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CU-Boulder PHYS 7450 - Polymer Coils

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VOLUME85, NUMBER 12 PHYSICAL REVIEW LETTERS 18SEPTEMBER2000Can Polymer Coils Be Modeled as “Soft Colloids”?A. A. Louis,1P. G. Bolhuis,1J. P. Hansen,1and E. J. Meijer21Department of Chemistry, Lensfield Road, Cambridge CB2 1EW, United Kingdom2Department of Chemical Engineering, University of Amsterdam,Nieuwe Achtergracht 166, NL-1018 WV Amsterdam, The Netherlands(Received 26 May 2000)We map dilute or semidilute solutions of nonintersecting polymer chains onto a fluid of “soft” par-ticles interacting via a concentration dependent effective pair potential, by inverting the pair distributionfunction of the centers of mass of the initial polymer chains. A similar inversion is used to derive aneffective wall-polymer potential; these potentials are combined to successfully reproduce the calculatedexact depletion interaction induced by nonintersecting polymers between two walls. The mapping opensup the possibility of large-scale simulations of polymer solutions in complex geometries.PACS numbers: 61.25.Hq, 61.20.Gy, 82.70.DdA statistical description of polymer solutions in complexgeometries, such as the colloid-polymer mixtures whichhave recently received much experimental attention [1–3],generally relies on a nanometer scale segment representa-tion of the polymer coils, a computationally very demand-ing task except in the special case of ideal (nonintersecting)polymers obeying Gaussian statistics [4]. This obviouslyfollows from the fact that, although the colloidal particlesmay reasonably be modeled by hard impenetrable spheresor other complex shapes lacking internal structure, eachpolymer coil involves L segments which must satisfy anonintersection constraint. It thus appears natural to at-tempt a mesoscale coarse graining, whereby polymer coilsinteract via effective pair potentials acting between theircenters of mass (CM). Since polymers can interpenetrate,the effective potential by共r兲 is expected to be soft, with arange of the order of the radius of gyration Rgof individualcoils. Such a coarse-grained description has been a long-time goal in the statistical mechanics of polymer solutions,dating back to the first attempts by Flory and Krigbaum [5]who employed mean-field theory to find an interaction forwhich the strength at overlap scales as by共r 苷 0兲⬃L0.2.Later, scaling arguments [6], field-theoretical renormaliza-tion group calculations [7], and simulations [8] confirmedthat the range of the interaction between two isolated poly-mer coils is of the order of Rg, but found that in the scalinglimit the strength by共r 苷 0兲 is independent of L and oforder kBT.In this Letter, we show that a meaningful “soft colloid”picture of polymer coils may be built on a coherent “firstprinciples” statistical mechanical foundation. We deriveboth the effective wall-polymer CM interaction bf共z兲,and the “best” local effective pair potential by共r兲 betweenpolymer CM’s for finite polymer concentrations. Thesepotentials are then applied to simulate bulk polymer so-lutions, as well as inhomogeneous polymers near a hardwall and polymers confined between two parallel wallsto extract the effective depletion potential between plates.The soft colloid approach turns out to be successful notonly in the dilute regime but also, perhaps more surpris-ingly, well into the semidilute regime. A related “soft par-ticle” picture has been applied to polymer melts and blends[9], but the corresponding phenomenological implementa-tion differs substantially from the present first principlesapproach.We consider a popular model for polymers in a goodsolvent [10], namely, N excluded volume polymer chainsof L segments undergoing nonintersecting self-avoidingwalks (SAW) on a simple cubic lattice of M sites, withperiodic boundary conditions. The packing fraction isequal to the fraction of lattice sites occupied by poly-mer segments, c 苷 N 3 L兾M, while the concentration ofpolymer chains is r 苷 c兾L 苷 N兾M. For a single SAWchain, the radius of gyration Rg⬃ Ln, where n ⯝ 0.6 isthe Flory exponent [10]. The overlap concentration rⴱ,signaling the onset of the semidilute regime, is such that4prⴱR3g兾3 ⯝ 1, and hence rⴱ⬃ L23n. We have carriedout Monte Carlo (MC) simulations for chains of lengthL 苷 100 and L 苷 500, and covered a range of concentra-tions up to r兾rⴱ⬃ 5. The pair distribution function g共r兲of the centers of mass was computed for several concen-trations; g共r 苷 0兲 is always nonzero, thus confirming the“softness” of the effective pair potential by共r兲. The lat-ter was then derived from g共r兲 by an inversion procedurebased on the hypernetted-chain (HNC) approximation clo-sure relation [11]:g共r兲 苷 exp兵2by共r兲 1 g共r兲 2 c共r兲 2 1其 , (1)where b 苷 1兾kBT, while c共r兲 is the direct pair correla-tion function, related to g共r兲 by the Ornstein-Zernike (OZ)relation [11]. To any given g共r兲 and density there corre-sponds a unique effective pair potential by共r兲, capable ofreproducing the input g共r兲, irrespective of the underlyingmany-body interactions in the system [12]; in a variationalsense this by共r兲 provides the best pair representation ofthe true interactions [13], and leads back to the true ther-modynamics via the compressibility relation [11]. Whilethe simple HNC inversion procedure would be inadequatefor dense fluids of hard-core particles, where more sophis-ticated closures or iterative procedures are required [13],2522 0031-9007兾00兾85(12)兾2522(4)$15.00 © 2000 The American Physical SocietyVOLUME85, NUMBER 12 PHYSICAL REVIEW LETTERS 18SEPTEMBER2000we are able to demonstrate the consistency of the HNC in-version in the present case [14]. If the resulting effectiveby共r兲, examples of which are shown in Fig. 1, are useddirectly in MC simulations, the calculated “exact” g共r兲for this effective representation coincides within statisti-cal errors with the g共r兲 derived from the simulation of thefull initial polymer segment model. In fact, the HNC clo-sure turns out to be quasiexact when applied to the simpleGaussian model [15] whereby particles interact via the po-tential by共r兲 苷 e exp关2a共r兾Rg兲2兴, which yields a rea-sonable fit to the effective pair potentials shown in Fig. 1.Even the much cruder random-phase approximation clo-sure, c共r兲 苷 2by共r兲, yields semiquantitatively accurateresults in the regime of interest [16,17]. Careful inspectionof Fig.


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