VOLUME85, NUMBER 1 PHYSICAL REVIEW LETTERS 3JULY2000Complex Phase Behavior Induced by Repulsive InteractionsE. Velasco,1L. Mederos,2G. Navascués,1P. C. Hemmer,3and G. Stell41Departamento de Física Teórica de la Materia Condensada, Universidad Autónoma de Madrid,E-28049 Madrid, Spain2Instituto de Ciencia de Materiales de Madrid, Consejo Superior de Investigaciones Científicas,E-28049 Madrid, Spain3Institutt for Fysikk, NTNU, N-7491 Trondheim, Norway4Department of Chemistry, State University of New York, Stony Brook, New York 11794-3400(Received 11 January 2000)For a solid in which the interactions have a hard core plus a simple soft repulsive tail we show, usinga perturbation theory, that the possible stable crystalline structures give rise to a rich phase behavior.We find two concomitant critical points each corresponding to phase transitions separating bcc and fccstructures, respectively, and the occurrence of a transition between fcc and bcc phases without change indensity. This novel phenomenology may be relevant to the behavior of some metallic systems, colloids,and to water.PACS numbers: 64.60.–i, 64.70.Dv, 64.70.KbIt has long been known that systems of particles inter-acting through pair potentials with a hard core plus softattractive and/or repulsive tails can possess a rich vari-ety of phase transitions [1,2] and thermodynamic anoma-lies [3]. The subject has gained renewed interest with thefinding [4] that a simple hard-sphere plus square-well (at-tractive) or shouldered (repulsive) potential of sufficientlyshort range, simulated on a computer, does indeed showan isostructural transition in the solid region. These stud-ies are particularly relevant since the interatomic poten-tials of some pure metallic systems (e.g., Ga and Ge) [5],metallic mixtures [6], electrolyte and associating systems,and colloidal systems can be approximately mapped ontoshouldered pair potentials. Such isostructural transitionshave been observed to occur in Cs and Ce [7] and thereis strong evidence for a high-density liquid-liquid transi-tion in K2Cs [8]. In this paper we present results, based ontheoretical studies, of the phase behavior of a whole familyof simple repulsive potentials which could describe moreclosely some of the real systems mentioned above. Ourstudy reveals a surprisingly complex phase behavior in thesolid region and a remarkable dependence of the latter onpotential parameters.Isostructural phase transitions may be understood interms of simple ground-state arguments which are ulti-mately based on a basic property of the pair potential,namely, its convexity, or lack thereof. It is therefore in-teresting to ask whether pair potentials, more flexible thanthe square-shouldered potential, and such that convexitymight be tuned at freedom, could provide new insight intothe nature of these phase transitions. Equally interestingis the possibility that more open, non-close packed struc-tures, typical of metallic systems, can be obtained withthese potentials. We have found new and interesting phe-nomenology in the phase diagrams, such as (i) a variety ofcritical points involving different crystalline phases (i.e.,bcc-bcc and fcc-fcc critical points); (ii) triple points where,e.g., two coexisting fcc phases do in turn coexist with athird bcc phase; and (iii) as potential parameters are con-veniently varied, a novel point, arising from the collapse ofa fcc-fcc critical point into a fcc-fcc-bcc triple point, wherebcc and fcc structures with identical densities coexist. Dif-ferent scenarios are also possible, as discussed below.Our potential models will be referred to as model Aand model B (Fig. 1). Both models consist of a hard-corepart characterized by a hard-sphere potential of diameters. Model A is a squared-shouldered potential but witha sloped, instead of flat, section. Model B is a square-shouldered potential with the addition of a linear ramp.The range of both potentials is taken to be fixed at 1.1s.Model A will be parametrized in terms of the ratio e1兾e,whereas in model B we will vary d1and d2, keeping theirsum fixed, d 苷 d11d2苷 0.1s [9].The theory used in our study has been described in detailelsewhere [10,11]. It is based on a perturbation approachfor the free energy F:F共r兲 苷 FHS共r兲 1 2pr2Z`sdr r2˜gHS共r; r兲fp共r兲 ,(1)where FHS共r兲 is the free energy of a hard-sphere system,r is the mean density, fp共r兲 is the perturbative interaction,FIG. 1. Model pair potentials f共r兲.122 0031-9007兾00兾85(1)兾122(4)$15.00 © 2000 The American Physical SocietyVOLUME85, NUMBER 1 PHYSICAL REVIEW LETTERS 3JULY2000and ˜gHS共r; r兲 is the angle-averaged two-particle distribu-tion function of the reference hard-sphere system. It waspointed out by Stell and co-workers that Eq. (1) increasesin accuracy with increasing density, becoming exact asr approaches its close-packed value [12]. Hence (1) isespecially appropriate to the high-density systems westudy here. A crucial step in using Eq. (1) is to input accu-rate thermodynamic [FHS共r兲] and structural [˜gHS共r; r兲]properties for the reference hard-sphere system. Forthe fluid we have used the usual Carnahan-Starling andVerlet-Weiss [13] approximations. In the case of the solidwe have used a simple free-volume approximation forFHS共r兲.For˜gHS共r; r兲 it is possible [14] to constructa very accurate expression by using the exact virial,compressibility, and normalization sum rules.Our numerical results for the phase diagrams of modelA are for values of e1in the range 0.30.6e. Figure 2contains phase diagrams in the T -r and P-T planes, show-ing the dependence of the phase behavior on the potentialparameter. For high values e1. 0.4e the phase diagramexhibits an isostructural fcc-fcc transition and an associ-ated critical point, such as found for the square-well po-tential. However, in addition to this transition, there existsFIG. 2. Phase diagrams of model A in the P-T (left column)and T-r (right column) planes for different values of e1. Fromtop to bottom: e1苷 0.6, 0.5, 0.4, and 0.3e.a second solid-solid phase transition, involving two stablebcc phases of different density. The bcc and fcc phasesare in turn separated by a first-order phase transition of theusual type. As e1is decreased, the fcc-fcc phase boundaryis shortened and eventually disappears as a stable transi-tion boundary at e1苷 eⴱ艐 0.35e. The weakening of thefcc-fcc transition is consistent with the idea that isostruc-tural phase
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