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J. Phys. Chem. B 2001, 105, 11662-11668
Quasi-Chemical Theory for the Statistical Thermodynamics of the Hard-Sphere Fluid† Lawrence R. Pratt,*,‡ Randall A. LaViolette,§ Maria A. Gomez,| and Mary E. Gentile| Theoretical DiVision, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, Idaho National Engineering and EnVironmental Laboratory, P.O. Box 1625, Idaho Falls, Idaho 83415-2208, and Department of Chemistry, Vassar College, Poughkeepsie, New York 12603 ReceiVed: April 23, 2001; In Final Form: July 24, 2001
We develop a quasi-chemical theory for the study of packing thermodynamics in dense liquids. The situation of hard-core interactions is addressed by considering the binding of solvent molecules to a precisely defined cavity in order to assess the probability that the cavity is entirely evacuated. The primitive quasi-chemical approximation corresponds to an extension of the Poisson distribution used as a default model in an information theory approach. This primitive quasi-chemical theory is in good qualitative agreement with the observations for the hard-sphere fluid of occupancy distributions that are central to quasi-chemical theories but begins to be quantitatively erroneous for the equation of state in the dense liquid regime of Fd3 > 0.6. How the quasichemical approach can be iterated to treat correlation effects is addressed. Consideration of neglected correlation effects leads to a simple model for the form of those contributions neglected by the primitive quasi-chemical approximation. These considerations, supported by simulation observations, identify a “break away” phenomena that requires special thermodynamic consideration for the zero (0) occupancy case as distinct from the rest of the distribution. An empirical treatment leads to a one-parameter model occupancy distribution that accurately fits the hard-sphere equation of state and observed distributions.
Introduction The quasi-chemical theory1-3 is a fresh attack on the molecular statistical thermodynamic theory of liquids. It is intended specifically to be appropriate in describing liquids of genuinely chemical interest. But in view of its generality, the quasi-chemical theory must be developed and tested for its description of the paradigmatic hard-sphere fluid. In addition to the conceptual point, these developments are expected to be helpful in subsequent applications of the quasi-chemical theory to real solutions. The foundational virtues of the hard-sphere fluid for the theory of liquids are widely recognized,4,5 and the interest in this system continues to evolve.6-11 Recent developments of the theory of hydrophobic effects,3,12-24 in addition the related quasi-chemical theory, have emphasized again the significance of packing issues in a realistic molecular description of complex liquids. This paper studies the hard-sphere fluid and develops default models with utility in recent information theory approaches.3,15-23,25 The idea for the development below is the following: The quasi-chemical approach defines a near neighborhood of a distinguished molecule and pursues a detailed description of the occupancy statistics for that neighborhood. At the same time, these approaches anticipate that a rough approximation will be satisfactory for the remainder of the volumesthe outer-sphere region. This is a natural idea for the cases where the interactions of the distinguished molecule with near neighbors would be strong or complicated as is common for chemical problems. But several difficulties arise in carrying out this idea for fluids †
Part of the special issue “Howard Reiss Festschrift”. * Corresponding author. ‡ Los Alamos National Laboratory. § Idaho National Engineering and Environmental Laboratory. | Vassar College.
without lattice idealizations. One complication is that liquids of common interest are dense, disordered materials and thus the “packing” issues are foundational. In the available quasichemical organization of the theory of solutions, these packing contributions are associated with reserving adequate space for chemical structures, and the descriptions of the packing effects ultimately are found predominately in the outer-sphere contributions. To tackle the packing issues means that the outer-sphere contributions must be considered in further detail. To use quasi-chemical structures to analyze those outer-sphere interactions, we adopt a trick. We consider a hypothetical solute that does not interact with other constituents of the solution. For such a solute, the inner-sphere and outer-sphere contributions must add to zero. The necessary characteristics of this hypothetical species can be adjusted so that the outer-sphere contributions are just the same as those of the hard-core solution species of first interest. For such a hypothetical solute, consequently, we can study the defined inner-sphere terms and get the balancing result for the desired outer-sphere contributions. A compact derivation requires several preliminary results, specifications of the potential distribution theorem, the expression of chemical equilibrium, and the quasi-chemical formulation. Additionally, the notation here is not elsewhere standardized because these ideas are unconventional. Beyond the present notice, we will attempt to avoid disruptive citations of appendix material. The reader can look into the appendices first for elaborative underpinnings. The plan of the paper is thus to collect the necessary preliminary results in Appendix A so that the conceptual argument needn’t be interrupted. Then we derive the new equation of state format, learn what we can by comparison of the primitive quasi-chemical approximation with Monte Carlo simulation results, study correlation contributions to propose an improved equation of state format, and finally examine how this improved format works.
