## 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)}(**r**_{1}, … , **r*** _{s}*) by

(6.51)

where the definition of the correlation function *g*^{(s)}(**r**_{1}, … , **r*** _{s}*) is given by

(6.52)

Taking logarithms on both sides, and then taking the gradient with respect to one of the particles, say *i*, one obtains

(6.53)

Since −∇* _{i}Φ* represents the force on molecule

*i*for a fixed configuration of

**r**

_{1}, … ,

**r**

*, the right-hand side is the mean force averaged over the configurations of all molecules*

_{s}*s*+ 1, … ,

*N*not in the fixed set 1, … ,

*s*. Therefore,

(6.54)

Since a force *F _{j}* on particle

*j*is always calculated from the corresponding potential

*Ψ*according to

*F*= −∂

_{j}*Ψ*/∂

**r**

*, the quantity*

_{j}*W*

^{(s)}(

**r**

_{1}, … ,

**r**

*) can be interpreted as potential of mean force (fixating*

_{s}*s*particles).

In particular, for *s* = 2 we have *W*(**r**_{1},**r**_{2}) ≡ *W*^{(2)}(**r**_{1},**r**_{2}), representing the *potential of mean force* between one particle held at position **r**_{1} and another held at position **r**_{2}. The pair correlation function *g*(**r**_{1},**r _{2}**) and the pair potential of mean force

*W*(

**r**

_{1},

**r**

_{2}) are thus related by

(6.55)

for isotropic systems. For *r* (= |**r**_{2} − **r**_{1}|) → ∞, *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

(6.56)

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

(6.57)

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)}

__(6.58)__

Since d_{ }exp(−*βϕ*)/d*r* = −*β* exp(−*βϕ*)d*ϕ*/d*r*, substituting __Eq. (6.58)__ in the pressure __Eq. (6.39)__ leads to the expression

__(6.59)__

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)

__(6.60)__

so that *g*(*r*) becomes *g*(*r*) = exp[−*βϕ*(*r*)]*y*(*r*) ≡ exp[−*βϕ*(*r*)](1 + *A*_{21}*ρ* + *A*_{22}*ρ*^{2} + …). In the same spirit, we write *g*^{(3)}(*r*) = exp[−*β*(*ϕ*_{12} + *ϕ*_{12} + *ϕ*_{12})](1 + *A*_{31}*ρ* + *A*_{32}*ρ*^{2} + …). Substituting these expansions in the Yvon–Born–Green (YBG) equation (Eq. [7.5]),

(6.61)

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

(6.62)

Integration yields

__(6.63)__

where *C* is the integration constant. Since for *r*_{12} → ∞, *g*(*r*) → 1, we obtain from *g*(*r*) = exp(−*βϕ*)(1 + *A*_{21}*ρ* + …) as the boundary condition *A*_{21}(*r*_{12}) = 0 for *r*_{12} → ∞. Using *f*(*r*) = exp[−*βϕ*(*r*)] ≡ *e*(*r*)−1, this boundary condition is fulfilled if we write

(6.64)

Introducing __Eq. (6.60)__ together with this result in __Eq. (6.59)__, one obtains again the virial expression

__(6.65)__

__(6.66)__

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__.

(6.67)

** Figure 6.12** The hard-sphere potential

*ϕ*

_{HS}, the overlap function

*A*

_{HS}, the correlation function

*g*

_{HS}and the potential of mean force

*W*

_{HS}as a function of

*r*in units

*σ*.

Hence, we have

(6.68)

(6.69)

The function *A*_{21}(*r*) deviates from zero only if both *f*(*r*_{12}) ≠ 0 and *f*(*r*_{23}) ≠ 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 *A*_{HS}, reads (__Figure 6.12__)

(6.70)

This implies that the correlation function *g*_{HS}(*r*) for hard-sphere fluid becomes

__(6.71)__

representing exp(−*βϕ*_{HS}). The second factor in __Eq. (6.71)__ results in a peak in g_{HS}(*r*) (__Figure 6.12__). The potential of mean force correlation *W*_{HS}(*r*) = −*kT* ln*g*_{HS}(*r*) reads

(6.72)

The net result is an effective attractive force for *σ* < *r* < 2*σ* (__Figure 6.12__). This attraction is thus purely a result of geometric restrictions. Since__ ^{4)}__ d exp(−

*βϕ*

_{HS})/d

*r*= −

*β*exp(−

*βϕ*

_{HS})d

*ϕ*

_{HS}/d

*r*=

*β*exp(−

*βϕ*

_{HS})

*δ*(

*r*−

*σ*), the pressure

*P*, given by

__Eq. (6.59)__, becomes

(6.73)

with *η* = π*σ*^{3}*ρ*/6 the packing fraction__ ^{5)}__. By using

__Eq. (6.66)__, we obtain

*B*

_{2}=

*b*≡ 2π

*σ*

^{3}/3 and

*B*

_{3}= 5

*b*

^{2}/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.

Problem 6.8*

Check the steps necessary to obtain __Eqs (6.63)__ and __(6.65)__.

Notes

__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)}(**r**_{1}) = *ρ*^{(1)}(**r**_{1} + **r**_{1}′) for any **r**_{1}′ (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**)_{ }d**r**_{1} = *N*, and on the other hand ∫*ρ*^{(1)}(**r**)_{ }d**r**_{1} = *C*∫_{ }d**r**_{1} = *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)*] = Δ*W*(*r*). Here, Δ*W*(*r*) represents of the potential of mean force in excess over the interaction potential *ϕ*(*r*). This permits the expansion *y*(*r*) = exp[Δ*W*(*r*)] = 1 + Δ*W*(*r*) + ½[Δ*W*(*r*)]2 + … .

__4)__ Note that because d*h*(*r* − *r*′)/d*r* = *δ*(*r* − *r*′), we have d*ϕ*_{HS}(*r* − *σ*)/d*r* = −*δ*(*r* − *σ*). Moreover, note that direct integration of __Eq. (6.39)__ cannot be done because *g*_{HS}(*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*^{+}) ≡ lim_{r}_{→σ+0} *g*(*r*), implying that *r* approaches *σ* from the positive side. Often, *g*(*σ*^{+}) is given as *g*(*σ*).

References

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.

Further Reading

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.