3359-6~) The martensitic units in ferrous alloys having Ms temperatures below room temperature are generally lenticular or plate-like in shape. Units that form at very low temperatures are usually thin and flat-sided, whereas those forming closer to room temperature tend to be more lenticular in morphology. Thin-plate martensite is completely internally twinned; the twinning is of the ( l l T ) { l l 2 } type, a common deformation twinning system in BCC crystals. Lenticular martensites are not fully internally twinned; the untwinned portion contains a relatively high density of lattice dislocations (having 1/2(1 1 1) Burgers vectors) which are often pure screw in character362)

234

P R O G R E S S IN M A T E R I A L S S C I E N C E 1.11

Fe-Mn-C martenelte$ 1.10

0 virgin

warmed

•

1.00 1.0e

al

1.07

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Fe-C m

1.03 1.02 1,01 1.00

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0.2

0.4

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1.2

1.4

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1.8

2.0

Carbon Content, wt pct FIo. 6. Axial ratio (c/a) vs carbon content for F e - M n - C martensites. (47) Open circles represent values for virgin martensites (at - 160°C), and solid circles represent values obtained after warming to room temperature. The dashed line indicates the variation in axial ratio for Fe--C martensites, (54) measured at room temperature.

In addition to lattice dislocations and {112} twins, an additional substructural feature was postulated by Roytburd and Khachaturyan (63) in order to explain anomalous axial ratios in Fe-Mn--C, F e - N i ~ , and Fe-AI-C martensites. Relevant observations reviewed by Kurdjumov and Khachaturyan (53) include: (1) Fe-Mn-C martensites containing 0.6 to 1.8 wt % C and formed at subzero temperatures exhibit a smaller axial ratio than the corresponding Fe-C martensites (Fig. 6), and the axial ratio increases upon warming to room temperature (but still remains smaller than c/a for Fe--C)347) Furthermore, martensites with such abnormally low axial ratios actually have different 'a' and 'b' parameters, i.e. the lattice is macroscopically orthorhombic. (2) Fe-AI--C and Fe-high Ni-C martensites formed at subzero temperatures are found to display abnormally high axial ratios and also have a _-__b. Here, the axial ratio tends to decrease upon warming to room temperature. Recent data for Fe--Ni--C alloys(6e are compared with data for Fe-C martensites (54) in Fig. 7. Roytburd and Khachaturyan rationalized these findings by suggesting that some carbon atoms occupy Oa and Ob sites as a result of [01I]{011}* twinning. While it is not meaningful to speak of such a twinning system in a BCC lattice, this twinning is possible for BCT in which the {011 } planes no longer possess mirror symmetry. Figure 8 shows that {011} twinning can *Due to tetragonality, it should be noted that the third Miller index (representing the c-axis) is not interchangeable with the other two indices when obtaining equivalent variants. This restriction is implied throughout this paper.

235

A G I N G OF F E R R O U S MARTENSITES

1.11

Fe-NI-C martenslte$ 1.10

0 virgin • warmed

0

/

1.09 1.00

1.07 (.1 0

o

1.00

rr 1.05 1.04 1.03 1.02

1.01 L

Fe-C martenslte$ • ¢*

1.00 0.0 0.2 0.4 0.0 0.8 1.0 1.2 1.4 1.0 C a r b o n Content, w t pet

1.8 2.0

FIG. 7. Axial ratio (c/a) vs carbon content content for Fe-Ni-C martensites having Ms temperatures between - 3 5 and -120°C3 s~) Open circles represent values for virgin martensites (at -180°C), and solid circles represent values obtained after warming to room temperature. The data at 1.81 wt % C are considered to be anomalous. The dashed line indicates the variation in axial ratio for F e ~ martensites,t~) measured at room temperature.

be described operationally by a shuffle displacement of carbon atoms of a/2(011), which moves the carbon atoms from Oc sites to Oa or Ob sites (Fig. 8 depicts the latter case). Roytburd and Khachaturyan suggested that if these {011} twins are thin, they would scatter X-radiation coherently and produce a diffraction pattern characteristic of an orthorhombic structure. Hence, such twinning provides a possible explanation for the abnormally low axial ratios and orthorhombic distortions measured in virgin Fe-Mn-C martensites. In contrast, the presence of much smaller volume fractions of such twins in Fe-AI-C and Fe-high Ni-C martensites would result in nearly tetragonal structures with maximum axial ratios. Experimental observations of planar features on {011} have been reported by Sachdev, ~24) Oka and Wayman, ~65'66)and Taylor e t al., (67) and the latter investigators were able to show that such features are indeed {011} twins. While {011 } twinning in ferrous martensites seems firmly established, the mechanism(s) by which this twinning occurs remains unresolved. Roytburd and Khachaturyan proposed that such twinning occurs during the martensitic transformation (as a lattice-invariant deformationS63'68)), and this could explain the anomalous lattice constants measured for some virgin martensites. Danil'chenko, ~69)on the basis of X-ray diffraction results from Fe-24Ni-0.5C martensite, concluded that the {011 } twinning takes place on warming to room temperature as a stress-relaxation mechanism. This could explain the changes in axial ratios that many

236

P R O G R E S S IN M A T E R I A L S S C I E N C E

REGIMEI (Retainedy 11"anMormetlon)

REGIMEIII (MartenelteTempering)

REGIME,II (Msrtenelte Aging) "~

/

|

/ lin-

T

-f m

o

o

Time or Temperature FIG. 8. Schematic representation of {011} twinning in BCT Fe-C martensites (after(24,53)). (a) This specific example of twinning on (011)transfers carbon atoms in O¢ sites to O~ sites. (b) Such micro-twinning can transform a tetragonal lattice to one which is macros6opically orthorhombic, containing 'domains' with orthogonal

FIG. 9. Schematic resistivity vs aging time/temperature curve identifying the regimes which occur during the aging and tempering of initially-virgin ferrous martensites (after Sherman et aL(S)).

'c'-axes.

martensites exhibit upon warming from subzero temperatures.* Perhaps it is reasonable to assume that {011} twinning can occur both during the martensitic transformation, and after the transformation as a stress-relieving mechanism. 3. KINETICS OF MARTENSITE AGING

3.1. Stages of Aging Changes in physical properties as measured by electrical resistivity, dilatometry, and calorimetry vs temperature and time have provided information on the kinetics of martensite aging, and certain systematic changes in these properties have led to the identification of specific regimes in the overall aging process. *Changes in c/a that occur upon warming to room temperature arc assumed of the three octabedral interstitial sublattices. Such changes may indicate the equilibrium values of the long-range order parameter (53) or may be driven by changes in axial ratios must be interpreted with caution since other important take place in martensites in the vicinity of room temperature.

to reflect changes in the population attainment of new thermodynamic internal stresses. (~) However, these structural changes (described later)

237

A G I N G OF F E R R O U S M A R T E N S I T E S

3.1.1. Electrical resistivity changes during aging Substantial changes in electrical resistivity can occur when ferrous martensites are aged or tempered, and the relative ease with which resistivity can be measured has led to extensive application of this technique. °-~2) The resistivity behavior on aging ferrous martensites can generally be divided into three regimes, (8) as illustrated schematically in Fig. 9: Regime I, an initial small, rapid drop in resistivity; Regime II, an increase to a peak and subsequent decrease; and Regime III, a continuation of the decrease, at a progressively slower rate, to a fully tempered plateau value. The Regime I resistivity decrease (Ap~) occurs well below room temperature (Fig. 10) and is associated with the transformation of some of the retained austenite in these alloys to martensite. Increases in specimen volume (~8)and saturation magnetization °°) accompany the Regime I resistivity decrease, as would be expected on the basis of retained austenite transformation. It has also been suggested (8) that other processes such as dislocation interactions or movements of carbon atoms trapped in high-energy positions (as a result of subzero-temperature martensitic transformation) occur within the time/temperature range of Regime I. However, these phenomena probably do not contribute significantly to the observed Regime I resistivity changes. As the processes taking place in Regime I are presumably largely diffusionless or internal-stress driven in nature, this regime has also been termed 'relaxation' .(70) The Regime II resistivity peak occurs on aging at temperatures in the vicinity of room temperature. The magnitude of the peak (Ap, in Fig. 9) generally increases with increasing carbon content (s) (Fig. 11), although the resistivity peak is apparently diminished or absent entirely in martensites with relatively high initial resistivity (e.g. no such aging peak was found 1.01

Fe-15NI-1C M s r t e n s l t e A~led i t -120°C

o 1.00

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i

i

i i i i i i

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10

i

I

IIIIll

I

. . . . . . .

100

'

10+00

. . . . . . .

10000

Time, s e c o n d s FIc. 10. Electrical resistivity (measured at - 196°C) vs time in Regime I for initially-virgin F e - 1 5 N i - I C martensite aged at - 120°C. (t°)

238

PROGRESS

IN MATERIALS

SCIENCE

$.00 [ Fe-NIoC Martensltea]

2.75

e

2.50 2.26 2.00

E

1.r5

:~

1.5o

?

•,~

1.25 1.00 0.75 0.50

f

• Wlnchell & Cohen • Taylor et al.

0.25 0.00 0.0

I

I

I

I

I

0.2

0.4

0.6

0.8

1.0

Carbon Content, wt pot FIG. 11. Magnitude of the Regime II electrical resistivity peak, Apu (measured at - 196°C), vs carbon content for initially-virgin F e - N i ~ martensites. (4'u~

in Fe-I.8Mn-I.8C (H> or Fe-18Ni-3Mo-0.4C martensites, (9) and the peak in Fe-25Ni4).7C martensite was reduced significantly by low-temperature deformation°2~). The onset of the resistivity peak (as well as the subsequent resistivity decrease) is delayed with increasing carbon content of the martensite, as reflected by the data in Fig. 12. Possible structural changes that take place in ferrous martensites during the time/temperature range of Regime II (to be discussed in the next section) include: (1) segregation of carbon atoms to lattice defects such as dislocations and subboundaries, t6'16'm~and (2) the formation of carbon clusters and a modulated structure *° t,26>consisting of alternating carbon-rich and essentially carbonfree layers. (r~ The shift of carbon atoms from normal interstitial sites (about which near-neighbor iron atoms are severely displaced from their normal positions) to lattice defects (i.e. sites in which neighboring iron atoms are already greatly strained) would presumably decrease electrical resistivity; t6~ hence, the resistivity increase occurring in Regime II is more likely to be dominated by clustering. A modulated structure is detected well before the appearance of carbide precipitates, and therefore Regime II is termed 'aging'. (7°) The initial carbon clustering and the structural modulations have been designated as the 'AI' and 'A2' substages of aging. (7mSection 4 deals

*The term 'carbon clusters' or 'clustering', as adopted in this paper, refers to a spontaneous build-up of high-carbon concentrations in localized regions that remain coherent with the resulting low-carbon matrix. The observed 'modulated structures' are regarded as a special case o f clustering in which the coherent high- and low-carbon regions are arranged in alternating layers or bands.

AGING

OF FERROUS

239

MARTENSITES

300

100

.E: gl @

Q.

