J. Phys. Chem. 1982,
86, 869-872
868
FEATURE ARTICLE Potential-Distribution Theory and the Statistical Mechanics of Fluldst B. Widom
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Department of Chemistry, Cornell University, Ithaca, New York 148S3 (Received: December 9, 1981; In Final Form: December 11, 1981)
By the potential-distribution principle of statistical mechanics the canonical average of the Boltzmann factor of the interaction potential as measured by a test particle is the ratio of the density to the thermodynamic activity. This principle has many extensions and applications both in uniform and in nonuniform fluids. In uniform fluids it leads to relations between the pair-correlation function and thermodynamic properties, while in nonuniform fluids it provides the most direct route to the mean-field theory of the density profiles of the interfaces in two-phase equilibrium and of the line of mutual contact in three-phase equilibrium.
It is my intention here to outline the ideas and major applications of what has come to be called the potentialdistribution theory (or “method”, or “theorem”) of statistical mechanics. It has proved very useful and has had applications in almost all areas of equilibrium statistical mechanics. The general principle has been known for nearly 20 years,1,2 and special cases of it for longer. Let us write the configurational partition function QN of a system of N particles in a volume V as Qn
=
^ V J\../vexp[-(Un-i
+
*)/kT\ diy-dr^H
(1)
elements of volume, [/N-i is the energy of interaction of particles 1.....N- 1 as a function of their configuration, and is the additional interaction energy due to the presence of the Nth particle (the “test” particle) at an arbitrary fixed point in the fluid. Equation 1 may
where the dr¡
are
then be rewritten QN
=
as
(2)
^V(N-l)!QN_1(e-*/*7’>
where (...) is a canonical average in the (N l)-particle system, in which the configurations of the N 1 particles are not influenced by the presence of the test particle. We have Qjv-i/Qjv = , the thermodynamic activity, and N/V -
-
=
p, the density; so
/
=
(3)
the basic form of our principle. The reason for the name “potential-distribution” theory is that, if ( ) is the potential distribution, that is, if ( ) dü' is the probability that the potential felt by the test + , then the canonical and particle is between of function of (...) may be evaluated as any average /... ( ) ¿ . Useful methods of approximation then result from approximations to ( ). Equation 3 has an interesting, recently discovered inverse.1 We write 23
=
(N =
=
1)!
-
(N-1)\V
Previously published in the Ned. Tijdschr. Natuurkd., A, 47, 4 H. Levelt Sengers.
(1981). Translated into Dutch by J. .
diY-dr^i
/vexPHt/N- *)/kT]
dTi~dTN
(Ñ^6» (4)
where ((...)) is an average different in kind from (...); it is a canonical average in the N-particle system, in which the particle that feels the potential is one of the N, and not a test particle, so that the configurations of the remaining N 1 particles are influenced by its presence. and N/V = p, so from (4) Again, Qn-i/Qn = -
\/p
=
((e*/kT))
(5)
the inverse of (3).
In a dense fluid of molecules that repel each other strongly at short distances expHk/fcT) is almost always very small, so it is difficult to apply (3) directly in the evaluation of the activity (or, equivalently, the configurational chemical potential µ = kT In ) from the molecular configurations that arise, say, in computer simulation; the configurations that ultimately give rise to a substantial value of p/\ occur only infrequently. The inverse (5), suffers from the opposite defect: exp( /kT) is almost always enormous, but there is a compensating, very small weighting factor in the average ((···)); and / must then be found as the result of a delicate cancellation, which is difficult to accomplish accurately. But techniques to overcome these handicaps have been developed, and these ideas, (3) especially, have now found valuable application in the evaluation of chemical potentials, even of dense fluids, from computer-generated configurations. Adams4 has done this for the fluid of hard spheres, and Romano and Singer6 for dense model fluids of molecules interacting with realistic forces; Powles6 has applied the method to (1) B. Widom, J. Chem. Phys., 39, 2808 (1963). (2) J. L. Jackson and L. S. Klein, Phys. Fluidis, 7, 279 (1964). (3) M. J. de Oliveira (with R. B. Griffiths), and later, independently, S. Shing (with K. E. Gubbins); private communications. (4) D. J. Adams, Mol. Phys., 28,1241 (1974). (5) S. Romano and K. Singer, Mol. Phys., 37, 1765 (1979). (6) J. G. Powles, Mol. Phys., 41, 715 (1980).
