Mean-field fluid behavior of the Gaussian core modelA. A. Louis, P. G. Bolhuis, and J. P. HansenDepartment of Chemistry, Lensfield Road, Cambridge CB2 1EW, United Kingdom共Received 4 July 2000兲We show that the Gaussian core model of particles interacting via a penetrable repulsive Gaussian potential,first considered by Stillinger 关J. Chem. Phys. 65, 3968 共1976兲兴, behaves as a weakly correlated ‘‘mean-fieldfluid’’ over a surprisingly wide density and temperature range. In the bulk, the structure of the fluid phase isaccurately described by the random phase approximation for the direct correlation function, and by the moresophisticated hypernetted chain integral equation. The resulting pressure deviates very little from a simplemean-field-like quadratic form in the density, while the low density virial expansion turns out to have anextremely small radius of convergence. Density profiles near a hard wall are also very accurately described bythe corresponding mean-field free-energy functional. The binary version of the model exhibits a spinodalinstability against demixing at high densities. Possible implications for semidilute polymer solutions are dis-cussed.PACS number共s兲: 61.20.Gy, 61.25.Hq, 83.70.HqI. INTRODUCTIONInteractions between atoms or molecules in simple fluidsinvariably contain a short-range repulsive component or hardcore, such that the local molecular structure is dominated byexcluded volume effects. This observation explains the suc-cess of simple models involving hard convex bodies in ex-plaining the structure and phase transitions in simple atomicor molecular fluids 关1兴. For example, the hard sphere modelhas been instrumental in understanding freezing of simplefluids 关2兴. The same success extends to somewhat more com-plex fluids such as liquid crystals, where hard ellipsoids orspherocylinders have been widely used to investigate theisotropic-to-nematic transition and other mesophases 关3兴.However, the situation is generally not as simple in complexfluids, where effective interactions between mesoscopic par-ticles are often of entropic origin. While excluded volumeeffects still dominate the interaction between compact colloi-dal particles, the effective forces between ‘‘soft’’ or fractalobjects of fluctuating shape, such as polymer coils or mem-branes, cannot be modeled by hard cores. Polymers in a goodsolvent form highly penetrable coils and it is by now wellestablished that the effective interaction between the centersof mass of two polymer coils, duly averaged over internalconformations, is finite for all distances, and decays rapidlybeyond the radius of gyration of the coils 关4–7兴. For twoisolated nonintersecting polymer chains, the effective pairpotential at zero separation of the centers of mass,v(r⫽0),is of the order of 2kBT for sufficiently long chains 关6,7兴, andis reasonably well represented by a Gaussian whose width isof order the polymer radius of gyration RG, as shown inFig. 1.We have recently shown that the general shape of theeffective pair potential remains roughly the same in diluteand semidilute solutions of self-avoiding random walk共SAW兲 polymers, and does not vary strongly with polymerconcentration 共see Fig. 1兲关8兴. The effective pair potentialmodel has been shown to accurately reproduce the structureand thermodynamics calculated from Monte Carlo 共MC兲simulations of solutions of SAW polymers over a wide rangeof concentrations 关8兴.Neglecting in the first instance the state dependence of theeffective potential, it seems hence worthwhile to examine theequilibrium properties of a fluid of ‘‘soft’’ particles interact-ing via a pair potential approximated by a simple Gaussianformv 共r兲⫽⑀exp冉⫺r2R2冊, 共1兲where⑀is the energy scale and R determines the width. TheFourier transform 共FT兲 isvˆ共k兲⫽3/2R3⑀exp冉⫺k2R24冊. 共2兲FIG. 1. Polymer center of mass potentialsv(r) from simula-tions of L⫽500 monomer SAW chains 关8兴 are compared to a best-fit Gaussian 共1兲, determined by fittingv(0) to fix⑀, andvˆ(0)to fix R. The potential for two isolated coils (→0) is well approxi-mated by a Gaussian potential with⑀⫽1.87, R⫽1.13RG. Thepotential in the semidilute regime关⬃4⫻3/(4Rg3)兴is approxi-mated by a Gaussian potential with⑀⫽2.16, R⫽ 1.45RG.PHYSICAL REVIEW E DECEMBER 2000VOLUME 62, NUMBER 6PRE 621063-651X/2000/62共6兲/7961共12兲/$15.00 7961 ©2000 The American Physical SocietySuch a ‘‘Gaussian core model’’ 共GCM兲 was in fact intro-duced some time ago by Stillinger 关9兴, who focussed on thelow-temperature regime⑀*⫽⑀/kBTⰇ1, where the modelexhibits hard-sphere-like behavior, and a reentrant fluid-solid-fluid phase diagram under compression below a thresh-old temperature. This work was further expanded by Langet al. 关10兴, who showed that the model remains fluid at alldensities when⑀*ⱗ100. They also demonstrated that for thismodel, the familiar hypernetted chain 共HNC兲 closure for thepair distribution function g(r) becomes exact in the highdensity limit, and that the random phase approximation共RPA兲 is remarkably accurate at high densities.In this paper we concentrate on the fluid phase of theGCM (⑀*⬍100), with a particular emphasis on the regimerelevant for polymer solutions (⑀*⯝2) 关8兴, for which thedilute regime corresponds to reduced densities*⫽R3ⱗ3/(4)⬇0.239, and the semidilute regime corresponds to*ⲏ3/(4) 关11兴共here⫽N/V is the number of Gaussiancore particles per unit volume兲. We shall successively con-sider the homogeneous fluid phase, the inhomogeneous fluidphase in the vicinity of a hard wall, and the possibility ofdemixing of binary Gaussian core systems.II. THE HOMOGENEOUS FLUID PHASEA. The thermodynamic stability of the GCM fluidWe consider a system of N particles interacting via aGaussian pair potential 共1兲, in a volume V. In the absence ofan infinitely repulsive core, the first question is that of ther-modynamic stability against collapse, i.e., the existence of awell defined thermodynamic limit. According to definition3.2.1. in Ruelle’s classic book 关12兴, the total interaction en-ergy VN, which can be built up of pair and higher orderpotentials, is stable if there exists a B⭓0 such thatVN共r1,...,rN兲⭓⫺NB 共3兲for all N⬎0 and all兵ri其in the phase space RN. Stabilityimplies
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