The concept of pair and direct correlations, and the Ornstein-Zernike integral equation schemes described in earlier sections, can be generalized without difficulty to multi-component liquids and polydisperse (colloidal) systems. We will discuss these generalizations in the following.
Partial correlation functions
Consider an atomic or colloidal liquid consisting of m components of spherical particles of diametersσα and partial number densitiesρα = Nα/V. We employ greek symbols, withα = 1, . . . , m, to label the m components which build up the liquid. The particles within each component are identical. We assume that the particle interactions can be described by pairwise additive forces. The following replacements
u(r) → uαβ(r) g(r) → gαβ(r) c(r) → cαβ(r)
h(r) → hαβ(r) =gαβ(r)−1 (144)
are needed for a generalization to am-component mixture. There arem(m+ 1)/2partial pair potentialsuαβ(r), radial distribution functionsgαβ(r), total correlation functionshαβ(r), and direct correlation functionscαβ(r)(withα, β ∈ {1, . . . , m}) necessary to characterize the fluid pair structure. Here, uαβ(r) is the pair potential of two particles belonging to component α andβ, respectively. Furthermore, gαβ(r)gives the relative conditional probability of finding anβ-type particle a distance r apart from a given α-type particle. The partial pair potentials are obviously symmetric in the two component indices, i.e. uαβ(r) = uβα(r). The remaining functions in Eq. (144) are thus also symmetric in their component indices.
The one-component OZ equation is replaced in mixtures by a set ofm(m+ 1)/2coupled OZ equations, one for eachhαβ(r). In case of a homogeneous and isotropic liquid mixture, these OZ equations are given by
hαβ(r) = cαβ(r) +
∑m γ=1
ργ
∫
dr′cαγ(|r−r′|)hγβ(r′) . (145)
Fourier-transformation gives the OZ equations inq-space:
hαβ(q) =cαβ(q) +
∑m γ=1
ργcαγ(q)hγβ(q) . (146) The total correlation function,hαβ, between two particles of componentsαandβis thus written as the sum of a direct correlation part,cαβ, and an indirect correlation part mediated through all other particles of componentsγ = 1,· · · , mwith relative density weightργ.
Since there are m(m+ 1)/2 unknown functions, cαβ(r), in the OZ equations one needs the same number of closure relations to obtain a complete set of integral equations determining the partial radial distribution functions. We quote here only the multi-component generalizations of the MSA relations. All the other integral equation schemes discussed previously can be generalized accordingly to mixtures. The multi-component MSA closure relations are [3]
cαβ(r) ≈ −uαβ(r)/kBT , r >(σα+σβ)/2. (147) Combined with the exact non-overlap conditions
hαβ(r) = −1, r <(σα+σβ)/2, (148) and the OZ equations, one obtains a closed set of integral equations for thehαβ(r). In place of a single static structure factor, there are nowm(m+ 1)/partial static structure factors related to the partial rdf’s by
Sαβ(q) =δαβ+ (ραρβ)1/2
∫
dreiq·rhαβ(r) , (149) and defined such thatSαβ(q→ ∞) =δαβ.
4 Effective Colloid Interactions
In the framework of many-component liquid state theory, I will outline the derivation of two widely used effective pair potentials, namely the potential due to Asakura-Oosawa [42, 43] and Vrij [44, 45], that describes the attraction of colloidal hard spheres induced by free polymer chains, and the electrostatic potential in Eq. (3) describing the repulsion of charged colloidal spheres in the presence of screening microions. The effective colloid potentials will be obtained by contracting the small particles (i.e., the polymer coils and microions, respectively) out of the description.
Consider a binary suspension ofN large colloidal spheres (component L) at positionsrN and a second component (S) of small particles in a macroscopic volume V. Suppose that the small particles and the solvent, the latter treated here as an incompressible, structureless continuum, are in osmotic equilibrium with a reservoir of small particles and solvent, through a semi-permeable membrane impenetrable to the large colloids. This fixes the chemical potential,µS, of the small particles rather than their number density. The system under consideration is most conveniently described in terms of a semi-grand canonical ensemble, where the number N
σp
σ
r
Fig. 18: Left: Asakura-Oosawa-Vrij (AOV) model of colloidal hard spheres in osmotic equilib-rium with a solution of ideal, freely overlapping polymer coils. Right: lens-shaped depletion zones overlap volume∆V2(r)(shadowed) of two colloidal spheres.
of large particles in the suspension volumeV, and the chemical potential of small particles are fixed.1 The semi-grand total free energy,F(N, V, T, µS), of the mixture is then given by [13,46]
e−βF = is the activity coefficient of a small particle at fixed reservoir chemical potential µS, and Λ andΛS are the thermal wavelengths of large and small particles, respectively. We have used the grand-canonical average over the set of position vectors,xNS, of small-particle systems of particle numberNS ranging from zero to infinity.
