PHYSICAL REVIEW B VOLUME 49, NUMBER 13 1 APRIL 1994-I

Dynamics uf ordering in alleys arith modulated phases

Bulbul ChakrabortyThe Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts OM$$

(Received 20 October 1993)

This paper presents a theoretical model for studying the dynamics of ordering in alloys which

exhibit modulated phases. The model is different from the standard time-dependent Ginzburg-Landau description of the evolution of a nonconserved order parameter and resembles the Swift-

Hohenberg model. The early-stage growth kinetics is analyzed and compared to the Cahn-Hilliard

theory of continuous ordering. The sects of nonlinearities on the growth kinetics are discussed

qualitatively and it is shown that the presence of an underlying elastic lattice introduces qualitativelyinteresting efFects. A lattice Hamiltonian capable of describing these effects and suitable for carryingout simulations of the growth kinetics is also constructed.

I. INTRODUCTION

One of the most fascinating problems in statistical me-

chanics is that of growth kinetics, the evolution of or-der following a quench &om an initial high temperaturestate to one below the transition temperature. Theo-retical studies of kinetics have mainly concentrated onunderstanding the behavior of models such as the time-dependent Ginzburg-I andau model or the kinetic Isingmodel. Binary alloys have been viewed as convenientmodel systems for experimental investigation of thesetheoretical predictions. 4' Many alloys, however, exhibitcomplex ordered phases including long-period, modu-lated structures which cannot be described by simpleantiferromagnetic Ising models. ~ The kinetics of orderingin these alloys should be significantly different from theusual models of ordering. These alloys can in turn pro-vide models for both theoretical and experimental studiesof a different class of kinetic phenomena.

This paper presents model partial differential equa-tions for describing growth kinetics in alloys with mod-ulated phases and analyzes the early-stage kinetics. Thekinetics of growth in the strongly nonlinear regime haveto be approached through solutions of the partial differ-

ential equations or through Monte Carlo simulation of adiscrete lattice Hamiltonian. Such a discrete Hamilto-nian, derived &om the same atomistic description thatleads to the partial differential equations, is also pre-sented, and the qualitative aspects of this model are dis-cussed.

In recent years, there has been a concerted effort tounderstand phase stability in metallic alloys based ona realistic description of the "electron glue" responsiblefor metallic cohesion. One such approach, which has re-cently been applied to equilibrium phase transitions inthe Cu-Au alloys with success, is based on the eÃec-

tive medium theory (EMT) of cohesion in metals. o EMTprovides a semiempirical scheme for writing down the en-

ergy of a system of electrons and ions with an arbitraryconfiguration. This is its major advantage over first-principles approaches which have to rely on specific sym-metries of the system such as (a) completely random

or (b) ordered periodic structures. 2 By its very nature,EMT is more approximate than the first-principles ap-proaches; however, it does provide a model Hamiltonianwhich takes into account the metallic character of thesealloys, describes the coupling of positional and configura-tion degrees of &eedom, ' and can deal with size effectsarising Rom the fact that the Ising spins are actuallyatoms which have different "sizes."

The EMT expression for the total energy for a givenconfiguration of ions provides the classical Hamiltoniandescribing the alloys. ' This Hamiltonian is a functionof both the positions of the atoms and their chemi-cal identity. The latter can be represented by an Isingvariable. So far, our studies of alloys have been basedon an EMT Hamiltonian which includes the Ising vari-ables and uniform lattice distortions; i.e. , volume andshape distortions have been included but not volume con-serving phonons. ' From studies of compressible Isingsystems, it is known that the coupling to uniform lat-tice distortions can change the nature of the Ising phasetransition. ' The most striking demonstration of thisis an Ising antiferromagnet on a deformable triangularlattice where the system was found to order, through afirst-order phase transition, into a striped phase accom-panied by a lattice distortion which has the antiferro-magnetic bonds shortened and the ferromagnetic bondslengthened. On a rigid lattice there is no ordering. Ourstudies of the ordering in CuAu, based on EMT with uni-form lattice distortions, showed that including the latticedistortions leads to a first-order transition from the dis-ordered, undistorted lattice to a "layered" phase wherethe face-centered-cubic (fcc) lattice has alternating Cu(spin up) and Au (spin down) planes stacked along the(100) direction and there is a tetragonal distortion (theI lo structure observed experimentallyr") which shortensthe antiferromagnetic bonds and stretches the ferromag-netic bonds. ' This observation implies that the CuAutransition is in the same class as the triangular latticeIsing antiferromagnet coupled to uniform lattice distor-tions. Part of the physics entering the EMT Hamiltonianfor Cu-Au can therefore be understood simply in termsof Ising antiferromagnets on frustrated but deformable

0163-1829/94/49(13)/8608(7)/$06-00 49 8608 1994 The American Physical Society

49 DYNAMICS OF ORDERING IN ALLOYS WITH MODULATED PHASES

lattices. The advantage of EMT is that the model is de-

rived from a microscopic description of a given alloy andaH the alloy-specific parameters are known. However,the mapping of the physics onto a simple model helps inunderstanding the physics of ordering in these alloys.

