Polymer induced depletion potentials in polymer-colloid mixturesA. A. LouisDepartment of Chemistry, Lensfield Road, Cambridge CB2 1EW, United KingdomP. G. Bolhuis and E. J. MeijerDepartment of Chemical Engineering, University of Amsterdam, Nieuwe Achtergracht 166,NL-1018 WV Amsterdam, The NetherlandsJ. P. HansenDepartment of Chemistry, Lensfield Road, Cambridge CB2 1EW, United Kingdom共Received 21 February 2002; accepted 12 April 2002兲The depletion interactions between two colloidal plates or between two colloidal spheres, inducedby interacting polymers in a good solvent, are calculated theoretically and by computer simulations.A simple analytical theory is shown to be quantitatively accurate for the case of two plates. A relateddepletion potential is derived for two spheres; it also agrees very well with direct computersimulations. Theories based on ideal polymers show important deviations with increasing polymerconcentration: They overestimate the range of the depletion potential between two plates or twospheres at all densities, with the largest relative change occurring in the dilute regime. Theyunderestimate the well depth at contact for the case of two plates, but overestimate it for twospheres. Depletion potentials are also calculated using a coarse graining approach which representsthe polymers as ‘‘soft colloids;’’ good agreement is found in the dilute regime. Finally, the effect ofthe polymers on colloid–colloid osmotic virial coefficients is related to phase behavior of polymer–colloid mixtures. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1483299兴I. INTRODUCTIONEffective potentials are a key to unlocking the equilib-rium behavior of many soft-matter systems.1,2The basic phi-losophy behind this coarse-graining approach is that the ini-tial effort in deriving these potentials is recouped when theyare input into the well oiled machinery of liquid state theory,3or when they are used in computer simulations.4An arche-typal example is the depletion potential, induced betweencolloidal particles by nonadsorbing polymers. Asakura andOosawa5first showed that a bath of such polymers, charac-terized by their radius of gyration Rg, induces an attractivedepletion interaction of range D⬇2Rgbetween two plates.Their calculation was exact for noninteracting polymers.Later, the same authors, and independently Vrij,6derived adepletion potential between two colloidal hard spheres 共HS兲by approximating the 共ideal兲 polymers as penetrable spheres.This is often termed the Asakura–Oosawa 共AO兲 model.A good example of the effective potential coarse-graining approach is the calculation of the phase behavior ofpolymer–colloid mixtures by Gast, Hall, and Russel,7andalso by Lekkerkerker et al.8and Meijer and Frenkel.30Theyfound, using an AO depletion potential approach, that thefluid–fluid phase line of colloids of radius Rcbecomes meta-stable with respect to the fluid–solid phase line if the sizeratio q⫽Rg/Rcis less than about 0.35, in qualitative agree-ment with experiments.9,10Their work demonstrates how anaccurate knowledge of the depletion potential can lead to agood understanding of the equilibrium behavior of colloidpolymer mixtures. The latter are important not only becauseof their relevance to many industrial and biological pro-cesses, but also because they form an important model sys-tem for equilibrium and nonequilibrium behavior in soft mat-ter science.Whereas the depletion interaction for ideal polymers isnow quantitatively understood, the understanding of thedepletion interaction induced by polymers with excludedvolume interactions is at best qualitative. Experiments on thephase behavior9,10and structure11,12of polymer–colloid mix-tures also show deviations from the simple AO model,13,14asdo direct measurements of the depletion potentials.15–17The-oretical attempts to directly calculate the depletion potentialsfor interacting polymers include scaling theory,18self-consistent field theory,19perturbation theory,20,21directsimulations,22RG theory,23,24as well an interesting new‘‘overlap approximation’’ method.25All these approachesshow significant deviations from ideal polymer behavior, butmany questions still remain. This is in contrast to binary HScolloid mixtures, where the deviations from the AO potentialcan now be quantitatively calculated with density functionaltheory 共DFT兲,26–28and the effects of nonideality on thephase behavior are fairly well understood.29The goal of ourpaper is to derive a theory of similar accuracy for the deple-tion potential induced by interacting polymers in a good sol-vent.Before we proceed, an important caveat is that the deple-tion potential becomes less relevant for phase behavior atlarger size ratios q, since many-body interactions becomeincreasingly important.13,14,30When qⰇ1, i.e., when the col-loids are much smaller than the polymers, other approaches,which treat the polymers on a monomer level, are more rel-evant. Examples include integral equation techniques,31scal-ing theory,32,33or renormalization group theory 共RG兲.34InJOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 4 22 JULY 200218930021-9606/2002/117(4)/1893/15/$19.00 © 2002 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.jspthis paper we concentrate on the regime where Rgis of theorder of Rcor smaller.We have recently performed a systematic study of theinsertion free-energy of a single colloidal particle in a bath ofinteracting polymers35共henceforth referred to as Paper I兲,and found important deviations from ideal polymer behavior.In particular, we found that the range of the depletion layerdecreases with increasing density, and that this effect ismost pronounced in the dilute regime, whereⱗ*; here*⫽43Rg3is the overlap density. In the semidilute regime,where/*⬎ 1, we found that the effects of the interactionscould be well described by scaling theory.33Since these one-body effects were not well captured by 共nonadditive兲 HS-likemodels, we do not expect straightforward extensions of theDFT techniques that work so well for binary hard HS mix-tures, to also perform well for two-body depletion potentialsin polymer–colloid mixtures.To calculate the depletion potentials, we use a similarapproach to that used in Paper I, a combination of scalingtheories and computer simulations.
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