B 2 Theories of Fluid Microstructures

G. N ¨agele

Institut f ¨ur Festk ¨orperforschung Forschungszentrum J ¨ulich GmbH

Contents

1 Introduction 2

1.1 Model systems and pair potentials . . . 3

2 Pair Distribution Function 6 2.1 Basic properties . . . 6

2.2 Potential of mean force . . . 9

2.3 Scattering experiments . . . 11

2.4 Thermodynamic properties . . . 14

3 Ornstein-Zernike Integral Equation Methods 18 3.1 Ornstein-Zernike equation and direct correlations . . . 18

3.2 Theory of critical opalescence . . . 20

3.3 Various closure relations . . . 23

3.4 Percus-Yevick solution for hard spheres . . . 29

3.5 Thermodynamic consistency and Rogers-Young closure . . . 33

3.6 Extension to mixtures . . . 37

4 Effective Colloid Interactions 38 4.1 Depletion-induced colloid attraction . . . 41

4.2 Effective electrostatic macroion potential . . . 43

5 Summary and Outlook 47

Extended version of the article appeared in: Soft Matter: From Synthetic to Biological Materials, 39th IFF Spring School 2008, Series: Key Technologies Vol. 31, Forschungszentrum J¨ulich Publishing (2008)

1 Introduction

Atomic liquids and colloidal fluids are distinguished from dilute gases by the importance of short-range correlations and particle collision processes, and from crystalline solids by the lack of long-range order. The distinction between a liquid and a gas is only of quantitative nature, since there is no change in spatial symmetry in going from the gas to the liquid phase.

The most simple liquid systems are monoatomic liquids consisting of spherically shaped atoms (argon or neon) or quasi-spherical molecules such as methane. The interactions between the atoms can be characterized by simple pair potentials which depend only on the distance between an atomic pair. Colloidal fluids consist of mesoscopically large colloidal particles, typically a few hundred nanometers in size, dispersed in a low-molecular solvent such as water. There are strong local correlations in the positions of these particles (”super atoms”) so that they form a colloidal liquid in the liquid (solvent). The omnipresence of colloidal dispersions in chemistry and biology, and the unsurpassed variety and tunability of their particle interactions explains their importance both in industrial applications and fundamental research. Well-studied exam- ples of simple colloidal fluids are suspensions of silica and plexiglass spheres in organic sol- vents, and aqueous suspensions of globular proteins or highly charged polystyrene latex spheres.

While the friction-dominated, diffusive dynamics of colloidal particles is quite different from the ballistic one of atomic liquids, atomic and colloidal liquids are quite similar in terms of their equilibrium microstructure, i.e., in terms of the average ordering of particles. There are just orders of magnitude differences in their characteristic length scales.

In the present lecture, I will discuss a versatile class of liquid state theory methods which al- low to determine theoretically the microstructural and thermodynamic properties both of atomic and colloidal liquids, from the knowledge of the particle interactions. These so-called integral equation schemes are based on the Ornstein-Zernike (OZ) equation. Some of the most relevant OZ schemes will be introduced, and we will explore their predictive power in comparison with computer simulations and experiment. The quantity of central importance calculated with in- tegral equation schemes is the radial distribution function,g(r), of an isotropic liquid, and the associated static structure factorS(q). The latter property is essentially the Fourier transform of g(r), and it can be measured using scattering techniques. The radial distribution function (rdf) quantifies the probability of finding a particle at some distance from any other one. It can be indirectly measured by radiation scattering experiments and, for very large colloidal particles, also directly using video microscopy.

Typical atomic and colloidal liquids of spherically shaped particles will be briefly characterized in section 1 in terms of their interaction potentials. Section 2 provides an overview on general properties ofg(r)andS(q), and on their relation to scattering experiments and thermodynamic properties. Section 3 introduces the fundamental Ornstein-Zernike equation and the associated concept of direct correlations. As a first application of the OZ equation, a mean-field type description of critical opalescence in near-critical liquids will be given. Various approximate closure relations are discussed which lead to closed integral equations forg(r). Particular atten- tion is paid to the hard-sphere fluid, since it serves as a reference system of many liquids, similar to the harmonic solid in solid-state physics. In the final part of section 3, the Ornstein-Zernike integral equation concept is generalized to fluid mixtures. In section 4, I will explain the con-

cept of effective colloid interactions in systems which include, aside from the large colloidal particles and in addition to the solvent, components of small particles such as free polymer chains or salt ions. In two important examples, I will describe how a coarse-grained, simpli- fied description is achieved, in the form of a one-component suspension of decorated colloidal particles, by integrating out the small-particles degrees of freedom. The examples discussed are a binary mixtures of colloidal hard spheres and small polymer chains, and a suspension of charged colloidal particles with added salt ions.

1.1 Model systems and pair potentials

In the following we exemplify pair potentials suitable for describing the pair forces acting in simple and colloidal fluids of spherical particles. Simple fluids of non-polar atoms or molecules will be considered first, followed by a discussion of pair interactions in suspensions of spherical colloidal particles.

The Lennard-Jones 12-6 potential [1, 2]

u(r) = 4ϵ [(σ

r )12

−(σ r

)6]

for r >0 (1)

provides a fair description of the interaction between pairs of rare-gas atoms such as in argon, krypton and xenon, and also of quasi-spherical molecules such as CH₄. A sketch of the potential curve is provided in Fig. 1. Two parameters characterize the potential: the collision diameterσ whereu(r) = 0, and the depth,ϵ, of the potential minimum atr = 2^1/6σ. The values ofσand ϵhave been determined for a large number of atoms using, e.g., atomic scattering techniques.

For argon, σ_Ar = 0.34nm and ϵ_Ar/k_B = 119.8K. The short-range repulsive part of the pair potential proportional tor⁻¹² represents approximately the electronic repulsion of two atoms.

