Phase behavior and structure of star-polymer–colloid mixturesJ. Dzubiella,a)C. N. Likos, and H. Lo¨wenInstitut fu¨r Theoretische Physik II, Heinrich-Heine-Universita¨tDu¨sseldorf, Universita¨tsstraße 1,D-40225 Du¨sseldorf, Germany共Received 18 January 2002; accepted 8 March 2002兲We calculate the phase diagrams of mixtures between hard-sphere colloids and star-polymers of armnumbers f⫽ 2,6,32 for different star-polymer–colloid size ratios 0.2⭐q⭐0.6 using an effectiveone-component description for the colloids in the presence of the stars. We map the fulltwo-component system onto an effective one-component system by inverting numerically theOrnstein–Zernike equation for binary mixtures, supplemented by the Rogers–Young closure, in thelow-colloid density limit. The free energy for the fluid and crystalline phase is calculated by usingboth hard-sphere perturbation theory and thermodynamic integration of simulation data. We findstable fluid–fluid demixing transitions for low arm numbers f⫽ 2,6 above a critical value of the sizeratio qcbelow preempted by a fcc-solid. For the linear polymer limit, f⫽ 2, the critical size ratio isfound to be qc⬇0.4, in agreement with other approaches to colloid-polymer mixtures. Increasingthe arm number, the region of stability of the demixing transition with respect to crystallization ofthe colloids shrinks, and qcgrows. A comparison between the one- and two-component descriptionsthat demonstrates the consistency between the two routes is also carried out. © 2002 AmericanInstitute of Physics. 关DOI: 10.1063/1.1474578兴I. INTRODUCTIONMulticomponent mixtures display an enormously richerphase behavior than one-component systems. A typical puresubstance consisting of spherically symmetric moleculeswithout internal degrees of freedom, displays a generic phasebehavior on the temperature-pressure plane that featuresthree phases: a gaseous and a liquid one 共if sufficientlystrong attractions between the molecules are present兲 and acrystal.1Moreover, the Gibbs phase rule2asserts that there isonly one point in the phase diagram at which these three canbe found in simultaneous coexistence with one another. Con-sequently, investigations of the bulk thermodynamics of one-component systems focus on the calculation of the freezing-and liquid-gas coexistence curves, as well as on the proper-ties in the neighborhood of the critical point associated withthe latter. In multicomponent mixtures, the additional free-dom provided by the flexibility of changing the concentra-tion of any of the constituent species at will, opens up thepossibility of various types of phase transitions, such as, e.g.,vapor–liquid, demixing, crystallization of any of the numberof the components, alloy formation, etc. Thereby, new topo-logical features in the phase diagram, including regions ofmultiphase coexistence, lines of critical points and criticalend points show up. It is therefore not much of a surprise thatthe structure and thermodynamics of multicomponent mix-tures are studied in much less detail than those of pure sub-stances.In soft matter physics, on the other hand, mixtures arethe rule, not the exception. To complicate matters even fur-ther, typical soft matter systems include components with avast separation of length scales, a feature that makes a true,multicomponent description of real systems unfeasible.3Onepossibility is to consider model mixtures and two examplesthat have been intensively investigated in the recent past4aremixtures of hard spheres 共colloids兲 and free, nonadsorbingchains on the one hand,5,6and the binary hard sphere mixture共BHS兲 of two species with a variable size ratio on theother.7–9Many of the theoretical investigations of thecolloid–polymer 共CP兲 mixture have been based on an effec-tive, one-component description of the hard colloids, forwhich an additional, attractive depletion potential is intro-duced after the polymer has been integrated out. This is thewell-known Asakura–Oosawa 共AO兲 model,10,11in which thepolymers are figured as penetrable spheres experiencing inaddition a hard-sphere 共HS兲 interaction with the colloids. Anumber of theoretical investigations on the AO model12–15have revealed that the system displays a demixing transitionthat accompanies the freezing of the hard colloids. However,the former becomes metastable with respect to the latter14forpolymer-to-colloid size ratios q⭐qc⬵0.45. For size ratiosq⬎qc, the system displays three phases: a colloid-poor/polymer-rich and colloid-rich/polymer-poor fluid, as well asa solid phase, in which the colloids form a fcc-crystallinearrangement with the polymers diffusing in it. However, forq⬍qc, a single, mixed fluid and a crystal phase exist. Thesefindings are in semi-quantitative agreement with experimen-tal results.16In the BHS system, two-componentsimulations17have shown that the demixing transition in thefluid phase is either metastable with respect to crystallizationor it is completely absent, depending on the size ratio.18Wenote that in all cases mentioned above, freezing refers to thelarge hard spheres only: the crystallization of both compo-nents and the associated formation of binary alloys takesplace at size ratios close to unity and its investigation bytheoretical methods is highly nontrivial.19–21a兲Electronic mail: [email protected] OF CHEMICAL PHYSICS VOLUME 116, NUMBER 21 1 JUNE 200295180021-9606/2002/116(21)/9518/13/$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.jspA theoretical understanding of the 共meta兲stability of thedemixing transition in two-component mixtures is providedby the depletion potential that effectively acts between thelarger components of the mixture when the smaller ones arethermodynamically traced out.22Depletion is caused by thefact that the small components have more free space avail-able to them when two large particles are brought close tocontact than when they are far apart. Hence, an entropic ef-fective attraction appears between the colloids. The proce-dure of tracing out the small components facilitates the the-oretical studies but it is subject to two strong constraintsarising from the definition of the effective interaction,22namely 共i兲 the overall thermodynamics of the mixture must,evidently, remain invariant in switching from one
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