The other ingredient in EMT which is different fromthe usual Ising model description of alloys is the size ofan atom. This size is defined in EMT as that radius ofa sphere within w'hich each atom is neutral. The size istherefore a configuration-dependent quantity which de-

pends on screening in a metallic environment. It hasbeen argueds that the modulated CuAu II phasei~ is aconsequence of such size effects and it will be shown herethat these size effects lead to Ising models with com-

peting interactions similar to axial next-nearest-neighborIsing (ANNNI) models. The CuAu II phase has a one-dimensional modulation of the order parameter along acubic direction and the period of the modulation, whichvaries as a function of composition, is ten lattice con-stants at the 50-50 composition. There is an alter-nate mechanism which could lead to modulated phasesin metallic alloys, and this is driven by Fermi-surfaceinstabilities. ii EMT, in its present form, cannot describesubtle Fermi-surface effects and may fail to describe mod-ulated phases in alloys where this is the dominant effect.The model of kinetics that will be presented in this pa-per can encompass both classes of modulated-phase al-

loys. The model will be derived for size-effect alloys suchas CuAu and the differences and similarities with Fermi-surface alloys will be discussed.

It should be noted that E[vP] also acts as a Lyapunov

functional, ~&" & 0.

Most descriptions of kinetics in alloys have been basedon the traditional @4 form of X[@].In contrast, our dis-cussion of kinetics will be based upon a X[@]derived fromthe microscopic EMT Hamiltonian.

We use as our prototype alloy the CuAu (50-50) al-

loy which exhibits modulated phases and, in equilibrium,has a first-order phase transition accompanied by a lat-tice distortion. The latter shows the relevance of thecoupling to phonons and uniform lattice distortions andraises questions about the description of metastable andunstable states and the concept of a spinoidal. The pres-ence of two first-order transitions, from the disorderedto the modulated CuAu II phase, followed by the tran-sition &om CuAu II to the zero temperature unmod-ulated CuAu I structure, ' leads to unusual quench-temperature dependence of the kinetics and a Swift-Hohenberg-like description of the early-stage kinetics.

The free-energy functional E[g] for the CuAu alloy hasbeen constructed by numerically calculating the mean-field free energy from the EMT Hamiltonians and we

will only brieBy discuss the procedure and the results inthis paper. The Hamiltonian includes uniform lattice dis-tortions and the Ising degrees of freedom, H((e ), (a, )),where s; are the Ising variable on lattice sites i, and the(e ) are the set of uniform lattice distortions. The par-tition function for this model can be written as

Z= II de B exp —F,e

II. MODEL OF KINETICS

Pattern formation and the kinetics of phase transi-tions are traditionally described by the time-dependentGinzburg-Landau model. 2 For a nonconserved scalar or-der parameter (Q), this model Langevin equation can bewritten as

BvP bP+ rI(r, t),

where E[Q] is the Ginzburg-Landau free-energy func-tional, and q(r, t) is a Gaussian noise term reflecting fluc-tuations of the heat bath:

(q(, t)) =0,(g(r, t)g(r', t')) = k~Tb(r —r')b'(t —t') .

i.e., one integrates over the lattice distortions and theorder parameter field g. The Ginzburg-Landau func-tional P[Q] is obtained by making a saddle-point ap-proximation to the e integrals, i.e., minimizing I" withrespect to these variables. In our numerical procedure,the mean-field free energy corresponding to a given setof Q and a given set of e was calculated from the EMTHamiltonian and this function was then minimized withrespect to the e . This is equivalent to making themagnetothermomechanics approximation. The mini-mization of E[g] with respect to g leads to the mean-fielddescription of the CuAu ordering transitions. In this pa-per, this Ginzburg-Landau functional is used to describethe kinetics of phase transitions in these alloys.

