VOLUME 83, NUMBER 2 P H Y S I C A L R E V I E W L E T T E R S 12 JULY 1999

Comment on “Origin of the Modulated Phasein Copper-Gold Alloys”

In a recent Letter, Paxton and Polatoglou [1] analyzedthe modulated phase in copper-gold alloys. They findno indication of particular stability of the M � 5 modu-lated structure at zero temperature, and the unmodulatedCuAu-I structure is found to be the ground state, in agree-ment with experiment. In simulations, based on an em-pirical “pair functional,” they observe “wetting” near theantiphase boundaries (APB’s). Making a “semiquantita-tive” estimate of the free energy gain from wetting, theyconclude that this is the key effect stabilizing the CuAu-IIstructure at high temperature.

In this Comment, we would like to point out that theircalculation of the free energy gain from wetting has flawswhich cast serious doubt on their conclusion.

We have two concerns regarding the estimate of thefree energy gain from wetting. First, the authors statethat they have ignored the temperature dependence ofthe internal energy. However, the entropy gain comesfrom wetting, which involves an increased number ofenergetically unfavorable nearest neighbor bonds, so thereis a directly related energy cost. Second, they use the idealmixing entropy to estimate the entropy gain. We suggestthat the wetting is likely to involve fluctuating antiphaseboundaries, as suggested by Ref. [2] and others, rather thanpurely random “errors” in the planes next to the APB, sothat the ideal mixing entropy is an overestimate.

In order to understand the effects of wetting in the modu-lated structure, we have performed Monte Carlo sim-ulations, including local atomistic relaxations, and weallowed the three lattice constants to fluctuate indepen-dently. We used a semiempirical, many-body potential[effective-medium theory (EMT)] which provides a rea-sonable model of the Cu-Au phase diagram [3].

From our simulations, we find that, at zero temperature,the energy of the CuAu-I structure is 1.51 meV per atomlower than the CuAu-II structure. This corresponds to ag�M�5� (in the notation of Ref. [1]) of 0.034 J�m2, whichappears similar to the energy difference shown for theirpair potential in Fig. 2(b) of Ref. [1].

CuAu-II structures near the transition temperatureshowed planes near the APB’s with significantly reducedaverage order parameters, in agreement with experimentalobservations and the Monte Carlo simulations of Paxtonand Polatoglou. We have analyzed the disorder in theAPB’s by computing the percentage of nearest neighborbonds that involve unlike neighbors, just for bonds inthe planes with reduced average order parameters. Thenumber of unlike nearest neighbor bonds is significantlyhigher than it would be for a plane in which randomerrors were inserted to produce the same average order

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parameter. This, and a visual inspection of the orderparameter patterns, suggests a fluctuating interface, ratherthan a purely random softening of the order parameter.

We also measured the energy cost of this wetting asabout 1 meV in our simulations, which is the same orderof magnitude as the entropy gain using the ideal mixingentropy which is an overestimate. The free energy gainfrom the wetting of the APB’s is, therefore, expected tobe much smaller than the estimate of Ref. [1].

In explaining how wetting alone could be the sourceof the modulated phase, the authors sketch an effectivefree energy in terms of APB’s, where the free energydifference between a phase with a density r of APB’s andone without APB’s is proportional to r�acg 2 TSmix� 1

Eint�r�, where Eint is positive and goes to zero as r goesto zero. In this picture, if Eint has a minimum at M � 5,then the first order transition from CuAu-I to CuAu-II canbe explained. This minimum in Eint is then choosing M,and is a key part of understanding the modulated phase.In the picture the authors suggest, Eint is monotonicallydecreasing as r goes to zero. In that case, the entropygain is indeed the sole source of the melting of the CuAu-Iphase, but in general the transition would be expected to becontinuous, rather than the observed first order transition.

In conclusion, our EMT results support the observationthat there is significant wetting of the planes near theAPB’s in the CuAu-II structure at high temperature, andthis will help destabilize the CuAu-I structure. Sucheffects are well known in models with ordered APBstructures such as the ANNNI model [4]. Although theANNNI model is not directly applicable to the CuAusystem, according to the LMTO calculations, we suspectthat the wetting is playing a similar role here. We find nobasis for the statement made in Ref. [1] that the wetting canexplain the origin of the periodic APB structure of CuAu-II and the first order transition between the CuAu-I and theCuAu-II phase.

David L. Olmsted and Bulbul ChakrabortyMartin Fisher School of PhysicsBrandeis UniversityWaltham, Massachusetts 02254

Received 9 April 1998PACS numbers: 61.72.Nn, 64.70.Kb, 81.30.Hd

[1] A. T. Paxton and H. M. Polatoglou, Phys. Rev. Lett. 78,270 (1997).

[2] D. de Fontaine and J. Kulik, Acta Metall. 33, 145 (1985).[3] Zhigang Xi et al., J. Phys. Condens. Matter 4, 7191

(1992); Bulbul Chakraborty and Zhigang Xi, Phys. Rev.Lett. 68, 2039 (1992).

[4] Per Bak and J. von Boehm, Phys. Rev. B 21, 5297 (1980).

© 1999 The American Physical Society