10 M 0

ro

re

•

p-

Ref,

•

• 11 • 7 & 2,3 • 5 • 4 0.1 O.O

~ 0.2

I 0.4

I 0.6

I 0.8

I 1.0

Temperature (°C) ~alna M e n u r e 26 80 23 30 30 I 1.2

-196 -196 23 30 - 196 I 1.4

I 1.6

1.8

Carbon Content, wt pct FIG. 12. A g i n g time at r o o m t e m p e r a t u r e required to reach the electrical resistivity p e a k in R e g i m e II vs c a r b o n content. (2-5,7,t~)

in considerable detail with the actual structural changes that take place in ferrous martensites during aging. The increase in resistivity to a peak value and its subsequent decline are not well understood, but similar resistivity peaks have been detected in a number of alloy systems in which clustering or 'pre-precipitation' phenomena o c c u r . (71) Chen eta/. 09'72) have shown that the mean-square displacement of iron atoms increases during aging as a result of carbon-atom clustering, and this may explain the initial resistivity increase in Regime II. (73) As discussed later, the subsequent decrease in resistivity is probably associated with the eventual coarsening of the aforementioned structural modulations. The monotonic decrease in resistivity in Regime III is associated with carbide precipitation and coarsening. The onset of transition carbide precipitation (i.e. the first stage of tempering, designated 'TI') marks the beginning of this regime. While it is difficult to determine precisely when the first stage of tempering begins, transition carbides are usually detected after aging at temperatures of 80 to 100°C. The actual variation with time of the electrical resistivity (p) of Fe-15Ni-IC martensite (ll) (measured at - 196°C) is plotted for a series of aging temperatures in Fig. 13. In this case, Regime I was essentially eliminated by a pretreatment at -120°C (Fig. 10), which allowed the isothermal transformation of retained austenite to run its course. Transition carbides were detected in this alloy by TEM after aging at 90°C for 1 h. This corresponds to a relative resistivity of roughly 1.0; thus, Regime III in this alloy is taken to begin at p/Po = 1. The

240

P R O G R E S S IN M A T E R I A L S SCIENCE 1.1

o Q.

1.0

Q.

>~ > = ¢@ O.g 0

no > ¢0 3

)

0.0

._)

0.7 0.01

0.1

1

10

100

Time, h o u r s

FIo. 13. Electrical resistivity (measured at - 196°C) vs time for initially-virgin F e - 1 5 N i - I C martensite aged at the indicated temperatures (°C) in Regimes II and III3 tI)

Regime II resistivity peak in this alloy occurs over the - 4 0 to + 100°C temperature range, indicating that martensite aging can be considered to take place mainly within this temperature range. Aging proceeds so sluggishly below - 4 0 ° C and so rapidly above 100°C that it cannot be detected outside this range within convenient times.

3.1.2. Rate-controlling mechanism: Carbon-atom diffusion The resistivity behavior offers clear evidence that the structural changes giving rise to the Regime II peak are thermally activated, since the time required to reach a given resistivity value decreases as the temperature is increased. Assuming that the kinetics of aging are diffusion-controlled, the following simple calculations provide some insight into the relative importance of the diffusion of iron atoms vs interstitial carbon atoms. The volume diffusion coefficients for carbon (obtained from a fit to the data of Smith (74)for the temperature range of 25-150°C) and iron (75) in a-ferrite are given by: I 97kJ/moleq 2D~ = (7 × 10 -5) exp ~/ m/s,

l

and O~e = (2 X 10 - 4 )

251 kJ/mole~ 2. ~ _Jm/s.

exp

To allow for diffusion of iron atoms along 'short-circuit' paths, consideration is also given to the pipe diffusion of iron atoms along dislocations: (76) D [ ~ = ( 3 x 10-5)exp[

1_

134kJ/mole-]_jm/s. 2,

AGING OF FERROUS MARTENSITES

241

The jump time tj for diffusing carbon or iron atoms in a body-centered lattice (or sublattice) can be estimated from the expression: tj = 62/(8D), in which 6 is the jump distance (taken to be x/3/2a0, where a0 is the lattice constant for ferrite [0.286 nm]) and the coefficient 8 is the coordination number. The results for temperatures of -50, 0, 50, and 100°C, given in Table 1, show that any significant diffusion of iron atoms can be ruled out over this temperature range. Even pipe diffusion is extremely sluggish, with a mean jump time of 2 ks at 100°C. On the other hand, the interstitial carbon atom is quite mobile at temperatures above about 0°C, and the jump time is only about 6 ms at 100°C. Clearly, these results signify that substantial diffusion of carbon atoms can be expected over the time/temperature range of martensite aging, and reaction-rate analyses (described below) indicate that carbon diffusion controls the rate of aging. In analyzing the kinetics of aging, it has usually been assumed that J o h n s o n - M e h l Avrami (77,78) kinetics are obeyed, and that any given fraction of the resistivity peak height represents the same microstructural 'state' (regardless of the actual time/temperature combination). Early investigations of Fe-C alloys by Roberts et a/. (14)and King and Glove142'3) focused on the volume and resistivity changes (respectively) occurring at and above room temperature (clustering and autotempering were not appreciated in those days). Activation energies ranging between 81 and 129 kJ/mole were reported as a result of that work. More recently, Mittemeijer and coworkers ~17:8~ have used dilatometry and calorimetry to measure the kinetics of aging in F e - I . I C martensite. They reported activation energies of about 80 k J/mole for segregation of carbon atoms to lattice defects, and also for carbon-atom clustering. However, such findings on binary Fe-C martensites must be interpreted with caution inasmuch as the M s temperatures for these alloys are typically well above room temperature, and hence appreciable autotempering can be expected in these martensites despite efforts to obtain rapid quenching rates after austenitizing. Sherman et al. (8> found that the activation energy on aging initially-virgin F e - N i - C martensites with 0.10 to 0.62 wt % C varied between about 75 and 100 kJ/mole; activation energies within this range have likewise been reported for the early stages of aging in Fe-25Ni-0.4C and Fe--15Ni-IC martensites. °l) A summary of activation energies published for aged martensites is presented in Table 2. This listing also includes results from M6ssbauer investigations of high-carbon Fe-C martensites. While there is considerable variation among these activation energies, the actual values generally fall within the range of those expected for carbon diffusion in martensite. (79~ Hence, it seems clear that the aging kinetics are controlled by carbon-atom diffusion. 3.2. Effects o f Carbon Content on Kinetics The Regime II resistivity peak has been observed to evolve more slowly with increasing carbon content of the martensite (Fig. 12). This suggests that carbon diffusion is more Table 1. Mean Jump Times (s) for Diffusion of Iron Atoms in Ferrite Temperature (°C) Atom - 50 0 50 Fe 3.2 × 1042 5.4 × 1031 2.0 × 1024 Fe(pipe) 8.3 x 1015 1.5 x 10I° 1.6 × 1 0 6 C 7.7 × 1 0 6 5.3 X 102 7.1 x 10 I

and Carbon 100 7.2 × 1018 2.0 x I03 5.6 X l0 -3

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P R O G R E S S IN M A T E R I A L S SCIENCE Table

Alloys

2. Published Activation Energies for Martensite Aging* Method

Fe-(0.68-1.43)C

dilatometry

0.19-1.50C steels 0.79C steel Fe--0.98C Fe--1.96C Fe--1.86C Fe-I.IC Fe-17Ni-(0.70-0.94)C Fe-Ni-(0.14).62)C Fe-25Ni-0.4C Fe-15Ni-IC

resistometry dilatometry dilatometry M6ssbauer spectroscopy M6ssbauer spectroscopy dilatometry/calorimetry dilatometry resistometry resistometry resistometry

Activation energy (kJ/mole) 109 81-129 86-132 96-118 65 90 79/83 59-126 75-100 78-103 92-112

References Roberts et al. ~14) Lement and Cohen t8~) King and Glovert2.~) Gerdien o~) Wilson and Owen ts2) G6nin and Flinn t35) DeCristofaro et a138°) Mittemeijer and coworkers 07'~8) Kerlins and Altstetter(5) Sherman et a13 s~ Taylor et al. tH) Taylor et al. (")

*The binary Fe42 alloys were probably autotempered to some extent; hence, these data should be interpreted with caution.

sluggish as the carbon content increases, consistent with Hillert's proposal (79) that the activation energy for carbon diffusion in tetragonal martensite is higher than that for diffusion in ferrite. This difference arises because of the lattice distortion created by the interstitial ordering in martensite. Since the magnitude of the tetragonal distortion increases with carbon content, so too should the activation energy for carbon-atom jumping. In fact, Hillert suggested that the activation energy for carbon diffusion should increase by about 14 kJ/mole per wt % C. On the other hand, Owen and coworkers (8°) have argued that while the local lattice distortions created by interstitial carbon atoms in BCC iron are large, the long-range displacements are relatively small and hence the activation energy for carbon diffusion in martensite should be similar to that in ferrite. In comparing the activation energies for Fe-25Ni~).4C and Fe-15Ni-1C martensites during the early stages of aging (i.e. half-way up the resistivity peak), an energy some 14 kJ/mole higher was found for the latter alloy3 H) Although this difference is somewhat larger than that expected from the Hillert correlation (7 k J/mole), the notion of an activation energy for diffusion which increases with carbon content is compatible with the observed trend of slower aging kinetics with increasing carbon content. A number of investigations have also noted that the activation energy for Regime II increases during aging. (5'8'1U5) For example, the activation energy in various Fe-Ni-C martensites increased from about 75 k J/mole at the early side of the resistivity peak, to about 100 k J/mole on aging just beyond peak38.H) Hillert ~79)has suggested that this increase may be the result of a change in rate-controlling mechanism, from carbon diffusion during the early stages of aging, to a different, unspecified mechanism having a higher activation energy. Alternatively, Sherman et al. (s) have proposed that this increase may be associated with carbon-cluster coarsening, in which the activation energy is raised by a binding energy that must be overcome as carbon atoms debond from small clusters and diffuse to larger clusters. Clearly, the actual atomic movements occurring during martensite aging are not fully understood. Further details relating to structural changes are examined in Section 4. 4. STRUCTURAL CHANGES DURING MARTENSITE AGING

4.1. Clustering vs Segregation-to-Defects o f Carbon Atoms The previous section described the electrical resistivity changes attending martensite aging, and the phenomenon of carbon-atom clustering was introduced. Such clustering has also been

A G I N G OF F E R R O U S M A R T E N S I T E S

243

confirmed by X-ray diffraction and Mfssbauer techniques, which provide more direct probes of the actual structural variations taking place during aging than do changes in 'bulk' physical properties such as electrical resistivity and specific volume. Precision X-ray diffraction measurements on Fe-18Ni-I.0C martensite°9) have shown that upon aging, the X-ray peak corresponding to the original virgin martensite decreases in intensity and increases in breadth without shifting position. This finding is consistent with carbon-atom clustering whereby the mean-square displacement of iron atoms (produced by the interstitial carbon atoms) increases during aging (causing the decrease in X-ray peak intensity as well as the aforementioned increase in electrical resistivity) while the mean iron-atom displacement remains constant (leaving the peak position unchanged). Mrssbauer investigations~4°) have also consistently confirmed that carbon-atom clustering takes place upon aging, as revealed by the increase in intensity of the spectral component attributed to those iron atoms with no near-neighbor carbon atoms. While carbon-atom clustering is firmly established as an important process that occurs upon warming virgin martensites, numerous investigations of Fe-C martensites (with Ms temperatures well above room temperature) provide evidence that carbon-atom segregation to lattice defects may also be an important process preceding carbide precipitation; ~6'~s'33) Chang ~34) has actually imaged (in a field-ion microscope) carbon atmospheres around dislocations and subboundaries in aged Fe-0.2C martensite. Various estimates have placed the amount of carbon that can be 'tied-up' by dislocations and subboundaries at about 0.1~).2 wt %.(6,83) Plastic deformation has been reported to result in the dissolution of carbides, ~s3~indicating that lattice defects are potent traps for carbon atoms. In spite of these findings, Olson and Cohen ~7°)argue that, while segregation to defects may occur during quenching of high-Ms martensites, the diffusion distances involved in clustering are so small that the more widely spaced substructural defects do not appreciably influence the processes taking place throughout the major volume fraction of the virgin martensite. Hence, while segregation of carbon atoms to defects is plausible, the early stages of martensite aging are likely to be dominated by carbon-atom clustering, especially when starting with virgin martensites. The extent of any interaction (e.g. competition) between clustering and segregation is unclear, and remains a topic for further study. 4.2. Development of Structural Modulations A detailed picture of the structural changes that evolve during martensite aging has gradually emerged from many investigations in which aged martensites have been studied by a variety of methods; TEM, APFIM, and Mrssbauer spectroscopy have proven to be especially powerful techniques. 4.2.1. Transmission electron microscopy Nagakura and c o w o r k e r s (26'2s) have conducted detailed TEM investigations of aging in Fe~2 martensites. They reported the development of diffuse electron scattering in Fe-1.1C martensite upon warming to - 3 ° C (virgin martensite was formed in situ at -73°C from retained austenite), and attributed this to the formation of small carbon-rich clusters. Olson and Cohen ~7°)have termed this the A1 substage of aging. This diffuse scattering was initially observed as 'spikes' emanating from fundamental reflections along directions near (011 ), and the direction shifted toward (012) as the warming proceeded. Similar diffuse scattering has also been reported in Fe-Mn-C, ~23,27)Fe-Ni--C, 0 l) and Fe-Cr-C ~22'23'3°)martensites after aging, although some small differences apparently exist in the actual directions along which the