The form of P[g] for CuAu is most conveniently writ-ten in momentum space:

I

F[g] = ) [o(T —To)+~(q)]4~@ ~+ —) g~, . . . @~, b(qi+q2+qs+q4) —— )4 Qx Q4

)

(4)

where

~(q) = «.'+ (« —~o) &o .

Here q~ is the magnitude of a wave vector in the plane

I

perpendicular to the CuAu ordering which is taken to bein the z direction.

Before discussing the znodel of kinetics that follows&om this Ginzburg-Landau functional, we brie8y dis-cuss the applicability of this model to equilibrium phase

8610 BULBUL CHAKRABORTY 49

transitions in CuAu. Because of the form of ur(q), theGinzburg-Landau functional X[/] is minimized by a one-dimensional periodic function, which describes a con-figuration comprised of a periodic array of antiphaseboundaries separating regions with g = +1. This is thesource of the CuAu II phase. The appearance of station-ary one-dimensional, periodic patterns is also a featureof the Lyapunov functional associated with the Swift-Hohenberg equation. ~s The isotropic form of ~(q) im-plies that the one-dimensional pattern is rotationally in-variant in both models. However, the CuAu II structureobserved in Monte Carlo simulations, ' based on EMT,and in experiments has the modulation locked along oneof the cubic axes, and there are lattice distortions ac-companying this transition. The unusual fourth-orderterm [third term in Eq. (4)] arises &om the couplingto uniform lattice distortions. This term is responsiblefor the CuAu I type ordering which, as discussed ear-lier, is in the same class as the Ising antiferromagneton an elastic triangular lattice. This type of orderinginvolves two broken symmetries, one corresponding tothe twofold Ising degeneracy and one corresponding tothe three possible strain directions on the lattice. TheCuAu II ordering involves a lattice distortion along thedirection of the one-dimensional modulation and an ad-ditional symmetry breaking in the choice of this direc-tion. It is expected that the coupling to the lattice is thesource of the commensurate CuAu II ordering. In theLandau theory, presented earlier, the modulation wasassumed to be along one of the cubic directions. The ef-fects leading to the choice of this direction, within mean-Geld theory, are currently being investigated. It is clearfrom our Monte Carlo simulations that the EMT Hamil-tonian can describe the lattice distortions accompanying

the modulation. '

The function u(q) can be measured in diffuse scat-tering experiments. ' The EMT model would predictan isotropic distribution in the plane perpendicular tothe ordering direction [cf. form of ~(q) in Eq. (4)] andthis isotropic distribution would have a peak displacedfrom the CuAu I superlattice Bragg peak by a distanceqo. Experimental measurements on CuAu show an essen-tially isotropic distribution with a peak at qp 2z/10awhere a is the lattice constant. The EMT estimate forqp is 2x/4a. This was, however, a very crude numer-ical estimate which was only meant to show that qo isnonzero. The essentially isotropic distribution of CuAuis in sharp contrast with difFuse scattering intensities inCu3Pd, where the distribution has well defined peaksalong the q~ and q„directions. These peaks have been ex-plained on the basis of a Fermi-surface mechanism. Theessentially isotropic distribution in CuAu suggests thata different mechanism is responsible for the modulatedphase. The small anisotropy seen in experimental difFusescattering measurements in CuAu would be dificult toobtain ft. om our numerical calculations based on EMT. Itcould also be that there is a Fermi-surface contributionwhich is missing &om EMT that makes the distributionanisotropic. Because of the small anisotropy, it is rea-sonable to investigate the isotropic model and discussthe consequences of the anisotropy. The isotropic modelis interesting in its own right since it implies the break-ing of a continuous symmetry at the CuAu II transition,where the system chooses a direction of modulation.

The above discussion shows that P[Q] is a good modelfor describing kinetics of these alloys. The kinetic equa-tion derived from the model E[g] is

(5)

where @(q,t) = exp[D(q)t], (6)

