11772
J. Phys. Chem. B 2001, 105, 11772-11777
Application of the Geometric Gibbs Equation: Toward an Exact Closure Condition† David S. Corti‡ School of Chemical Engineering, Purdue UniVersity, West Lafayette, Indiana 47907-1283 ReceiVed: May 3, 2001; In Final Form: July 27, 2001
The geometric Gibbs equation describes how the available space and corresponding surface area of a singlecomponent hard particle fluid varies with the system density. When a closure condition is introduced, i.e., an additional equation describing how the surface area depends on the available space, the geometric Gibbs equation reduces to a second-order differential equation indicating how the available space varies with the system density. Solution of this new equation provides another route to the determination of the chemical potential and pressure of the hard particle fluid. The simplest proposed closure condition yields the properties of fully penetrable spheres. A modified closure condition is suggested, and its connection to thermophysical properties is derived. An extension of the exact form of the closure condition for the one-dimensional hard rod fluid yields a reasonably good approximation of the properties of the hard sphere fluid at low density, and is found to be the required form for densities above the freezing density. The simple form of the closure condition and its connection to bulk properties may be used to help suggest future closure relations.
I. Introduction Hard sphere systems, due to the nature of the hard particle intermolecular potential, can be completely described by geometric quantities. The connection between thermophysical and geometric quantities, labeled statistical geometry,1 was first anticipated by Boltzmann,2 and later extended over the years by others.3-10 The statistical geometry of hard particle systems has yielded significant insights into the properties of hard particle systems. For example, various relations have been derived which place rigorous constraints on the behavior of various geometric quantities across the fluid-solid transition exhibited by both single-component and multicomponent hard particle systems.11-13 The two most important quantities of statistical geometry are the available space, Vo(d), and its corresponding surface area. Vo(d) is defined as the space available to insert another hard particle of diameter d into the system, and So(d) is the corresponding surface area that separates the available space from the unavailable space V - Vo (where V is the volume that the system occupies). These quantities are illustrated in Figure 1. Several exact geometric relations describing hard particle systems have been derived and have proved successful in furthering our understanding of these systems.1,5 For example, for a single-component D-dimensional hard sphere system, the chemical potential, µ, is given by3
µ ) kT ln
D
NΛ Vo(σ)
(1)
where Λ is the de Broglie wavelength, k is Boltzmann’s constant, T is the system temperature, N is the number of particles occupying the volume V, and Vo(σ) is the ensemble average of the available space to insert another hard particle of diameter σ. Also, the pressure, P, of the single-component hard † ‡
Part of the special issue “Howard Reiss Festschrift”. E-mail:
[email protected].
Figure 1. Two-dimensional representation of an instantaneous configuration of N hard spheres within a volume V. Each hard sphere has a diameter of size σ, represented by the solid inner circle. The dashed circles of diameter 2σ enclose the space into which the center of another sphere of diameter σ cannot enter. The volume outside the extended dashed boundary is the available space Vo(σ), and the extended boundary itself has area So(σ). The hard cores (solid circles) cannot extend beyond the boundaries of the system volume V.