The longer-ranged van der Waals attraction between two atoms at a distanceris described by ther⁻⁶ part.

u(r)

- ε

σ

2

σ

[nm]

r

Fig. 1:Lennard-Jones pair potential describing simple atomic liquids.

The most simple pair potential one can think of is the potential between hard spheres of diameter σ, i.e.

u(r) =

{ ∞ for r < σ

0 for r > σ . (2)

While there exists no atomic fluid of hard atoms, colloidal suspensions of hard spheres are realized within good approximation, by coated polymethyl-methacrylate (PMMA) spheres dis- persed in a refractive index-matched non-polar solvent such as cyclohexane. The coating con- sists of a thin layer, as compared to σ, of adsorbed polymer chains (cf. Fig. 2). The polymer brush gives rise to a short-range repulsion between the colloidal spheres which counterbalances the remnants of the van der Waals attraction. The sizes of the colloidal hard spheres are in the range of several hundred to a few thousand nanometers.

a

r

Fig. 2: Model of colloidal hard spheres: PMMA spheres of radius a = σ/2 in a non-polar solvent with surface-grafted, short polymer hairs.

PMMA particles in a non-organic solvent are a paradigm for sterically stabilized dispersions.

A well-studied example of charge-stabilized colloidal dispersions are polystyrene latex spheres dispersed in a polar solvent such as water [3,4]. The latex particles acquire a high surface charge through the dissociation of ionizable surface groups. Each colloidal particle is surrounded by a diffuse layer of oppositely charged counterions, which are monovalent in the simplest case.

Overlap of the electric layers of two colloidal macroions leads to an electrostatic repulsion which counteracts the van der Waals attraction and prevents the particles from irreversible ag- gregation (cf. Fig. 3). In subsection 4.2, I will show that the screened electrostatic repulsion between two charged colloidal spheres dispersed in a solvent of static dielectric constantϵ is approximately described by the effective pair potential

βu_el(r) =L_BZ²

( e^κa 1 +κa

)2

e^−κr

r , r > σ (3)

which is the repulsive electrostatic part of the celebrated Derjaguin-Landau-Verwey-Overbeek (DLVO) potential [4]. Here, Z is the an effective or renormalized surface charge number of a spherical colloidal particle of radiusa = σ/2, β = 1/(k_BT), andL_B = e²/(ϵk_BT)is the so- called Bjerrum length. This length is the characteristic distance, for a pair of elementary charges e, where their Coulomb interaction energy is equal to the thermal energy k_BT. For water at room temperature, L_B = 0.71 nm. In case of a large and strongly charged colloidal particle (macroion), the effective charge number in Eq. (3) can be substantially smaller than the bare macroion charge as defined in a more refined many-component Primitive Model description of spherical macroions and microions (cf. subsection 4.2).

The Debye-H¨uckel screening length,κ⁻¹, in a closed system is given by κ² = 4πL_B∑

αρ_αZ_α²

1−Φ (4)

B

u(r)

k T ^{u (r)}

) r ( u

0

σσσσ r

Fig. 3: Left: Electrostatic and van der Waals potential contributions to the total effective pair potentialu(r)(red curve). Right: colloidal macroions with counterions and coions.

where the sum is taken over all types of microions, i.e., surface-released counterions and salt ions, of number densities ρ_α and charge numbers Z_α. The factor 1/(1−Φ)corrects for the free volume accessible to the microions, owing to the presence of colloidal spheres, where Φ = (4π/3)ρa³ is the colloid volume fraction and ρthe colloid number density. Notice here that the range and strength of the potential can be controlled by adding or removing small ions, and by changing the solvent or temperature.

The total effective pair potential,u(r), of charge-stabilized colloidal particles is the sum,u(r) = u_el(r) + u_vdW, of u_el(r) and the attractive van der Waals pair potential, u_vdW(r). The van der Waals attraction between two identical colloidal spheres can be described approximately by [4, 5]

uvdW(r) =−A_{ef f} 6

[ 2a²

r²−4a² +2a² r² + ln

(

1−4a² r²

)]

r > σ (5) withu_vdW(r) ∼ −(A_{ef f}σ⁶/36)/r⁶ for large r, and u_vdW(r) ∼ −(A_{ef f} σ/24)/(r−σ)near the contact distancer =σ(cf. Fig. 3). Clearly, the divergence of uvdW(r)at contact distance, arising from the assumption of ideally smooth spheres surfaces with precisely two surface atoms overlapping, is unrealistic.

Electrodynamic retardation and non-pairwise additivity effects on the dispersion forces are in- corporated to some extent in the effective Hamaker constantA_{ef f}. For non-metallic colloidal spheres,A_{ef f} is typically of the order of a fewk_BT [4, 5]. Van der Waals forces between iden- tical particles are always attractive. Non-identical particles can repel each other however, when the dielectric susceptibility of the solvent is in between that of the two particles.

For dispersions of highly charged colloidal particles at lower salt content, u_vdW(r) becomes completely masked by the electrostatic partu_el(r). In this case, one frequently refers to the col- loidal particles, with the associated diffuse layer of associated microions, as Yukawa spheres, since their microstructural properties are determined by the Yukawa-like, exponentially screened Coulomb potentialu_el(r).

It is often assumed that the potential energy, U(r^N), of aN-particle liquid system can be ap-

proximated by a sum of pair interactions U(r^N)≈

∑N i<j

u(|r_i−r_j|) =

∑N i<j

u(r_ij), (6)

for any configurationr^N ={r₁,· · · ,r_N}of position vectors{r_i}pointing to the particles cen- tres. The quality of this pairwise-additivity assumption depends on the choice of u(r), and on how certain many-body aspects (e.g., non-additive dispersion forces, influence of solvent molecules, electrostatic screening et cetera) are approximately included in the pair potential.

