Liquid-State Physical Chemistry: Fundamentals, Modeling, and Applications (2013)
6. Describing Liquids: Structure and Energetics
6.6. The Potential of Mean Force
It is useful to define a quantity W(s)(r1, … , rs) by
where the definition of the correlation function g(s)(r1, … , rs) is given by
Taking logarithms on both sides, and then taking the gradient with respect to one of the particles, say i, one obtains
Since −∇iΦ represents the force on molecule i for a fixed configuration of r1, … , rs, the right-hand side is the mean force averaged over the configurations of all molecules s + 1, … , N not in the fixed set 1, … , s. Therefore,
Since a force Fj on particle j is always calculated from the corresponding potential Ψ according to Fj = −∂Ψ/∂rj, the quantity W(s)(r1, … , rs) can be interpreted as potential of mean force (fixating s particles).
In particular, for s = 2 we have W(r1,r2) ≡ W(2)(r1,r2), representing the potential of mean force between one particle held at position r1 and another held at position r2. The pair correlation function g(r1,r2) and the pair potential of mean force W(r1,r2) are thus related by
for isotropic systems. For r (= |r2 − r1|) → ∞, W(r) → 0 and g → 1. From W(r) = −kT ln g(r), we see that, providing that βW(r) << 1, g(r) permits the expansion
For low density ρ, W(r) → ϕ(r) because the two molecules considered are no longer affected by the other molecules. Hence, at low density g(r) reduces to
and there is a unique correspondence between ϕ(r) and g(r). At higher density one writes, using the background correlation function y(r),3)
Since d exp(−βϕ)/dr = −β exp(−βϕ)dϕ/dr, substituting Eq. (6.58) in the pressure Eq. (6.39) leads to the expression
To obtain P as a function of ρ, we need the expansion for y(r) in ρ reading (and indicating but not using second-order terms)
so that g(r) becomes g(r) = exp[−βϕ(r)]y(r) ≡ exp[−βϕ(r)](1 + A21ρ + A22ρ2 + …). In the same spirit, we write g(3)(r) = exp[−β(ϕ12 + ϕ12 + ϕ12)](1 + A31ρ + A32ρ2 + …). Substituting these expansions in the Yvon–Born–Green (YBG) equation (Eq. [7.5]),
we obtain for both sides of the YBG equation a polynomial in the density ρ. The next step is to equate equal order terms in ρ, and for the first-order terms in ρ this leads to
where C is the integration constant. Since for r12 → ∞, g(r) → 1, we obtain from g(r) = exp(−βϕ)(1 + A21ρ + …) as the boundary condition A21(r12) = 0 for r12 → ∞. Using f(r) = exp[−βϕ(r)] ≡ e(r)−1, this boundary condition is fulfilled if we write
Introducing Eq. (6.60) together with this result in Eq. (6.59), one obtains again the virial expression
Example 6.1: The hard-sphere fluid
To illustrate these concepts, let us apply them to a hard-sphere fluid with a potential given by Figure 6.12.
Figure 6.12 The hard-sphere potential ϕHS, the overlap function AHS, the correlation function gHS and the potential of mean force WHS as a function of r in units σ.
Hence, we have
The function A21(r) deviates from zero only if both f(r12) ≠ 0 and f(r23) ≠ 0 and for a hard-sphere fluid represents the volume of penetration of two spheres of radius σ/2 at distance r. Stereometry learns that this overlap function, labeled AHS, reads (Figure 6.12)
This implies that the correlation function gHS(r) for hard-sphere fluid becomes
representing exp(−βϕHS). The second factor in Eq. (6.71) results in a peak in gHS(r) (Figure 6.12). The potential of mean force correlation WHS(r) = −kT lngHS(r) reads
The net result is an effective attractive force for σ < r < 2σ (Figure 6.12). This attraction is thus purely a result of geometric restrictions. Since4) d exp(−βϕHS)/dr = −β exp(−βϕHS)dϕHS/dr = β exp(−βϕHS)δ(r−σ), the pressure P, given by Eq. (6.59), becomes
with η = πσ3ρ/6 the packing fraction5). By using Eq. (6.66), we obtain B2 = b ≡ 2πσ3/3 and B3 = 5b2/8; these results have already been derived in Chapter 5.
