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 = 21/6σ. The values ofσand ϵhave been determined for a large number of atoms using, e.g., atomic scattering techniques.
For argon, σAr = 0.34nm and ϵAr/kB = 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
1/6σ
[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
βuel(r) =LBZ2
( eκa 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/(kBT), andLB = e2/(ϵkBT)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 kBT. For water at room temperature, LB = 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 = 4πLB∑
αραZα2
1−Φ (4)
B
u(r)
k T u (r)
el) r ( u
vdW0
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 ρα and charge numbers Zα. 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) = uel(r) + uvdW, of uel(r) and the attractive van der Waals pair potential, uvdW(r). The van der Waals attraction between two identical colloidal spheres can be described approximately by [4, 5]
uvdW(r) =−Aef f 6
[ 2a2
r2−4a2 +2a2 r2 + ln
(
1−4a2 r2
)]
r > σ (5) withuvdW(r) ∼ −(Aef fσ6/36)/r6 for large r, and uvdW(r) ∼ −(Aef 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 constantAef f. For non-metallic colloidal spheres,Aef f is typically of the order of a fewkBT [4, 5]. Van der Waals forces between iden-tical particles are always attractive. Non-ideniden-tical 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, uvdW(r) becomes completely masked by the electrostatic partuel(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 potentialuel(r).
It is often assumed that the potential energy, U(rN), of aN-particle liquid system can be
ap-proximated by a sum of pair interactions U(rN)≈
∑N i<j
u(|ri−rj|) =
∑N i<j
u(rij), (6)
for any configurationrN ={r1,· · · ,rN}of position vectors{ri}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(rN) 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
PN(rN) = e−βU(rN)
ZN(V, T) , (7)
with configurational integral ZN(V, T) =
∫
dr1· · ·drN e−βU(rN)=
∫
drNe−βU(rN) , (8) is the probability density that the centers of the N particles are at the positions r1,· · · ,rN. 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(rN) (9) of finding any set of n particles at a specified configuration rn = {r1,· · · ,rn}, 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,· · · ,rn) =ρ(n)N (r1+t,· · · ,rn+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,· · · ,rn)≈
∏n i=1
ρ(1)N (ri) =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,gN(r1,r2), in theN V T-ensemble as
gN(r1,r2) := ρ(2)N (r1,r2)
ρ(1)N (r1)ρ(1)N (r2) , (12) so thatgN(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 andgN are functions of the separationr=r12only. Then,
gN(r) = ρ(2)N (r)
ρ2 = N(N−1) ρ2
∫
dr3· · ·drN PN(rN) (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 overgN(r)andu(r), provided that the particles interact by pairwise additive forces (cf. subsection 2.4).
As an important observation, we note thatρgN(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(rN) =N −1 (14)
since∫
dr12=V−1∫
dr1dr2. Likewise, Eq. (14) can be rewritten as 1 +ρ
∫
V
dr [gN(r)−1] = 0 . (15)
Notice for the canonical rdf thatgN(r → ∞) = 1−N−1, since there are N −1particles left aside from the one atr = 0. For largeN, we can identifygN(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 ×gN(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 gN(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→∞gN(r)≡lim
∞ gN(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 gN(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
1FS’02_5
g(r)
Fig. 4:Theg(r)of a Lennard-Jones liquid (left), and a fluid of hard spheres (right).