Typically, u(r) is density and temperature dependent. The dependence of the (effective) pair potential on the thermodynamic state is a remnant of microscopic degrees of freedom which have been averaged out on the level of coarse-graining where the potential applies. Colloidal systems where U(r^N) is exactly pairwise additive are scarce. Examples of these exceptional cases are ideal hard spheres, and suspensions of spheres with a specific short-ranged depletion attraction that describes the integral influence of free polymers added to the system (cf. subsec- tion 4.1).

Under the premise of Eq. (6), the microstructural properties of the fluid and, to some extent, the thermodynamic properties are solely expressible in terms ofu(r) and its associated radial distribution functiong(r). The latter is the most simple and most relevant example for a reduced distribution function and will be discussed in the following.

2 Pair Distribution Function

In this section we discuss salient properties of g(r), and of its associated Fourier transform pair S(q), referred to as the static structure factor. From knowing g(r), one can calculate macroscopic thermodynamic properties and analyze the local microstructure. Furthermore, the knowledge of static pair correlations is an essential ingredient in the theory of diffusion and rheology of simple and complex fluids. A major task of the liquid state theory is therefore to calculate g(r) from the information on the particle interactions, and to determine from it scattering functions and thermodynamic properties.

2.1 Basic properties

The concept of reduced distribution functions has proven to be extremely useful in liquid state theory. Consider a system ofN identical spherical particles in a volume V at temperatureT, i.e., a canonicalN V T ensemble. The function

P_N(r^N) = e^−βU(rN)

Z_N(V, T) , (7)

with configurational integral Z_N(V, T) =

∫

dr₁· · ·dr_N e^−βU(rN)=

∫

dr^Ne^−βU(rN) , (8) is the probability density that the centers of the N particles are at the positions r₁,· · · ,r_N. It provides far more information than necessary for the calculation of scattering properties and

thermodynamic functions. What is really needed are the reduced distribution functions for a small subset ofn≪N particles irrespective of the positions of the remainingN −nparticles.

To this end we introduce then-particle distribution function [1, 6]

ρ⁽ⁿ⁾_N (rⁿ) =N(N −1)· · ·(N −n+ 1)

∫

drn+1· · ·drNPN(r^N) (9) of finding any set of n particles at a specified configuration rⁿ = {r₁,· · · ,r_n}, regardless of how these n identical particles have been labelled. Of major importance are the reduced distribution functions of ordern = 1,2. For a homogeneous system

ρ⁽ⁿ⁾_N (r₁,· · · ,r_n) =ρ⁽ⁿ⁾_N (r₁+t,· · · ,r_n+t) (10) for an arbitrary displacement vector t. Then ρ⁽¹⁾_N = ρ = N/V is equal to the average par- ticle number density, ρ, and ρ⁽²⁾_N (r1,r2) = ρ⁽²⁾_N (r1 −r2) depends only on the vector distance r₁₂=r₁−r₂ (to see this chooset=−r₂).

The correlation lengthξ(T)is a characteristic distance over which two particles are correlated.

For fluids,ξ is typically of the range ofu(r)or larger to some extent, but under certain con- ditions (i.e., near a liquid-gas critical point) it can become extremely large. For a n-particle cluster with large mutual distancesrij =|ri−rj| ≫ξ, andN ≫1,

ρ⁽ⁿ⁾_N (r₁,· · · ,r_n)≈

∏n i=1

ρ⁽¹⁾_N (r_i) =ρⁿ (11) since these particles are then uncorrelated. To describe pair correlations in a fluid relative to a classical ideal gas of uncorrelated particles at the same density and temperature, we define the pair distribution function,g_N(r₁,r₂), in theN V T-ensemble as

g_N(r₁,r₂) := ρ⁽²⁾_N (r₁,r₂)

ρ⁽¹⁾_N (r₁)ρ⁽¹⁾_N (r₂) , (12) so thatg_N(r₁,r₂) → 1forr₁₂ → ∞. If the system is isotropic as well as homogeneous (i.e., no spatially varying external force field, no crystalline state),ρ⁽²⁾_N andg_N are functions of the separationr=r₁₂only. Then,

gN(r) = ρ⁽²⁾_N (r)

ρ² = N(N−1) ρ²

∫

dr3· · ·drN PN(r^N) (13) is denoted as the radial distribution function (rdf). It plays a central role in one-component fluids, since it is indirectly measurable by radiation scattering experiments. Moreover, thermo- dynamic quantities can be written as integrals overg_N(r)andu(r), provided that the particles interact by pairwise additive forces (cf. subsection 2.4).

As an important observation, we note thatρg_N(r)is the average density of particles a distance rapart from a given one. In fact, integration ofρg(r)over the system volume leads to

ρ

∫

drgN(r) = N(N−1) ρ

∫ dr12

∫

dr3· · ·drNPN(r^N) =N −1 (14)

since∫

dr₁₂=V⁻¹∫

dr₁dr₂. Likewise, Eq. (14) can be rewritten as 1 +ρ

∫

V

dr [g_N(r)−1] = 0 . (15)

Notice for the canonical rdf thatg_N(r → ∞) = 1−N⁻¹, since there are N −1particles left aside from the one atr = 0. For largeN, we can identifyg_N(r)as the conditional probability of finding a particle a distancer from a given one. To see this we use that in a homogeneous system, dr/V is the single-particle probability of finding the given particle in the volume ele- mentdr. Then,dr/V ×g_N(r)is the (unconditional) joint probability for two particles being separated by the distance r, with one of them located inside dr. The sum (i.e., the integral) of the joint probability over all accessible volume elements must give the value one. Precisely this is stated in Eq. (14). The unconditioned single-particle and two-particle joint probabilities can attain values between zero and one only. The conditional probability g_N(r), on the other hand, is the ratio of two unconditioned probalities and can thus attain values also larger than one.

