1. INTRODUCTION2. RPA AND HNC3. SCALING PROPERTIES4. PHASE DIAGRAMS5. PAIR STRUCTURE AND CLUSTERING6. CONCLUSIONAPPENDIX A: SOLUTION OF THE VIRIAL ROUTE IN RPAAPPENDIX B: ANALYTIC SOLUTION OF THE RPA-C BINODAL IN THE SYMMETRIC CASEACKNOWLEDGMENTS10150022-4715/03/0300-1015/0 © 2003 Plenum Publishing CorporationJournal of Statistical Physics, Vol. 110, Nos. 3–6, March 2003 (© 2003)Phase Separation of Penetrable Core MixturesR. Finken,1J.-P. Hansen,1and A. A. Louis11Department of Chemistry, University of Cambridge, Cambridge CB2 1EW, UnitedKingdom; e-mail: [email protected] October 8, 2001; accepted November 16, 2001A two-component system of penetrable particles interacting via a gaussian corepotential is considered, which may serve as a crude model for binary polymersolutions. The pair structure and thermodynamic properties are calculatedwithin the random phase approximation (RPA) and the hypernetted chain(HNC) integral equation. The analytical RPA predictions are in semi-quantita-tive agreement with the numerical solutions of the HNC approximation, whichitself is very accurate for gaussian core systems. A fluid-fluid phase separation ispredicted to occur for a broad range of potential parameters. The pair structureexhibits a nontrivial clustering behaviour of the minority component. Similiarconclusions hold for the related model of parabolic core mixtures, which isfrequently used in dissipative particle dynamics (DPD) simulations.KEY WORDS: Gaussian core potential; phase separation; random phaseapproximation.1. INTRODUCTIONDemixing of binary or multicomponent mixtures is a very common phe-nomenon observed in a broad range of molecular fluids,(1)polymer solu-tions and blends,(2, 3)or colloidal dispersions.(4, 5)Phase separation isgenerally associated with differences in the attractive interactions betweenparticles of different chemical species. In polymer solutions these differ-ences are usually embodied in the Flory q-parameter,(2)which controls thecompetition between the entropy of mixing and the total interactionenergy, at least within a mean-field picture. On the other hand, in multi-component colloidal systems like binary dispersions involving colloidalparticles of very different sizes, or mixtures of colloidal particles and non-adsorbing polymer, phase separation can be driven by purely repulsive,excluded volume interactions. By mapping the initial multi-componentsystem onto an effective one-component system involving only the biggercolloidal particles, the largely entropy-driven demixing can be understoodin terms of attractive depletion interactions induced between the large par-ticles by the smaller species (the ‘‘depletant’’).(5)It should however be keptin mind that the initial bare interactions are purely repulsive, albeitstrongly non-additive, as in the highly simplified Asakura–Oosawa modelfor colloid-polymer mixtures.(6)In the case of fully additive hard spheremixtures, phase separation has been predicted for sufficiently large sizeratios,(7)but it is now generally believed that the fluid-fluid demixing ismetastable, and preempted by freezing.(8)A significant degree of positivenon-additivity of the core radii Rmn(whereby R12=(R11+R22)(1+D)/2,with D >0) is required to observe a stable demixing transition in the fluidphase.(9)Effective interactions between the centres of mass of fractal objects,like linear polymer coils,(10–13)star polymers(14)or dendrimers,(15)obtainedby averaging over individual monomer degrees of freedom, are now knownto be very ‘‘soft.’’ More specifically the effective pair potential divergesonly logarithmically for overlapping star polymers,(14)while remainingfinite, of the order of 1−2kBT, for linear polymers in good solvent.(10–13)This observation has stimulated the investigation of simple models, likefinite repulsive step potentials,(16)or the gaussian core potential,(17–19)whichwas first introduced by Stillinger, in a somewhat different context,(20)namelyv(r)=E exp(−r2/R2), (1)where E is the energy scale, while R determines the range of the effectivepotential. It is worth stressing that the ‘‘gaussian core’’ model is unrelatedto the ‘‘gaussian molecule’’ model, which was extensively studied byMichael Fisher and collaborators.(21)In the latter model it is the Mayerf-function, rather than the pair potential, which has a gaussian shape.Simple, penetrable particle models are also widely used in highlycoarse-grained simulations of large-scale phenomena within the so-called‘‘dissipative particle dynamics’’ (DPD) method.(22, 23)In DPD, effectiveinteractions between penetrable fluid ‘‘particles’’ are frequently modelledby a simple parabolic potential:(23, 24)v(r)=˛E(1−r/R)2;r<R0; r \ R.(2)It has recently been realized that binary mixtures of particles with pene-trable cores, which interact via generalizations of the gaussian and parabolic1016 Finken et al.potentials (1) and (2), involving different energy scales E and radii R for thevarious species, may phase-separate over appropriate ranges of theseparameters. Spinodal instability was first shown to occur for the gaussiancore model (1) within the random phase approximation (RPA)(18)andbinodals as well as interfacial properties were then calculated within thesame approximation.(19)Similarly Gibbs ensemble Monte Carlo simulationshave very recently shown that binary systems of soft particles interactingvia the parabolic potential (2) phase separate beyond a critical degree ofenhanced repulsion between particles of different species, in agreement withFlory-like mean field considerations.(24)In this paper we systematically extend our earlier results for the gaus-sian core model(18)and investigate the range of validity of the RPA bydetailed calculations of the pair structure, thermodynamics, and the result-ing phase coexistence curve within the much more accurate hypernettedchain (HNC) approximation. The break-down of the RPA is quantified inthe physically relevant regime where rR34 1 and E 4 kBT, which wouldcorrespond to the cross-over from dilute to semi-dilute regimes of theunderlying binary polymer solution. The RPA continues to provide reliablefirst estimates at higher densities.2. RPA AND HNCThe model under consideration is the binary gaussian core model(GCM) already introduced in refs. 18 and 19. It consists of N1particles of‘‘radius’’ R1and N2particles of ‘‘radius’’ R2in a volume V. The totalnumber density is
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