10.1021/jp011525w CCC: $20.00 © 2001 American Chemical Society Published on Web 09/11/2001
Statistical Thermodynamics of Hard-Sphere Fluid
J. Phys. Chem. B, Vol. 105, No. 47, 2001 11663 TABLE 1: “Hit-or-Miss” Monte Carlo Estimates,26 as Described in Appendix B, of ln Kn(0) for Unit Diameter Hard Spheres and Disks, Respectivelya n
spheres (3d)
disks (2d)
1 2 3 4 5 6 7
1.43 1.54 0.566 -1.47 -4.68 -9.17 -15.5
1.14 0.71 -1.19 -5.24 -13.8 -
a K1(0) ) 4π/3 (π) and K2(0) ) 17π2/36 (3x3π/8). The sample size was 24 G-configurations, and the results are believed to be accurate to the number of significant figures given.
with λ a mean field factor that achieves the self-consistency condition Figure 1. Example of an AS7 cluster considered in the text. The nucleus (A) is visible in the center. Each of the ligands (S) overlaps the nucleus but no other ligand.
Interestingly, though some of these basic considerations are regarded as “preliminary,” eq 2 (or 5) and the formal identification of the equilibrium ratio eq A.4 have not been given before and have wide generality. 2. Quasi-Chemical View of the Solvation Free Energy of Hard-Core Solutes Let’s consider a solute A that does not interact with the solvent S molecules at all. We will consider formation of ASn complexes and Figure 1 depicts such a cluster. The interaction contribution β∆µA is zero, and the inner- and outer-sphere contributions (eq A.5) balance to give
ln[〈〈
∏j (1 - bAj)〉〉0] ) ln x0
(1)
But the left side here involves a test particle average for a molecule that rigidly excludes the solvent from the region defined by the indicator function bAj. If that region is taken as defining a physically interesting molecule pair excluded volume, then the left side of eq 1 gives the negative of the excess chemical potential for the hard-core solute defined by bAj. This is an example of the well-known relation for hard-core solutes ln p0 ) -β∆µHC with “HC” denoting “hard core”.3,12-23 We next tackle the right side of eq 1 from a quasi-chemical perspective as in eq A.10 and consider “chemical” equilibria for binding of S molecules to the A molecule. Of course, there is no interaction between the A molecule and the solvent molecules. The binding is just the occupancy by solvent molecules of the cavity defined by bAj. Combining these considerations gives
β∆µHC ) ln[1 +
KmFSm] ∑ mg1
(2)
The Km values are typically computationally demanding but well-defined. The evaluation of Km(0) will require few-body integrals over excluded volumes, as discussed in Appendix B. The primitive quasi-chemical approximation is
β∆µHC ≈ ln[1 +
∑ Km
FS λ ]
(0)
mg1
m m
(3)
∑n nKn(0)FSnλn ) 〈n〉∑n Kn(0)FSnλn
(4)
This amounts to an extension of the Poisson distribution for use in an information theory procedure.21 〈n〉 ) FSK1(0) is the expected occupancy of the volume stenciled by bAj. Thus, the multiplicative factors of FS in xn(0) ∝ Kn(0)FSn are augmented by a self-consistent mean field λ. (This point has a twist for the non-profound one dimensional problem: This primitive quasichemical approximation is exact for the one dimensional “hard plate” system. But as a distribution without evaluation of the mean field factors, this distribution is an exceedingly weak theory. Evaluation of the mean field produces the exact answer because the number of statistical possibilities is only 2. The situation for the continuum analog, the Poisson distribution, is different. It is not accurate as a distribution, and the analogous information theory interpretation also gives an approximate result for the one dimensional hard plate system.) For reuse below, we summarize the technical results of this argument for hard-core solutes, writing
1
〈〈e-β∆UHC〉〉0 ) 1+
(5)
∑ K mF S
m
mg1