Eqs. (14) and (15) are valid for a finite system of fixedN, without fluctuations in the number of particles. In order to be independent of the specific statistical ensemble used in calculating static properties of a macroscopically large system, it is understood that the thermodynamic limit (T-limit, for short) of macroscopically large systems, i.e. N, V → ∞withρ=N/V kept fixed, is taken at the end of each calculation. Then

g(r) := lim

N,V→∞g_N(r)≡lim

∞ g_N(r) (16)

denotes the ensemble-independent radial distribution function,g(r), of a macroscopic system.

Let us summarize general properties of theg(r)for a fluid, which follow from the definition of g_N(r)in Eq. (13):

g(r) ≥ 0, g(r→ ∞) = 1 (17)

g(r) ≈ 0, for βu(r)≫1 (18)

g(r) = e^−βu(r)+O(ρ) (19)

g(r) continuous foru(r)piece-wise continuous (20) The typical behavior of g(r) for an atomic liquid with a soft pair potential of Lennard-Jones type, and for a hard-sphere fluid is sketched in Fig. 4. Regions ofrwithg(r) > 1(g(r) < 1) have a larger (lower) probability of finding a second particle from a given one atr= 0, than for an ideal gas at the sameT andρ. The main features are a small-rregion whereg(r) = 0owing to strong repulsive forces exerted by the particle at the origin, and several peaks representing increasingly diffuse shells, for increasingr, of next neighbors, second next neighbors and so on. This shell layering manifests the granularity (non-continuum nature) of the fluid. The oscillations in g(r) decrease in amplitude with increasing r. Eventually, g(r) approaches its asymptotic value one forr > ξ(T, ρ). The oscillations in g(r)become more pronounced with increasing ρ. Whereas the Lennard-Jones g(r) is continuous at any r, the hard-sphere g(r) jumps from zero to values ≥ 1at r = σ, due to the singular nature of the hard-sphere u(r).

For hard spheres, g(r) = θ(r−σ) +O(ρ) according to Eq. (19). Hard-sphere systems are athermal (i.e.,T-independent) since the probability for a given particle configuration is either zero or one, independent of β, depending only on whether two or more spheres do overlap or not.

1 2

r/ σσσσ

FS’02_5

g(r)

Fig. 4:Theg(r)of a Lennard-Jones liquid (left), and a fluid of hard spheres (right).

2.2 Potential of mean force

There is a remarkable relation betweeng(r)and the so-called potential of mean force,w(r). In the T-limit,w(r)is defined in terms ofg(r)by [2]

g(r) =: e^−βw(r) (21)

or

w(r₁₂) =−k_BT lng(r₁₂) =−k_BT lim

∞

[ ln

∫

dr₃· · ·dr_Ne^−βU(rN)+ ln (V²

Z_N )]

. (22) To reveal the physical meaning ofw(r), we take the derivative of−wwith respect to the position vectorr₁of a particle 1:

−∇1w(r12) = lim

∞

∫ dr₃· · ·dr_N

(−_∂r^∂U₁) e^−βU

∫ dr₃· · ·dr_Ne^−βU =⟨

−∇1U(r^N)⟩

1,2. (23)

The quantity on the right-hand-side can be interpreted as the force on particle 1, if we hold particle 2 fixed, and average over the positions of all the other particles (as denoted by⟨· · · ⟩_1,2).

Thus,w(r)is the potential for this force, and it is therefore called the potential of mean force.

Likewise,w(r)can be interpreted as the reversible work required in a process where two par- ticles in the system are moved together, at constantN, V andT, from infinite separation to a relative distancer. To see this explicitly, recall from statistical mechanics that

F(r₁₂) = −k_BTln

∫

dr₃· · ·r_Ne^−βU(rN)+F₀(N −2, V, T) (24) is the Helmholtz free energy of a(N −2)-particle system in the presence of two fixed spheres at a distancer₁₂. The spheres 1 and 2 influence the system through their excluded volume and longer-ranged interactions. Here,F₀ is an irrelevant configuration-independent part of the free energy. Consequently

−∂w(r₁₂)

∂r12

=−∂F(r₁₂)

∂r12

(25)

and

w(r₁₂) = F(r₁₂)−F(∞) , (26) sincew(r)has been defined such thatw(r → ∞) = 0. The first and second laws of thermody- namics tell us for a reversible small change of state that

dF =−SdT −pdV +µdN+δW_rev . (27) This implies that dF = δW_rev = −∇12w(r₁₂)· dr₁₂ is the reversible (maximal) work done by the N −2particle system to achieve an infinitesimal displacement, dr₁₂, of the two inner boundary particles at fixed temperature, system volume and particle number.

The functionw(r;ρ, T)is in general a considerably more complicated object than the mere pair potentialu(r), since it involves the effects of particles 1 and 2 on the configurations of the other particles. It is only in the limitρ→0that

w(r)→u(r) i.e. g(r)→e^−βu(r), (28)

as will be shown further down. A sketch ofw(r)for hard spheres at finite density is shown in Fig. 5. As seen, two hard spheres effectively attract each other at distances r ≤ 1.5σ. This many-body effect arises from an unbalance of forces on the two spheres when the gap between the two particles is depleted from the other ones. Depletion interactions of this kind occur in any system with strong excluded volume interactions. This is the reason why depletion effects have attracted considerable interest in the past few years (see also section 4).

FS’02_8

0

σ

w(r) g(r)

r 1

FS’02_8

2 σ r <

Fig. 5:Potential of mean force of hard spheres (left) and depletion attraction (right).