244

PROGRESS IN MATERIALS SCIENCE

diffuse scattering develops. It should be noted that in initially-virgin Fe-Mn-C and Fe-Cr-C martensites studied by Kurdjumov et a/., t23) the aforementioned diffuse scattering was only observed after warming (e.g. after 30 min at 10°C or 5 min at 20°C), in agreement with the results of Nagakura and coworkers on Fe--I.1C martensite. The diffuse electron scattering in initially-virgin Fe-15Ni-IC martensite consists of spikes along the (023) directions, ~l~ as shown in Fig. 14. The [100] diffraction pattern (Fig. 14(a)) was obtained after aging only 26min at room temperature, and corresponds to a time/temperature combination at the early part of the electrical resistivity peak for this alloy (Fig. 13 in Section 3). A diffraction pattern obtained from a thin foil oriented such that the incident beam is parallel to the martensite c-axis does not exhibit the diffuse scattering observed for the [100] (or equivalent [010]) orientation. However, slight tilting from exact [001] incidence results in the diffraction pattern shown in Fig. 14(b), in which diffuse spikes from four equivalent (023) variants intersect the reflecting sphere, producing what appears to be 'satellite' spots about the fundamental reflections. This result, in which diffuse scattering occurs only along the four (023) directions (and not along the (302) or (320) directions), is a direct consequence of the tetragonal symmetry of virgin martensite. That is, the carbon ordering (Zener order) present in virgin martensite leads to the preferential selection of certain crystallographic variants for the diffuse scattering. Moreover, this finding shows that the tetragonal symmetry of virgin martensite persists after its warming (although there is X-ray diffraction evidence of slight changes in the tetragonality of virgin martensite as it is warmed from liquid nitrogen temperature~2°'64~). Electron micrographs of aged martensites generally display a characteristic 'tweed-like' or modulated structure. In Fe-15Ni-IC martensite (Fig. 15), this structure consists of fine-scale striations (spaced on average about 1.5 nm in this case) lying along directions perpendicular to the diffuse scattering spikes (i.e. the modulations occur along the four (023) directions). Optical diffraction patterns obtained directly from TEM images reproduce the diffuse scattering exhibited in the electron diffraction patterns (Fig. 16), confirming that the diffuse scattering arises from the spatial periodicities associated with the fine-scale striations contained in the image. Figure 17 is a schematic representation of the four modulation variants in aged Fe-C martensites, as proposed by Kusunoki and Nagakura326) Nagakura and coworkers propose that the structural modulations actually result from a stress-induced alignment t84)of individual clusters which are initially formed in a random distribution. Olson and Cohen ~7°)refer to this aging substage as 'A2'. However, as discussed later, it is also possible that the modulations actually start to build up from the very beginning of aging. Results for Fe-C martensites, ~26) combined with data for Fe-Ni-C and Fe--Mn-C martensites, ~1~) show that the initial wavelength of the structural modulations generally decreases with increasing carbon content (Fig. 18), although the wavelength becomes essentially independent of carbon content above about 1.4 wt %, at a value of about 1 nm. It should be noted, however, that the data for Fe-C alloys were obtained from autotempered martensites, and should be interpreted with some care. In some cases, well-defined 'satellite' spots develop at the tips of the diffuse scattering that is present in electron diffraction patterns of aged martensites. Kusunoki and Nagakura ~s4) reported that such spots emerge from high-carbon F e ~ martensites after aging at temperatures slightly above room temperature (about 60-80°C). Apparently, in these martensites the structural modulations initially contain a range of periodicities (and give rise to diffuse scattering as described above), but continued aging results in 'amplification' of a narrow band of wavelengths and the development of comparatively distinct spots. However, no such satellite spots have been reported for initially-virgin Fe-Ni-C martensites.

A G I N G OF F E R R O U S M A R T E N S I T E S

245

Q

FIG. 14. (a) [100], and (b) [001] electron diffraction patterns obtained from initially-virgin Fe--15Ni-IC martensite after aging for 0.5 and 6 h, respectively, at room temperature.~H) The insert shows an enlargement of the TT0 diffraction 'spot'. High-resolution T E M images of room-temperature-aged F e - C (29) and F e - A I - M n - C (aS) martensites have indicated that iron atoms are displaced by neighboring interstitial carbon atoms along (01 l ) directions. Similar displacements have also been reported in elastically anisotropic cubic alloys in which tetragonal distortions produced by solute-atom dusters are apparently accommodated by shear-like displacements of the host atomsfl 6~ As discussed in

246

P R O G R E S S IN MATERIALS SCIENCE

FIG. 15. Dark-field transmission electron micrograph of initially-virgin Fe-15Ni-IC martensite, aged 5.5 hr at room temperature,o°)

Section 3, such iron-atom displacements and the attendant electron scattering produced by carbon-atom clustering are responsible for the electrical resistivity increase in Regime II. Further aging at room temperature leads to a gradual coarsening of the modulations. For example, after 50 days (1200 hr) at room temperature, the wavelengths for Fe-25Ni4).4C and Fe-15Ni-IC martensites increase to about 6-10nm. Although the physics of electrical conduction in systems exhibiting pre-precipitation phenomena are not yet well understood, the modulation coarsening is probably the cause of the electrical resistivity decrease in the latter portion of Regime II. For diffusion-controlled coarsening, the structural wavelength increases with time as follows: 2" - 2~ = k(t - to) (6) where 2 is the wavelength at time t, 2o is the wavelength at time to, and n and k are constants. The wavelength exponent takes on a value of 3 for classical Lifshitz-Slyozov-Wagner~s7,s8) (LSW) coarsening behavior, so that during the late stages of coarsening a 2 ~ t 1/3law prevails. A coarsening law of this form has been found experimentally for modulated structures in spinodally decomposed Cu-Ni-Fe alloys~89)and in numerical simulations of the later stages of spinodal decomposition.°°'91) However, data for initially-virgin Fe-25Ni-0.4C and Fe-15Ni-IC martensitesoo when fitted to eq. 6 (continuous curves in Fig. 19), yield values for the time exponent of 0.24 and 0.39, respectively. The deviation in these experimental exponents from the ideal value of 1/3 is not well explained at present, but other experimen-

A G I N G OF F E R R O U S M A R T E N S I T E S

FIG. 16. Dark-field transmission electron micrographs of initially-virgin (a) Fe-15Ni-IC and (b) Fe-25Ni-4).4C martensites and their associated electron and optical diffraction patterns (EDP and ODP, respectively), ta°)The latter was obtained from the circled region in each micrograph. The electron beam direction and imaging reflection were [100]/020 and [001]/1 l0 in (a) and (b), respectively, JPMS 3 6 / I ~ - I

247

248

P R O G R E S S IN M A T E R I A L S S C I E N C E

FIG. 16(b)

tal t92) and theoretical ~93'94~investigations have also noted departures from the LSW value in systems undergoing continuous transformations under combined phase separation and coarsening conditions.

A G I N G OF F E R R O U S MARTENSITES

c

o

249

~2),,

cE

FIG. 17. Cluster model of the modulated structure in aged FeqE martensites, as proposed by Kusunoki and Nagakura. (26)The individual clusters are described to be less than 1 nm in size, arranged in domains within which fluctuations in carbon concentration in {102} planes are shown schematically by the projection along the a- or b-axis.

Another interesting observation on very high-carbon Fe--C ~22'2s) and Fe-Mn-C martensites~n'24"27) concerns the appearance of additional diffuse 'superstructure' spots in electron diffraction patterns when these martensites are aged at or slightly above room temperature (Fig. 20). The superstructure spots, which usually indicate splitting along the c-axis of the modulated structure, have been attributed to the formation of a coherent Fe4C configuration with a long-period arrangement of interstitial carbon atoms along the c-direction. (25)Krauss (95) has suggested that this ordering be designated the A3 substage of aging. Such secondary carbon ordering during aging is further addressed in Section 4.2.3. 4.2.2. Atom -probe field-ion microscopy Information on the fine-scale fluctuations of carbon concentration due to carbon clustering has been obtained recently by APFIM. Miller, Smith, and coworkers °~-33) examined Fe-Ni-0.4C martensites and, surprisingly, found regions of carbon enrichment (up to 7.8 at. %) in as-quenched virgin martensite. Aging at room temperature and at 40°C resulted in carbon-enriched regions (which tend to image darkly in the FIM) with compositions approaching 10 at. % C. At the same time, the carbon concentration in the 'matrix' regions decreased. More recently APFIM analysis of Fe-15Ni-IC martensite has been performed by actually probing along the (011) directions (the closest low-index direction to the (023) modulations in this alloy)3 "'34) Composition profiles obtained from virgin martensite and after aging 1580 hr at room temperature are presented in Fig. 21. While the profile for virgin martensite (Fig. 21(a)) exhibits no systematic variation in carbon concentration, the profile obtained

250

PROGRESS

IN MATERIALS

SCIENCE

0 Kusunokl & Naoakura •

Taylor et al.

_E

° e

1

0 0

I 0.5

I I

I 1.5

I 2

2.5

Carbon Content, wt pet FIG. 18. Initial modulation wavelength in ferrous martensites vs carbon content. (",26)

after aging (Fig. 21(b)) indicates regular variations in carbon concentration with a wavelength of about 30 atomic layers (one layer corresponds to approximately 50 ions), showing that the modulated structures which develop in aged martensites consist of alternating highcarbon and low-carbon bands. The average peak amplitude finally reached is about 11 at. % C, and the carbon concentration in the low-carbon regions is estimated to be about 0.2 at. %. Atom-probe results for a series of room-temperature aging treatments on Fe-15Ni-IC martensite are summarized in Fig. 22, in which the minimum and maximum compositions (averaged over 100 ions) are plotted as a function of aging time. Each solid point is the averaged composition of either the high-carbon or the low-carbon regions; the bars on the data points represent the ranges of the extreme values. The composition of the high-carbon regions initially rises rapidly and levels off at about 11 at. % C, corresponding to a carbon concentration of approximately FesC. The apparently continuous increase in composition amplitude is significant, and is very suggestive of a classical spinodal decomposition mechanism in which the composition of a new phase gradually attains its equilibrium (or metastable equilibrium) value. Additionally, the fine-scale, aligned modulated structure in aged ferrous martensites is typical of spinodally evolved structures found in elastically anisotropic alloys. Further atom-probe studies on aged Fe-25Ni-0.2C, (H'34) Fe-25Ni-0.4C, (13) and Fe1 . 8 5 C (11'34) martensites showed about 10-11 at. % C in the high-carbon regions of the modulated structure in these alloys, in good agreement with the above findings for the spinodally decomposed Fe-15Ni-1C martensite, as would be expected for compositions lying within the spinodal gap.