sphere system can be expressed as7,11
P σ So(σ) )1+ FkT 2D Vo(σ)
(2)
where F ) N/V and So(σ) is the ensemble average of the surface area separating the available and unavailable space created by the attempted insertion of a particle of diameter σ. These relations are valid for both the hard sphere fluid and crystal.7 Previously, Reiss and co-workers11,12,14 derived a form of the Gibbs-Duhem equation
F
(∂µ∂F) ) (∂P∂F) T
10.1021/jp011669a CCC: $20.00 © 2001 American Chemical Society Published on Web 09/22/2001
T
(3)
Application of the Geometric Gibbs Equation
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11773
specialized to hard spheres at thermodynamic equilibrium. By substituting eqs 1 and 2 into the above relation, they derived a connection between the two fundamental geometric parameters underlying the hard sphere equation of state. By introducing the quantities
Vo ) Vo(σ)/N
(4)
so ) So(σ)/N
the so-called geometric Gibbs equation11,12,14 was obtained:
(
so -
)
(
)
2DVo dVo 2DVo dso Vo - Vo ) so + σ dF dF F σ
(5)
where Vo and so are functions only of the density F. This equation restricts the behavior of the statistical geometric quantities over the entire density range of hard particle fluids (including the solid phase) and therefore provides a strict consistency test on various proposed solutions of how Vo and so vary with density. Some of the consequences of the behavior of Vo and so as required by the geometric Gibbs equation have already been explored.11,12,14 II. Geometric Gibbs Equation: Toward an Exact Closure Condition The geometric Gibbs equation provides an exact description of the behavior of the available space and the corresponding surface area as the density of the hard particle system changes. Suggested relations describing both Vo and so, if thermodynamically consistent, must satisfy the geometric Gibbs equation. If, on the other hand, the dependence of the surface area upon the available space were known, the geometric Gibbs equation would reduce to a differential equation describing how the available space itself (or the surface area itself) depends on density. Solution of this new differential equation, which automatically satisifes the required thermodynamic consistency between Vo and so, would provide a new route to obtaining the chemical potential and the pressure of the hard particle fluid. The connection between Vo and so, which we call a closure condition of the geometric Gibbs equation, is of course not known. The purpose of this paper, however, is to suggest and investigate some approximate closure conditions. Future work will consider the derivations of more precise connections between Vo and so, which should improve our understanding of hard particle systems and lead to the development of more accurate equations of state. An initial closure condition is suggested by the following known relation between Vo(r) and So(r), where r now represents the radii of all the dashed spheres in Figure 1. Consider the augmentation of the radii of all the dashed spheres in Figure 1 by an infinitesimal amount dr. For a fixed configuration of the hard particles, the differential change in the available space is equal to11
dVo ) -So dr
(6)
〈So(r)〉 ) -
dVo(r) ) -So(r) dr
(7)
The above relation is valid for any configuration of the hard particles, so the ensemble average of the surface area, denoted by 〈‚‚‚〉, is given by
dVo(r) dr
(8)
Note that the available space and surface area in eqs 1 and 2 are already ensemble averages. The angular brackets denoting ensemble averaging were dropped for convenience. If eq 8 is evaluated at r ) σ, and both sides are divided by the particle number N, we find that
〈so(σ)〉 ) -
〈 〉 dVo(r) dr
r)σ
(9)
The case for which r ) σ corresponds to the attempted insertion of another hard particle of diameter σ, so that 〈so(σ)〉 is equivalent to so in eqs 4 and 5. In the thermodynamic limit (where only one configuration becomes important), the above expression reduces to11
〈so(σ)〉 ) -
(
)
d〈Vo(r)〉 dr
r)σ
(10)
Although eq 10 is exact, it is not relevant to the geometric Gibbs equation. The derivative in eq 10 is with respect to the radius of the exclusion sphere, and not with respect to the system density. To make a connection between r and F, we first change the derivative with respect to r evaluated at r ) σ to a derivative with respect to the hard core diameter σ, i.e.
