The thermochemical implications of deformation twinning and martensite transformations

Progressin MaterialsScienceVol. 36, pp. 149-166,1992 Printed in GreatBritain.All rightsreserved.

0079-6425/92$15.00 © 1992PergamonPressplc

THE THERMOCHEMICAL IMPLICATIONS OF DEFORMATION T W I N N I N G A N D MARTENSITE TRANSFORMATIONS John

W. Cahn

Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, M D 20899, U.S.A. Akstraet--Shear transformations, such as deformation twinning, pseudotwinning, and martensites, can result in predictableand reversiblerearrangementsof the local atomic environments and the crystal symmetries. By manipulating the composition and local order of the parent phase, a degree of control can be achieved over the structure of the sheared phase. Unusual crystal structures can be created with interestingimplications for thermochemical,mechanical and physical properties, and alloy theory.

1. INTRODUCTION Thermodynamics began with a study of heat engines, not materials3 ~'2) The Carnot efficiency for the conversion of heat into mechanical work is actually independent of the nature of the working fluid. That measurement of heat added and work done on the working fluid could give valuable insight into the properties, reactions, and equilibria of chemicals, solutions, and other phases, came half a century later. °) Chemical thermodynamics hardly ever considers mechanical work, although chemical engines have a long history. Even the recently developed engines using martensite memory alloys are Carnot engines with a solid working 'fluid'. The shear transformation in these and other alloys seems to be near enough to thermodynamic reversibility that chemical thermodynamic data can be obtained from classical 'engine' data, heat, and work, even about non-equilibrium states. In this paper we explore the chemical thermodynamic information about structures that can be obtained from measurement of mechanical work and heat effects in shear transformations, (4) and whether these non-equilibrium structures can provide information that is useful in the statistical mechanical modeling of alloys. (5) Diffusionless shear transformations, such as the martensite transformations, mechanical twinning, and pseudotwinning, result in a predictable relative motion of the atoms. This relative motion can be described by a homogeneous deformation matrix plus a set of shear waves called shuffles, which taken together lead to a correspondence between the atom positions in the original and final crystal structures. (6) The atomic positions in the martensite, or in the twin or pseudotwin, are determined by the structure and atomic order of the original alloy phase, and the correspondence. This orderly mechanical rearrangement of the atoms creates correlations that would never occur by purely diffusional processes in a phase, and produces structures that can differ from the equilibrium structures in energy and symmetry. The homogeneous deformation that restores lattices and some simple crystal structures to rotated versions of themselves is called deformation twinning. (7-9)For some crystal structures, additional inhomogeneous deformations, called shuffles, are required to restore the structure. 149

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In an alloy the shears always rearrange local configurations of the species and result in a distortion that alters the symmetry3l°) For alloys that have superlattice structures, a deformation that would have restored the disordered structure can result in a change of crystal structure3 H-~3) When there is a clear change in crystal structure, the process has sometimes been called pseudotwinning and even a martensite transformation. Quasitwinning might be more accurate. Pseudotwinning is used by crystallographers with several other meanings3~4) In this paper we will call twinning any deformation that recreates the same phase, ignoring the changes in local order and symmetry that will always occur in any but a fully disordered or ideally ordered version of the alloy. We will call pseudotwinning any deformation of a superlattice that results in a change of the superlattice, but would be a twinning of the disordered version of the crystal. In this way the twinning and pseudotwinning energy changes can be understood in terms of rearrangements of atoms on the basic structure (called 'lattice' in many papers) alone. Martensite transformations are those that result in a change of basic (disordered) crystal structure, and have, in addition, the energy of the change in local arrangements. Diffusive motion is assumed absent during the transformation, and any subsequent diffusion will be treated as a separate phenomenon, often called tempering. Tempering is the diffusional relaxation of the newly formed phase toward equilibrium, and will give insight and additional quantitative evaluation about the non-equilibrium aspects of the sheared phase. The shear processes create unusual initial states (that are as ordered as the original austenites or untwinned crystals but with highly non-equilibrium configurations) for following the kinetics and thermodynamics of the various intermediate states of subsequent ordering processes. In many of these systems, because the sheared state is one of high energy, the reverse shear transformation, giving back the original state, occurs easily and almost reversibly with small reductions in loading. This gives rise to true elastic behavior with large strains that has been called a pseudo-elasticity.°5) In ideal pseudo-elastic behavior of a single crystal, the initial loading is elastic with a constant modulus; at a critical stress the transformation occurs, resulting in a large strain at constant stress. If the stress is reduced below that critical stress, the transformation reverses, and the strain is recovered. In an ideal polycrystalline pseudo-elastic material, this transformation strain is spread over a range of stresses due in part to the variations in resolved stress among the various grains; again the strain is recovered on unloading. In most systems there is a significant difference between the stresses for the transformation and its reverse, giving rise to a dissipation that is measurable from the hysteresis loop. The transformed state is thus only an equilibrium state in the presence of the applied stress, and in the absence of diffusion, slip, or creep. Except for the shear transformations these configurations would be inaccessible for study. We will examine the possible use of these configurations in testing assumptions made in solution models about the energies of various configurations. Such observations could provide stringent tests for the validity of thermodynamic and kinetic models. In this paper we examine the shear transformations that alter the local or long range order of the original crystals by rearranging the atoms in a predictable way to generate alloys with unusual non-equilibrium configurations. We will focus on their thermodynamic properties, obtainable through the measurement of heat and work during the shear, and the consequences of those on the formulation of statistical mechanical calculations and on the mechanical properties of these materials.

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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).

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