This combines eq 2 and the potential distribution theorem for this problem. 3. Primitive Quasi-Chemical Approximation We can give a simple demonstration of the quantitative results of the primitive quasi-chemical theory by considering the hard disk (2d) and hard-sphere fluids (3d). Table 1 gives Monte Carlo estimates of the Kn(0) for those cases. The predicted distributions xn for two densities are in Figures 2 and 3. Equation of state results β∆µ(F) for these systems predicted by this primitive quasi-chemical theory are shown in Figures 4 and 5. The primitive quasi-chemical approximation is remarkably successful in all qualitative respects, particularly in view of its simplicity. The predicted occupancy distributions such as shown in Figure 3 are remarkably faithful to the data. Nevertheless, the equation of state predictions begin to incur progressively serious quantitative errors at liquid densities Fd3 > 0.6 (Figure 5). 4. Test of the Equilibrium Ratios As a direct check on the primitive quasi-chemical mechanism, we can focus on the approximation Kn ≈ Kn(0). It is then natural
11664 J. Phys. Chem. B, Vol. 105, No. 47, 2001
Figure 2. For the hard-sphere fluid at FSd3 ) 0.277, comparison for n e 5 of the Poisson distribution (solid curve) with primitive quasichemical distribution (dashed curve) implemented with the information theory constraint on the first moment ∑nxn ) 4πrSd3/3. The dots are the results of Monte Carlo simulation,22 as discussed in Appendix D. The primitive quasi-chemical default model depletes the probability of high-n and low-n constellations and enhances the probability near the mode.
Pratt et al.
Figure 5. β∆µ(F) for the three-dimensional hard-sphere fluid on the basis of the primitive quasi-chemical approximation (dashed line). The solid line is the prediction of the Carnahan-Starling equation of state, taken as the accurate basis for comparison, and the dashed-dotted line is the first virial coefficient approximation.
prescribed density. We now consider how to go beyond that mean field description. One idea is to extract the features of the summand of eq 5 that would give purely geometric weighting and then to analyze what remains. To this end, we define ζ ) exp(β∆µ) and consult the formal identification of the equilibrium ratios eqs A.4. Thus, we can rewrite eq 5 as
ζ)1+
∑ x0/mKm(0)ζmFSm
(6)
mg1
with
x0/n ≡ Figure 3. As in Figure 2 but for FSd3 ) 0.8. The error bars indicate the statistical uncertainty by showing the 67% confidence interval.
〈〈e-β∆Un〉〉0
(7)
〈e-β∆U〉0
The remarkable eq 6 is formally exact and has not been given before. The correlation factors x0/m might, in principle, be investigated on the basis of simulation data and information theory analysis. That is likely to a specialized nontrivial activity except for the lower density cases where the primitive quasichemical approximation is satisfactory. 5.1. Iterating the Quasi-Chemical Analysis. Alternatively, the quasi-chemical rules suggest natural theoretical approximation for the equilibrium ratios given formally by eq A.4. Applying the rule eq 5, for n > 0
Kn ) 〈〈e-β∆Un〉〉0Kn(0)ζn+1 )
Kn(0)ζn+1 1+
Figure 4. β∆µ(F) for the two-dimensional hard disk fluid on the basis of the primitive quasi-chemical approximation (dashed lined). The ReeHoover 3,3 Pade´ approximant27 is the upper solid line, and the lower solid line is the first virial coefficient approximation.
Kn(0)ζn+1
≈ 1+
to consider the ratios xn/x0 ) KnFSn. Consideration of these ratios corresponds to shifting the curves of Figures 2 and 3 so that the initial point is at the common value (0,1). A specific example is shown in Figure 6. Compared with this normalization, it is clear that the observed equilibrium ratios Kn are greater than the ideal ratios Kn(0). 5. Correlations A point of view here is that the geometric weighting with the λ’s of eq 3 establishes a mean field that adapts to the
Km/nFSm ∑ mg1
∑ Km/n
(8) (0)
FS
m
mg1
The Km/n can be understood by considering the chemical equilibrium
AS′nSm)0 + mS h AS′nSm
(9)
i.e., the original AS′n cluster is the solute, and it provides a nucleus for a constellation of m S particles, different in type for the S′ species. How to address the calculation of the Km/n(0) is discussed in Appendix C.