(drd )
r)σ
f
d dσ
(11)
The derivative with respect to σ varies the hard core diameter and exclusion sphere simultaneously. Consequently, both derivatives in eq 11 describe variations in the exclusion sphere when the radius of the exclusion sphere is equal to σ. We therefore propose the following connection between the available space and the surface area:
so ) -
dVo dσ
(12)
where so and Vo (as in eq 2) now represent 〈so(σ)〉 and 〈Vo(σ)〉, respectively. Given that the properties of hard particle systems (with different values of N, V, or σ) become equivalent when expressed in terms of various reduced parameters, we introduce the following reduced density F*:7
F* )
NσD ) FσD V
(13)
Now, eqs 6-12 were obtained by considering systems in which the system volume V and the total number of particles N were held fixed. Thus, a change in the diameter σ of each hard particle, while keeping N and V constant, is equivalent to a change in the reduced density F* of the system. In other words
(dF* dσ )
N,V
or
〈 〉
) DFσD-1 )
DF* σ
(14)
If we now replace the derivative with respect to σ in eq 12 with a derivative with respect to F*, we have
so ) -
dVo dF* DF* dVo )dF* dσ σ dF*
(15)
where the explicit reference to holding N and V constant for
11774 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Corti
each derivative has been dropped. If the following reduced variables are now introduced
so/ ) so/σD-1
(16)
Vo/ ) Vo/σD and then substituted into eq 15, we find that
[
so/ ) -D Vo/ + F*
]
dVo/ dF*
(17)
This relation provides us with an additional connection between so and Vo that can be substituted into the geometric Gibbs equation. Before doing so, the geometric Gibbs equation, eq 5, must first be rewritten in terms of the various reduced parameters. Unlike eq 12, however, the geometric Gibbs equation was obtained by considering changes in the density F at constant hard core diameter σ. Thus, dF* ) σD dF, and so eq 5 is expressed in terms of the reduced available space, reduced surface area, and reduced density as follows:
(so/ - 2DVo/)
dVo/ dso/ Vo/ / - Vo/ ) (s + 2DVo/) (18) dF* dF* F* o
If eq 17 is now substituted into eq 18, we find that
F*Vo/
d2Vo/ d(F*)
- F* 2
( )
dVo/ 2 (Vo/)2 )0 dF* F*
(19)
Since Vo f 1/F as F f 0, let us define a new variable y ) F*Vo/ ) FVo that approaches unity as the density goes to zero. The variable y is simply the volume fraction of the available space since y ) FVo ) Vo/V. From eq 1, we also see that the excess chemical potential is related to y via βµex ) βµ - βµid ) -(ln y), where β ) 1/kT and µid ) kT ln FΛD is the chemical potential of an ideal gas at the same temperature and density. When rewritten in terms of y, eq 19 reduces to
yy′′ - (y′)2 ) 0
(20)
where y′ and y′′ represent the first and second derivatives of y with respect to the reduced density. The boundary conditions for the above differential equation are generated as follows. As the density of the hard particle fluid approaches zero, the available space is given by Vo ) V - VDN, where VD is the volume of a D-dimensional sphere with a radius equal to σ. Thus, in the low-density limit, Vo ) (1/F) - VD so that y ) 1 FVD. Consequently, y(0) ) 1 and y′(0) ) - VD. With these two boundary conditions, the above differential equation has the following solution
y ) F*Vo/ ) exp(-VD/F*)
(21)
which implies via eq 17 that
so/ ) -Dy′ ) DVD/ exp(-VD/F*)
(impenetrable) fluids. What did the closure condition, eq 12, fail to take into account? Although eq 8 is exact for fixed positions of particle centers, the replacement of the derivative with respect to the outer exclusion sphere with the derivative with respect to the hard core diameter implicitly assumes that no two sphere centers are within distances σ and σ + dσ of each other. The derivative with respect to the hard core diameter, despite changing the radius of the exclusion sphere, also varies the diameter of the hard core. Thus, an increase in the diameter of the hard core by dσ, while maintaining the fixed positions of the particles, causes two particles to overlap if their centers are separated by a distance between σ and σ + dσ. Hence, it is not surprising that the closure condition