We wish to calculate the effective electrostatic interaction between two charge-stabilized col-loidal spheres of radiusa =σ/2and chargeZe, immersed in a supporting electrolyte solution (cf. Fig. 20). For this purpose, we model the colloidal particles, the counterions of chargezce dissociated from the colloidal particle surfaces, and the added salt ions as uniformly charged hard spheres dispersed in a solvent described as a structure-less, uniform continuum of dielec-tric constantϵ. This is the so-called Primitive Model which is frequently used as a model for electrolytes and charge-stabilized colloidal dispersions. It is primitive in the sense that the struc-ture of the solvent and particle surfaces, and polarization effects are completely disregarded.
Fig. 20:Primitive Model of charged colloidal hard spheres (macroions) of radiusaand charge Zeand point-like counterions and coions (microions).
The surface-released counterions and added salt ions (i.e., the microions) are much smaller than the colloidal particles (macroions). Therefore we can assume all microions to be point-like, i.e., of zero diameter. For simplicity, and although we want to study a system with added salt, let us restrict for the time being to a two-component system of macroions and point-like,
surface-released counterions. Overall charge neutrality requires then that
ρ1Z +ρczc= 0 , (162)
where, typically,|zc| = 1and|Z| ≫ |zc|. According to Eq. (146), there are thus three coupled Ornstein-Zernike equations
hαβ(q) = cαβ(q) +ρ1cα1(q)h1β(q) +ρ2cα2(q)h2β(q) (163) for components 1 and 2, i.e., for the colloidal macroions of densityρ1, and for the point-like counterions of densityρ2 =ρc. The subscriptcis used here to label the counterions, and not the colloidal particles. The three OZ equations include information on macroion-macroion (11), macroion-counterion (12) and counterion-counterion (22) pair correlations. However, we are only interested here in the macroion-macroion correlations (11). To eliminate explicit reference to the counterions, we define an effective one-component Ornstein-Zernike equation for the macroions alone as
h11(q) =cef f(q) +ρ1cef f(q)h11(q) . (164) By demandingh11to be the same as in the original two-component OZ equations, the effective direct correlation function,cef f(q), is determined by
cef f(q) =c11(q) + ρcc12(q)2
1−ρcc22(q) . (165)
It is thus related to all three partial direct correlation functions. Eq. (164) is formally identical with the OZ equation of a genuinely one-component system. It describes the microstructure of particles 1 (macroions) immersed in a bath of particles of component 2 (counterions). The counterions do not appear explicitly in Eq. (164). Their effects are hidden incef f(q).
Supposecef f(q)and its Fourier-inverse,cef f(r), would be already known from Eq. (165). We can associate an effective pair potential,uef f(r), describing the interaction of two ’counterion-dressed’ macroions from noting that
cef f(r) =−βuef f(r) for r→ ∞ , (166) valid at large pair separation. In fact, according to a general theorem due to Henderson [56], at a given state point(ρ1, T), there is a one-to-one correspondence,
g11(r;ρ1) ⇔ uef f(r;ρ1), (167) between a given pair distribution functiong11(r)and an effective one-component system with exactly pairwise interactions described by a pair potential uef f(r;ρ1), that exactly reproduces g11(r)irrespective of the underlying many-body interactions. In general, however, the knowl-edge ofuef f(r)alone is insufficient to give higher-order distribution functions such asg111(3)(r,r′), and to obtain the volume term. The potential appearing in the asymptotic relation in Eq. (166) is identical to the uniquely determined effective pair potential associated withg11(r). The form of uef f(r) may be different at different state points (densities) and must be re-established in each case. For a given pair distribution function and state point, the associated effective pair po-tential can be determined by an inversion procedure using, e.g., an appropriate one-component OZ closure relation whereu(r)is now the searched-for quantity, or using computer simulations
or density functional theory schemes [57, 58].
At non-zeroρ1 > 0, uef f(r;ρ1) includes in general higher-order contributions to U11ef f in an averaged way, and agrees therefore neither with eu2(r), nor with the potential of mean force associated withg11(r). In general,
ρlim1→0uef f(r;ρ1) = lim
ρ1→0w11(r;ρ1). (168) However, salt-free macroion suspensions are an exemption to the zero-density rule in Eq. (168) since, in three dimensions and due to Eq. (162), the counterions cease to screen the Coulomb interaction between a pair of macroions whenρ1 →0. A salt-free macroion suspension is thus not weakly coupled even in the infinite dilution limit (see [47] for a lucid discussion of this subtle point).
