Density profiles and surface tension of polymers near colloidal surfacesA. A. Louis,a)P. G. Bolhuis, and E. J. MeijerDepartment of Chemistry, Lensfield Rd., Cambridge CB2 1EW, United Kingdom, and Department ofChemical Engineering, University of Amsterdam, Nieuwe Achtergracht 166, NL-1018 WV Amsterdam,NetherlandsJ. P. HansenDepartment of Chemistry, Lensfield Rd., Cambridge CB2 1EW, United Kingdom共Received 4 December 2001; accepted 6 March 2002兲The surface tension of interacting polymers in a good solvent is calculated theoretically and bycomputer simulations for a planar wall geometry and for the insertion of a single colloidal hardsphere. This is achieved for the planar wall and for the larger spheres by an adsorption method, andfor smaller spheres by a direct insertion technique. Results for the dilute and semidilute regimes arecompared to results for ideal polymers, the Asakura–Oosawa penetrable-sphere model, and tointegral equations, scaling and renormalization group theories. The largest relative changes withdensity are found in the dilute regime, so that theories based on noninteracting polymers rapidlybreak down. A recently developed ‘‘soft colloid’’ approach to polymer–colloid mixtures is shown tocorrectly describe the one-body insertion free-energy and the related surface tension. © 2002American Institute of Physics. 关DOI: 10.1063/1.1473658兴I. INTRODUCTIONBinary mixtures of polymers and colloidal particles invarious solvents are the focus of sustained experimental andtheoretical efforts, both because they constitute a challengingproblem in Statistical Mechanics of ‘‘soft matter,’’ and be-cause of their technological importance in many industrialprocesses. One of the most striking aspects of polymer–colloid mixtures, namely the depletion interaction betweencolloids induced by nonadsorbing polymer was recognizednearly 50 years ago.1More recently, the importance of thepolymer depletant in determining the phase behavior of themixtures was realized,2and much recent experimental workwas devoted to the phase diagram,3–5structure,6,7interfaces,8and to the direct measurement of the effectiveinteractions.9–11On the theoretical side, most efforts haveconcentrated on impenetrable spherical colloids, while vari-ous models and theoretical techniques have been investigatedfor the description of the nonadsorbing polymer coils. Themodels include noninteracting 共ideal兲 polymers,1,12,13poly-mers represented as penetrable-spheres,14–17and interactingpolymers coarse-grained to the level of ‘‘soft colloids.’’18–21Monomer level representations of polymer chains, like theself-avoiding walk 共SAW兲 model, appropriate in good sol-vent, have been considered within polymer scalingapproaches,22–24renormalization group 共RG兲 theory,25–29andfluid integral equations.30,31While many effects for the simplest case of colloidsmixed with noninteracting polymer are quantitatively under-stood, the behavior of the experimentally more relevant caseof polymers with excluded volume interactions is at bestunderstood on a qualitative basis; a quantitatively reliabletheory is still lacking. Clearly, to construct such a theory forfinite concentrations of colloidal particles, one must first un-derstand how interacting polymer coils distribute themselvesaround a single spherical colloid of radius Rc. This problemis addressed in the present paper using a combination ofMonte Carlo 共MC兲 simulations and scaling theories to deter-mine the key quantities, which are the monomer or center-of-mass 共cm兲 density profiles(r) of SAW polymers arounda single impenetrable sphere, as well as the resulting surfacetension. If Rgdenotes the radius of gyration of the polymers,these quantities clearly depend on the ratio q⫽ Rg/Rc,which controls the curvature effects. The limit q→0, corre-sponding to a polymer solution near a planar wall, will beexamined first, before considering the case of finite q. Thecomplete theory for the opposite limit, qⰇ 1, will be the sub-ject of a future publication, although we show some prelimi-nary results here. Throughout this work we focus on thedilute and semidilute regimes22,32of the polymers, where themonomer density c is low enough for detailed monomer–monomer correlations to be unimportant; the melt regime,where c becomes appreciable, will not be treated here.The surface tension of a polymer solution surrounding asphere is macroscopically defined by considering the immer-sion of a single hard colloidal particle into a bath of nonad-sorbing polymer. Because this immersion reduces the num-ber of configurations available to the polymers, resulting inan entropically induced depletion layer around the colloid,there is a free energy cost F1for adding a single sphere tothe polymer solution which naturally splits into volume andsurface termsF1⫽ ⌸共兲43Rc3⫹ 4Rc2␥s共兲. 共1兲The first term in Eq. 共1兲, describes the reversible workneeded to create a cavity of radius Rcin the polymer solu-tion. Since the osmotic pressure ⌸() of a polymer solutiona兲Electronic mail: [email protected] OF CHEMICAL PHYSICS VOLUME 116, NUMBER 23 15 JUNE 2002105470021-9606/2002/116(23)/10547/10/$19.00 © 2002 American Institute of PhysicsDownloaded 06 May 2003 to 198.11.27.12. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jspin the dilute and semidilute regimes is quantitatively knownas a function of polymer concentrationfrom RGcalculations,25this volume term is well understood. Theproblem of a quantitative description of a single colloid in apolymer solution thus reduces to understanding the secondterm, which defines the surface tension␥s(), i.e., the free-energy per unit area that is directly related to the creation ofthe depletion layer. It is customary to relate the surface ten-sion␥s() around a sphere to the surface tension␥w()near a planar wall, by expanding in powers of the ratioq⫽ Rg/Rc␥s共兲⫽␥w共兲⫹1共兲q⫹2共兲q2⫹ O共q3兲, 共2兲which is expected to be most useful when q is not too large.The coefficientsi() control the curvature corrections.They are analogous to the Tolman corrections in the macro-scopic case.33,34The paper is organized as follows: The case of a singleplate or hard wall immersed in a polymer solution is dis-cussed in Sec. II, where we report results for density profiles(z) at
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