Capillary condensation and interface structure of a model colloid ...

CAPILLARY CONDENSATION AND INTERFACE ...PHYSICAL REVIEW E 68, 061404 2003A. Zero-dimensional limitIn this section we derive the zero-dimensional 0D Helmholtzfree energy for the three-component system of the AOcolloid-polymer mixture in contact with quenched hardspheres. This 0D free energy is used as an input to constructthe fundamental measure theory in the following section.Here, we give only a brief derivation, a more extensive versionwith more comments is given in Refs. 43,51. The essentialingredient is that we need to perform the so-called‘‘double average’’ which refers to the statistical average overall fluid configurations and subsequently over all matrix realizations.To that end we consider a 0D cavity which eitherdoes or does not contain a matrix particle. Hence, the 0Dpartition sum is that of a simple hard-sphere fluid,¯m1z¯m ,3FIG. 1. Sketch of the ternary mixture of mobile colloids dark,mobile polymers transparent, and immobile matrix particlesgray. The polymer coils can freely overlap. There are three modelparameters, i.e., the packing fraction of matrix particles ( m ), andtwo size ratios qR p /R c and sR m /R c , where R i is the radius ofparticles of species i. The packing fractions of colloids and polymers, c and p , respectively, are the thermodynamic parameters.N i /V, where N i is the total number of molecules of speciesi and V is the system volume. All of these components aremodeled as hard bodies, meaning that they cannot overlapbut otherwise do not interact with each other, except for thepolymer-polymer interaction, which is taken to be ideal, seeFig. 1. Consequently, when r is the mutual distance, the pairpotentials becomeu ij r if rR iR j0 if rR i R jfor i, jc,p,m except for i jp, 1and concerning the polymer-polymer interaction, this simplybecomesu pp r0 for all r. 2As all interactions are either hard core or ideal, the phasebehavior is governed by entropic packing effects and thetemperature T does not play a role. The only thermodynamicparameters are the colloid and polymer packing fractions c 4R c 3 c /3 and p 4R p 3 p /3, respectively. The remainingmodel parameters are two size ratios qR p /R c andsR m /R c and the packing fraction of matrix particles ( m4R m 3 m /3). It has to be mentioned that due to the factthat the polymers can freely overlap, the ‘‘polymer packingfraction’’ can easily be larger than one Fig. 1. The mixtureof hard spheres with these last-mentioned ideal polymersi.e., without the matrix particles is called the AO mixture15,16.III. DENSITY FUNCTIONAL THEORYwhere z¯m exp(¯m) is the fugacity of the hard spheres.Further, 1/k B T with k B Boltzmann’s constant and ¯m thechemical potential. The irrelevant prefactor scales with thevanishing volume of the cavity but has no effect to the finalfree energy and will not be discussed further. In general, weuse an overbar to refer to quantities of 0D systems. With thegrand potential, ¯mln ¯m , the average number of matrixparticles is ¯mz¯m¯m /z¯mz¯m /(1z¯m).Next, we consider the colloid-polymer mixture in contactwith the matrix in zero dimensions. If the cavity is occupiedby a matrix particle, no colloid, or polymer can be present.On the other hand, if there is no matrix particle, it can eitherbe empty, occupied by a single colloid, or an arbitrary numberof polymers. Hence,1 matrix particle in cavity¯ 4z¯cexpz¯p no matrix particle in cavity,where z¯c and z¯p are the colloid and polymer fugacities, respectively.Then, the contribution ln ¯ to the grand potentialshould contain the appropriate statistical weight for eachof the cases, i.e., z¯m /¯m for the first and 1/¯m for the second,¯ lnz¯cexpz¯p. 51z¯mAverage particle numbers are again readily obtained via ¯iz¯i¯/z¯i for ic,p not for m). The Helmholtz freeenergy can then be calculated using a standard Legendretransformation F¯¯ ic,p ¯i ln(z¯i), and we obtain forthe excess part, F¯exc F¯ ic,p ¯iln(¯i)1,F¯exc ¯c ,¯p ;¯m1¯c¯p¯mln1¯c¯m¯c1¯mln1¯m.This result can be shown to be equal from that which wouldbe obtained using the so-called ‘‘replica trick’’ 50.B. Fundamental measure theoryFMT is a nonlocal density functional theory, in which theexcess part of the three-dimensional free energy F exc is expressedas a spatial integral over the free energy density ,F exc i r dr „n i r….67061404-3