On assuming pairwise additive interactions so that

−∇1U(r^N) = −∑

i>1

∇1u(r_1i), (29) the force law forw(r)in Eq. (23) can be rewritten as

−∇1w(r12) =−∇1u(r12)−ρ

∫ dr3

[g⁽³⁾(r₁,r₂,r₃) g(r₁₂) −1

]

∇1u(r13). (30) Here,

g⁽³⁾(r1,r2,r3) := lim

∞

ρ⁽³⁾_N (r₁,r₂,r₃)

ρ³ (31)

is the triplet distribution function of an isotropic and homogeneous liquid. Furthermore, g⁽³⁾ depends only on the two pair separationsr₁₂,r₁₃, and the angle betweenr₁₃andr₁₂. The mean force on particle 1, in presence of particle 2 at a distancer₁₂, is thus the sum of the direct in- teraction between 1 and 2, and the interaction of 1 with a third particle atr₃, weighted by the factorg⁽³⁾(r₁,r₂,r₃)/g(r₁₂). The latter gives the probability of finding a particle at r₃, given that there are particles atr₁ andr₂. The equation given above is the lowest order one in the so-called Yvon-Born-Green (YBG) hierarchy of equations relating reduced equilibrium proba- bility density functions of consecutive order. We see now that Eqs. (19) and (28) are limiting cases of Eq. (30), forρ →0.

To first order in ρ, the triplet distribution function factorizes according to g⁽³⁾(r₁,r₂,r₃) = g(r12)g(r13)g(r23), since the probability of finding three particles simultaneously within the correlation distanceξ is of orderρ². The substitution of this factorization approximation into Eq. (30) results into a closed, non-linear integro-differential equation for g(r) which can be solved numerically for a given u(r). This so-called Born-Green integral equation, however, gives poor predictions for theg(r)of dense liquids [2]. More reliable schemes to calculateg(r) based on the Ornstein-Zernike equation will be discussed in section 3.

2.3 Scattering experiments

In the following, I briefly explain how pair correlations can be measured indirectly through radiation scattering. A scattering experiment will have to probe distances of the order of the particle sizes and next neighbor distances, which are ˚Angstroms in case of atomic liquids, and fractions of microns in case of colloidal dispersions. Therefore, X-rays and neutrons are used for atomic liquids, whereas colloids can be probed by light scattering as well as small-angle neutron and synchrotron radiation scattering.

A schematic view of a scattering experiment is shown in Fig. 6. Monochromatic radiation of wavelengthλ impinges on a fluid sample, and is scattered at an angleϑinto a distant detector which measures the average intensity,I(q), of scattered neutrons or photons.

i (t) r

detector k

k

laser

q ϑ

Fig. 6:Schematic light scattering setup.

For single and quasi-elastic scattering from spherical particles [2, 7],

I(q)∝ ⟨N⟩P(q)S(q) (32)

whereq = (4π/λ) sin(ϑ/2)is the modulus of the scattering wave vectorq=k_f −k_i, and⟨N⟩ is the average number of particles in the illuminated volume part of the sample. The so-called form factor P(q), normalized as P(0) = 1, contains information on the scattering material distribution inside a particle, i.e., information on the particle size and form.

The most relevant quantity in Eq. (32) including information on inter-particle correlations is called the static structure factorS(q). Its statistical mechanical definition forq >0reads

S(q) = lim

∞

⟨ 1 N

∑N l,p=1

e^{iq·[rl−rp]}

⟩

= lim

∞

⟨

√1 N

∑N l=1

e^iq·rl

2⟩

≥0 (33)

where⟨· · · ⟩denotes an equilibrium ensemble average. The static structure factor is the T-limit of the autocorrelation function,

S(q) = lim

∞ SN(q) = lim

∞

⟨ 1

N ρqρ₋q

⟩

N

, (34)

of theq-th Fourier component,

ρ_q =

∑N l=1

e^iq·rl−N δ_q,0 , (35)

of microscopic density fluctuations of N particles. The prefactor 1/N renders S(q) into an intensive quantity. We have denoted the structure factor of the finite-volume NVT ensemble as S_N(q), and periodic boundary conditions have been used.

By expanding the double sum into self,l=p, and distinct,l̸=p, parts, it can be shown that S(q) = 1 +ρ

∫

dre^iq·rh(r) = 1 + 4πρ

∫ _∞

0

dr r²h(r)sin(qr)

qr , (36)

where,h(r), with

h(r) = g(r)−1 (37)

is called the total correlation function. Notice thath(r → ∞) = 0, and S(q → ∞) = 1. As a result, the static structure factor determines the Fourier transform,h(q), ofh(r). Since Fourier transforms are one-to-one mappings,S(q)can be inverted to determineh(r)and thusg(r):

g(r) = 1 +F⁻¹

[S(q)−1 ρ

]

(r) = 1 + 1 2π²ρ r

∫ _∞

0

dq q sin(qr)[S(q)−1]. (38) In principle, this would require to measureS(q)for all wave numbers q where S(q) exhibits significant oscillations. In light scattering experiments on colloids, this is usually not feasible since the largestqvalue accessible is limited byq_max = 4π/λ, which corresponds to backward scattering.

We have used here the following basic result on Fourier transforms: supposeh(r)is an isotropic function, withh(r)→0sufficiently fast forr→ ∞. The three-dimensional Fourier transform, h(q), ofh(r)is then defined as

h(q) := F{h(r)}=

∫

dre^iq·rh(r) = 4π

∫ _∞

0

dr r²h(r)sin(qr)

qr (39)

where the third equality follows from performing two angular integrals in spherical coordinates.

Fourier inversion leads then to h(r) =F⁻¹{h(q)}= 1

(2π)³

∫

dqe^−iq·rh(q) = 1 2π²

∫ _∞

0

dq q²h(q)sin(qr)

qr . (40) For notational simplicity, Fourier transformed functions are distinguished from their real-space counterparts by their argumentqonly.