I 2

f[v/r] = — g (r)dr ——~ g (r)dr

~

+— g (r)dr.4 ( ) 6

III. EARLY-STAGE KINETICS

The early-stage kinetics of CuAu, following a quench,is defined by a linear stability analysis of the disor-dered, g = 0 state. This implies neglecting the bf/8@term in Eq. (5). The linear theory is very different fromthe usual model of a nonconserved order parameter.The difference is evident from an analysis of the solutionto the linearized equations without the noise term. Theeasiest route to this solution is through transformationto momentum space where the different modes decoupleand the solutions for Q(q) are

where D(q) = —a(T —Tp) —ur(q). This solution showsthat modes with D(q) greater than zero grow exponen-tially with time. Equation (6) is the generalization of theCahn-Hilliard solutions describing continuous orderingto alloys with modulated phases. If none of the eigenval-ues D(q) are greater than zero, then there is no contin-uous ordering and the disordered state is either stable ormetastable. In contrast to the Cahn-Hilliard solutions,the fastest growing mode has a finite wave vector qo andthe growth rate depends only on the magnitude of thewave vector and not its direction. This is reminiscent ofCahn-Hilliard description of spinodal decomposition butthere the wave vector of maximum growth depends on thetemperature ' whereas in our model the linear dynam-ics chooses a unique length scale which is a charactensticof the system. Since the interesting pattern formationstake place in the plane perpendicular to q, we will an-

49 DYNAMICS OF ORDERING IN ALLOYS WITH MODULATED PHASES 8611

alyze the equations in the q, = 0 plane. This linearizedCuAu equation is then identical to the linearized ver-

sion of the two-dimensional, stochastic Swift-Hohenbergequation

= ( —(1+V' )') Q(r, t),

O0)oo

oN 0

oo

II

~ \~ ~

/ r ~ el.' r

OO e + ~

ClO

I

OOi0

I

0.1 0.2 0.3 0.4 0.5

q(8m/a)

FIG. 1. The linear dispersion relation D(q) vs ]qz~. Thedispersion is isotropic in the ~q~~ plane. The solid line rep-resents D(q) at the temperature Tz ——To + qo /a, the dottedline corresponds to the temperature To, and the dashed lineto an intermediate temperature where a narrow band of wavevectors are linearly stable.

if the control parameter e is identified with qo a(T—Ts)—and the natural wave vector qp is set to 1. The Swift-Hohenberg equation is used to model pattern formationin Rayleigh-Benard convection and has been studied ex-tensively both in one and two dimensions. 2s

The linear dispersion relation D(q) is plotted in Fig.1 for various temperatures. Three diferent temperatureregimes can be identified. For T ) To + qo4/a, all modesdecay and there is no continuous ordering; this is theregime where the disordered state is metastable and or-dering takes place by nucleation. For (To+qo ) (T ( To,a band of wave vectors centered around qp grows, and atT = Tp the q = 0 mode becomes unstable. The q = 0structure corresponds to the unmodulated CuAu I phase.In the Cahn-Billiard description of spinodal decomposi-tion, the growth rate at q = 0 is always zero, and inits description of continuous ordering, the q = 0 modealways has the fastest growth rate. It should be remem-bered that the linear dispersion relation is isotropic inthe plane perpendicular to q, and all modes with a givenmagnitude of q grow at the same rate. This is expected togive rise to an interesting morphology of the early-stageordering process. ~

Experimental studies of kinetics are based on the struc-ture factor. The structure factor obtained &om the lin-earized equation at a temperature within the second tem-perature regime is shown in Fig. 2(a). The plot showsthe evolution of the structure factor as a function of time.The peak of the structure factor remains stationary as asmall band of wave vectors centered around this peakgrows as a function of time. The modes outside thisband are seen to decay with the growth of order. At alower temperature, the structure factor at q = 0 wouldstart to grow, but the growth rate predicted by the lin-ear theory always has its maximum at q = qp. The in-

EI~~o I I

0 0.1 0.2 0.3 0.4 0.5

q(zn/a)

O0

I )

II

II

\

I~ ~

~ ~

.r .r r~ ~

I I I

0.1 0.2 0.3 0.4 0.5

q(2m/a)

FIG. 2. (a) The evolution in time of the structure factorobtained from the linear theory without the noise term. Theplots show S(q) at a temperature T such that Ti ( T ( To.The solid line denotes the earliest time and the dashed-dottedline the latest time. (b) The evolution in time of the structurefactor predicted by the linear theory with the noise term in-

cluded. The parameters used are those appropriate for CuAu.The symbols are the same as in (a).