by K. f
fv"Sv “PHWW
Published 1982 American Chemical Society
870
The Journal
of Physical Chemistry,
Vol. 86, No. 6, 1982
Widom
calculate the liquid-vapor coexistence line; and Shing and Gubbins7 have made an important advance by combining the ideas of the potential-distribution theory with the techniques of umbrella sampling in Monte Carlo calculations. Further results along the lines of (3) are also known1,2 and have proved useful. For example, if the fluid is a mixture of two or more components a, b,..., then (3) applies separately to each species; thus
Pa/
(6) (expHka/AD) etc. the potential felt by a test particle that is a =
where , is molecule of species a while the average (...) is over the configurations of all the molecules of the mixture. As another example, in a one-component system of molecules in which the energy of interaction is the sum of the energies of interaction of pairs (“pairwise additive”), the mean
potential-energy density u
u is1
(1/2)\( 6-*/*7’)
=
(7)
A further relation of this kind yields the radial distribution function g(r) or the closely related pair-distribution and pair-correlation functions. The properties of these functions were expounded in the historic review by de Boer,8 9who showed also how essential is their role in the theory of fluids. From its definition8 (according to which pg(r) is the mean local density at the distance r from a molecule), and by an argument like that which led to (3), g(r) is found1,2 to be given by
g(r)e*^kT
( / ) 2attr(r/)p(r +
r') diy
(20)
functional of the density distribution. If p(r) changes over distances of order we will still have Xh«[p(r)]
ß [ „ ( )/ ]
in mean-field approximation,
(21)
in (16)—(18), but where = p(r), while the X without any indicated argument is still the spatially uniform thermodynamic activity. If p(r) also changes little over the range of attr(r0[p(r +
=
-
\
(23)
µ
-
f
2µ()d
*sr>c
(24)
positive constant. These are van der Waals’ famous formulae for determining the density profile of the liquid-vapor interface. Near the critical point we may expand µ( , ) µ in powers of p pc and T Tc, the deviations of p and T from their values at the critical point, and truncate the expansion after the leading terms. Thus a
-
µ( , )
p(z)
-
Pc
=
-
-
µ
-
=
-A(TC
From this and (23) tangent profile16 [A(TC
we
-
T)(p
-
pc)
+ B(p
pc)3
(25)
obtain the classical hyperbolic-
D/B]1/2 tanh ([A(TC
-
-
-
T)/2m]V2z] (26)
We may similarly analyze the lattice-gas model we defined earlier. Let -e (with e > 0) be the energy of interaction of molecules in neighboring cells; and let each cell at any height z have c'neighbors above it, at the height z + 1, likewise c'neighbors below it, at z 1, and then c 2c'neighbors in its own layer at height z, so that the total coordination number is again c. (Note that the present z is dimensionless; it is the height measured in units of the height of one cell.) The model excludes multiple occupancy of cells, so the potential felt by a test particle is +oo when there is a molecule of the fluid in the same cell with it. Then (19) becomes -
as
the configurational chemical potential µ
rewrite (21)
µ[ ( ), ]
=
is the second moment of -0attr(r) m
( )/
^ ) evaluated at the local p
now Xh,[p(r)] is
m
d2p(z)/dz2
-
little
=
-
>
(18)
=
871
Pi are
(17)
^( )ß-2 »/*
=
Vol. 86, No. 6, 1982
Equation 22 is a functional equation for the density distribution p(r), which is what we aimed for. In an interface p(r) is understood to be p(z), and then (22) is to
is van der Waals’s a. The probability may be identified from (15) as p/ for a fluid of hard spheres without attraction; call it /Xhs, where Xh, = ^( ) is to be evaluated at the density p of our model fluid. Then the equation of state of the fluid in this approximation is
,)
of Physical Chemistry,
r')
-
p(r)] diy
—
=
[1
[1 -
-
p(z)](exp(-*attr/feT)>,
p(z)]
exp^^^/ftT]
µ
(22)
Here p[p(r),T] kT In X[p(r),T] with \(p,T) given by (18), so it may be interpreted as the chemical potential of the fluid hypothetically constrained to be homogeneous at a density p(r) equal to the local density in the inhomogeneous fluid; while the µ with no indicated arguments is just the spatially uniform chemical potential of the inhomogeneous fluid. =
(14) C. A. Leng, J. S. Rowlinson, and S. M. Thompson, Proc. R. Soc. London, Ser. A, 352, 1 (1976). (16) B. Widom, J. St at. Phys., 19,663 (1978); C. Varea, A. Valderrama, and A. Robledo, J. Chem. Phys., 73, 6266 (1980).
(28)
the factor 1 p(z) being the probability that the test particle find an arbitrary cell at z empty. (This p is dimensionless, being the number density in units of the reciprocal cell volume.) Equation 27 is exact, and eq 28 is the mean-field approximation. We again have given by (20), which for the present model gives us ’iw(z) = -y)> and pc(x,y), where x and y are the two spatial coordinates perpendicular to the direction of the threephase line. (Clearly, nothing varies in the direction parallel to that line, which is also the direction that is simultaneously parallel to all three of the two-phase interfaces.) We
then take (6) in the form
Pa(x.y)Aa
=
(expHFg/kT))x