λ

d = π 2 / k ϑ

ϑ

= q k

Fig. 7: First-order Bragg diffraction from a sinusoidal particle density wave of wavenumber k= 2π/d. Here,ϑis the scattering angle, andλthe wavelength of light in the solvent.

For an intuitive understanding of scattering from a fluid-like particle system we show in the fol- lowing that a measurement at a particularqcorresponds to diffusive first-order Bragg diffraction from a spatially sinusoidal particle density fluctuation aroundρ=⟨N⟩/V of the form

ρ_mk(r, t) = exp{−k²D_ct}ρ⁰_k exp{−ik·r} , (41) provided thatk = q, with a real-valued amplitude factorρ⁰_k. At each instant of time, a super- postion of such density fluctuations is thermally induced with wave vectorskof varying length and direction. Viewed from a coarse-grained level of resolution, the superposition of density waves

ρ_m(r, t) = ∑

k

ρ_mk(r, t) , (42)

is the general solution of the macroscopic diffusion equation

∂

∂tρ(r, t) = D_c∇²ρ(r, t) , (43) describing large-scalemeandensity fluctuations (as indicated by the overline) characterized by a collective diffusion coefficientD_c. For interacting particles, D_c can be substantially differ- ent from the diffusion coefficient of an isolated particle. By linearity of the diffusion equa- tion, also the real and imaginary parts of ρ(r, t) are solutions. A special density fluctuation of wavenumberk decays with the time constantτ_k = 1/(k²D_c). A fluctuation of larger wave- lengthd= 2π/krequires a longer time for its relaxation since the particles diffuse over a longer

distanced.

Consider now the scattering of a monochromatic radiation beam of wave vectorqfrom such a melange of density fluctuations. Note that Eq. (34) can be readily recast as

S(q) = lim

∞

⟨1 N

∫

V

drexp{iq·r}ρ_m(r)

2⟩

, (44)

with the microscopic particle density fluctuation ρ_m(r) = ∑

j

δ(r−r_j)−ρ . (45)

On identifyingρ_m(r)with its coarse-grained average ρ_m(r,0), one notices from Eq. (44) that S(q)is non-zero only for a density fluctuation of special orientation and wavelength such that k = q. The planes of constant phase of this sinusoidal density fluctuation are oriented at an angleϑ/2with respect to the incident radiation, withϑdenoting the scattering angle. Diffraction from these planes is governed by the Bragg condition (see Fig. 7)

2dsin(ϑ/2) =m λ (46)

of constructive interference of radiation scattered from two equal-phase neighboring planes, whereλis the radiation wavelength within the scattering medium. In this equation, we have to setm = 1since, different from an ideal crystal with sharply defined lattice planes, higher order Fourier components are absent for anexactlysinusoidally varying density fluctuation of infinite spatial extent. Indeed, using k = 2π/d andk = q in Eq. (46) with m = 1, we recover the equation

q = 4π

λ sin(ϑ/2), (47)

which relatesqfor quasielastic scattering from a fluid-ordered colloidal system to the scattering angle and radiation wavelength. Finally, we emphasize that the length scale, d = 2π/q, of probed density fluctuations is inversely proportional toq.

2.4 Thermodynamic properties

There exist various routes through which thermodynamic properties of the liquid can be related to integrals involvingg(r). In the following, we review the three most common ones.

The energy equation

E =⟨T_kin⟩+⟨

U(r^N)⟩

= 3

2N K_BT +1 2ρ N

∫ _∞

0

dr4πr²g(r)u(r) (48) expresses the internal energy, E, of a one-component N-particle system with pair-wise addi- tive U(r^N) in terms of u(r) and g(r). The internal energy is the sum of a kinetic ideal gas part,(3/2)N k_BT, and an interaction or excess part,⟨

U(r^N)⟩

. The latter can be understood on physical grounds as follows: For each particle out ofN, there are 4πr²ρ g(r)drneighbors in a spherical shell of radius r and thicknessdr, and the interaction energy between the central particles and these neighbors isu(r). Integration from0to∞gives the interaction energy part

ofE, with the factor1/2included to avoid double counting of particle pairs.

The pressure equation,

p=p_id+p_ex =ρk_BT − 2π 3 ρ²

∫ _∞

0

dr r³g(r)u^′(r) , (49) relates the thermodynamic pressure,p, of a liquid with pairwise additiveU to an integral over g(r)and the derivative,u^′(r), of the pair potential. Here,p_id =ρk_BT is the kinetic pressure of a classical ideal gas. The excess pressure contribution,p_ex, due to particle interactions can be de- rived along the same lines as the energy equation using, e.g., the classical virial equation [1, 8].

For a repulsive pair potential whereu^′(r)<0,p_intis a positive pressure contribution originating from the enhanced thermal bombardment of container walls by the mutually repelling particles.

For a non-pairwise additive potential energy which can be decomposed as U(r^N) =

∑N i<j

u(r_ij) +

∑N i<j<k

u₃(r_ij,r_ik) +· · · , (50) in terms of two-body, three-body and higher-order interaction terms, the energy and pressure equations generalize to [10]

lim∞

(E_ex N

)

= ρ 2

∫

dru(r)g(r) + ρ² 6

∫ ∫

drdr^′g⁽³⁾(r,r^′)u₃(r,r^′) +· · · (51) and

p_ex

p_id =−ρ 6

∫

drr g(r)βu^′(r)− ρ² 18

∫ ∫

drdr^′r∂β u₃(r,r^′)

∂r g⁽³⁾(r,r^′) +· · · (52)

The compressibility equation , χ_T

χ^id_T = lim

q→0S(q) = 1 +ρ

∫

dr[g(r)−1] , (53)

relates the isothermal compressibility,χ_T, of a one-component system, defined by χ_T :=−1

V (∂V

∂p )

T

= 1 ρ

(∂ρ

∂p )

T

(54) to an integral involving onlyg(r). Here,χ^id_T = (ρk_BT)⁻¹is the compressibility of an ideal gas.