251

A G I N G OF F E R R O U S MARTENSITES

/

14 O Fe-28NI-0.40(TEM)

C] Fo-18NI-10(TEM) 12

• Fe-18NI-1G(APFIM)

10

O

t,-

O

:E

O

4

O

OCD 0

" o

uP o

0

I

0.1

I

0oo

[3

illllll

I

1

I I .....

I

........

10

I

100

,

.......

I

1000

. . . . . . .

10000

Time, h FIG. 19. Modulation wavelength vs time at room temperature for initially-virgin Fe--25Ni-0.4C and Fe-15Ni-IC martensites/t~ Curves represent fit of eq. 6 in the modulation-coarsening regime.

The rather large difference in carbon concentration between the high- and low-carbon products of spinodal decomposition in aged martensites requires that the strains necessary to maintain their coherency must be substantial. In the precision X-ray work of Chen and Winchell on an Fe-18Ni-I.0C martensite, ~9) a very broad peak from regions exhibiting 'negative tetragonality' (c/a < 1) was detected after aging at temperatures between about 50 and 100°C. This peak was originally attributed to elastically distorted volumes between early transition carbide precipitates. However, it now seems more likely that this peak arises from the severely strained low-carbon regions that evolve during coherent spinodal decomposition of the martensite. In order to assess the temperature dependence of the decomposition behavior, specimens of initially-virgin Fe-15Ni-IC martensite have been examined after aging above room temperature. (11'34)Banded FIM images were again obtained, and dark-imaging areas containing less than 15 at. % C were gradually replaced by regions containing up to 25 at. % C, signifying the onset of transition carbide precipitation, as the aging temperature was increased to 150°C. The observed coexistence of carbides and regions with less than 15at. % C demonstrates that the decomposition process taking place at room temperature precedes the precipitation of carbides at temperatures at least as high as 150°C. This result is consistent with the electrical resistivity behavior presented earlier for this alloy (Fig. 13), in which a resistivity peak associated with carbon-atom clustering (and the development of structural modulations) occurs at temperatures at least as high as 125°C.

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FI6. 20. 'Superlattice' reflections (arrows) in electron diffraction patterns obtained from high-carbon martensites: (a) [100] pattern from F e - I . 8 M n - I . 8 C martensite °~ (aged at room temperature for 8 days); (b) [110] pattern from Fe-l.62C martensite C28)(aged at 60°C for 1 hr).

4.2.3. Mrssbauer spectroscopy There is still considerable difference of opinion in the interpretation of complex M6ssbauer spectra from aged martensites. For present purposes, the hyperfine magnetic field is probably the most reliable Mrssbauer parameter; this quantity provides information regarding the local magnetic moment of iron atoms in their various environments. The iron atoms with

AGING OF FERROUS MARTENSITES 15

253

Oh

atl C

OF IOttS

lO000

il

t5

1580 h

atl C

t0000 b

F]G.21. Compositionprofilesobtainedfrominitially-virginFe-15Ni-IC martensiteby atom-probing along (011) directions: (a) virgin martensite;(b) after aging at room temperature for 1580h.(It) near-neighbor carbon atoms (along the c-axis) are severely displaced from their normal BCC lattice positions, and consequently these iron atoms show the smallest hyperfine magnetic fields. While this effect has been generally recognized,(4°) opinions have differed with regard to how the group (or groups) of iron atoms having larger hyperfine magnetic fields should be interpreted. Kaplow and coworkers(36'37) identified three components in M6ssbauer spectra from splat-quenched Fe-l.86C alloys. One component was attributed to iron atoms essentially unperturbed by carbon atoms, while the other two components were present with an apparent population ratio of 1: 3 and a ratio of hyperfine magnetic fields of 3 : 2. From the similarity of these results to those of the two Fe-atom environments in the well-established ),'-Fe4N structure, an isomorphous FeaC structure (Fig. 23) was proposed as a reasonable model for the evolved high-carbon phase in aged Fe42 martensites. (36) Although some electron diffraction data (described in Section 4.2.1) have also been interpreted in terms of such a structure, the carbon concentration of 20 at. % required for this phase is very different from the 11 at. % C measured by APFIM in aged martensites. Subsequent work by Ino e t a/. (39) o n Fe-C martensites containing 1.5-7.2 at. % C identified three M6ssbauer components in aged martensites. One component with a low hyperfine magnetic field attributed to iron atoms having two near-neighbor carbon atoms along the c-axis (which would correspond to 'face' atoms in an ideal FCC Fe4C structure) was extremely weak and was only detected after aging at 90°C. Accordingly, for the high-carbon

254

PROGRESS IN MATERIALS SCIENCE

15 Fe-15Ni-1C

10

¢= O

o

o

0

500

1000

1500

2000

Ageing time, hours

F16. 22. Range of maximum and minimum carbon concentrations (averaged over 100 ions) in initially-virginFe-15Ni-IC martensite vs aging time3n) product of aging, the authors proposed an ordered, non-stoichiometric Fe4Cx (x ,~ 1) structure exhibiting considerable deviation from Fe4C stoichiometry. G6nin (41)has recently performed M6ssbauer spectroscopy on an Fe-9.5 at. % C martensite and decomposed the spectrum of aged martensite into seven components. He proposed a complex 'carbon multiplet' model for the various carbon atom configurations in aged martensite, and concluded that the high-carbon product of martensite aging can be described as a monoclinic structure with an approximate composition of Fe6C. (96) In an effort to reconcile the apparent disparity among M6ssbauer investigations, a reassessment of available results has been carried out (1') based primarily upon published values for the hyperfine magnetic fields for iron atoms in their various environments, as well as on previous data for well-established structures in the F e N system. Five M6ssbauer components were identified for aged Fe42 martensites and an Fet6C2 structure was proposed isomorphous with the ~"-Fel6N2 nitride that is known to precipitate in F e N martensitesfl 7) This structure, which can be derived from the Fe 4C structure through systematic removal of one-half of the carbon atoms, is illustrated in Fig. 23. In this interpretation, the three components originally identified by Kaplow et al. are assigned to the three types of iron sites in the Fel6N 2 structure: (1) those with one carbon near neighbor, (2) those with one carbon next near neighbor, and (3) those with no carbon near neighbors or next near neighbors; these atoms are designated as 'C', 'D', and 'E', respectively, in Fig. 23 (consistent with the components identified by Ino et al.(39)). The two remaining components are assigned to iron atoms essentially unperturbed by carbon atoms and to occasional iron atoms that would have two near-neighbor carbon atoms in an imperfect at" structure (corresponding with the weak M6ssbauer component exhibiting a small hyperfine magnetic field reported by some investigators(39,41)). A recent theoretical study of the relative stability of interstitial phases based on bodycentered iron has been published by Sato et alfl s) Using a molecular-orbital-based discrete

255

A G I N G OF F E R R O U S M A R T E N S I T E S

or" [1UUJ a, a"

tnuj F,4c

[110] F,4o

Fe atoms •

• 0 •

vaoant 0 o sites C atoms In ~"-FeleC 2 C atoms In Fe4C

Fro. 23. Schematic representation of the Fe4C and Fel6C 2 structures (the lattice correspondence with low-C ferrite is also indicated). The FemsC2 structure isomorphous with ~t"-Fet6N2 is obtained from Fe4C by systematic removal of one-half the carbon atoms. See text for iron-atom designations according to neighboring carbon atoms.

variational cluster method, the 'eigenstrains' defining the local relaxations of iron atoms around nitrogen and carbon atoms were computed by maximization of total bond order. These calculations indicated that the most stable ground-state structure in both F e N and Fe-C systems is the n"-Fe~6N2 structure. Given this finding, as well as (a) the generally similar behavior of carbon and nitrogen in iron, (b) the Fes C composition of the high-carbon product of martensite aging measured by APFIM, and (c) the known intersolubility of carbon and nitrogen in Fe,6(C,N)2 carbonitrides which precipitate in Fe-C-N ferrite as reported by Ferguson and Jack, °9) it seems quite reasonable to expect that a similar structure should form upon aging Fe-C martensites. Henceforth, we may designate the high-carbon product of spinodal decomposition in aged Fe-C martensites as ~t". However, there is no evidence as yet that F e N martensites undergo a spinodal decomposition. 4.3. Thermodynamics of Martensite Aging The thermodynamics of 'Zoner ordering' in Fe-C martensites was treated briefly in Section 2.3. According to the model developed by Khachaturyan<52) based upon pairwise elastic JPMS 36/|-4--I*

256

P R O G R E S S IN M A T E R I A L S SCIENCE

700

eoo

t

0-0 /

// ?o.esI

/

800

0.5

]

400

i~J

800

,t-

eoo

110.75

100

0

-100

I

I

I

I

10

20

80

40

CarbonContent,at.pet

50

FIG. 24. Helmoltz free energy (F) as given by eq. 7 vs carbon concentration for body-centered Fe--C solutions at 25°C. Curves for several values of the Zener-order parameter (q) are shown.

interstitial-atom interaction energies, the Helmoltz free energy, F, of the Zener-ordered Fe--C solution is given by: F(c,~l) = N21(O)c/2 + 3N•2(0)C2q 2 + kTN{2c(1 - q) ln[c(1 - r/)] + c(1 + 2~/) In[c(1 + 2n)]

+ 2[1 - c(1 - r/)] ln[1 - c(1 - q)] + [1 -- c(1 Jr- 2q)] ln[1 -- c(1 + 2q)]},

(7)

in which N is the total number of iron atoms, 2~(0) is an energy characteristic of the disordered solution (calculated to be 91.8 eV for BCC iron(52)), and the other terms have their previous meanings (see Section 2.3). Free-energy curves for several values of the order parameter are plotted in Fig. 24, and attention is drawn to that for fully Zener-ordered (17 = 1) solutions. The second term on the right side of eq. 7 defines the decrease in internal energy due to the ordering and grows rapidly in magnitude as the carbon concentration increases. The result is an inflection in the free-energy curve at a very small carbon concentration (not apparent on the scale of Fig. 24), i.e. spinodal decomposition becomes feasible. These inflections, or 'spinodals', are the loci of O:F/Oe2= O, and (differentiating eq. 7 twice) are given by: [ 2(l-q), 622(0)q 2 + k T c[l - c(1 - q)]

1+2~/ 1 c[1 - c(1 + 2q)] = 0,

(8)

257

A G I N G OF F E R R O U S M A R T E N S I T E S 0.5 0.4 0.3 0.2 0.1 0

co

0

o

-0.1

-0.2 -0.3 -0.4 -0,6 -0.4

-0.$

-0.2

-0.1

0

0.1

0.2

0.3

0.4

kOlOao/2rr FIG. 25. Locus of wavevectors in the [100] zone which are spontaneously amplified during the initial decomposition of Zener-ordered body-centered Fe-C solid-solutions at 25°C (52) (contours for three carbon contents are shown). The axes are numbered in units of dimensionless wavenumber, where ko~o and k00~ are reciprocal lattice vectors and a0 is the lattice constant for BCC Fe.