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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11665
Figure 6. ln[(xn/x0)qca/(xn/x0)sim] vs n comparing for the hard-sphere fluid the primitive quasi-chemical approximate populations with those observed by Monte Carlo simulation for Fd3 ) 0.8. This normalization focuses on the relative sizes of Kn and Kn(0) suggesting that Kn > Kn(0) even after the maxent reweighting. The variations are modest except for the change between n ) 0 and n ) 1. The error bars indicate the statistical uncertainty by showing the 67% confidence interval. In the middle of the distribution, the statistical uncertainty is mostly from the uncertainty in the denominator factor x0.
It is still helpful to focus on the populations even though more coefficients are involved now. To do this, we consider
ζ)1+ζ
∑ 〈〈e
-β∆Um
〉〉0Km FS ζ (0)
m m
(10)
mg1
and, to accommodate the additional factor of ζ multiplying the terms m g 1, rearrange so that
〈〈e-β∆U 〉〉0Km(0)FSmζm ∑ mg1 m
ζ)1+
1-
∑ 〈〈e
Figure 7. Equation of state with the empirical correlation model, eq 14. (lower panel) The solid line is the Carnahan-Starling equation of state, and the dashed line is the model of eq 14. (upper panel) Discrepancy: the empirical correlation model (eq 14) less the Carnahan-Starling value. The mean absolute discrepancy against CarnahanStarling equation of state is about 1%, and the maximum discrepancy is less than 3%, nearly as good a conformance to the Carnahan-Starling model as that of the model to simulation data. When the final empirical parameter was fitted using only FSd3 e 0.3, the mean absolute discrepancy hardly changed, but the maximum discrepancy doubled.
(11) -β∆Um
〉〉0Km FS ζ (0)
m m
mg1
This last equation is significant particularly because it suggests that a principal consequence of correlations can be a uniform reweighting of all coefficients m g 1. Strikingly, that is exactly the suggestion of Figure 6. We can use this insight to push the argument further: the fact that the primitive quasi-chemical populations for m g 1 are correct relative to each other means that the quantities 〈〈e-β∆Um〉〉0 are nearly exponentially dependent on m. For, in the first place, when the density is high, almost all the population is in the center of the distribution, and the Lagrange multipliers are negligibly affected by the relative reweighting of the m ) 0 term. Then the alteration of the original geometric weighting is irrelevant. In the second place, when the density is sufficiently low, these correlation factors are also irrelevant. So we can accurately write
ζ ≈ 1 + A(FS)
Km(0)FSmλm ∑ mg1
(12)
x0 “breaks away” from the rest of the distribution and requires individual consideration when the density is high enough that x0 is sufficiently small due to correlation effects. Nevertheless
A(FS) ≈
ζ-1 ζ0 - 1
(13)
where ζ0 is the primitive quasi-chemical approximate value. Thus, when the primitive quasi-chemical approximation is sufficiently accurate, the difficulty of evaluating the corrections should be much reduced. 5.2. Empirical Treatment of Correlations in the QuasiChemical Analysis. Though it would be interesting to calculate correlation corrections on the basis of eq 8 and Appendix C, a simpler, empirical approach suffices for our present purposes.
Figure 8. Predicted occupancies with the empirical correlation model, eq 14, for FSd3 ) 0.8. Compare to Figure 3. Again the solid line is the Poisson distribution.
