1063-7761/02/9504- $22.00 © 2002 MAIK “Nauka/Interperiodica”0710 Journal of Experimental and Theoretical Physics, Vol. 95, No. 4, 2002, pp. 710–713.From Zhurnal Éksperimental’no œ i Teoretichesko œ Fiziki, Vol. 122, No. 4, 2002, pp. 820–823.Original English Text Copyright © 2002 by Ryzhov, Stishov. It has been known for many years that the system ofhard spheres only experiences a phase transition at ahigh density, when σ ≈ l , where l is the hard spherediameter and l = ( V / N ) 1/3 is the average interparticle dis-tance ( V is the system volume and N is the number ofparticles). This transition corresponds to the ordering ofthe centers of gravity of the particles and can be calledan order–disorder transition, or crystallization. In caseof hard particles of different shapes such as hard rods,ellipses, and discs, a number of orientational phasetransitions can occur in accordance with a hierarchy ofcharacteristic lengths defined by particle shapes. A newsituation arises when an extra interaction of a finiteamplitude ε is added to the system of hard particles. Asknown from the van der Waals theory, a negative valueof ε inevitably causes an instability of the system in acertain range of densities and generally leads to a first-order phase transition with no symmetry change (theorder parameter characterizing this transition is simplythe density difference of the coexisting phases, ∆ρ = ρ 1 – ρ 2 ). This situation is almost universal and indepen-dent of the interaction length.Much less is known about the case where the inter-action parameter ε has a positive value. The simplestexample of an interaction of that kind is the so-calledrepulsive step potential (Fig. 1):(1)In what follows, the system of particles interacting viapotential (1) is called the system of “collapsing” hardspheres [1]. Systems of this type are studied in relationto anomalous melting curves, isostructural phase tran-sitions, transformations in colloid systems, etc. (see,Φ r()∞, r σ,≤ε, σ r σ1,≤<0, r σ1.>= e.g., [2–5]). A general conclusion derived from numer-ous studies of the system is that the repulsive interac-tion of finite amplitude and length results in the meltingcurve anomaly and the isostructural solid–solid phasetransition. The latter is a first-order phase transition andcan end in a critical point, because there is no symmetrychange across the phase transition line. The existenceof a phase transition of that type is a direct consequenceof the form of the interparticle interaction, and we seeno particular reason why it cannot occur in a fluidphase. FLUIDS A Liquid–Liquid Phase Transition in the “Collapsing” Hard Sphere System ¶ V. N. Ryzhov a, * and S. M. Stishov a, b a Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia b Los Alamos National Laboratory, Los Alamos, NM 87545, USA*e-mail: [email protected] Received May 6, 2002 Abstract —A liquid–liquid phase transition is discovered in a system of collapsing hard spheres using the ther-modynamic perturbation theory. This is the first evidence in favor of the existence of that kind of phase transi-tion in systems with purely repulsive and isotropic interactions. © 2002 MAIK “Nauka/Interperiodica”. ¶ This article was submitted by the authors in English. (a) Φσ r (b) Φσ 1 r ε Fig. 1. (a) The hard-sphere potential with the hard-spherediameter σ 1 . (b) The repulsive step potential; σ is the hard-core diameter, σ 1 is the soft-core diameter, and ε is theheight of the repulsive step.JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 95 No. 4 2002 A LIQUID–LIQUID PHASE TRANSITION IN THE “COLLAPSING” HARD SPHERE SYSTEM 711 Despite the growing interest in the possible poly-morphic phase transitions in liquids and glasses (see,e.g., [6] for recent reviews), the nature of differentphases that can be found in dense (and possibly meta-stable) liquids is still puzzling. In recent years, experi-mental evidence of such features of the phase diagramas liquid–liquid transitions and polymorphismappeared for a wide range of systems including water,Si, I, Se, S, C, P, etc. [6]. The complexity of the phasediagrams in these substances may result from complexinteractions depending on the intermolecular orienta-tions. At the same time, exploring the possibility thatsimple fluids interacting through isotropic potentialsmay exhibit a similar behavior represents a seriouschallenge for theorists.The possibility of the existence of a liquid–liquidphase transition drastically depends on the shape of theinterparticle potential. After the pioneering work byHemmer and Stell [2], much attention has been paid toinvestigating the properties of the systems with the so-called core-softened potentials—the potentials thathave a negative curvature region in their repulsive core.It has been shown that, depending on the parameters ofthe potentials, waterlike thermodynamic anomalies andthe second critical point can be observed in this system[6–10]. It is widely believed, however (see, e.g., [7, 8]),that the existence of a fluid–fluid transition must berelated to the attractive part of the potential. In thispaper, we show that the purely repulsive step potentialin Eq. (1) is sufficient to explain a liquid–liquid phasetransition and the anomalous behavior of the thermalexpansion coefficient.We apply the second-order thermodynamic pertur-bation theory for fluids to this problem. The soft core ofpotential (1) (Fig. 1b) is treated as a perturbation withrespect to the hard sphere potential (Fig. 1a). In thiscase, the free energy of the system can be written as[11, 12](2)where ρ = V / N is the mean number density, β = 1/ k B T , u 1 ( r ) is the perturbation part of the potential u 1 ( r ) = Φ ( r ) – Φ HS ( r ), Φ HS ( r ) is the hard sphere singular poten-tial, and g HS ( r ) is the hard sphere radial distributionfunction, which is taken in the Percus–Yevick approxi-mation [13]. In the same approximation, the compress-ibility can be written as [12](3)We note that the actual small parameter in expan-sion (2) is the ratio ε /( k B T ), and therefore, the perturba-tion scheme used in this paper works very well at highFFHS–NkBT------------------12---ρβ u1r()gHSr()rd∫=–14---ρβ2kBT∂ρ∂P------0u1r()[]2gHSr()r,d∫kBT∂ρ∂P------01
View Full Document
Unlocking...