From Example 6.1 it becomes clear that the hard-sphere correlation function is not just a step function, as one might expect naively (Figure 6.5), but rather shows an increased value near the reference molecule by the geometric restrictions imposed on the coordination shell, leading to an effective attraction. Briefly summarizing, the correlation function g(r) on average should yield a value of 1. Since g(r) = 0 for r < σ, there should be a region where g(r) > 1. Because the potential of mean force W(r) = −kT ln g(r), the region where g(r) > 1 leads to W(r) < 0.
Check the steps necessary to obtain Eqs (6.63) and (6.65).
1) Remember that we use the word “molecule” as a generic term for atoms, ions, and molecules.
2) More formally, we require translational invariance, i.e. ρ(1)(r1) = ρ(1)(r1 + r1′) for any r1′ (obviously not too close to the wall). This is only possible if ρ(1) = C with C a constant. Since we have on the one hand ∫ρ(1)(r) dr1 = N, and on the other hand ∫ρ(1)(r) dr1 = C∫ dr1 = CV, we obtain CV = N or ρ(1) = N/V.
3) Also known as cavity function, as it describes the distribution of cavities in a hard-sphere fluid. Note that taking logarithms results in ln g(r) = −βϕ(r) + ln y(r) or ln y(r) = ln g(r) + βϕ(r) or ln y(r) = −β[W(r) − ϕ(r)] = ΔW(r). Here, ΔW(r) represents of the potential of mean force in excess over the interaction potential ϕ(r). This permits the expansion y(r) = exp[ΔW(r)] = 1 + ΔW(r) + ½[ΔW(r)]2 + … .
4) Note that because dh(r − r′)/dr = δ(r − r′), we have dϕHS(r − σ)/dr = −δ(r − σ). Moreover, note that direct integration of Eq. (6.39) cannot be done because gHS(r) is discontinuous at r = σ but can be done using Eq. (6.59) since y(r) is continuous at that point.
5) The abbreviation g(σ+) is defined by g(σ+) ≡ limr→σ+0 g(r), implying that r approaches σ from the positive side. Often, g(σ+) is given as g(σ).
1 de With, G. (2006) Structure, Deformation and Integrity of Materials, Wiley-VCH Verlag GmbH, Weinheim. This book provides a concise discussion on the structure, binding and defects of solids.
2 (a) Bernal, J.D. (1964) Proc. R. Soc. A280, 299; (b) Frost, H.J. (1982) Acta Metall., 30, 889.
3 Finney, J.L. (1970) Proc. R. Soc. A319, 279.
4 Coppens, P. (1997) X-Ray Charge Densities and Chemical Bonding, Oxford University Press.
5 Pings, C.J. (1968) X-Ray Charge Densities and Chemical Bonding (eds H.N.V. Temperley, J.S. Rowlinson, and G.S. Rushbrooke), North-Holland, Amsterdam, p. 389.
6 (a) Bernal, J.D. and Mason, J. (1960) Nature, 188, 910; (b) Bernal, J.D., Mason, J., and Knight, K.R. (1962) Nature, 194, 957; (c) For further elaboration, see Bernal, J.D. (1964) Proc. R. Soc., A280, 299.
7 (a) Scott, G.D. (1960) Nature, 188, 908; (b) Scott, G.D. (1962) Nature, 194, 956.
8 (a) Woodcock, L.V. (1971) Chem. Phys. Lett. 10, 257; (b) Woodcock, L.V. (1972) Proc. R. Soc., A328, 83.
9 Eisenstein, A. and Gingrich, N.S. (1942), Phys. Rev., 62, 261.
10 Yarnell, J.L., Katz, M.J., Wenzel, R.G., and Koenig, S.H. (1973) Phys. Rev., A7, 2130.
11 Yogi, T. (1995) J. Phys. Soc. Jpn, 64, 2886.
12 Dore, J.C. (1990) Il Nuovo Cimento, 12D, 543.
Friedman, H.L. (1985) A Course on Statistical Mechanics, Prentice-Hall, Englewood Cliffs, NJ.
Hansen, J.-P. and McDonald, I.R. (2006) Theory of Simple Liquids, 3rd edn, Academic, London.
McQuarrie, D.A. (1973) Statistical Mechanics, Harper and Row, New York.