This is because the discrepancies seen in Figure 5 are substantial but not problematic and, therefore, the required A(FS) is simple. In particular, the form
Km(0)FSmλm) ∑ mg1
β∆µ ) ln(1 + e7.361FS4
(14)
conforms accurately to the Carnahan-Starling equation of state; see Figure 7. The sole adjustable coefficient in eq 14 was obtained by fitting on the basis of eq 13 in order to minimize the discrepancy with the Carnahan-Starling equation of state. The occupancies predicted by this empirical model are depicted in Figure 8. 6. Concluding Discussion Our first goal was to work out how the quasi-chemical theory, a fresh attack on the statistical thermodynamic theory of liquids, applies to the paradigmatic hard-sphere fluid. The second goal was to work out theoretical approximation procedures that might assist in describing dense liquids of nonspherical species. The new fundamental results here apply generally to “hard-core”
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molecular models. The primitive quasi-chemical approximation, the procedure for iterating the quasi-chemical analysis, and the recognition of the “break away” phenomenon of Figure 6 are likely to be helpful in understanding packing in dense molecular liquids, beyond the hard-sphere fluid. For the hard-sphere system specifically, we have obtained a simple occupancy model, eq 14, that is likely to be helpful in a variety of other situations. One situation is the description of packing restrictions when the quasi-chemical theory is used to treat genuinely chemical interactions, for example, in the study of hydration of atomic ions in water.28,29 The issue of “context hydrophobicity” associated with many molecular solutes, including molecular ions, in water can also be addressed on the basis of quasichemical calculations and the developments here. Another situation of current interest is the theory of primitive hydrophobic effects that has recently been reborn.3,12-23 A historical view has been that the initial issue of hydrophobic effects was the hydration structures and thermodynamics following from volume exclusion by nonpolar molecules in liquid water. The balance of attractive forces that might produce drying phenomena was a secondary concern, except that “drying” was always present in the scaled particle models.30,31 With the convincing clarification of the first of these problems,3,12-23 the issue of drying phenomena has been now taken up more enthusiastically.3,32-36 In this context, we note that the striking success of the two-moment information models and the Pratt-Chandler theory3,12-23,37 is due, in part, to a fortuitous balance of a “Gaussian” approximation in the theory21 and a compensating disregard for drying possibilities;38 both of these compensating approximations are expected to be benign for small molecule solutes. (Another factor is that the dimensionless densities relevant for the first hydrophobic applications are not as large as those in the most difficult cases considered here.) One ingredient in a better understanding of this situation is a careful solution to the case where drying phenomena are entirely absent. That ingredient is better in hand with the results above. Appendix A: Preliminary Results A.1. Potential Distribution Theorem. The potential distribution theorem1,2,17,39 may be expressed as
Fσ ) 〈〈e-β∆U〉〉0zσ(qσ/V)
(A.1)
where Fσ is the density of molecules of type σ (the constituent under consideration), zσ ) exp(βµσ) is the absolute activity of that species, qσ is the single molecule partition function for that species, and V is the volume. The double brackets 〈〈‚‚‚〉〉0 indicate the average over the thermal motion of a molecule of type σ and the solution under the conditions of no interaction between them, and the averaged quantity is the Boltzmann factor of those interactions not present in the averaging but expressing the effects of actual interest. The average indicated here is the ratio of the activity of an isolated solute, FσV/qσ, divided by the absolute activity, zσ, of the actual solute. Thus
βµσ ) ln
[
〈〈e
VFσ -β∆U
]
〉〉0 qσ
(A.2)
This is a formal result to the extent that evaluation of the quantities on the right side typically will involve nontrivial calculations on many-body systems. An important notational point will serve to close this specification. The double brackets in the formulas above are suggested by the observation that averages over the distributions
of two decoupled systems are required here. The subscript zero serves to enforce that prescription that these subsystems be decoupled for the averaging. The double-bracket notation has been previously used for a similar average but with the full anticipated coupling between the two subsystems under discussion and without the subscript zero.40 Thus, the subscript zero is significant. Here the double brackets have a renewed significance also since it will be important that this approach is operationally relevant to flexible molecular species. A.2. Chemical Equilibrium. The traditional chemical thermodynamic consideration of a chemical transformation such as