In the second equation, I have introduced the effective interaction energy,
ULLef f(rN;V, T, µS) =ULL(rN) + Ω(rN;V, T, µS), (151) of theN large particles in the mixture. It is the sum of the bare potential contribution,ULL, and a grand-free energy contribution,Ω, with
Ω(rN;V, T, µS) =−kBT ln of an inhomogeneous dispersion of small particles in the external field ofN large particles fixed at positionsrN. Similar to the discussion of Eq. (30), we can convince ourselves that
F1(rN) =
1Different ensembles become equivalent in the T-limit, and one can change then from one description to the other by a Legendre transformation. However, it is conceptually advantageous to select that ensemble compatible with the experimentally controlled variables and the physical constraints.
is the mean force on a large sphere1, for the positions of theN large particles fixed, and grand-canonically averaged over the positions of the small particles. Here again, the total potential en-ergy in the mixture is denoted byU(rN;xNS). Hence,ULLef f plays the role of a state-dependent N-particle potential of mean force. OnceULLef f is known, any configurational average over a function,f(rN), of large-particle positions, such as the colloidal pair distribution function, can be formally expressed as a one-component average involvingULLef f only. However,ULLef f is state-dependent, since it includes a free-energy contribution,Ω, of the small particles. Therefore, and even for a pairwise-additiveULL, ULLef f contains in general the whole sequence of many-body contributions.
In the framework of the semi-grand canonical treatment, the ULLef f of a homogeneous system can be uniquely decomposed in a sum of independent many-body contributions [13],
ULLef f(rN;V, T, µS) =Ue0(N, V, T, µS) +
∑N i<j
e
u2(rij;µS) +
∑N i<j<k
e
u3(rij,rik;µS) +· · · (154)
Here,eu2(rij;µS)is the grand-free energy-change required to bring two colloidal spheres in the system from infinity to positionsr1 andr2. Likewise,ue3(rij,rik;µS)is the free energy change to bring three infinitely distant particles to(r1,r2,r3), minus the sum of the three pairwise work contributions, and so on. The many-body terms are constructed such that eun → 0, whenever the distance of a pair of particles in a n-cluster is very large. It is crucial to realize that, pro-vided the LS and SS interactions embodied in ULS and USS are short-ranged as compared to the LL interactions, one can expect that the three-body and higher-order contributions toULLef f are quite small. This allows then to map the mixture on an one-component system of large par-ticles interacting pairwise via the state-dependent pair potentialue2. Incidentally, if the degrees of freedom of the large particles would be integrated out instead of the small ones, then the resultingUSSef f(x)would include non-negligible many-body contributions of any order, which renders such a contraction rather useless.
The decomposition ofULLef f in Eq. (154) includes a configuration-independent contributionUe0. This state-dependent, so-called volume term has no bearing on the microstructure and thus on gLL(r), but it contributes to thermodynamic properties like the system pressure. In certain cases, the volume term may influence the phase behavior. A case in point are low-salt suspensions of charged colloidal particles where, due to the electro-neutrality constraint and the infinite range of the bare Coulomb forces,Ue0 reveals a nonanalytic dependence on the colloid concentration (see, e.g., [47–50]).
For a binary mixture of neutral particles, the thermodynamically extensive volume term is given by [13]
Ue0 =−V ΠS(µS) +Nue0(µS), (155) whereΠS(µS)is the osmotic pressure of small particles in the system void of colloids, which is equal to the reservoir polymer pressure, andeu0(µS)is the grand-free energy change arising from inserting a single colloidal particle to the polymer solution. Since Ue0/V is here linearly dependent onρL =N/V, it only adds a constant to the pressure and chemical potential of the effective one-component system, with zero influence on its phase behavior.
4.1 Depletion-induced colloid attraction
The effective pair potential,ue2(r), and the volume term can be straightforwardly calculated for the Asakura-Oosawa-Vrij (AOV) model of colloidal hard spheres (L = c) of diameterσ = 2a and density ρc = N/V, in osmotic equilibrium with a solution of small and non-adsorbing free polymer chains (S = p) (see Fig. 18). In this idealizing model, originally discussed by Vrij [44, 45], the polymer coils are simply described as hard spheres of diameter σp = 2ap, as far as their interactions with the colloidal spheres is concerned, but with respect to their mutual interactions they are treated as ideal point particles, i.e., as freely overlapping spheres.