in which the equilibrium order parameter (which satisfies the condition aF/~rl = 0) was defined by eq. 5. At room temperature, an inflection actually occurs at a composition of about 0.2 at. % C. The overall shape of the free-energy curves in Fig. 24 indicates that the equilibrium two-phase state that results from the spinodal decomposition is very low-C iron and the stoichiometric compound FeC. Khachaturyan extended his thermodynamic analysis to calculate the wavevectors which are spontaneously amplified at the onset of spinodal decomposition. The locus of such wavevectors in the [100] zone at 25°C is plotted for carbon contents of 0.8, 1.0, and 3.0 at. % in Fig. 25; these contours define the angular dependence of the minimum spontaneously amplified wavelength. The overall shape of the contours (as determined by the elastic constants for BCC iron) and the increase in wavenumber (i.e. decrease in wavelength) with increasing carbon content are in qualitative agreement with experiment. In particular, the calculations give wavelength minima along directions near (023), in excellent agreement with experimental results on the modulated structure in initially-virgin Fe--Ni-C martensites. While the Khachaturyan model predicts a spinodal reaction with the correct habit, it none the less has some shortcomings. Foremost among these is the assumption that the equilibrium ground state is a mixture of essentially pure BCC iron and the compound FeC. The Fe4C or ct"-Fe~6C~ structures, for which longer-range atomic interactions are important in determining structural stability, are not treated. Also, the calculated minimum spinodal wavelengths generally come out shorter than those measured experimentally. For example,

258

P R O G R E S S IN MATERIALS SCIENCE

~ ~

\

disordered,, \

-50

Zener ordered

@

3

>, -100

\\

®

m ® @

~"""oooordered

\\

-15o

-200

0

0.05

0.1

0.15

0.2

0.26

Atom Fraction C FIG. 26. Schematic free-energy curves for disordered, Zener-ordered, and ~t"-ordered body-centered Fe-C solutions.

the calculated minimum wavelength for a carbon content of 3.0 at. % (0.66 wt %) is less than 1 nm (Fig. 25), whereas the wavelength measured experimentally for this carbon content is actually about 1.5 nm (Fig. 18). A picture of phase relations in the Fe-C system more consistent with the most recent experimental results ~H~is illustrated by the free-energy curves in Fig. 26. The curve for the disordered (cubic) solution represents the free-energy of mixing function derived by Kaufman and Nesor tl°°) for BCC Fe-C alloys and has positive curvature over the entire composition range. The curve for the Zener-ordered solution was obtained from the Kaufman-Nesor function by adding the free-energy change for ordering, as prescribed by the Khachaturyan model. This curve has positive curvature initially (not evident on the scale of Fig. 26), but quickly develops negative curvature, reaching a minimum at the stoichiometric FeC composition. A more appropriate total free-energy function is illustrated schematically by the continuous curve in Fig. 26, reflecting a secondary ordering component which stabilizes the 0t" structure. The latter hypothetical ordered free-energy curve contains two inflections, denoted by X] and X 2, which define the compositional limits between which spinodal decomposition can occur. We note that the reaction is predicted to proceed via a 'conditional spinodal '0°') mechanism since Zener-order is required to produce the initial compositional instability. The ~" composition of the high-carbon regions that ultimately develop appears to result from the secondary ordering imposed by longer-range carbon-carbon repulsions. These two corn-

AGING OF FERROUS MARTENSITES

259

ponents of ordering are consistent with a metastable phase diagram similar in form to that of the Fe-AI system(~°~). The room-temperature phase boundaries (which atom-probe measurements would actually place at 0.2 and 11 at. % C) are denoted by X~ and X~ as indicated by the common-tangent construction. A quantitative description of the thermodynamics of this system could be obtained from a synthesis of the models of Khachaturyan ~52) and Sato et al. ~9s) A full description of the coherent two-phase state would also prescribe elastic energy contributions as treated by Cahn and Larch6 (1°2), but is beyond the scope of the current analysis. The relatively large coherency strains in this system may offer a fertile experimental approach for testing various aspects of coherent equilibria.

5. MECHANICAL BEHAVIOR DURING MARTENSITEAGING

5.1. Strength Properties Although there is a voluminous literature on the mechanical behavior of ferrous martensites, particularly in connection with hardened and tempered steels, relatively little of that vast scientific and technological attention has been focused on virgin martensites and their subsequent aging. It will be evident from the nature and difficult accessibility of virgin martensites, portrayed in the foregoing sections of this review paper, that the determination of mechanical properties presents special experimental problems in the avoidance of prior aging or autotempering. For such reasons, it is easier to measure strength properties with compressive, rather than tensile, specimens when dealing with virgin martensites. In addition, mechanical properties as typically determined by bulk measurements can be significantly affected by the existence of retained austenite, which is often present in substantial amounts because of the incompleteness of the martensitic transformation in ferrous alloys that happen to be appropriate for forming virgin martensites. Flow-strength values, for instance, have been corrected for the influence of retained austenite by extrapolating measurements on a series of martensite/austenite mixtures (obtained by cooling to selected subambient temperatures) to 100% martensite. An example of this procedure for aged (3 hr at 0°C) initially-virgin Fe-Ni-C martensites is illustrated in Fig. 27. t4) The extrapolated flow-stress values for these aged martensites, as well as for the corresponding unaged martensites, are plotted against carbon concentration in Fig. 28, and show that the strength of martensite depends primarily on the carbon content and secondarily on the aging treatment. ~4) Moreover, the strengthening increment due to aging clearly increases with the carbon content, and may reach the order of a 25% contribution over and above the strength of unaged martensite. This factor is undoubtedly at play, at least to some extent, in the quench-hardening of conventional steels that normally have their martensitic-transformation ranges above room temperature. It should be noted that the flow stresses plotted in Figs 27 and 28 were determined at 0°C and therefore, in the light of the aging kinetics presented in Section 3, the designated 'unaged' state had actually undergone some inadvertent aging before and during the testing, even though the martensites in these Fe-Ni-C alloys were initially virgin when formed at subzero temperatures. This potential complication, which becomes still more troublesome if the testing is carried out at room temperature, was later circumvented by conducting the strength measurements at - 196°Cfl ) The results are summarized in Fig. 29 for two Fe-Ni-C alloys; the concomitant changes in electrical resistivity are included for correlation with the structural changes discussed in Sections 3 and 4.

260

P R O G R E S S IN MATERIALS SCIENCE

2000

pf~f

1800

a

G.

18oo

~E @ 0

~omooMtlon8 1400

WI~,%NI 80.5 28.8 28.8 18.7

1200

# o

wt % C 0.02-O. 28 0.40 -0.82--

1000

m

Qo

8OO

CO o

0oo

n-

400

200

0

10

20

80

40

00

80

70

80

00

100

Volume Percent Mertenelte Fx6. 27. Flow stress in compression at 0.6% plastic strain of initially-virgin Fe-Ni-C martensite--austenite mixtures after aging 3 hr at 0°C; measured at 0°C. (4) The carbon and nickel contents shown produce an Ms temperature of about -35°C.

Starting with virgin martensite, there is a very slight increase in the - 196°C strength as the aging temperature (1 hr isochronal treatments) is raised to the vicinity of - 100 to - 50°C; this change is accompanied by a small decrease in resistivity and can be identified with Regime I where some retained austenite transforms isothermally to martensite. The conspicuous increase in strength on aging above 0°C is a manifestation of the spinodal decomposition in Regime II as denoted by the attendant resistivity peak. It is evident from Fig. 29, however, that the strength and resistivity changes have rather different sensitivities to the underlying compositional amplitude and wavelength variations that characterize the spinodal decomposition. This is not surprising inasmuch as such properties depend intrinsically on quite different solid-state phenomena. The strengthening peak on aging is relatively broad compared to the resistivity peak because aging above approximately 100°C allows Stage I tempering (Regime III) to set in, i.e. e-carbide precipitation (referred to as the T1 stage of tempering) begins to replace the spinodally generated structure by means of a heterogeneous nucleation-and-growth process. °°3) The resulting precipitation hardening evidently 'takes over' from the spinodal strengthening and thereby extends the strengthening range to higher temperatures. In contrast, the resistivity begins to fall off earlier in the spinodal regime because of its sensitivity to coarsening of the compositional modulations, and it decreases even faster when the precipitation stage begins to remove solute carbon atoms from the modulated structure. The high- and low-carbon regions of the compositional modulations evolve coherently in the spinodal process, and the resulting coherency strains can be extremely large. This state

261

A G I N G OF F E R R O U S M A R T E N S I T E S 2200

Fe-NI-C Martensite: Testing Temperature - O'C

2000 1600 1600

:

14oo 1200

O~ 1000 800 o n"

60O 4OO 2O0 0 0

,I

I

I

I

I

I

I

I

I

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0

Carbon Content, wt pet FIG. 28. Flow stress in compression at 0.6% plastic strain vs carbon content of initially-virgin Fe-Ni-C martensites unaged and aged (3 hr at 0°C); measured at 0°C. (4)

of affairs must play a noteworthy role in the strength and brittleness of aged martensites, although these properties cannot be treated in depth at the present time. Nevertheless, one can now understand why tempering in the Stage I carbide-precipitation regime can provide appreciable stress relief while the strength is being effectively maintained. Among the papers on spinodal strengthening in the literature, three treatments are considered germane to the aging of initially-virgin martensites. °°*-~°6) These papers focus on the interaction between dislocations and periodic strain fields in relation to the amplitude and wavelength of compositional modulations. Two such derived relationships are as follows (simplified for present purposes): Az = K1A2(2/F), due to Cahn (1°4) A'r = K2AS/3(A/F) 2/3, due to Ardell (1°5)

(9) (10)

where Az is the increment of critical resolved shear stress attributable to the periodic strain, A is the compositional amplitude, 2 is the wavelength, F is the dislocation-line tension, and K 1 and K2 are constants. In both equations, the spinodal strengthening is seen to increase with increasing amplitude and wavelength, but with decreasing line tension. The latter two effects appear to arise because dislocation-line flexibility is important in the general case of mixed dislocations. With increasing 2/F, segments of the dislocations are more likely to locate the energy troughs in the periodic structure, thus favoring greater resistance to motion.

262

P R O G R E S S IN M A T E R I A L S S C I E N C E 32

IFe-NI-C Martenllltee I

I

so f.l E

elm-O.4OO..mlw, ......

I

.lli.ll

2e

) ~NI-O.~'ac J Q ..... . r e . e e,,,

'~II

24 = n-

"i

2000

'i',,e

~ 000 0

1800

|

1400

"'""n 2o r~

" ' I I . . . . . "0

le

14NI-0,10C

03 1:~00 O

,rr

1000 -200

I

I

I

I

I

,,

I

-lOO o lOO 200 $oo 40o A g i n g / T e m p e r i n g T e m p e r a t u r e , *C

FIG. 29. Flow stress in compression at 0.6% plastic strain and electrical resistivity o f 0.26 and 0.40 wt % C initially-virgin Fe--Ni--C martensites; measured at - 1 9 6 ° C after aging/tempering I hr at indicated temperatures. (9)

The circumstance that wavelength coarsening as well as amplitude development contributes to spinodal strengthening suggests that, in accordance with the time dependences of amplitude and wavelength in Figs 19 and 22 (Section 4), the strengthening of the initially-virgin martensites may undergo two separate, but overlapping, stages vs time of aging. Attention is given to this possibility in Section 5.3 where the microhardness of individual martensitic plates is measured as a function of aging time. 5.2. Ductility Changes An early paper due to Pietik/iinen°°7) offers convincing evidence that, just as initially-virgin martensites undergo strengthening on aging, they also become much more brittle. In fact, by comparison, virgin martensite is surprisingly ductile, as shown by the tension-test data in Fig. 30. (1°7) Here, the property changes of interest, as measured at -65°C, include the effects of both aging and tempering. The strengthening and accompanying embrittlement that take place during aging are quite evident. The true fracture stress of the virgin martensite is particularly high and reflects the very large reduction of area at fracture. These tensile specimens contained about 10% retained austenite prior to testing, possibly enhancing the ductility of the virgin martensite somewhat, but this is not likely to have obscured two key points that should be emphasized here: (1) virgin martensite is ductile, even in tension; and (2) this ductility is essentially lost on aging, when the spinodal decomposition occurs.