yields solutions that describe a system of fully penetrable particles. Equation 12 is valid only for very dilute systems, and clearly fails for systems at higher densities. An improved closure condition, one that accounts for the possible overlap of neighboring particles, must be used instead. Note, however, that to my knowledge the above derivation represents a new approach to determining the volume fraction and surface area of fully penetrable spheres. Although the results of this system are well-known, the relations of statistical geometry, i.e., the geometric Gibbs equation and the simple closure condition suggested by eq 12, provide another simple route to obtaining the properties of fully penetrable spheres. To account for those configurations in which particles may be within a center-to-center distance between σ and σ + dσ, we introduce the following modified closure condition:
(22)
where VD/ ) VD/σD (which equals 2, π, and 4π/3 in one, two, and three dimensions, respectively). The above solutions are known to be the volume fraction and surface area of a system of fully penetrable particles,15 i.e., randomly centered or spatially uncorrelated spheres, and so are poor descriptions of the geometric properties of hard particle
so ) -
dVo f dσ
(23)
where f is some function of density. The correction factor f is not known a priori, and so eq 23 may be considered a definition of f. Nevertheless, there is reason to believe that, in general, the value of f lies only between zero and unity. Consider the oversimplified case in which the volume V is occupied by only two hard spheres (D ) 3). If the centers of the hard spheres are separated by a distance z, the available space to insert another hard particle of diameter σ is given by
4π 3 πz3 σ - πσ2z + 3 12 8π 3 )V- σ 3
Vo ) V -
σ e z < 2σ (24) z g 2σ
and the surface area is equal to
So ) 4πσ2 + 2πσz ) 8πσ
2
σ e z < 2σ
(25)
z g 2σ
Thus, for a fixed distance z > σ we verify that So ) -dVo/dσ. When the spheres are in contact, such that z ) σ, we find that Vo ) V - 27πσ3/12, So ) 6πσ2, and So ) -(24/27)(dVo/dσ). One may therefore expect the correction factor f, which also accounts for the possible contact between three or more neighboring particles, to be less than unity. Future work will be concerned with a derivation of various approximations of f based on similar geometric ideas and on the form of eq 23. Although at this point eq 23 merely serves as a definition of the function f, we can still determine a connection between f and various thermophysical properties of hard particle systems. If eq 23 is rewritten in terms of the appropriate reduced variables, we have
Application of the Geometric Gibbs Equation
[
so/ ) -D Vo/ + F*
]
dVo/ f dF*
Now βµex ) -(ln F*Vo/) so that
F*
( ) ∂βµex ∂F*
)-
T
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11775
[
(26)
] ( )
/
dVo 1 ∂βPex V / + F* ) / o dF* ∂F* Vo
T
(27)
where the last equality follows from the Gibbs-Duhem equation (eq 3) with βPex ) βPσD - F*. Consequently
( )
(28)
F*so 2DVo
(29)
so/ ) DfVo/
∂βPex ∂F*
T
since eq 2 implies that
βPex )
f is therefore related to the excess pressure via the following relation:
f)
2βPex F*(∂βPex/∂F*)T
(31)
(32)
As F* f 0, we see that f f 1. Since g(σ+) for a hard particle system is a monotonically increasing function of the density, we conclude that 0 e f e 1. g(σ+) and its derivative typically diverge at some close-packed structure, either at one of several close-packed random configurations for the fluid phase17,18 or at the density of closest packing for the crystal phase. These close-packed structures correspond to rigidly jammed configurations which cannot be further compressed. (Unlike the closest packing for the crystal, rigidly jammed amorphous packings contain a small number of loose rattlers within the jammed matrix.19) The pressure diverges for these jammed systems, implying that the correction factor f goes to zero at some limiting density. We demonstrate below that the value of f does indeed lie between zero and one for the one-dimensional hard rod fluid. If eq 26 is substituted into the geometric Gibbs equation and the change of variables to y ) F*Vo/ is again performed, we obtain the following second-order differential equation:
yy′′ - (y′) + yy′R(F*) ) 0 2
∫0F*exp[-∫0R(ξ) dξ] d}
y ) exp{-VD/
(33)
where
∫0F*exp[-∫0R(ξ) dξ] d
(36)
(37)
providing an interesting connection between the excess chemical potential and the pair correlation function at contact. If the above equation is substituted into the Gibbs-Duhem relation, we also find that
∫0F* exp[-∫0R(ξ) dξ] d
βPex ) VD/
(38)
in which βPex ) 0 for F* ) 0. Finally, the closure condition provides a connection between y and so, revealing that
∫0F*R() d}] × F* [exp{-VD/ ∫0 exp[-∫0 R(ξ) dξ] d}]
so/ ) DVD/f[exp{-
(39)
All the above expressions are exact, if f is rigorously known. To test the validity of the above relations, let us consider the one-dimensional hard rod fluid. The pair correlation function at contact is known exactly20 and is equal to g(σ+) ) 1/(1 F*), where for the hard rod fluid F* ) Fσ and 0 e F* e 1 (F* ) 1 corresponds to the close-packed density of the hard rod fluid). Thus, the correction factor f is equal to
f)
2(1 - F*) 2 - F*
(40)
revealing that 0 e f e 1, in which f ) 1 at F* ) 0 and f ) 0 at F* ) 1. From eq 34, the above expression for f implies that
R(F*) ) F*(df/dF*) + 2f - 2 R(F*) ) F*f
(35)
Our choice of boundary conditions implies that the above expression for y is valid only for the hard particle fluid. Choosing boundary conditions appropriate for the solid phase would yield an expression for y valid for the hard particle crystal. (Unlike the fluid phase, however, the relevant boundary conditions for the solid are not known.) If the correction factor f is known, the volume fraction y can be determined. Since f is related to the derivative of the pair correlation function at contact via eq 32, eq 36 provides a direct connection between the available space Vo/ and g(σ+). Equation 36 also indicates that
βµex ) VD/
where B2/ ) B2/σo is the second virial coefficient, one finds, upon substituting the above equation into eq 30, that f is also given by
2 f) 2 + F*[d ln g(σ+)/dF*]
d ln(-z) + R(F*) dF*
where we have made use of the fact that z is always negative. Given that y(0) ) 1 and y′(0) ) -VD/, so that z(0) ) -VD/, we obtain the following exact solution for y in terms of the function f:
(30)
Again we see that a geometric quantity, in this case the correction factor f, is related to a bulk thermodynamic property. The above relation for f can also be rewritten in terms of the pair correlation function at contact, g(σ+). Noting that16
βPex ) B2/(F*)2g(σ+)
z′ + zR(F*) ) 0 )
F* - 3 (2 - F*)(1 - F*)
(41)
(34)
Introducing the new variable z ) y′/y allows us to rewrite eq 33 as
The function R is always negative; R ) -3 at F* ) 0 and R f -∞ as F* f 1. Substituting R into eqs 37-39 leads to the following results:
11776 J. Phys. Chem. B, Vol. 105, No. 47, 2001
[
y ) (1 - F*) exp -
F* 1 - F*
βµex ) -[ln(1 - F*)] +
Corti
]
F* 1 - F* (42)
(F*)2 βP ) 1 - F* ex
[
so/ ) 2 exp -
F* 1 - F*
]
g(σ+) )
The above relations match all previously known results for the hard rod fluid.8,20 We conclude this section by considering an extension of f for higher dimensional fluids based on the simple form of the correction factor for the hard rod fluid. Since the limiting density of the hard rod fluid is F* ) 1, we propose the following general form of f for the D-dimensional hard particle fluid:
f)
2(Fc/ - F*)
(43)
2Fc/ - F*
where Fc/ is the close-packed density of the D-dimensional fluid. Recent work17,18 shows that a unique randomly close-packed state does not exist; various jammed states of differing degrees of order, or disorder (however appropriately defined), and with different densities can be generated. Nevertheless, for our present purposes, we still refer to the notion of a single random closepacked density (an approximate value for the hard sphere fluid is Fc/ ≈ 1.2221). Note that the recent work of Speedy22 on the hard sphere glass transition suggests that eq 43 may not be an appropriate description of f for densities beyond the glass transition (where Fgt/ ≈ 1.12). Consequently, eq 43 does not account for the glass transition in the hard sphere fluid (the existence of the hard sphere glass transition22 is still an unresolved issue23,24). Future work will consider how the form of f must be modified to describe the hard sphere glass transition. Equation 43 still satisfies the condition that 0 e f e 1, where f f 1 as F* f 0 and f f 0 as F* f Fc/. The given choice of f implies that
