10.1098/rsta.2000.0809Interactions and phase transitions of colloidaldispersions in bulk and at interfacesBy H. Low e n, E. All ahy ar ov, J. Dzubiella, C. v o n Fe rb e r,A. J usufi, C. N. L ikos a n d M. H en iInstitut fur Theoretische Physik II, Hei nrich-Heine-Universitat Dusseldorf,D-40225 Dusseldorf, GermanyRecent progress in the theory and computer simulation of e¬ective interactions andphase transitions of colloidal dispersions is reviewed. Particular emphasis is puton the role of the discrete solvent in determining the e¬ective interaction b etweencharged colloids, bulk ®uid{®uid phase separation of star-polymer{colloid mixtures,and on interfacial freezing transitions of steric ally stabilized colloids on patternedsubstrates.Keywords: e® ective interaction; charged colloids; star polymers;phase separation; surface f reezing1. IntroductionColloidal dispersions represent excellent model systems with a clear separation oflength-scales between microscopic degree s of freedom (such as solvent particles,counter- and salt-ions, monomers of grafted polymer chains, etc.) and the mesocopi-cally sized colloidal particles. Typically, one is only interested in colloidal propertiessuch as, for example, structural corre lations or phase transitions of the colloid al par-ticles. In this case, only thermodynamic averages with respect to the microscopicdegrees of freedom are needed, resulting in e¬ective interactions between the col-loids. In this paper, we apply this concept to di¬erent situations of colloidal science,highlighting recent progress in the theory and compu ter simulation of e¬ective inter-actions between charged colloidal suspensions, as well as mechanisms of ®uid{®uidphase separation in mixtures of colloids and polymers. Finally, we also address inter-facial freezing transitions induced by a periodic substrate pattern.The paper is organized as follows. The role of a molecular solvent on the e¬ectiveinteractions is emphasized in x2. The e¬ective interactions in star-polymer{colloidmixtures and their impact on ®uid{®uid phase separation are brie®y sketched in x3.Finally, we describe recent results on surface freezing of neutral colloidal particles ontopographically structured templates in x4, and conclude in x5.2. In° uence of a granular solvent on the e® ectiveinteraction between charged colloids(a) General remarks: modelling on di® erent levelsBasically, the theoretical model for the description of charged colloidal particles canbe done on ve di¬erent levels (for a recent review, se e Hansen & Lowen (2000)).Phil. Trans. R. Soc. Lond. A (2001) 359, 909{920909c® 2001 Th e Royal Society910 H. Lowen and othersspYukawaparticlescolloidscounteriondensity fieldscq++-------------++++++++++++++++++ppolyioncounteriondielectricbackground ofthe solventqp----++ss++----------------++++--+++---++------hard-spheresolvent particledielectricbackgroundpqc+++++ssÔ++ÔÔÔÔÔÔÔÔÔ-----++++++--+++----------+ÔÔÔÔÔÔÔÔÔÔÔ+ÔÔÔdipolar solventparticle(a) (b)(c) (d)(e)Figure 1. Di® erent levels for the modelling of spherical charged colloidal particles. (a) level 1,DLVO; (b) level 2, PB; (c) level 3, PM; (d) level 4, hard-sphere solvent model; (e) level 5, dipolarsolvent. For further ex planation, see the text.The higher the level, the more realistic the model is, and, at the same time, the morecomplicated the computational e¬ort. Any higher level includes the lower levels asspecial cases. This is shown schematically in gure 1.(1) The simplest approach is the linear counterion screening theory, which resultsin an analytical Yukawa pair p otential for the e¬ective interaction between col-loids, as given by the electrostatic part of the c elebrated Derjaguin{Landau{Verwey{Overbeek (DLVO) th eory (Derjaguin & Landau 1942; Verwey & Over-beek 1948). In the absence of added salt, this potential, as a function of thedistance between a colloidal p air, readsV (r) =q2pexp(¡ µ(r ¡ ¼p))(1+12µ¼p)2° r; (2.1)with an inverse Debye{Huckel screening length µ = 4º »cq2c=° kBT . Here, qp(qc) is the polyion (counterion) charge, ¼pis the polyion radius, and ° is thedielectric constant of the solvent. Furthermore, »cis the counterion bulk num-ber density and kBT is the thermal en ergy. On this level, only the colloidalparticles are considered explicitly.(2) The next level is the nonlinear Poisson{Boltzmann (PB) approach, whichincludes a full treatment of the counterion entropy but still works on a mean- eld level. By a suitable linearization, one recovers the DLVO theory as aspecial case. On the PB level, the colloids and the averaged counterion density eld are considered explicitly.Phil. Trans. R. Soc. Lond. A (2001)Colloidal dispersions 911(3) In the `primitive’ model (PM), one treats the counterions explicitly, arrivingat the two-component mo del of strongly asymmetric electrolytes. The numberdensity of the counterions is linked by global charge neutrality to the polyioncharge and number density. This approach includes full counterion correlations.Ignoring these, the PM reduces to the PB level. The solvent, on the other hand,only enters via a continuous dielectric background. In detail, the input pairpotentials areVij(r) =1 for r612(¼i+¼j);qiqj=° r otherwise;(2.2)with (ij) = (pp), (pc), (cc), and ¼cdenoting the diameter of the counterions.(4) If the solvent is treated explicitly, but only crudely modelled as neutral hardspheres of diameter ¼s, one arrives at the so-called hard-sphere solvent model(HSSM). This level includes the discrete molecular structure of the solvent (itsgranularity), but ignores its polarizability, as well as all its multipole moments(Kinoshita et al. 1996). Still, th e dielectricity of the solvent is treated as acontinuous background. In the limit ¼s! 0, the solvent is a decoupled idealgas and one gets the PM as a special case. The interactions on this level arethe same as in (2.2), but now with (ij) = (pp), (pc), (ps), (cc), (cs), (ss), wherethe solvent is neutral, qs² 0.(5) Finally, one may describe the polar solvent with a permanent dipole moment. Asuitable model is hard-sphere dipoles, the so-called dipolar solvent model (Lado1997; Weis 1998) or a Stockmayer liquid (Groh & Dietrich 1994, 1995). On thislevel, the screening of the Coulomb i nteractions (i.e. the dielectric constant ° )is an output and not an input. With quantum chemistry, one may even reacha higher level with a full
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