Explicitly, the partial pair potentials in the mixture are ucc(r) = ∞, r < σ ucc(r) = 0, r > σ
ucp(r) = ∞, r <(σ+σp)/2 ucp(r) = 0, r >(σ+σp)/2
upp(r) = 0, r≥0 . (156)
The grand free energy of the ideal gas of ’polymers’ in the system volume V in presence ofN hard spheres fixed at positionsrN, and in contact with a reservoir of ideal polymers, is given by [51]
Ωid(rN;V, µp) = −Πidp (µp)Vf ree(rN). (157) Here, theT-dependence has been hidden for conciseness. According to Fig. 18,
Vf ree(rN) =V −N4π
3 (a+ap)3+
∑N i<j
∆V2(rij) +
∑N i<j<k
∆V3(rij,rik) +· · · (158) is the free volume in the mixture accessible to the polymer sphere centers. Moreover,Πidp = kBT zp is the osmotic pressure in the reservoir of the ideal-gas ’polymers’ at the given reservoir activityzp, which for ideal polymers that do not interact among themselves is equal to the reser-voir polymer density,ρrp, withµp = kBT ln(
ρrpΛ3p)
. Note here that the polymer concentration in the reservoir is in general different from the one in the mixture (see Fig. 18 and [51,52]). The reason why the reservoir osmotic pressure of polymers appears in Eq. (157), and not the os-motic polymer pressure in the mixture, is that work has to be done against the polymer pressure of the reservoir, when the free volume in the system decreases due to a change in the positions of the colloidal spheres.
Each colloidal sphere excludes a volume (4π/3) (a+ap)3 from the polymer centers. When two or more exclusion (depletion) volumes overlap, more free volume becomes available for the polymers with a corresponding loss in the grand-free energy. The overlap volumes,∆V2,
∆V3, and so on of two, three and more excluded volumes correct for this gain in free volume.
In summary, the suspension can lower its free energy by the clustering of colloidal particles, which implies that the spheres feel an effective attractive interaction commonly referred to as depletion attraction. For the present model, where only temperature-independent excluded vol-ume effects are operative, this attraction is purely entropic, i.e., the loss in free energy is due to the gain in entropy caused by overlapping depletion zones.
For size ratiosq = σp/σ <2/√
3−1 ≈0.155, three-body and higher-order excluded volume overlaps are exactly zero.2 When the ratioqis increased beyond this threshold, three-body and higher-order interaction terms come successively into play. If three excluded volume zones can overlap, the total gain in three volume is smaller than would be estimated on basis of the pair interactions alone. Thus, three-body interactions lead to a repulsive correction to the depletion interaction. The four-body interactions, in turn, are attractive again.
Quite notably, for q < 0.155, the AOV model of a colloid-polymer mixture is mapped on an effective one-component colloid system with exactly pairwise additive effective interactions.
On noting thatUccef f =Ucc+ Ωid, and using Eq. (158), we obtain the result for the effective colloid interaction energy, with a purely attractive AOV depletion pair potential given by [44, 45] for σ < r < (1 + q)σ. The AOV pair potential vanishes for non-overlap distances, r >
(1 +q)σ, of two excluded-volume spheres. Fig. 19 shows βuAOV(r) for various (reduced) osmotic polymer pressures,Πred = βΠidp σ3, at a fixedq = 0.1(left), and for various polymer-colloid size ratios at fixedΠred = 50(right).
The hard-core potential associated withσis denoted byuHC(r). It prevents the colloid spheres from overlapping. The volume term is seen to be linear in the colloid concentration. Therefore, it has no effect on the phase behavior of the colloids in the mixture. Sinceµp is externally con-trolled, the AOV potential can be considered as state-independent. The range of the attractive AOV potential in this simplifying model is equal toσp, independent of the polymer concentra-tion, but its depth can be controlled by changing the density and thus the osmotic pressure of the polymer solution in the reservoir.
For interacting (self-avoiding) polymers, which deform at non-zero concentrations, the range and depth of the depletion potential is smaller than for the ideal polymers considered here. In addition, there is a slight repulsive barrier in front of the attractive well, caused by the corre-lations between the interacting polymers. At high enough polymer concentration, depletion-induced attraction can cause the suspension to phase-separate into colloid-poor and colloid-rich phases. Depending on the range of attraction, the colloidal particles in the colloid-rich phase can be either in liquid-like arrangements for longer-ranged attractions, or in a crystalline state (for very short-ranged attractions) [52]. For very short-range attractions of range typically less than 5%of the colloid diameter, a first-order iso-structural solid-solid transition is predicted in simulations [54, 55].
To make contact with subsection 2.2, note that the potential of mean force associated with the rdf of large particles,wLL(r) = −kBT lngLL(r), contains in general many-body contributions.
2The valueq= 2/√
3−1characterizes a small polymer sphere that fits exactly into the space in between three colloid spheres touching each other in an equilateral configuration (see [53]).
red 50 Π = 10
30 q=0.1
red 50 Π = q=0.3
0.2 0.1
Fig. 19: AOV depletion potential in units ofkBT for values of reduced osmotic polymer pres-sure,Πred=βΠidp σ3, as indicated (left), and polymer-colloid size ratios,q =σp/σ, as indicated (right).
Different fromue2(r), its definition involves also an average over the positions of all large parti-cles with the exception of two kept fixed at the distancer. Therefore, it reduces to eu2(r)only in the zero- concentration limit of the large particles.