A self-consistent integral equation study of the structure andthermodynamics of the penetrable sphere fluidMaria-Jose Fernaud and Enrique LombaInstituto de Quı´mica Fı´sica Rocasolano, CSIC, Serrano 119, E-28006 Madrid, SpainLloyd L. LeeSchool of Chemical Engineering and Materials Science, University of Oklahoma, Norman,Oklahoma 73019-0628共Received 26 August 1999; accepted 13 August 1999兲The penetrable sphere fluid consists of a system of spherical particles interacting via a potential thatremains finite and constant for distances smaller than the particle diameter and is zero otherwise.This system, which was proposed sometime ago as a model for micelles in a solvent, hasrepresented so far a remarkable challenge for integral equation theories which proved unable tocorrectly model the behavior of the two-body correlations inside the particle overlap region. It isshown in this work that enforcing the fulfillment of zero separation theorems for the cavitydistribution function y(r), and thermodynamic consistency conditions 共fluctuation vs virialcompressibility and Gibbs–Duhem relation兲, on a parametrized closure of the type proposed byVerlet, leads to an excellent agreement with simulation, both for the thermodynamics and thestructure 共inside and outside the particle core兲. Additionally, the behavior of the integral equation athigh packing fractions is explored and the bridge functions extracted from simulation are comparedwith the predictions of the proposed integral equation. © 2000 American Institute of Physics.关S0021-9606共00兲51102-7兴I. INTRODUCTIONIn recent years it has become clear that classes of mate-rials such as solutions of certain types of colloidalparticles—a particular example of what is nowadays knownas ‘‘soft matter’’—can be modeled via ultrasoft potentials.1It is characteristic of this type of potentials that the excludedvolume effects are relatively small, the particles being highlypenetrable. This is the case of star polymer solutions for lowarm numbers,2,3a system in which the dissolved polymerdoes not seem to undergo crystallization 共‘‘liquid–solid’’transition兲 for any concentration.An extreme case, and perhaps the simplest, of ultrasoftpotentials is the penetrable sphere model,4closely connectedwith Stillinger’s Gaussian core model.5Both potentials illus-trate what is known as bounded interactions, since they re-main finite at zero separation. On the other hand, the ultra-soft potential recently proposed by Likos et al.1to model starpolymer solution exhibits a logarithmic singularity at zeroseparation. The penetrable sphere model was first proposedby Marquest and Witten4to explain the crystallization ofcopolymer mesophases. On the basis of ground-state calcu-lations, and the assumption of single-site occupancy, theseauthors concluded that an interaction of this type might giverise to a stable simple cubic phase in equilibrium with thefluid 共i.e., with the disordered suspension of colloidal par-ticles兲 in accordance with experimental evidence. More re-cently, however, a detailed study by Likos, Watzlawek, andLo¨wen6has shown that this is not the case when multiple siteoccupancy is allowed for, and thus one has to look for someother causes to explain the stability of the simple cubicphase, as for instance the influence of the many body inter-actions.Nevertheless, Likos et al. have raised an important pointin their study. It turns out that the integral equation approxi-mations usually employed in the liquid state theory, like thePercus–Yevick 共PY兲 equation, hypernetted chain approxima-tion 共HNC兲, or the Rogers–Young 共RY兲 hybrid closure, ei-ther fail to reproduce the structure of the fluid 共in particularfor distances smaller than the particle size兲 or completelylack of a solution for many states of interest 共from moderateto high concentrations of copolymer兲. In particular, in thecase of the PY integral equation, this failure is easy to un-derstand, since it is mostly suited for repulsive interactions,i.e., the larger the excluded volume region where the pairdistribution function satisfies g(r)⫽ 0, the more accurate theapproximation. However, the reason for the failure of theHNC is not so obvious. On the other hand, self-consistentapproximations like the RY integral equation which interpo-lates between PY and HNC, in some cases either lack asolution or lead to disparate results, since as will be shown inthis paper, consistency between virial and bulk compressibil-ity turns out to be insufficient to guarantee a physicallymeaningful solution in systems in which the particle exclu-sion hardly plays any role.From the above arguments it is clear that an integralequation that intends to describe the behavior of penetrableparticles at small or zero separation will have to take intoaccount some local consistency property of the functions de-scribing the fluid structure at zero separation, going beyondsimple thermodynamic consistency relations that are basedon the use of quantities derived from the integration through-JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 2 8 JANUARY 20008100021-9606/2000/112(2)/810/7/$17.00 © 2000 American Institute of PhysicsDownloaded 07 May 2003 to 198.11.27.21. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jspout the space of two-particle correlation functions. Preciselyin order to take into account the correct behavior of the cor-relation functions inside the core region 共even in systemswith strong core repulsions兲, Lee7proposed some years ago azero-separation 共ZSEP兲 closure for the Ornstein–Zernike共OZ兲 integral equation which makes use of the zero separa-tion limit of the cavity function y(12) (y(12)⫽ exp(u(12))g(12), where u(12) is the pair potential andg(12) the pair distribution function兲 in addition to other ther-modynamic consistency conditions. This ZSEP closure wasexplored in detail for Lennard-Jones systems by Lee, Gho-nasgi, and Lomba8who showed that imposing the fulfillmentof the zero-separation theorems for the cavity function inconjunction with the Gibbs–Duhem relation 共chemical po-tential vs pressure兲 and the consistency between virial pres-sure and bulk compressibility leads to an excellent agree-ment for all correlation functions, including those like thecavity and bridge function which are non zero inside theoverlap region. It is obvious that the ZSEP integral equationis an
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