INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTERJ. Phys.: Condens. Matter 14 (2002) 7551–7561 PII: S0953-8984(02)38821-0Phase behaviour of mixtures of colloidal spheres andexcluded-volume polymer chainsDGALAarts1,RTuinier2and H N W Lekkerkerker11Van’t Hoff Laboratory, Debye Research Institute, University of Utrecht, Padualaan 8,3584 CH Utrecht, The Netherlands2Forschungszentrum J¨ulich, Institut f¨ur Festk¨orperforschung, Weiche Materie,D-52425 J¨ulich, GermanyE-mail: [email protected] 1 July 2002Published 9 August 2002Onlineatstacks.iop.org/JPhysCM/14/7551AbstractWe study the phase behaviour of mixtures of colloidal spheres and polymersthat have an excluded-volume interaction dispersed in a (background) solventusing the concept of free volume theory. The depletion layer thickness iscalculated from the negative adsorption of polymer segments around a sphere.The correlation length and thermodynamic properties of the excluded-volumeinteracting polymer chains in solution are taken into account by using resultsfrom the renormalization group theory. For small polymer–colloid size ratiosthe difference from an ideal description of the polymers is small, while for largersize ratios the gas–liquid coexistence region shifts in the direction of higherpolymer concentrations and at the same time the liquid–crystal coexistenceregion becomes more extended. Both the gas–liquid region and the gas–liquid–crystal region become less extended. These features are compared toexperiment.1. IntroductionMixtures of colloids and non-adsorbing polymers display a rich phase behaviour, involvingcolloidal ‘gas’ (poor in colloid, rich in polymer), colloidal ‘liquid’ (rich in colloid, poor inpolymer) and colloidal ‘crystal’ phases (rich in colloid,poor in polymer). This phase behaviourfinds its origin in the interaction between colloidal particles in a sea of polymers. Betweentwo particles the interaction was first described by Asakura and Oosawa [1, 2], Vrij [3] andJoanny et al [4] who showed that there isanosmotic imbalance pushing the particles togetherif they are within a certain distance of each other. Subsequent calculations for the phasebehaviour of colloidal spheres and polymers in a ‘background’ solvent based on perturbationapproaches were performed by Gast et al [5] and Vincent and co-workers [6, 7]. Theseapproaches successfully identified that the topology of the phase diagram depends on the0953-8984/02/337551+11$30.00 © 2002 IOP Publishing Ltd Printed in the UK 75517552 DGALAartset alpolymer-to-colloid size ratio q = Rg/Rc,with Rgthe polymer’s radius of gyration and Rctheradius of the colloid. For a concise review of this early work, see [8]. In these approaches thepolymer partitioning between coexisting phaseswas not taken into account. This issue wasfirst addressed by Lekkerkerker [9] and developed in collaboration with Peter Pusey and co-workers using the concept of free volume theory [10]. It is especially successful in explainingwhy one should find a three-phase coexistence region instead of a three-phase coexistenceline.Extensive computer simulations [11–13] and exact solutions in one dimension [14–16] validate the free volume approximation. It qualitatively and, for polymers much smallerthan the colloids, even quantitatively predicts the correct phase behaviour as can be seenby comparison with experiments done in the laboratory of Peter Pusey [17–21]. The mainlimitation of the original free volume approach is that it considers the polymers as ideal.Recently, other theoretical approaches to describe colloid–polymer mixtures were explored,aiming at a better description of the polymer [22–25] (for a review, see [26]). In this paper weextend the free volume theory to describe the phase behaviour of mixtures of colloidal spheresand polymer chains with excluded-volume interactions.In section 2 we will briefly explain the thermodynamic framework needed to calculatephase behaviour using the free volume theory. The theory for ideal polymers and colloidalspheres of Lekkerkerker et al [10] and the resulting phase diagrams will be discussed insection 3. Inthisapproach polymers are modelled as penetrable hard spheres with a radius Rg.Therefore, the polymers are ideal and the depletion layer thickness around a colloid is equalto Rg.However, when using the polymer density profile around a sphere [27, 28] and replacingthis profile by astepfunction, becomes a function of the curvature q (=Rg/Rc) as will alsobe shown in section 3. In section 4 we make the transition from ideal polymers to polymerswith excluded-volume interactions using results from renormalization group (RG) theory [29].We will use an expression from Hanke et al [30] to incorporate curvature effects. In section 5the resulting phase diagrams are presented and compared to the original free volume theoryand to experimental phase diagrams. We will summarize and conclude with our findings insection 6.2. Free volume theoryThe natural thermodynamic potential to use when calculating the phase behaviour of colloid–polymer mixtures is the semi-grand canonical potential [9]. The colloids are treatedcanonically, while the polymers are treated grand canonically as is schematically shownin figure 1. The solvent is treated as background. The semi-grand canonical potential(N, T, V ,µp) can be written as(N, V, T ,µp) = F(N, V, T ) −µrp−∞Npdµrp, (2.1)in which F(N, V, T ) is the Helmholtz free energy of a pure hard-spheredispersion and dependson the number of colloidal particles N,thesystem volume V and the temperature T .Thereservoir is filled with polymers up to a final chemical potential of polymers µrp,resulting inNppolymers being pushed into the system. To calculate Npas a function of µrpthe followingassumption is made:Np(µrp) = nrp Vfree0= nrpαV, (2.2)saying that Npis equal to the number density of polymers in the reservoir with nrptimes thefree volume of the unperturbed system Vfree0.Thisfree volume is equal to the free volumePhase behaviour of mixtures of colloidal spheres and excluded-volume polymer chains 7553reservoirsystemN,V,T,µpFigure 1. Aschematic representation of the semi-grand canonical scheme. The reservoir is filledwith polymer and connected to the system via a semi-permeable membrane. The system containsboth polymers and colloids and its free volume is the total system volume V minus the volume ofthe depleted
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