where So(q) is the structure factor in equilibrium at thetemperature prior to the quench. The inclusion of theCook term introduces minor modifications of the struc-ture factor which are illustrated in Fig. 2(b). These plotsshow that an experimental investigation of the early stagekinetics of CuAu in this temperature regime should showa peak growing at a, finite wave vector (measured with re-spect to the superlattice Bragg peak) and there shouldbe a ring of wave vectors at which the structure factoris a maximum. The question of whether the early-stagedescription is ever valid in these systems can only be an-swered after a detailed study of the complete nonlinearequations or from Monte Carlo simulations of the appro-priate model Hamiltonian. It has been argued that thelength of time over which the early-stage kinetics remainsvalid is given approxiinately by 1/D(qo) and depends log-arithmically on the range of interaction. In our model,we have an effective infinite-range four-spin interactionarising from the coupling to the lattice distortions andthis might imply that the range of validity of early-stage

elusion of the Gaussian noise term changes the form ofthe structure factor (generalization of the Cahn-Hilliard-Cook theory ) and can be written as

S(q, t) = Ss(q) exp[2D(q)t] + (exp[2D(q)t] —1},Ic@T

2D q

8612 BULBUL CHAKRABORTY 49

kinetics will be significantly enhanced in these systems.The evolution of the lattice distortion as a function oftime is a novel feature of these systems and experimentalinvestigations should shed light on the coupling betweenthe evolution of order and the evolution of the latticedistortions.

IV. DISCUSSION OF NONLINEAR EFFECTS

The effect of nonlinearities may change the kineticsin various ways. In the simplest case, the nonlinearitiesmainly come in to dampen the exponential growth. Somesimple considerations of the nonlinearities in our modelindicate that there can be more significant differences be-tween the linear and nonlinear dynamics. The periodicsolutions to which the g = 0 state becomes linearly un-

stable have wave vectors lying in the range [kq, k2], where

k1,2 (qo' + g[qo —a(T —To)]}'

For temperatures close to but less than Tq ——To + qo4/a,the width of the band of wave vectors grows as ~T T~

~

~2. —As the temperature approaches To, the q = 0 mode be-comes unstable and below To the width grows as (T —To)for T close to To. This band of wave vectors, chosen bythe linear dynamics, usually defines the range in whichthe stationary periodic solution is found at a given tem-perature; i.e., one could examine the periodic solutionslying in the range [k2, kq], and the solution with thelowest free energy becomes the selected pattern. How-

ever, because of the negative fourth-order term in P[g],there is a first-order transition from the disordered tothe ordered periodic phase. This implies that, in a giventemperature range, periodic solutions which are outsidethe band predicted by the linear dynamics could arisethrough a nucleated process and could become the sta-tionary solutions. The condition that an order parameterof the form Q(r) = g exp(iq r) is a solution to

nonlinear terms of our model equations if we are carefulin keeping the umklapp terms involving nonzero recip-rocal lattice vectors in Eq. (4). However, it might bemore profitable to investigate these effects directly in aneffective lattice model.

Numerical solutions of the full nonlinear Swift-Hohenberg equations have shown that the lamellar pat-terns exhibit phases which are akin to nematic and smec-tic ordering in liquid crystals. In CuAu alloys, thelamellar patterns have a length scale comparable to theunderlying lattice and an analogy may be drawn betweenthese systems and nematic ordering in the presence ofan underlying lattice. 0 The late-stage growth laws inSwift-Hohenberg systems is expected to be different fromusual models of nonconserved order parameters since thewave vector defining the modulation has a continuoussymmetry. 6 It would be interesting to study the late-stage growth laws in our model which has the extra fea-ture of coupling to the lattice.

V. LATTICE HAMILTONIAN

We have argued that the study of nonlinear dynamicsis CuAu-like alloys may be best approached through adiscrete lattice Hamiltonian. With this in mind, a latticeHamiltonian has been constructed which has the salientfeatures of the EMT Hamiltonian when applied to or-

dering on a deformable lattice. The model is an Isingantiferromagnet on an elastic lattice with an unusual pairinteraction term which arises &om the size difference be-tween the two chemical species making up the alloy. Thesource of this term in EMT is the density-dependent co-hesive energy. When expanded in terms of the spinvariables, a part of this energy gives rise to a term inthe Hamiltonian which is an antiferromagnetic interac-tion between two atoms which share a common nearestneighbor. This interaction depends on the bond lengthsand therefore can be different along different lattice di-rections in the presence of deformation. The strengthof the antiferromagnetic interaction depends on the sizedifference of the two atoms and is zero for atoms of thesame size. This model can be written as