nAA + nBB h nCC + nDD
(A.3)
with the formal result of eq A.2 leads to the formal expression
K≡
FCnCFDnD FAnAFBnB
( (
)( )(
qC V ) q A 〈〈e-β∆UA〉〉0 V 〈〈e-β∆UC〉〉0
nC
nA
) )
qD V q B 〈〈e-β∆UB〉〉0 V
〈〈e-β∆UD〉〉0
nD
nB
(A.4)
This should be compared to the textbook result for ideal gas systems.41 That comparison shows that the single molecule partition functions are multiplicatively augmented by the test particle averages. But otherwise, the structure of this important result is unchanged. The conclusion here is that the equilibrium ratios are well-defined objects though formal to the extent that nontrivial computational effort would be required to evaluate them on the basis of molecular information. A.3. Quasi-Chemical Theory. The quasi-chemical development starts from consideration of a distinguished molecule in the solution and seeks to evaluate the chemical potential on the basis of events occurring within a defined inner sphere. For a species of type A, that definition is codified by specifying a function bAj that is equal to one (1) when solution molecule j is inside the defined region and zero (0) otherwise. Our starting point can be2
∏j (1 - bAj)〉〉0]
β∆mA ) ln x0 - ln[〈〈e-β∆UA
(A.5)
where x0 is the fraction of A solute species with zero (0) neighbors in the defined region. ∆UA is the interaction energy of the solvent with the solute A that is treated as a test particle. The potential distribution theory perspective on eq A.5 is
∏j (1 - bAj)〉
x0 ) 〈
∏j (1 - bAj)〉〉0
〈〈e-β∆UA )
〈〈e-β∆UA〉〉0
(A.6)
The first, or chemical term, of eq A.5 can be analyzed with chemical concepts associated with the reactions
ASn)0 + nS h ASn
(A.7)
Here the indicated complexes are composed of n solvent (S) molecules within the defined region. Remember that the A molecule is a “distinguished” solute molecule considered at the lowest nonzero concentration.1 The fractional amount of A
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J. Phys. Chem. B, Vol. 105, No. 47, 2001 11667
species with a given solvation number n can be described by a chemical equilibrium ratio
Kn )
FASn
(A.8)
FASn)0FSn
deviates from the goal of bounding these thermodynamic quantities that was pursued by Reiss and Merry.45 Appendix B: Calculation of the Kn(0) for Hard Spheres and Hard Disks The Kn(0) values sought for reaction eq A.7 are given by
The Fσ values are the number densities and, in particular, FS is the bulk number density of solvent molecules since the distinguished A molecule is infinitely dilute. This notation permits the normalized re-expression
KnFSn
xn ) 1+
(A.9)
∑ KmFS
Kn(0) )
(B.1)
(qS/V)nqASn ) 0
(see eq A.4). For this problem, qS ) V/ΛS3, with ΛS being the thermal deBroglie wavelength for S, but these momentum integrals cancel perfectly and are irrelevant as usual. Therefore
m n
mg1
∫A d3r1 ... ∫A d3rn ( ∏ e(i,j))
n!Kn(0) )
(B.2)
j>i)1
Since this yields
x0-1 ) 1 +
∑ KmFSm mg1
(A.10)
the original eq A.5 can be re-expressed as
β∆µA ) -ln[1 +
qASn
KmFSm] - ln[〈〈e-β∆ ∏(1 - bAj)〉〉0] ∑ mg1 j U
(A.11)
The virtue of these rearrangements is that the natural first approximation is
x n ≈ xn
(0)
Kn(0)FSn
) 1+
∑
(A.12) Km(0)FSm
mg1
The Kn(0) are equilibrium ratios for the chemical reaction A.7 in an ideal gas. This formulation and the approximation of eq A.12 are closely related2 to the quasi-chemical (or clustervariation) approximations of Guggenheim,42 Bethe,43 and Kikuchi.44 This approach should have greatest utility where the chemical balances of eq A.7 are dominated by inner-sphere chemistry that can be captured with computations on clusters. Such chemical interactions are often much larger than the outer-sphere contribution, the right-most term of eq A.5. But that outer-sphere contribution remains and cannot be neglected forever. An interesting example based on simulation of liquid water was discussed recently.3 There the x0 was estimated from molecular dynamics results, and the remainders, the outer-sphere contributions to β∆µ, were positive, suggesting domination of those outer-sphere contributions by the packing constraints studied here. A principal goal of the present work is the development of a reasonable approach for describing the packing issues necessary for treating those outer-sphere contributions. Reiss and Merry45 and Sotocampos, Corti, and Reiss46 analyzed population relations analogous to eq A.9 but with activities appearing in the place of densities and with coefficients, here the equilibrium ratios Kn, appropriately different. The additional formal point here is the replacement of the activity by the density that permits the identification of the Kn in eq 2 and then further permits consideration of the mean field treatment eq 3 on the basis of an information theory constraint when Kn(0) will be used. At this stage, the quasi-chemical approximation achieves a particularly primitive character and
