Simple fluids with complex phase behaviorGianpietro Malescio and Giuseppe PellicaneDipartimento di Fisica, Universita`di Messina and Istituto Nazionale Fisica della Materia, 98166 Messina, Italy共Received 26 May 2000; published 22 January 2001兲We find that a system of particles interacting through a simple isotropic potential with a softened core is ableto exhibit a rich phase behavior including: a liquid-liquid transition in the supercooled phase, as has beensuggested for water, a gas-liquid-liquid triple point, a freezing line with anomalous reentrant behavior. Theessential ingredient leading to these features resides in the presence of two effective radii in the repulsive core.The potential investigated appears appropriate for a class of spherical polymeric micelles recently investigated.DOI: 10.1103/PhysRevE.63.020501 PACS number共s兲: 61.20.Gy, 61.25.Hq, 64.70.⫺pPredicting the phase behavior for a given interparticle in-teraction is a central problem in statistical physics. The issueis relatively well assessed for pure systems interactingthrough a variety of radially symmetric pair potentials, suchas hard-spheres, Lennard-Jones systems, inverse-power po-tentials, etc. The complexity of the phase diagram increasesfor pure substances characterized by complex interactionsdepending on the intermolecular orientation, e.g., water,C,S,Ga, Se,Te,I2,Cs,SiO2, etc. 关1兴. For the above materialssomewhat exotic features appear in the phase diagram, suchas fluid-fluid or liquid-liquid transitions, polyamorphism, andmultiple crystalline structures. Exploring the possibility thatsimple fluids interacting through suitable isotropic potentialsmay exhibit similar behaviors represents a major challenge.The importance of such model systems is not limited to pro-viding a better understanding of the components of the inter-action responsible for the above phase behaviors. They mayalso represent an adequate description for systems having acompletely different nature: the suspensions of colloidal par-ticles dispersed in a fluid medium. In fact, in these systems,through the experimental control of particle and solventproperties 关2兴, it is possible to generate ‘‘nonstandard’’ ef-fective pair interactions similar to those investigated in thisRapid Communication.Such features of the phase diagram as fluid-fluid transi-tions, polyamorphism, etc. may be related to the competitionbetween expanded and compact structures. This suggests thatthe potential should possess two equilibrium positions 关3兴.The most obvious form with such a feature is one with twowells. Such potentials were shown to give rise to waterlikethermodynamic anomalies, though the presence of a newcritical point could not be directly observed 关4兴. Anotherform of interparticle interaction which could produce differ-ent equilibrium positions is that in which there is a region ofnegative curvature in the repulsive core: these so-calledsoftened-core potentials were proposed by Hemmer and Stell关5兴 who argued that they might produce an additional transi-tion, if a first already exists. Recently, through a mixednumerical–mean field type calculation, it was found that apotential consisting of a softened-core plus an infinite rangevan der Waals attractive term may give rise to a secondcritical point 关6兴. Very recently, molecular dynamics simula-tion showed for a softened-core potential with an attractivewell evidence of a transition between two fluid phases in thesupercooled region 关7兴.The purpose of this article is to report the findings of astudy of the phase behavior of a system of particles interact-ing through a potential with a softened-core and an attractivewell. Our analysis, based mainly on thermodynamically self-consistent 共TSC兲 integral equations for fluids 关8兴 and partlyon Monte Carlo 共MC兲 simulations, shows the existence of aliquid-gas critical point in the stable fluid phase and of aliquid-liquid critical point in the supercooled region. Theliquid-gas and liquid-liquid coexistence lines meet in a gas-liquid-liquid triple point. Moreover, the behavior of thefreezing line, estimated through one-phase criteria 关9–11兴,such as the Hansen-Verlet 共HV兲 rule 关10兴 and the entropiccriterion based on the analysis of residual multiparticle en-tropy 关11兴, is consistent with the existence of multiple crys-talline structures in the solid phase. Unlike previous studies关12,13兴, we show that a microscopic theory, directly linkingthe behavior of the system to the form of the interparticlepair potential, predicts for a simple fluid the existence of aliquid-liquid critical point and of a gas-liquid-liquid triplepoint.The chosen potential has a repulsive part Vrep(r) consist-ing of a hard core of radius r0⫽and a repulsive squareshoulder of height⑀and radius r1⫽ 2.5, plus an attractivecomponent Vattr(r) having the form of a square well ofdepth 1.25⑀extending from r1⫽ 2.5to r2⫽ 3关13兴.Toreach a thorough comprehension of the role played by thedifferent components of the potential we first study its purelyrepulsive part, and then consider the effect of adding theattractive component.Let us consider a system of particles interacting throughthe potential Vrep(r). We study its structural and thermody-namical properties using the TSC Roger-Young 共RY兲 inte-gral equation 关14兴. Figure 1 illustrates the structure factorS(k) for different densities, at a constant temperature, ascalculated within the above theory and through MC simula-tions 关15,16兴. We note the unusual behavior of the first peaksof S(k) which, as the density increases, progressively riseand fall 共with the exception of the third peak, which growsmonotonously with the density兲. This anomalous structurefactor recalls in its essential features that which is observedin dense star polymer solutions 关16兴. The ‘‘rising and fall-ing’’ of the peaks of S(k) reflects the turning on and off,upon the density increasing, of different effective lengthscales. When T andare sufficiently small, the soft core ispractically impenetrable and the particles behave as hardRAPID COMMUNICATIONSPHYSICAL REVIEW E, VOLUME 63, 020501共R兲1063-651X/2001/63共2兲/020501共4兲/$15.00 ©2001 The American Physical Society63 020501-1spheres of radius r1.AsT andincrease, more and moreparticles penetrate the soft core until this becomes scarcelyinfluent and the system is essentially equivalent to an
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