R)
F* - 3Fc/
(44)
(2Fc/ - F*)(Fc/ - F*)
which when substituted into eq 36 predicts that
( )
y) 1-
F* Fc/
[
Fc*VD*/2
exp -
VD/Fc/F*
(
]
2(Fc/ - F*)
)
Fc/VD/ VD/Fc/F* F* ln 1 - / + βµ ) 2 Fc 2(Fc/ - F*)
(45)
(46)
and
βPex )
VD/Fc/(F*)2 /
2(Fc - F*)
1 - η/2 (1 - η)3
(48)
where η ) πF*/6 is the packing fraction. This form of g(σ+) yields the following expression for the correction factor f:
f)
2(1 - η)(2 - η) 4-η
(49)
Although different from eq 43, eq 49 still retains a simple form. This suggests that the search for an exact, or at least highly accurate, expression for f may be feasible. The CS expression for f predicts that f approaches zero at the unphysical value of η ) 1 (η ≈ 0.7405 corresponds to the densest possible packing fraction of identical spheres17). In addition, the CS equation increasingly diverges from simulation data for packing fractions larger than ηf ) 0.494, the density at which the hard sphere fluid freezes into the crystalline phase.21 This information implies that the nature and functional form of g(σ+) must be different in the density range between freezing and random close-packing. In fact, the value of [g(σ+)]-1decreases almost linearly from freezing to close-packing.25,23 This suggests that g(σ+) can be well represented by the following form:21
ηc - η f for ηf e η e ηc ηc - η
g(σ+) ) gf(σ+)
(50)
where gf(σ+) is the contact value at the freezing packing fraction, ηf, and ηc is the packing fraction at the random close-packed limit. This form of the contact value yields the following expression for the correction factor f:
f)
2(ηc - η) 2(Fc/ - F*) ) 2ηc - η 2F / - F*
(51)
c
This expression for y implies that the excess chemical potential and the excess pressure are given by ex
and 2π/3 in one, two, and three dimensions, respectively), yet eqs 46 and 47 yield poor predictions at moderate to high densities. The Carnahan-Starling (CS) equation of state16 is known to provide accurate predictions of the pressure and chemical potential of the hard sphere fluid (D ) 3). The contact value of the pair correlation function as predicted by the CS equation is given by21
(47)
These relations provide reasonable estimates of βµex and βPex at low densities; eq 47 predicts the correct value of the second virial coefficient, B2/ ) B2/σD ) VD//2 (which equals 1, π/2,
which is exactly the same expression for f suggested in eq 43. Again we note the work of Speedy,22 which also showed that [g(σ+)]-1 decreases almost linearly from the freezing density, but the slope of [g(σ+)]-1 was found to be discontinuous at the glass transition. For densities between the glass transition and close-packing, [g(σ+)]-1 still continues to decrease almost linearly with density but with a slope that has increased in magnitude by about 50%. How the form of f needs to be modified to account for this discontinuity will be the focus of future investigations. Now eq 33 can be solved for y with the above expression for f, but only for densities between the freezing density and random close-packing. In general, the solution is represented by
y ) y(Ff/) exp
{
y′(Ff/) y(Ff/)
∫FF*exp[-∫F R(ξ) dξ] d /
f
/
f
}
(52)
where y(Ff/) and y′(Ff/) are the values of y and its first derivative, respectively, at the freezing density, Ff/. Using eq 43, y is therefore given by
Application of the Geometric Gibbs Equation /
y ) y(Ff )
[ ] Fc/ - Ff/
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11777
(y′(Ff/)(Fc/-Ff/)2)/(y(Ff/)(2Fc/-Ff/))
Fc/ - F*
exp
×
[
y′(Ff/)
F*(Fc/ - Ff/)2
y(Ff/) (Fc/ - F*)(2Fc/ - Ff/)
]
(53)