1s

(u —u')'D(q) &-

8m

H=) J(l —ee )) s;s, +) M pe epCk (2j) aP

(10)compared to the condition for linear stability, D(q) & 0.For example, the q = 0 state becomes the stable station-ary solution at a temperature higher than its instabilitytemperature To. The role of nonlinearities is different inthis model than in the Swift-Hohenberg model. Sincethe negative fourth-order term is a consequence of thecoupling to the lattice, ' the investigation of nonlineardynamics has to take proper account of this coupling.The other crucial difFerence between this model and theSwift-Hohenberg model is that the underlying lattice inCuAu introduces a second length scale comparable to thelength scale chosen by the linear dynamics (27r/qo) andmay be instrumental in locking in the direction of mod-ulation. These effects are, in principle, all there in the

The parameters can all be obtained from EMT; givena specific alloy, however, we will treat this as a generalmodel with arbitrary parameters. This should then forma generic, minimal model for alloys where Fermi-surfaceefFects do not dominate. The first two terms in Eq.(10) describe an Ising model coupled to uniform strainthrough the coupling constant e and an elastic Hamilto-nian defined by the tensor M p. This model for J & 0 hasbeen analyzed on the triangular lattice and gives riseto a first-order transition kom a disordered to a stripedphase. The last term in the Hamiltonian is the unusualterm arising from size efFects. Since K & 0, this term

49 DYNAMICS OF ORDERING IN ALLOYS WITH MODULATED PHASES 8613

favors a spin con6guration where, for each site, the sumof the nearest neighbor spins along a given direction addsup to zero. In terms of atoms, this says that when atomsof different sizes are present, it is favorable to have a bigatom and a small atom as nearest neighbors. Neitheran antiferromagnetic nor a striped phase meets the opti-mum demand of the K term and a mean-field argumentshows that, on a triangular lattice, modulated phases oc-cur beyond a critical value of K. This is reminiscent ofthe ANNNI model. The one crucial feature which dis-tinguishes the current model &om the ANNNI model isthat all order-disorder transitions are first-order becauseof the coupling to the lattice and makes it a more at-tractive model for describing alloys. Qualitatively, thesource of the modulation can be inferred from looking atthe striped phase of the triangular lattice antiferromag-net which has alternating chains of up and down spinsand the sum over nearest neighbor spins for each direc-tion adds up to 2. If a modulation is introduced along theferromagnetic direction, such that after N spins, the spinup and spin down chains are interchanged, then the spinsat the domain boundaries have nearest neighbor con6g-urations which do satisfy the K term requirement. Thereason for the choice of the ferromagnetic direction hasits origin in the preferred lattice distortion. Therefore,in spite of the K term being isotropic, a unidirectionalmodulation occurs. This scenario is very similar to theCuAu II ordering and investigations on the fcc latticeare currently underway. At this stage, we would like topresent this model as an interesting variation on com-peting interactions models and suggest that the study ofkinetics of this model would lead to novel features whichcan be tested through experimental studies of alloys.

Fermi-surface eHects. In these alloys, the q dependenceof the second-order term in X[vPI is not isotropic; i.e. ,

~(g) has well defined minima along directions chosen bythe Fermi-surface nesting vectors. The linear dynamicsin these alloys chooses a length scale and a set of direc-tions and the symmetry breaking that takes place is notcontinuous but discrete, Ising like. Also, the role of thelattice distortions is expected to be minimal and there-fore the nonlinearities may play a simpler role and thelate-stage kinetics may follow the same scaling laws as inthe usual models.

A complete description of ordering kinetics in al-loys has to take into account the effect of phononsand local lattice distortions. Allowing for Buctuationsin bond lengths also leads to an additional size-effect-derived term in the lattice Hamiltonian. Under cer-tain circumstances, this can give rise to a long-rangepair interaction. The EMT prediction for this strain-mediated pair interaction term and its effects on kineticsare currently being investigated.

In conclusion, we have presented a model for describ-ing kinetics of ordering in alloys which have modulatedphases. The linear theory shows that the kinetics is simi-lar to that of Swift-Hohenberg models, but the nonlinearterms are distinctly different &om the Swift-Hohenbergequations. A lattice Hamiltonian which captures theessential features of the microscopic model of these al-loys has been constructed and simulations based on thismodel should be helpful in understanding both the staticsand dynamics of ordering in these alloys.

ACKNO% LEDGMENTS

VI. CONCLUSION

As mentioned at the beginning of the paper, there is aclass of modulated alloys where the modulation is due to

The author wishes to thank W. Klein, Karl Ludwig,Duane Johnson, Frank Pinski, and Mohan Phani formany helpful conversations. This work was supportedin part by NSF Grant No. DMR-9208084.

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8614 BULBUL CHAKRABORTY 49

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