The compressibility equation is more general than the energy and pressure equations, since it re- mains valid even when the interparticle forces are not pairwise additive. According to Eqs. (36) and (53), χ_T can be determined experimentally from measuring S(q) in the long-wavelength limitq→0, i.e., forqξ ≪1.

The compressibility equation can be derived in the grand canonical ensemble representing an open system, at constantV andT, which allows for fluctuations in the particle number. This is perfectly appropriate sinceS(q)is related to the intensity of quasi-elastically scattered radiation.

The radiation beam samples only a fraction of the system volume, and in this sub-volume the number of particles, while macroscopically large, fluctuates.

To derive the compressibility equation, we need thus the definition of the radial distribution function in the grand canonical ensemble. For a givenN, we know from Eq. (13) that

ρ²_Ng_N(r) = N(N −1)

∫

dr₃· · ·r_NP_N(r^N)

= N(N −1)

∫

dr₃· · ·r_NP_N(r^N)1 V

∫

dr₁dr₂δ(r−r₁₂)

= N(N −1)

V ⟨δ(r−r₁₂)⟩_N = 1 V

⟨ _N

∑

i̸=j

δ(r−r_ij)

⟩

N

(55) where⟨· · · ⟩_N denotes the fixedN, canonical ensemble average, andρ_N =N/V. The canonical gN(r)has been formulated by the most right equality in terms of an ensemble average invoking Dirac δ functions. To obtain the grand-canonical rdf, denoted by g_V(r), we merely have to replace the canonical average by the grand canonical one, denoted by⟨· · · ⟩_gc:

⟨(· · ·)⟩_N =

∫

dr_NP_N(r^N)(· · ·) → ⟨(· · ·)⟩_gc :=

∑∞ N

P(N)⟨(· · ·)⟩_N . (56) We have introduced here the grand canonical probability, P(N), of finding a system with ex- actlyN particles. The grand-canonical pair distribution function is thus given by

ρ²g_V(r) := 1 V

⟨ _N

∑

i̸=j

δ(r−r_ij)

⟩

gc

=

∑∞ N=2

P(N)ρ²_Ng_N(r) (57)

withρ=⟨N⟩_gc/V. Note here thatg_V(r) = g_N(r) +O(1/⟨N⟩_gc). For a fixedN, we know from Eq. (14) that

ρ²_N

∫

drg_N(r) = N(N −1)

V . (58)

For an open system of volumeV, this leads to ρ²

∫

dr [gV(r)−1] = ⟨N(N −1)⟩_gc

V − ⟨N⟩²_gc V =ρ

[⟨N²⟩_gc− ⟨N⟩²_gc

⟨N⟩_gc − 1 ]

(59) The variance of the number fluctuations in an open, macroscopic system is related to the isother- mal compressibility by the thermodynamic relation [6, 8]

ρkBT χT =

⟨(

N − ⟨N⟩_gc)2⟩

gc

⟨N⟩_gc . (60)

This completes our derivation of the compressibility equation, on noting thatg(r) = lim_∞g_V(r) in the thermodynamic limit of ⟨N⟩_gc → ∞ and V → ∞ with ρ fixed. The contradiction between the compressibility equation and the canonical ensemble result in Eq. (15) is only apparent. On first sight the latter might suggest thatlim_q→0S(q) = 0is exactly valid. However,

Eq. (15) applies only to a closed, finite-volume system with zero particle number fluctuations.

Notice further, forN andV finite and fixed (at given densityρ), that physically allowed wave numbers are restricted toq > V^−1/3: It makes no sense to consider particle density fluctuations of wave lengths (∼ q⁻¹) larger than the system size. Therefore, the thermodynamic limit of a macroscopic system should be performed first, followed by the limitq→0. In summary,

limq→0S(q) = lim

q→0lim

∞ S_N(q)̸= lim

∞ S_N(q= 0) = 0 , (61)

where SN(q) is the NVT static structure factor defined in Eq. (34). In the integral of the compressibility equation which relates to a thermodynamic quantity, one must use the radial distribution function in the thermodynamic limit (see also [9]).

The particles of a liquid near the triple point of gas-liquid-solid coexistence are densely packed so thatχ_T is very small. Furthermore, the compressibility is nearly zero for a classical crystal nearT = 0, since there are hardly any vibrations of the atoms around their equilibrium posi- tions. In contrast, χ_T diverges at a critical point which is the terminal point of the gas-liquid coexistence line. The divergence is accompanied by a long distance tail inh(r)which causes the phenomenon of critical opalescence observed in light scattering studies near critical points.

We will study critical opalescence in section 3.2. Summarizing, we note χ_T

χ^id_T ≈





0, fluid near triple point 0, ideal crystal

∞, fluid at critical point .

(62) The reader should be warned that Eqs. (51) and (52) forE andp, and also the compressibility equation as stated in Eq. (53), apply to state-independent, i.e., (ρ, T)-independent pair inter- actions only. Special considerations and care are required for state-dependent pair potentials u(r;ρ, T) and, more generally, for a state-dependent U(r^N) [11–17]. Effective potentials of this kind occur in size-asymmetric colloidal mixtures, and in liquid metals, in the process of av- eraging out the particle degrees of freedom associated with all but a single component of ’large’

particles, which may be colloidal particles in the case of colloid-polymer mixtures, or metal ions in case of liquid metals. The mixture is hereby mapped onto a one-component system of pseudo-particles, or decorated particles, whose interactions are governed by an effective, state- dependent interaction energyU^{ef f}(r^N). This mapping on an effective one-component system will be explained in section 4, and two prominent examples will be discussed.

