Fluctuation Solution Theory - Fluctuation Theory of Solutions - page 12

the integral is over all space and the product is over all types of species ( s ) in the
mixture (Hill 1956). Here, n s is the number of molecules of type s in the n particle
probability distribution. Hence, we can evaluate the following integrals related to
the singlet and doublet distributions,

∫

() =

ρ

() 1

rdr N

1 1 i

i

(1.34)

∫∫

( )

() 2

ρ rr dr dr , = NN − δ N

ij 12 12 i j i j i

Various combinations of the above integrals are commonly encountered in statistical
thermodynamic theories of solutions. The most relevant is given by,

∫∫

()

( ) −

()

() ()

()

2 1 1

ρ rr , ρ r ρ r dr dr = δδδ NN − δ N (1.35)

ij 12 i 1 j 2 1 2 i j ij i

and is identical in form to the integrals appearing in the theory of imperfect gases and
the McMillan-Mayer theory of solutions (McMillan and Mayer 1945).
A set of grand canonical distribution functions g ij () can then be defined for species
i and j by,

()

() =

() ()

( ) =

() (, )

1 1 2 2

ρ r gr c () = ρ r r , cg rr (1.36)

i 1 i i 1 i ij 12 ij ij 12

which then provide expressions for integrals over the pair ( n = 2) distribution functions,

rdrV NN
NN

δδ

∞

− δ

∫

() −

i j

() 2 2 ij

GG == 4 π gr 1 = (1.37)

ij ji

ij

c

0

i j

where we have integrated over the position of the central particle and only con-
sider the scalar interparticle distance, r = ∙ r 2 - r 1 ∙ . At this point we have related the
particle number fluctuations to integrals over radial distribution functions (RDFs)
in the grand canonical ensemble. FST does not require information on the angular
distributions for pairs of molecules—these are averaged out in the above expressions
(see Chapter 6). The RDFs correspond to distributions obtained in a solution at the
composition of interest, after averaging over all the remaining molecular degrees of
freedom. A typical RDF and the corresponding integral are displayed in Figure 1.1.
The G ij s are known as Kirkwood-Buff integrals (KBIs) and are the central compo-
nents of FST (Kirkwood and Buff 1951).
The KBIs quantify the average deviation, from a random distribution, in the dis-
tribution of j molecules surrounding a central i molecule summed over all space.
In this respect they are more informative than the particle number fluctuations as
they can then be decomposed and interpreted in terms of spatial contributions—
using computer simulation data, for example. They clearly resemble the integrals