As expected, y f 0 as F* f Fc/ (note that y′(Ff/) < 0). Although the value of y(Ff/) could be estimated from the CS prediction for f, we cannot determine the value of y′(Ff/). The different forms for f for densities below and above the freezing density suggest that y′ is discontinuous at the freezing point, preventing us from obtaining an estimate for y′ from an expression for y that is valid below Ff/. Nevertheless, eq 53 suggests a functional form for describing the volume fraction, and the surface area (via eq 26) of the (metastable) hard particle fluid at densities above the freezing density. III. Concluding Remarks We have proposed various closure conditions for the geometric Gibbs equation. The simplest closure relation, which allowed for the overlap of the hard particles, yielded expressions for the available space and the corresponding surface area of fully penetrable spheres. To account for the possible overlap of particles, a modified closure condition was introduced that included a correction factor f. The correction factor f was found to be related to the pair correlation function at contact. Consequently, f ranged from a value of unity at zero density to a final value of zero at some limiting close-packed density (i.e., 0 e f e 1). A form of f based on a simple extension of the exact result for the one-dimensional hard rod fluid yielded a reasonable equation of state of the hard sphere fluid at low densities. The simple form for f, however, was found to be consistent with the behavior of the pair correlation function at contact for a hard sphere fluid at densities between the freezing density and the density at random close-packing. The connection between f and the pair correlation function may be used to suggest other forms for f. In addition, the simple form of f, as obtained from both the hard rod and the CarnahanStarling equations of state, implies that the development of accurate approximations of the exact behavior of f may be feasible. Future work should also involve the derivation of f from geometric considerations, determining how the surface area
and available space are directly related (given various configurations in which two or more particles are in contact). Statistical geometry has so far yielded important insights into the behavior of hard particle fluids. The above relations, some of which are new, should also improve our understanding of hard particle systems. Acknowledgment. I am grateful to the Shreve Trust of the Purdue Research Foundation and to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research. An anonymous reviewer is also thanked for bringing the results of ref 22 to my attention. It is a privilege to contribute this paper in honor of Howard Reiss. References and Notes (1) Reiss, H. J. Phys. Chem. 1992, 96, 4736. (2) Boltzmann, L. Lectures on Gas Theory; Dover Publications: New York, 1995. (3) Reiss, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959, 31, 369. (4) Speedy, R.J. J. Chem. Soc., Faraday Trans. 2 1977, 73, 714. (5) Reiss, H. In Statistical Mechanics and Statistical Methods in Theory and Application; Landman, U., Ed.; Plenum: New York, 1977; Section IV. (6) Speedy, R. J. J. Chem. Soc., Faraday Trans. 2 1980, 76, 693. (7) Speedy, R. J. J. Phys. Chem. 1988, 92, 2016. (8) Speedy, R. J.; Reiss, H. Mol. Phys. 1991, 72, 999. (9) Speedy, R. J.; Reiss, H. Mol. Phys. 1991, 72, 1015. (10) Corti, D. S.; Bowles, R. K. Mol. Phys. 1999, 96, 1623. (11) Reiss, H.; Hammerich, A. D. J. Phys. Chem. 1986, 90, 6252. (12) Reiss, H.; Schaaf, P. J. Chem. Phys. 1989, 91, 2514. (13) Bowles, R. K.; Corti, D. S. Mol. Phys. 2000, 98, 429. (14) Schaaf, P.; Reiss, H. J. Chem. Phys. 1990, 92, 1258. (15) Torquato, S.; Lu, B.; Rubinstein, J. Phys. ReV. A 1990, 41, 2059. (16) McQuarrie, D. Statistical Mechanics; Harper & Row: New York, 1976. (17) Torquato, S.; Truskett, T. M.; Debenedetti, P. G. Phys. ReV. Lett. 2000, 84, 2064. (18) Truskett, T. M.; Torquato, S.; Debenedetti, P. G. Phys. ReV. E 2000, 62, 993. (19) Lubachevsky, B. D.; Stillinger, F. H. J. Stat. Phys. 1990, 60, 561. Speedy, R. J. J. Phys.: Condens. Matter 1998, 10, 4185. (20) Helfand, E.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1961, 34, 1037. (21) Torquato, S. Phys. ReV. E 1995, 51, 3170. (22) Speedy, R. J. Mol. Phys. 1998, 95, 169. (23) Rintoul, M. D.; Torquato, S. J. Chem. Phys. 1996, 105, 9258. (24) Williams, S. R.; Snook, I. K.; van Megen, W. Phys. ReV. E 2001, 64, 021506. (25) Song, Y.; Stratt, R. M.; Mason, E. A. J. Chem. Phys. 1988, 88, 1126.