The notation ∫A d3rk indicates the three-dimensional spatial integral over the volume of the A-ball, a sphere of radius 1. The indicated integrand is thus 3n dimensional. The integrand is zero (0) if |ri -ri| < 1 (overlap) for any (ij) and one (1) otherwise. Thus, the integral can be estimated by sampling n-point uniform placements in the A-ball and scoring the fraction of such placements that are free from overlaps between the n unit diameter S-spheres. This approach fails for n larger than those presented in Table 1. But larger clusters were not observed in our simulation of the fluid, so our approach is satisfactory. The analogous two-dimensional procedure was used for the hard-disk results. AppendixC: Calculation of the Km/n(0) In contrast to Appendix B, here the ratio sought is
Km/n(0) )
qASn′Sn
(C.1)
(qS/V)mqASn′
corresponding to the reaction eq 9. Again, the explicit factors of V, the momentum integrals, and the factor of n! all cancel perfectly so that m
m!Km/n(0) ) 〈
∫AS′ d3r1 ... ∫AS′ d3rm ( ∏ e(i,j))〉 n
n
(C.2)
j>i)1
Here the notation ∫ASn′ d3rk indicates an integral over the excluded volume of an AS′n complex to an S ligand. The SS excluded volume, the integrand, is the same as before. But the structure of the AS′n complex fluctuates and the volumes obtained for specific structures are averaged over these fluctuations. The brackets 〈...〉 indicate the average over the structures of the isolated AS′n complex. This averaging is permitted and governed by the nontrivial denominator that appears in eq C.1. Operationally, the calculation can be much as in Appendix B except for (a) averaging by way of a Metropolis Monte Carlo calculation for the n ligand spheres in the star AS′n and (b) random placements of the m additional points are into a sphere of radius two (2) since that would fully enclose any conformation of the cluster. Appendix D: Calculation of xn for the Hard-Sphere Fluid The probability that there are n - 1 points in a sphere of radius r, xn-1(r), can be obtained from the distribution,
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4πrSr2Dn(r), of the distance r to the nth nearest neighbor of an arbitrary point. The probability that there are no more than n 1 molecules in the void is equal to the probability that the nth nearest neighbor is at least r away from the void center n-1
xm(r) ) 4πrS∫r Dn(y)y2 dy ∑ m)0 ∞
(D.1)
Isobaric-isothermal Monte Carlo can be used to calculate Dn(r). xn(r) for a range of r can be obtained from the distributions Dn(r). To increase the accuracy of the estimated Dn(r) for rarely observed r, small and large, a specific point in the simulation volume was chosen, and the sampling probability was reweighted by [4πrSrj2e-4πFSrj3/3 + C]-1, where rj is the distance from the chosen point to the jth nearest center for each configuration and C is an empirically chosen, dimensional constant. This importance sampling
〈[4πrSrj2e-4πFSr /3 + C]δ(r - rn)〉 3
4πrSr Dn(r) ) 2
〈[4πrSrj2e-4πFSrj /3 + C]〉 3
(D.2)
is based upon the idea that D1(0)(r) ) e-4πFSr3/3 is the function47 appropriate for a random distribution of spheres. This idea attempts to make the observed distribution of the distance to the nearest particle more nearly uniform. The constant C was included to avoid an unbounded weighting function. The denominator of eq D.2 is just a normalizing factor on the distribution. Isobaric-isothermal ensembles of 108 and 256 hard spheres were sufficient. The Carnahan-Starling equation was used to find the βp needed for a hard-sphere simulation at each specific density. Acknowledgment. Work at LANL was supported by the U.S. Department of Energy, under Contract W-7405-ENG-36 and the LANL LDRD program. Work at INEEL was supported by the Office of Environmental Management, U.S. Department of Energy, under DOE-ID Operations Office Contract DEAC07-99ID13727. Work at Vassar was supported by the Camille and Henry Dreyfus Faculty Start-Up Program and Vassar College. We thank John Weeks (Maryland) for helpful comments on a draft of this manuscript. References and Notes (1) Pratt, L. R.; LaViolette, R. A. Mol. Phys. 1998, 94, 909. (2) Pratt, L. R.; Rempe, S. B. In Simulation and Theory of Electrostatic Interactions in Solution. Computational Chemistry, Biophysics, and Aqueous Solutions; Pratt, L. R., Hummer, G., Eds.; AIP Conference Proceedings 492; American Institute of Physics: Melville, NY, 1999; p 172. (3) Hummer, G.; Garde, S.; Garcı´a, A. E.; Pratt, L. R. Chem. Phys. 2000, 258, 349. (4) Chandler, D.; Weeks, J. D.; Andersen, H. C. Science 1983, 220, 787. (5) Stell, G. In The Wonderful World of Stochastics A Tribute to Elliot W. Montroll; Shlesinger, M. F., Weiss, G. H., Eds.; Studies in Statistical Mechanics XII; Elsevier Science Publishers: New York, 1985; p 127. (6) Altenberger, A. R.; Dahler, J. S. Phys. ReV. E 1996, 54, 6242.
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