263

A G I N G OF F E R R O U S M A R T E N S I T E S 2000

1000

cO

v

1800 el Q"

1700

2000

:f

w t

B o ft.

18oo 1800 ®

1600

¢/)

140o

2

1200

g

-SO

1o0o

-2o ;~ -10

E

rr

-0

80 50

-~ 40 ~

30

0

20

-~

lO 0 -10o

=

0

10o

200

:SO0

Aging/Tempering Temperature, oC Fie. 30. Tensile properties of an Fe-24Ni-I.6Si-0.32C initially-virgin martensitic alloy (containing 10% retained austenite); measured at - 6 5 ° C after aging/tempering 1 hr at indicated temperatures. Numbers of microcracks shown were encountered after tensile testing, o°7)

The propensity to microcracking of the martensite as a result of tensile testing increases sharply during aging, as depicted in Fig, 30. These cracks were found to initiate at martensitic interfaces, often at the point of contact by another martensitic unit, and then propagate either across the martensite or along the interfaces. However, no such cracks were detected in the virgin martensite. One may safely conclude that the reported microcracking was caused by embrittlement of the martensite on aging, in conjunction with the operative stresses. Pietik/iinen and his colleagues have continued to investigate the interplay of embrittlement and microcracking of aged martensites, °°8'1°9) all generally confirming the earlier findings. In Ref. 108, virgin martensites were metallographically polished at liquid nitrogen temperature and then electrolytically etched at - 6 0 ° C in order to reveal the presence or absence of microcracks without mechanical-property testing. No microcracking of the martensite was found in the virgin condition, whereas microcracking did ensue merely by aging at room temperature. In this case, it was the internal stresses induced by the martensitic transformation itself, together with the loss of ductility on aging, that led to the microcracking. Another novel experiment involved a complex sequence of subzero cooling and aging treatments to yield a mixture of distinguishable virgin and aged martensites; only the latter exhibited any microcracks, o°s)

264

PROGRESS

IN M A T E R I A L S

SCIENCE

By virtue of the ductility of virgin martensite, it can be plastically deformed to a substantial degree before aging. (]2) When this is done with an Fe-25Ni-0.7C alloy by rolling at -60°C, the electrical resistivity peak on subsequent aging at room temperature decreases, and almost vanishes with 60% prior cold deformation. This interesting effect has been linked to a reduction in the tetragonality of the virgin martensite due to shifting of carbon atoms from their initial Zener-ordered c-sites to a- and b-sites, as a result of the low-temperature plastic deformation312) It may be inferred that the cold-deformed martensite becomes essentially cubic, and then spinodal decomposition does not ensue during aging. This intriguing possibility warrants further detailed study with respect to the accompanying structural and property changes.

5.3. Microhardness Tests In order to monitor indications of strength changes more closely during the aging of initially-virgin martensites, while avoiding possible second-order contributions from retained austenite, Hartfield (]3) conducted microhardness tests on individual martensitic plates in an Fe-13Ni-1C alloy, using a 25-g load on a diamond pyramid indentor. Such indentations were performed at room temperature, requiring about 10 min after each aging treatment. Figure 31 offers preliminary results for room-temperature aging; here the time taken for the room-temperature hardness indentations is of minor consequence. At least 50 microhardness readings were averaged for each aging time. Hardness values softer than one standard deviation below the average were discarded, on the supposition that such indentations were probably affected by adjacent austenite. Even so, as indicated in Fig. 31, considerable scatter remained, undoubtedly reflecting the different orientations and sizes of the martensitic plates being indented. The hardness trend on aging shown by Fig. 31 is roughly consistent with the strength changes discussed in Section 5.1; after aging for only 1 h at room temperature, the initially-virgin martensite is still quite soft compared to the aged hardness peak. However, 10oo I Fe-13Ni-1C Msrtenlllte]

850 -r O. a

oe¢B 3:

/

900 880 8o0 750 70o

8BO 800

I

o.1

I

I llllll

I

1

I

I .....

I

........

10

I

100

........

I

1000

........

10000

Tlmo, h F1G. 31. Microhardness at room temperature of individual initially-virgin Fe--13Ni-IC martensitic plates vs time of aging at room temperature; bars denote plus-and-minus one standard deviation of the observed microhardness values, o3)

AGING OF FERROUS MARTENSITES

265

these microhardness measurements reveal some interesting effects, not shown by the flowstress observations: (1) During the aging period leading up to the spinodal hardness peak, there is some hint of two substages which may correspond to a subtle differentiation between compositional clustering and subsequent modulation development. This sequence has been previously referred to as substages A1 and A2. (7°) On the other hand, the timing of the microhardness substages in Fig. 31 is found to correlate, respectively, with the amplitude build-up and modulation coarsening in Figs 19 and 22 (Section 4), and both of these structural changes contribute to spinodal strengthening as noted in Section 5.1. Hence, it is now worth considering that the substages of aging A1 and A2 may be manifestations of the time dependences of amplitude development and wavelength increase rather than of clustering and modulation formation. As it happens, the available experimental evidence is currently too limited for settling this difference in interpretation. (2) At later aging times in Fig. 31, there is a clear separation between two microhardness peaks where Regimes II and III overlap. This distinction is not apparent in the flow-stress measurements reported up to now (Sections 5.1 and 5.2), possibly for lack of sufficient strength testing vs time of aging at appropriate temperatures. The distinct drop in microhardness between the two peaks has not been previously encountered. Conceivably, it could result from the recently explained displacive nature of e-carbide precipitation in Stage I tempering (T1) of Fe-Ni-C martensites.(1°3) In other words, a form of transformation plasticity may be operative here in which a displacive structural change is deformation-induced by the microhardness indentation, thus interjecting a softening component under special conditions. These preliminary findings suggest that much more detailed microhardness testing could prove quite fruitful for tracing the stages of mechanical-property changes during aging and tempering of initially-virgin martensites, especially if such measurements were conducted at subzero temperatures in order to avoid uncontrolled aging. This approach to the martensite aging process would have the advantage of permitting numerous measurements at closely selected aging times and temperatures.

5.4. lnternal Friction Internal-friction measurements have been employed in recent years to study the aging of initially-virgin martensitesf 1°-H3) but such property changes are still extremely difficult to interpret. An example is given in Fig. 32* for a highly dislocated lenticular (Fe-20Ni-0.7C) martensite and an internally twinned thin-plate (Fe-20Ni-I.2C) martensite. The plotted data were obtained on heating at about I°C per min starting at -190°C with virgin martensite and retained austenite. The difference in the general level of the two curves is due to the difference in the amounts of martensite formed on cooling to - 190°C: 80% in alloy A and 27% in alloy B. It appears, at least tentatively, that both alloys exhibit variations of the same internal-friction phenomena, denoting structural changes as a function of rising temperature on aging from the initially-virgin state. The first peak (or stage) in Fig. 32 occurring at approximately - 170°C is associated with Regime I, i.e. the transformation of some retained austenite on heating. Evidently, this transformation is smaller in extent for the Fe-20Ni-I.2C alloy because of its higher carbon content and, hence, greater austenitic stability. The peaks setting in at higher temperatures, *Figure 32 was kindlysuppliedby Prof. J. Pietikiiinenof Helsinki Universityof Technology,and will appear in the doctoral thesis of Mr Y. Liu. These curves are a later version of their previouslypublished results.(~3~

266

P R O G R E S S IN M A T E R I A L S SCIENCE 0.01

0.000-

~

N

alloy A Fe-2ONI-O7C

0.000

f-

0.007

-

0.000-

o ---~ 0.006-

,,r o

.I:: o.0o4-

e 0.008-

/ /

0.002-

0.001

alloy B F.-2ONI-1.2C

~

-

0 -228

-2()0

"170

"1;0

"125

"100

"70

Temperature,

"00

"20

0

ill

0'0

70

°C

FIG. 32. Internal friction of initially-virgin Fe-20Ni-0.7C (alloy A) and Fe-20Ni-I.2C (alloy B) martensitic alloys on heating at a rate of about l°C/min from -190°C. 80% martensite in (A), lenticular and highly dislocated; 27% martensite in (B), thin-plate and internally twinned, (Private communication, J. Pietik/iinen and Y. Liu.)

ranging from about - 100 to -75°C, probably reflect the spinodal decomposition in Regime II, with the differences in magnitudes between alloys A and B being attributable to the large difference in martensite contentsl Other published internal-friction vs temperature curves for dislocated lenticular martensites (113) reveal double peaks in Regime II, suggesting that the spinodal decomposition may be taking place in two substages. As expected, all these internal-friction manifestations are irreversible; they do not appear during a second heating.(113) It must be emphasized that reported internal-friction phenomena are very hard to unravel because of possible influences arising from many types of interfacial and dislocation motions, made even more complicated by uncertain degrees of carbon pinning. Much more has to be learned about the operative internal-friction mechanisms in order for this interesting property to shed new light on the aging of martensite.

6. CLOSURE

A pictorial summary of the aging of initially-virgin martensites, as developed in this survey, is presented in Fig. 33. The designated terminology is intended to highlight the main structural changes in relation to the accompanying changes in electrical resistivity and mechanical strength. The principal phenomenon of aging takes place in Regime II, where the martensite

267

A G I N G OF F E R R O U S M A R T E N S I T E S

- R E G I M E II AGING (8plnodal D e c o m p o s i t i o n )

A1

A2 (A3)

Increasing Amplltuds

4.0

~ REGIME III - TEMPERING (Carbide Precipitation)

T1,T3

Increasing Wavelength

l w

>

roll=

M O D U L A T I O N S

b3 G) (l)

imm

n" m

Clusters

kKi] bL~-i]

P

¢I O

i m

t,. 4,,o

O m

U.l In

O A

V (o b3 (I) tm

(/)

CT

j= O U.. m

T i m e or T e m p e r a t u r e FIG. 33. Summary of the structural changes that occur during aging and tempering of initially-virgin martensites, in relation to the trends in electrical resistivity (p) and flow stress (a). Aging is identified with spinodal decomposition of the martensite, which gives rise to the Regime II resistivity peak as well as an increase in strength. The respective increases in amplitude (Ac) and wavelength (2) of the carbon-concentration modulations that develop coherently are designated as substages AI and A2; possible secondary ordering of the carbon atoms constitutes substage A3. Alternatively, substages A 1 and A2 have also been associated with the formation of randomly distributed carbon clusters and subsequent evolution of the modulated structure by stress-induced alignment of these clusters. Carbide precipitation--e-carbide in Stage T1 and cementite in Stage T3--marks the end of aging and the subsequent course of tempering. Mechanical strength passes through a maximum near the onset of T1, while resistivity continues to decrease in Regime III. Structural changes are illustrated by electron micrographs obtained during the aging of initially-virgin Fe-Ni-C martensites ~'°) (magnifications for these micrographs are all the same).

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decomposes spinodally into a modulated structure of coherent high- and low-carbon regions. Regime I is not shown in this diagram because it is associated with retained-austenite transformation, and is not a part of the martensite-aging process p e r se. On the other hand, Regime III is included here inasmuch as it constitutes the end of aging as well as the onset of the subsequent stages of tempering--in particular, e-carbide precipitation in Stage T1 and cementite precipitation in Stage T3. The high-carbon regions produced by the spinodal decomposition approach the composition Fel6C2 which appears to be quite similar to the well-known ct"-Fel6N2 phase in the F e N system. Coherency with the adjacent low-carbon regions leads to relatively large elastic strains, thereby contributing to the brittleness of aged martensite and also setting the stage for the displacive e-carbide precipitation in Regime III. The unusual state of coherency strains in this system may offer a fertile approach for testing various aspects of coherent equilibria and for modeling the underlying thermodynamics. Carbon-atom segregations to defects are not necessarily ruled out, but the indications are that they do not play a substantive role in the aging of initially-virgin martensites.