We can easily generalize Eqs. (163) and (165) to a more than a two-component ionic system by including the effect of point-like added salt ions of number density ρs and charge number zs. For an overall electro-neutral system,
∑
s
ρszs = 0 , (169)
where the sum extends over all components of salt ions.
Our task is to calculate first all partial direct correlation functionscαβ of the mixture, from solv-ing the coupled Ornstein-Zernike equations with appropriate closure relations. In a next step, cef f(r) can be determined using the many-component extension of Eq. (165), with uef f(r) deduced from its long-distance behavior. To determine cef f(r), we will use here the many-component linear MSA closure because of its analytical simplicity.
The direct correlation functions of point-like salt and counterions (i.e., componentsα > 1) are approximated in the MSA by
cαβ(r) = −uαβ(r)/kBT , r >0 (170) where
βuαβ(r) =LBzαzβ
r , r >0 , (171)
andzα ∈ {zc, zs}. Fourier transformation leads to
cαβ(q) =−4πLBzαzβ
q2 . (172)
Notice in this context that in the limit of point-like ions, or for ionic systems at infinite dilution, the MSA reduces to what is known in electrolyte theory as the Debye-H¨uckel (DH) approxi-mation. When dealing with direct correlation functions of charged particles, which have short-range (e.g., excluded volume) interaction contributions aside from the long-short-range Coulomb interactions, it is helpful to partition cαβ(r) into a short-range part, csαβ(r), and a long-range Coulomb part according to
cαβ(r) =csαβ(r)−LBzαzβ
r , (173)
since it holds quite generally that
cαβ(r)≈ −βuαβ(r) = −LBzαzβ
r r→ ∞ . (174)
This partitioning into long-range and short-range parts in combination with the MSA closure for the point-like microions leads to
cef f(q) =cs11(q) + ∑ where we have introduced the Debye-H¨uckel screening length,κ−1, with
κ2 = 4πLB
The screening parameter is determined by the concentrations and charges of all microion com-ponents in the system.
The large-rasymptotic behavior ofcef f(r)is determined by the factor[κ2+q2]−1in Eq. (175), sincecs1α(q)andcs11(q)are Fourier transforms of short-range functions. Fourier inversion leads thus to the intermediate result
with a yet undetermined interaction strengthg. The magnitude ofg is determined by the pre-factor of[κ2+q2]−1 which includes the so far unspecified colloid-microion direct correlations cs1α(q). Thus, the effective macroion pair potential is asymptotically of the Yukawa-type, with a screening parameter determined by the Debye-H¨uckel relation in Eq. (176). The microions distribute themselves around the macroions to screen the colloid-colloid Coulomb repulsion.
The only approximation used to obtain the asymptotic result in Eq. (177) has been the DH approximation for the microionic direct correlations.
To determine g analytically, we apply the MSA closure additionally to the colloid-colloid and colloid-microion direct correlation functions:
In MSA, the short-range direct correlation parts are thus zero outside the overlap region, i.e.
cs11(r) = 0forr > σ = 2aandcs1α(r) = 0, forr > a. As a consequence, the MSA predicts the microion-colloid short-range direct correlation part to be a linear function of the colloid charge number. This linearity inZ is exactly valid for the true cs1α only in the limit of weak particle charges. For larger macroion charges, the MSA underestimates the accumulation of counterions close to the surface of a colloidal macroion.