In many practical situations, a mixture of large colloidal particles and small colloidal or poly- meric particles, dispersed in a solvent is in osmotic equilibrium with a reservoir of the small particles and solvent by means of a semi-permeable membrane impermeable to the large parti- cles. Since this fixes the chemical potentials,{µ^′}, of the small particles and the solvent, it is most convenient to treat them grand-canonically in order to deriveU^{ef f}(r^N;{µ^′}), so that, in case of a phase separation,U^{ef f} remains in the same form in both phases [18]. It is remarkable, that the osmoticcompressibility of the non-contracted-out large particles component is deter- mined by their pair correlations alone, independent of the large-small and small-small particle correlations, and independent of the so-called volume term (see section 4), through theosmotic compressibility equation of Kirkwood and Buff [19]

1 k_BT

(∂Π

∂ρ )

T,N,{µ^′}

= 1

S(q →0) . (63)

Here,Πis the osmotic pressure of large particles relative to the reservoir of small particles and solvent, andS(q)is the structure factor describing the large-large particle correlations. More- over, the{µ^′}denote the externally controlled chemical potentials of the small particles and sol- vent in the reservoir. To obtain the thermodynamics and phase behavior of the whole mixture, however, information on the pair correlations of all particulate components is required, includ- ing the large-small and small-small ones. Eq. (63) can be obtained from the multi-component extension of the compressibility equation in Eq. (53) in combination with the thermodynamic Gibbs-Duhem relation for the osmotic pressure of the large particles.

3 Ornstein-Zernike Integral Equation Methods

We proceed to discuss theoretical methods which allow to calculate theg(r)andS(q)of dense liquids from a given pair potential. All these methods are based on the so-called Ornstein- Zernike (OZ) equation, initially introduced by Ornstein and Zernike (1914) in their investiga- tions of critical opalescence in near-critical liquids. The OZ equation introduces the direct cor- relation function,c(r), as a very useful concept. Closed integral equations determiningg(r)can be derived from the OZ equation, whenc(r)is additionally related in some physically appealing approximation tog(r)andu(r). These additional relations are known as closure relations. We will introduce various closure relations, and discuss their merits and shortcomings.

3.1 Ornstein-Zernike equation and direct correlations

The Ornstein-Zernike equation of a homogeneous and isotropic system is given by [1]

h(r₁₂) =c(r₁₂) +ρ

∫

dr₃c(r₁₃)h(r₂₃) . (64) It introduces the direct correlation function,c(r), as a new function and can be viewed as the definition ofc(r)in terms of the total correlation function,h(r) = g(r)−1, of two particles a distancer =r₁₂apart. Eq. (64) can be recursively solved forh(r₁₂)to give

h(r12) = c(r12) +ρ

∫

dr3c(r13)c(r23) +ρ²

∫

dr3dr4c(r13)c(r24)c(r34) +O(c⁴) . (65) This leads to the following physical interpretation of the OZ equation: the total correlations be- tween particles 1 and 2, described byh(r₁₂), are due in part to the direct correlations,c(r₁₂), of these particles but also to an indirect correlation propagated by direct correlations via increas- ingly large numbers of intermediate particles.

Once information is available aboutc(r)in form of a closure relation involving u(r), the OZ equation can be viewed also as a closed integral equation forh(r). Some information on c(r) derives from Eq. (64) in the low-density limitρ→0wherec(r)→h(r). With Eq. (19) follows then

c(r)→f(r)for ρ→0 , (66)

where

f(r) :=e^−βu(r)−1 (67)

is called a Mayer-f function. It follows then that

c(r) = −βu(r) (68)

forr → ∞. Without proof we note that the long-distance asymptotic result (68) holds true for a wide class of pair potentials even at finite densities. The range ofc(r) is thus comparable with that of u(r), and the fact that h(r) is generally longer ranged than u(r) can be ascribed to indirect correlation effects. One word of warning: we refer here and in what follows to one-component liquids of electrically neutral particles. Ionic fluids must be distinguished from neutral fluids in that the effect of screening (cf. subsection 4.2) in such systems is to cause h(r)to decay exponentially at larger, whereasc(r)still has the range of the infinite Coulomb potential and therefore decays asr⁻¹. In principle, ionic liquids consist of at least two charged components of opposite sign to enforce overall charge neutrality.

To relatec(r)toS(q), we slightly rewrite the OZ equation as h(r) = c(r) +ρ

∫

dr^′c(|r−r^′|)h(r^′) (69) usingr = r₁₂, r^′ = r₂₃ and |r−r^′| = r₁₃. Fourier transformation of both sides of the OZ equation leads to

h(q) = c(q) +ρ c(q)h(q) (70)

or

ρ h(q) = ρ c(q)

1−ρ c(q) , (71)

wherec(q)is the three-dimensional Fourier transform of c(r). From noting that S(q) = 1 + ρ h(q), we obtainS(q)in terms ofc(q):

S(q) = 1

1−ρ c(q) ≥ 0, (72)

from which we learn thatρ c(q)≤1.

In the derivation of Eq. (70), we have employed the convolution theorem of Fourier transfor- mation theory. The convolution (german: ”Faltung”),f₁∗f₂, of two integrable functionsf₁(r) andf₂(r)is defined as

(f₁∗f₂)(r) :=

∫

dr^′f₁(r^′)f₂(r−r^′) =

∫

dr^′f₂(r^′)f₁(r−r^′) . (73) The convolution theorem states that

∫

dre^iq·r(f₁∗f₂)(r) = f₁(q)f₂(q) (74) i.e., the Fourier transform of the convolution of two functions is equal to the product of their Fourier transforms.

Using Eq. (72), we finally obtain the compressibility equation in terms ofc(q):

1

ρ k_BT χ_T = 1−ρ c(q→0) = 1−4πρ

∫ _∞

0

dr r²c(r) . (75)