ACKNOWLEDGEMENTS

The authors wish to express their appreciation to Susan E. Hartfield (now Mrs Susan E. Hartfield-Wiinsch and a doctoral candidate at the University of Michigan) for permission to include her unpublished microhardness data on which Fig. 31 is based. These findings are taken from her master's thesis on 'The Effect of Aging on the Mechanical Behavior of Fe-Ni-C Martensites', MIT (February, 1988). Special thanks are also due to Prof. J. Pietik/iinen of Helsinki University of Technology for his helpful communications on internal-friction measurements and his permission to use the data in Fig. 32 obtained from the ongoing research of doctoral candidate Y. Liu.

REFERENCES I. P. G. WINCHELL, The structure and mechanical properties of iron-nickel-carbon martensites, PhD Thesis, Massachusetts Institute of Technology (1958). 2. H.W. KING and S. G. GLOVER,A resistometric study of the first stage of tempering in plain carbon steels, J. Iron and Steel Inst. 193, 123 (1959). 3. H. W. KING and S. G. GLOVER,Influence of alloying elements on the first stage of tempering in high-carbon steels, J. Iron and Steel Inst. 196, 281 (1960). 4. P. G. WINCHELLand M. COHEN, The strength of martensite, Trans. A S M 55, 347 (1962). 5. V. KERLINSand C. ALTSTETTER,Kinetics of the initial stage of decomposition of low M s iron-nickel-carbon martensites, Trans. A I M E 227, 94 (1963). 6. G. R. SPEICH, Tempering of low-carbon martensite, Trans. A I M E 245, 2553 (1969). 7. A. M. SHERMAN, The structure and aging of martensite, PhD Thesis, Massachusetts Institute of Technology (1972). 8. A. M. SHERMAN,G. T. ELDISand M. COHEN,The aging and tempering of Fe-Ni-C martensites, Metall. Trans. .4 14A, 995 (1983). 9. G.T. ELDISand M. COHEN,Strength of initially virgin martensites at - 196°C after aging and tempering, Metall. Trans. A 14A, 1007 (1983). 10. K. A. TAYLOR,.4ging phenomena in ferrous martensites, ScD Thesis, Massachusetts Institute of Technology (1985). 11. K. A. TAYLOR, L. CHANG, G. B. OLSON, G. D. W. SMITH, M. COHEN and J. B. VANDERSANDE, Spinodal decomposition during aging of Fe--Ni~ martensites, Metall. Trans. A 20A, 2717 (1989). 12. K. ULLAKtO, M. NmmN'ENand J. PmTxr,~r~EN, Effects of low temperature deformation on the lattice structure and carbon distribution in Fe--Ni~2 martensites, Proc. lnt. Conf. on Martensitic Transformations--1989 (1COMAT-89), p. 191, Materials Science Forum, 56--58 (1990). 13. S. E. HARTFmLD, The effect o f aging on the mechanical behavior of F e - N i - C martensites, SM Thesis, Massachusetts Institute of Technology (1988).

A G I N G OF F E R R O U S M A R T E N S I T E S

269

14. C. S. ROBERTS,B. L. AVERBACHand M. CONEN, The mechanism and kinetics of the first stage of tempering, Trans. A S M 45, 576 (1953). 15. H. GERDmN, A dilatometric examination of the kinetics of the initial reaction on isothermal annealing of martensite (in German), Arch. Eisenh. 30, 673 (1959). 16. E. J. MI3~rE~,mIJERand F. C. VAN DOORN, Heat effects of preprecipitation stages on tempering of carbon martensites, Metall. Trans. A 14A, 976 (1983). 17. E. J, MITTE~,mUER,LIU CX-mN~,P. J. VAN DER SCHAAE,C. M. BRAKMANand B. M. KOREVAAR,Analysis of nonisothermal transformation kinetics; tempering of iron-carbon and iron-nitrogen martensites, Metall. Trans. A 19A, 925 (1988). 18. LIu CrmNG, C. M. BRAKMAN,B. M. KOREVAARand E. J. MITTEM~LmR,The tempering of iron-carbon martensite; dilatometric and calorimetric analysis, Metall. Trans. A 19A, 2415 (1988). 19. P. C. CHiN and P. G. WINCI~LL, Martensite lattice changes during tempering, Metall. Trans. A l l A , 1333 (1980). 20. M. HAYAKAWA,M. TANIGAMIand M. OKA, Low temperature aging of the freshly formed martensite in an Fe-Ni-C alloy, Metall. Trans. A 16A, 1745 (1985). 21. I. R. ENTIN,V. A. SOM~NKOVand S. SH. SHIL'SHTEIN,Atomic and magnetic structure of martensite in hardened steel, Soy. Phys. Dokl. 17, 1021 (1973). 22. V. I. IzoTov and L. M. UTEVSKIY,The structure of the martensite crystals of high-carbon steel, Phys. Met. Metallogr. 2~1), 86 (1968). 23. G. V. KURDVUMOV,A. V. SUYAZOVand M. P. USIKOV, The occurrence of short-range order in martensite in high-carbon manganese and chromium steels, Soy. Phys. Dokl. 15, 1067 (1971). 24. A. K. SACI-IDEV,The substructure and aging of high-carbon ferrous martensites, ScD Thesis, Massachusetts Institute of Technology (1977). 25. S. NAGAKURAand M. TOYOSHIMA,Crystal structure and morphology of the ordered phase in iron-carbon martensite, Trans. J I M 20, 100 (1979). 26. M. KUSUNOK1 and S. NAGAKURA, Modulated structure of iron-carbon martensite studied by electron microscopy and diffraction, J. appl. Crystallogr. 14, 329 (1981). 27. Y. TANAKAand K. S~nMIZU,Carbide formation upon tempering at low temperatures in Fe-Mn-C alloys, Trans. J I M 22, 779 (1981). 28. S. NAGAKURA,Y. HIROTSU, M. KUSUNOKI,T. SUZUKI and Y. NAKAMURA, Crystallographic study of the tempering of martensitic carbon steel by electron microscopy and diffraction, Metall. Trans. A 14A, 1025 (1983). 29. Y. NAKAMURAand S. NAGAKURA,UHV-HR-1 MV electron microscope observation of modulated structure and precipitated carbide in tempered iron-carbon martensite, Proc. 7th Int. Conf. High Voltage Electron Microsc., p. 221 (1983). 30. B. P. J. SANDVIKand C. M. WAYMAN,Direct observations of carbon clusters in a high-carbon martensitic steel, Metallography 16, 429 (1983). 31. M. K. MILLER,P. A. BEAVENand G. D. W. SMITH, A study of the early stages of tempering of iron-carbon martensites by atom probe field ion microscopy, Metall. Trans. A 12A, 1197 (1981). 32. M. K. MILLER,P. A. BEAVEN,G. D. W. SMITH and S. S. BRENNER, Proc. Int. Conf. on Solid--Solid Phase Transformations (edited by H. I. Aaronson, D. E. Laughlin, R. F. Sekerka and C. M. Wayman), p. 863, TMS-AIME, Warrendale, PA (1982). 33. M. K. MILLER, P. A. BEAVEN,S. S. BRENNERand G. D. W. SMITH, An atom probe study of the aging of iron-nickel-carbon martensite, Metall. Trans. A 14A, 1021 0983). 34. L. CHANG, Tempering of martensite, PhD Thesis, University of Oxford, Oxford (1985). 35. J. R. GENIN and P. A. FLINN, M6ssbauer effect study of the clustering of carbon atoms during the room-temperature aging of iron-carbon martensite, Trans. A I M E 242, 1419 (1968). 36. W. K. CHOOand RoY KAPLOW,M6ssbauer measurements on the aging of iron-carbon martensite, Acta Metall. 21, 725 (1973). 37. N. DEC1LISTOFAROand R. KAPLOW, Interstitial atom configurations in stable and metastable F e N and Fe-C solid solutions, Metall. Trans. A gA, 35 (1977). 38. V. N. GRIDNEV,V. G. GAVRILYUK,V. V. NEMOSHKALENKO,YU. A. POLUSHKINand O. N. RAZUMOV,M6ssbauer study of the structure of iron-carbon martensite, Phys. Met. Metallogr. 43(3), 109 (1977). 39. H. INO,T. ITO, S. NASUand U. GONSER,A study of interstitial atom configuration in fresh and aged iron-carbon martensite by M6ssbauer spectroscopy, Acta Metall. 30, 9 (1982). 40. R. KAPLOW,M. RON and N, DECRISTOEARO,M6ssbauer effect studies of tempered martensite, Metall. Trans. A 14A, 1135 (1983). 41. J.-M. R. GENIN, The clustering and coarsening of carbon multiplets during the aging of martensite from M6ssbauer spectroscopy: The preprecipitation stage of epsilon carbide, Metall. Trans. A ISA, 1371 (1987). 42. V. ROSATO,Comparative behavior of carbon in B.C.C. and F.C.C. iron, Acta Metall. 37, 2759 (1989). 43. H. I. AARONSON,H. A. DO,nAN and G. M. POUND, Thermodynamics of the austenite-~proeutectoid ferrite transformation. I, Fe-C alloys, Trans. A I M E 236, 753 (1966). 44. F. E. FUJITA,T. MORIYA and H. INO, Low-temperature phase and tetrahedral site of carbon in iron-carbon martensite, Proc. Int. Conf. Sci. Technol, Iron and Steel (1970), p. 1273, Iron and Steel Inst. Japan (1971).