Using these properties of the MSA closure leads, in the limitρ1 →0, to the final result βuef f(r) =− lim
ρ1→0cef f(r) =LBZ2
( eκa 1 +κa
)2
e−κr
r , r > σ , (180) for the effective macroion pair potential at infinite dilution, as quoted already in Eq. (3). Note here thatκis non-zero for a finite salt ion concentration. Within the linearization approximation of weak particle charges,
g11(r) = e−βw11(r) ≈1−βw11(r), (181) which is consistent with the DH-MSA approximation of direct correlations, uef f(r) can be identified, for non-zero salt content, with the macroion - macroion potential of mean force w11(r). From the mean-spherical approximations made in its derivation, one might expect that this form of the effective pair potential applies only at long distancesr, and for weakly charged colloidal particles. However, it has been empirically found that the range of applicability of Eq. (180) can be substantially extended whenZ is treated not as the bare macroion charge, but as an effective or dressed colloid charge smaller than the bare one. In this way one accounts approximately for the stronger screening by counterions close to strongly charged macroion surfaces. Methods to determine the effective macroion charge by means of so-called cell-model and jellium approximations are discussed in [59, 60]. For an interesting alternative derivation of the effective macroion pair potential, and volume energy, of charged colloidal spheres based on linear response theory, see [61, 62]. This work gives also some justification for the free-volume correction factor1/(1−Φ)in the screening constantκappearing in Eq. (4).
5 Summary and Outlook
The aim of my lecture has been to give an introduction to integral equation theories of the liq-uid state. These theories are based on the Ornstein-Zernike equation with its concept of direct correlations. The OZ based integral equation schemes provide a versatile framework for pre-dicting microstructural and thermodynamic properties of simple and colloidal liquids, from the knowledge of the pair potential. Moreover, I have explained how effective interaction potentials of colloidal particles can be obtained from averaging out the degrees of freedom of the small particle components, e.g., polymers or microions.
Due to the approximate nature and, usually, non-perturbative character of integral equation schemes, one can not make decisive a priori statements on their accuracy. The accuracy of a specific scheme depends in general on the range and the attractive or repulsive nature ofu(r), the system dimensionality, and on the degree of thermodynamic consistency. The extent of ther-modynamic inconsistency is particularly influential when critical phenomena are studied. The existence and location of the spinodal instability line, e.g., depends on fine details of the un-derlying integral equation scheme. For a more recent discussion of liquid-gas criticality in the context of the OZ integral equation approximations, see [63]. Partially self-consistent integral equation schemes such as the RY approximation, in which different routes to the same thermo-dynamic properties are enforced, are superior to standard integral equation schemes, however, for the price of a somewhat larger numerical effort. Yet, even this enlarged numerical effort is usually quite small as compared to computer simulation studies, which makes the integral
equation schemes well suited to study liquid state properties over a broad range of system pa-rameters.
The importance of OZ integral equation schemes goes well beyond the calculation ofg(r)and S(q)from a given pair potential. In combination with powerful density functional theory meth-ods [64–66] and perturbation schemes, one can study first-order liquid-solid phase transitions, and calculate structural properties of inhomogeneous fluids in external electric or magnetic fields, and near confining walls or liquid-gas interfaces (e.g., layering and wetting phenomena).
Furthermore, integral-equation-calculated distribution functions play an important role as static input to theories dealing with the dynamics of fluids.
Integral equation approaches form also the basis of inversion schemes to deduce information on the (effective) pair potential from an experimentally determined g(r)or S(q). From knowing g(r), c(r)can be determined using the OZ equation. The pair potential of the effective one-component system (cf., Eq. 167) follows then directly, but approximately, from using a closure relation such as the HNC closure. I should also add that the OZ equation for two-particle cor-relation functions, that has been discussed in this lecture, can be generalized to higher-order correlation functions using functional calculus methods [67]. The OZ equations for higher-order correlation functions, in turn, are the basis for extended integral equation schemes that allow to calculate, e.g., the triplet distribution functiong(3)(r,r′)[68].
My lecture has been restricted to fluids of spherical particles with spherically symmetric pair interactions. The integral equation schemes and their closure relations have been broadened in the past to deal also with fluids of non-spherical rigid and flexible particles, such as more complex molecules, rod-shaped viruses and polymers. For an example, the so-called refer-ence interaction-site model (RISM) method has been used to calculate the site-site distribution function of rigid molecules, whose interactions are modelled by an interaction-site potential.
The molecule is hereby represented by a discrete set of interaction sites located at the places of the atomic nuclei [1]. The site-site distribution of non-rigid molecules like polymer chains and polyelectrolytes has been successfully determined on the basis of the polymer reference interaction-site model (PRISM) [69]. Furthermore, the microstructure of liquids adsorbed in a porous medium, such as in a gel or in an arrested particle matrix, can be approximately predicted using so-called replica Ornstein-Zernike equations [70, 71].
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