Depletion interactions in suspensions of spheres and rod–polymersY.-L. Chen and K. S. SchweizerDepartments of Chemical Engineering and Materials Science & Engineering,and Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801共Received 12 March 2002; accepted 19 April 2002兲Liquid-state integral equation methods are employed to study the thermodynamic and structuralproperties of ideal and repelling rigid rods mixed with hard spheres in the limits when one of thespecies is dilute. The role of rod aspect ratio and sphere/rod size asymmetry is explored over a widerange of system parameters encompassing the colloid, nanoparticle, and crossover regimes. Novelpredictions are found for the polymer 共sphere兲 mediated depletion potentials and second virialcoefficients of particles 共rods兲 in dense polymer 共sphere兲 suspensions. The adequacy of the closureapproximations employed is tested by comparison with available numerical calculations and morerigorous theories in special limits. The liquid-state theory appears to be accurate for all propertiesin the nanoparticle regime and for the insertion chemical potential of needles and spherocylinders.However, it significantly underestimates depletion attractions effects in the colloidal regime of shortrods and large spheres due to nonlocal entropic repulsion effects between polymers and particles notcaptured by the classic Percus–Yevick approximation. © 2002 American Institute of Physics.关DOI: 10.1063/1.1485071兴I. INTRODUCTIONPolymer–colloid suspensions are ubiquitous in diversefields of science and engineering.1–3Recent years have wit-nessed a resurgence of interest in the fundamental physicalbehavior of such systems which have a large number of in-dustrial and biological applications.4,5Both spherical ‘‘col-loids’’ 共micelles, proteins, microgels, quantum dots, nanopar-ticles,...兲 and polymers 共synthetic or biological兲 can vary inglobal size from nanometers to microns. Recently, the devel-opment and application of a microscopic statistical mechani-cal approach to the equilibrium structure, thermodynamics,and phase behavior of such mixtures has been pursued.5–8Itis based on a fusion and generalization of liquid-state inte-gral equation theory of spheres and polymers 共polymer ref-erence interaction site model, or ‘‘PRISM,’’ theory9–11兲.Sofar, only high-entropy flexible chain systems have been con-sidered. However, there are many experimental realizationsof rigid 共or nearly so兲 polymer rods including biologicalmacromolecules 共e.g., tobacco mosaic virus, cytoskeletonprotein F-actin兲, synthetic polymers 关e.g., polybenzyl-glutamate 共PBLG兲, Kevlar, conducting polymers兴, and othersupramolecular objects 共e.g., carbon nanotubes, wormlikemicelles兲 with aspect ratios varying from o(101)too(103).The thickness of colloidal rods can be small, comparable to,or even larger than the ‘‘particles.’’ Often the rod polymersare used as additives to control the flow properties, structure,and/or competing equilibrium phase transitions 共e.g., crystal-lization兲 and nonequilibrium processes 共gelation, vitrifica-tion兲 of particle suspensions.3–5,12–16On the other hand,dense polymer materials ‘‘filled’’ with hard colloidal ornanoparticles are also of great scientific and industrial inter-est, as are true composite polymer–particle mixtures.17Asakura and Oosawa 共AO兲 were the first to study deple-tion phenomenon in mixtures containing macromolecules ac-counting only for entropic effects.18In their analysis, pureexcluded volume interactions between particles of differentsizes can induce an entropic attraction between the particles.In particular, for a binary solution containing spherical par-ticles with diameter D and random coils with radius of gy-ration Rg, the polymers were treated as phantom spheres thatare allowed to fully penetrate other polymers but not thecolloids. The attractive AO depletion potential in the limitDⰇRgis given byU共r兲kBT⫽再⫺关共D⫹2Rg兲3⫹ r3/2⫺ 3r共D⫹2Rg兲2/2兴/6,D⬍r⬍ D⫹2Rg0, r⬎D⫹ 2Rg,共1兲where r is the interparticle center-to-center distance andisthe polymer number density. This approach generally over-estimates the depletion effect when nondilute amounts ofpolymers are present in the mixture due to its neglect ofinterpolymer repulsions, and it is not applicable in the nano-particle or semidilute polymer concentration regimes. Intensetheoretical efforts are ongoing to develop better and moregeneral theories for such flexible coil-basedmixtures.5–8,19–25Far fewer theoretical studies have treated particle mix-ture systems containing rigid rod-like polymers. The AO ap-proach has been used to consider the induced depletion in-teraction between flat plates in a solution of dilute rodlikemacromolecules of length L and width d in the LⰇd regime.This knowledge can be used to determine the intermoleculardepletion potential between two large spheres of surface-to-surface separation h within the Derjaguin approximation(DⰇ L,h) with the result26JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 3 15 JULY 200213510021-9606/2002/117(3)/1351/12/$19.00 © 2002 American Institute of PhysicsDownloaded 22 Sep 2004 to 192.38.100.242. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspUs共h兲⫽⫺kBTLR2K1共h/L兲K1共h/L兲⫽⫺共/6兲共1⫺h/L兲3, for DⰇ L. 共2兲This AO prediction has been compared to exact numericalevaluations previously and shown to be in good agreement inthe Derjaguin limit.3,26More sophisticated approaches haverecently appeared which relax the low polymer densityrestriction,27,28but do not extend beyond the DⰇL ‘‘extremecolloid’’ or the LⰇ d ‘‘thin rod’’ limits and generally consideronly the two particle problem. Mean-field statistical thermo-dynamic approaches have also been developed that addressthe latter issue, but neither the former limitation nor the roleof polymer–polymer interactions.22,29The goal of the present paper is to initiate study of rigidrod–colloid suspensions within the microscopic PRISMframework. The focus is on excluded volume interactionscorresponding to a purely entropic mixture. The two funda-mental limiting cases are studied: 1 or 2 particles dissolvedin a rod polymer solution, and 1 or 2 rigid rods dissolved ina hard sphere fluid. Section II briefly describes the
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