270

P R O G R E S S IN M A T E R I A L S SCIENCE

45. F. E. FUJITA, On the lattice deformation in martensite transformation of steel, Metall. Trans. A 8A, 1727 (1977). 46. L. I. LYSAKand B. I. NIKOLIN,Position of carbon atoms in the lattices of e', e, and K' martensite, Phys. Met. Metallogr. 22(5), 82 (1966). 47. L. I. LYSAK,YA. N. VOVKand Yu. M. POLISHCHUK,Influence of carbon concentration on the crystal structure of r ' martvnsite, Phys. Met. Metallogr. 23(53, 123 (1967). 48. L. I. LYSAK, S. P. KONDRAT'YEVand Yu. M. POLISHCHUK,Dependence of the crystal structure of r'- and ~t-martensites on carbon concentration, Phys. Met. Metallogr. 36(3), 85 (1973). 49. J. W. FLOCKEN, Application of the method of lattice statics to carbon interstitials in g-iron, Phys. Rev. B 4, 1187 (1971). 50. M. COHEN and C. M. WAYMAN,Fundamentals of martensitic reactions, in: Metallurgical Treatises (edited by J. K. Tien and J. F. Elliott), p. 445, TMS-AIME, Warrendale, PA (1981). 51. E. C. BAIN, The nature of martensite, Trans. A I M E 70, 25 (1924). 52. A. G. KHACHA'rURYAN,Theory o f Structural Transformations in Solids, John Wiley, New York (1983). 53. G. V. KURDJUMOVand A. G. KHACHATURYAN,Nature of axial ratio anomalies of the martensite lattice and mechanism of diffusionless ~--,~t transformation, Acta Metall. 23, 1077 (1975). 54. C. S. ROBERTS, Effect of carbon on the volume fractions and lattice parameters of retained austenite and martensite, Trans. A1ME 197, 203 (1953). 55. L. CHENG, B. I~JYrGER, TH. H. DE KEUSER and E. J. MITTEMEIJER,Lattice parameters of iron-carbon and iron-nitrogen martensites and austenites, Scr. Metall. et Mater. 24, 509 (1990). 56. C. ZEt,rElt, Kinetics of the decomposition of austenite, Trans. AIME 167, 550 (1946). 57. C. ZENER, Theory of strain interaction of solute atoms, Phys. Rev. (Set. 2) 74, 639 (1948). 58. S. B. l~N and S. T. WANG, A theoretical analysis of the spinodal decomposition in Fe-C martensite during aging stage of tempering, Metall. Trans. A 19A, 2427 (1988). 59. G. R. SPEICH and W. C. LESLIE,Tempering of steel, Metall. Trans. 3, 1043 (1972). 60. M. KEHOE and P. M. KELLY,The role of carbon in the strength of ferrous martensite, Scr. Metall. 4, 473 (1970). 61. L. A. NORSTROM,On the yield strength of quenched low-carbon lath martensite, Scand. J. Metall. 5, 159 (1976). 62. R. L. PATTERSONand C. M. WAYMAN,The crystallography and growth of partially-twinned martensite plates in Fe-Ni alloys, Acta Metall. 14, 347 (1966). 63. A. L. RoYrnURD and A. G. KHACI~TURYAN, Interstitial atoms and the crystallographic mechanism of martensite transformation in steels, Phys. Met. Metallogr. 30(6), 68 (1970). 64. S. KAJIWARAand T. KIgucHl, On the abnormally large tetragonality of martensite in Fe-Ni~2 alloys, Acta Metall. 39, I123 (1991). 65. M, OKA and C. M. WAYMAN,{ll0} twinning in Bet martensite, Trans. A I M E 242, 337 (1968). 66. M. OgA and C. M. WAYMAN,Electron metallography of the substructure of martensite in high-carbon steels, Trans. A S M 62, 370 (1969). 67. K. A. TAYLOR,G. B. OLSON, M. COHEN and J. B. VANDERSANDE, {011} twinning in Fe-Ni-C martensites, Metall. Trans. A 20A, 2739 (1989). 68. A.G. KHACHATURYAN,A. F. RUMYNINAand G. V. KURI)JUMOV,Axial ratio abnormalities and morphology of martensitic crystals in steels, Proc. Int. Conf. on Martensitic Transformations--1979 (ICOMA T- 79), p. 71, MIT Press, Cambridge, MA (1979). 69. V. YE. DANIL'CHENKO,Martensite twinning in quenched steel, Scr. Metall. 23, 2101 (1989). 70. G. B. OLSONand M. COHEN,Early stages of aging and tempering of ferrous martensites, Metall. Trans. A 14A, 1057 (1983). 71. P. L. ROSSITER,Application of electrical resistivity to investigation of atomic and magnetic microstructures, Metals Forum 8, 204 (1985). 72. P. C. CHEN, B. O. HALL and P. G. WINCHELL,Atomic displacements due to C in Fe-Ni42 martensite, Metall. Trans. A l l A , 1323 (1980). 73. D. W. HOI~MAN and M. COHEN, Static displacements and the electrical resistivity of interstitial alloys, Acta Metall. 21, 1215 (1973). 74. R. P. SMITH, The diffusivity and solubility of carbon in alpha-iron, Trans, A I M E 224, 105 (1962). 75. F. S. BUF~NG3"ON,K. HIRANOand M. COHEN, Self-diffusion in iron, Acta Metall. 9, 434 (1961). 76. N. A. GJOSTEIN,Diffusion, pp. 241-274, ASM, Metals Park, OH (1973). 77. W. A. JOtlNsoN and R. F. MF.HL, Reaction kinetics in processes of nucleation and growth, Trans. A I M E 135, 416 (1939). 78. M. AVRA~,II,Kinetics of phase change. I-III, J. chem. Phys. 7, 1103 (1939); ibid. 8, 212 (1940); ibid. 9, 177 (1941). 79. M. HILLERT, The kinetics of the first stage of tempering, Acta Metall. 7, 653 (1959). 80. N. DECRXSTOFARO,R. KAPLOWand W. S. OW~N, The kinetics of carbon clustering in martensite, Metall. Trans. A 9A, 821 (1978). 81. B. S. LEI~NT and M. COHEN,A dislocation-attraction model for the first stage of tempering, Acta Metall. 4, 469 (1956).

A G I N G OF F E R R O U S M A R T E N S I T E S

271

82. F. G. WILSONand W. S. OWEN, First-stage tempering of low-chromium hypereutectoid steels, J. Iron and Steel Inst. 203, 590 (1965). 83. D. KALISHand M. COHEN, Structural changes and strengthening in the strain tempering of martensite, Mater. Sci. Eng. 6, 156 (1970). 84. M. KUSUNOKI and S. NAGAKURA, Incommensurate direction of the modulated structure in iron-carbon martensite, Trans. JIM 25, 517 (1984). 85. S. KAJIWARA,S. UEHARAand Y. NAKAMURA,Structure images of unaged martensite containing a high carbon content, Phil. Mag. Lett. 60, 147 (1989). 86. P.J. FILLINGI-IAM,H. J. LEAMYand L. TANNER,Simulation of electron transmission images of crystals containing random and periodic arrays of coherency strain centers, Electron Microscopy and Structure of Materials (edited by G. Thomas et al.) p. 163, University of California Press, Berkeley, CA (1972). 87. I. M. LIFSHITZand V. V. SLYOZOV,The kinetics of precipitation from supersaturated solid solutions, J. phys. chem. Solids 19, 35 (1961). 88. C. WAGNER,Theory of aging of precipitates through dissolution and re-precipitation (Ostwald ripening), Z. Elektrochem. 65, 581 (1961). 89. E. P. BUTLER and G. THOMAS, Structure and properties of spinodally decomposed Cu-Ni-Fe alloys, Acta MetalL 18, 347 (1970). 90. D.A. HUSE, Corrections to late-stage behavior in spinodal decomposition: Lifshitz-Slyozov scaling and Monte Carlo simulations, Phys. Rev. B 34, 7845 (1986). 91. R. TORAL, A. CHAKRABARTIand J. D. GtrNTON, Numerical study of the Cahn-Hilliard equation in three dimensions, Phys. Rev. Lett. 60, 2311 (1988). 92. P. HAASEN,The early stages of the decomposition of alloys, Metall. Trans. A 16A, 1173 (1985). 93. J. S. LANGER, M. BAR-ON and H. D. MmLER, New computational method in the theory of spinodal decomposition, Phys. Rev. A 11, 1417 (1975). 94. J. S. LANGER and A. J. SCHWARTZ, Kinetics of nucleation in near-critical fluids, Phys. Rev. A 21, 948 (1980). 95. G. KRAUSS, Tempering and structural change in ferrous martensitic structures, Proc. Int. Conf. on Phase Transformations in Ferrous Alloys (edited by A. R. Marder and J. I. Goldstein) p. 101, TMS-AIME, Warrendale, PA (1984). 96. J.-M. R. GENIN, Discussion of 'spinodal decomposition during aging of Fe-Ni4S martensites' and structure of the Fe6C carbide, Metall. Trans. A 21A, 2083 (1990). 97. K. H. JACK, The occurrence and the crystal structure of ct"-iron nitride; a new type of interstitial alloy formed during tempering of nitrogen-martensite, Proc. R. Soc. (Lond.) 208A, 216 (1951). 98. A. SATO,Y. WATANABEand T. MURA, Octahedral defects in a BCC lattice examined by lattice theory, J. phys. chem. Solids. 49, 529 (1988). 99. P. FERGUSONand K. H. JACK, Quench-aging and strain-aging of nitrogen-ferrite, Heat Treatment '81, p. 158, The Metals Society, London (1983). 100. L. KAUFMANand H. NESOR, Coupled phase diagrams and thermochemical data for transition metal binary systems--IV, CALPHAD 2, 295 (1978). 101. S. M: ALLENand J. W. CAaN, Mechanisms of phase transformations within the miscibility gap of Fe-rich FeA1 alloys, Acta Metall. 24, 425 (1976). 102. J. W. CAHN and F. LARCHE,A simple model for coherent equilibrium, Acta Metall. 32, 1915 (1984). 103. K.A. TAYLOR,G. B. OLSON,M. COHENand J. B. VANDERSANDE,Carbide precipitation during stage I tempering of Fe-Ni42 martensites, Metall. Trans. A 20A, 2749 (1989). 104. J. W. CAHN, Hardening by spinodal decomposition, Acta Metall. 11, 1275 (1963). 105. A. J. ARDELL, Precipitation hardening, Metall. Trans. A 16A, 2131 (1985). 106. W. S. OWEN and M. GRUJICIC, The strength of Fe-Ni~2 aged martensite, Proe. Int. Conf. on Martensitic Transformations--1986 (ICOMAT-86), p. 419, Japan Inst. of Metals (1986). 107. J. PIET1K.Z,INEN, Effects of the aging of martensite on its deformation characteristics and on fracture in Fe Ni-Si-C steel, J. Iron and Steel Inst. 206, 74 (1968). 108. T. VILO and J. PIETIK,g,INEN, Microcracks in F e - N i ~ martensites, Proc. Int. Conf on Martensitic Transformations--1979 (ICOMAT-79), p. 721, MIT Press, Cambridge, MA (1979). 109. J. PIETIIO~INEN,Acoustic emission and microcracking during aging, Proc. Int. Conf. on Martensitic Transformations--1986 (ICOMAT-86), p. 425, Japan Inst. of Metals (1986). 110. P. GREGOIRE,P. PAHUTA,C. DAGBERT,J. GALLAND and L. HYSPECKA, Aging and tempering processes in martensite of an Fe-Niq2 alloy, Proc. Int. Conf. on Martensitic Transformations--1986 (ICOMA T-86), p. 40l, Japan Inst. of Metals (1986). 111. M. KAJATKARIand J. PIETIK~IINEN,Internal friction and deformation behavior of Fe-Ni-C martensite, Proc. Int. Conf. on Martensitic Transformations--1986 (ICOMAT-86), p. 442, Japan Inst. of Metals (1986). 112. C. PRIOULand L. HYSPECKA,Carbon dislocation interaction in virgin Fe-Ni~C martensite, Proc. Int. Conf. on Martensitic Transformations--1986 (ICOMAT-86), p. 337, Japan Inst. of Metals (1986). 113. Y. LIU, K. ULLAKKO,V. GAVRILJUK,V. NADtrrov and J. PtErIIOIINEN,Internal friction behaviour of F e - N i ~ alloys during low temperature ageing, Proc. Int. Conf. on Martensitic Transformations--1989 (ICO MA T-89), p. 207, Materials Science Forum, 56-58 (1990).

272

P R O G R E S S IN M A T E R I A L S S C I E N C E

114. S. NASU, T. TAKANO,F. E. FUJITA, K. TAKANASHI,H. YASUOICAand H. ADACm, M6ssbauer and NMR studies of iron-carbon martensite, Hyperfine Interactions 28, 1071 (1986). 115. T. V. DOUDZE and A. F. SrtEVAK1N, t3C N M R study of carbon distribution in Fe-C, Fe-Ni-C and Fe-Mn-C martensite, Hyperfine Interactions 59, 333 (1990).

NOTE ADDED IN PROOF The technique of nuclear magnetic resonance (NMR) has recently been applied to study the site-occupation of the interstitial carbon atoms in freshly quenched as well as aged martensites. (m'llS) Fujita and coworkers, 014) based on t3C domain-wall enhanced spin-echo spectra from Fe-C alloys (which were probably autotempered), still maintain that some carbon atoms occupy tetrahedral sites in martensite immediately after quenching. In contrast, Dolidze and Shevakinu 15)conclude that all the carbon atoms occupy octahedral sites in initially-virgin Fe-7.SNi-l.6C and Fe-3.6Mn-l.6C martensites. In martensites aged at 50°C both investigations found evidence of carbon-atom clustering in octahedral sites; Fujita and coworkers indicate that the NMR spectra of aged martensite are consistent with the formation of ct", while Dolidze and Shevakin find carbon-atom configurations that are not consistent with this structure.