Strategies for Evaluating Thermodynamic Properties of Lattice

J. Phys. Chem. B 1999, 103, 5869-5880

5869

Strategies for Evaluating Thermodynamic Properties of Lattice Mixtures of Differently-Sized Particles and Chain Molecules: Application to Partitioning Data Involving Alkanes Charles D. Eads Miami Valley Laboratories, The Procter & Gamble Company, P.O. Box 538707, Cincinnati, Ohio 45253-8707 ReceiVed: NoVember 23, 1998; In Final Form: May 17, 1999

Simple approximate strategies for treating thermodynamic properties of lattice mixtures of differently sized particles and chain molecules are described. Based on these strategies, expressions for chemical potentials and partition coefficients are derived. These expressions account in a simple way for effects of particle size and chain length. In the limit of long chains and for single-cell solute molecules, the resulting expression for chemical potential reduces to the Flory-Huggins expression. However, for shorter chains such as alkanes and for large globular solutes, the predicted contribution of chain entropy to the chemical potential differs from the Flory-Huggins treatment. The results are used to interpret published partitioning data on xenon in alkanes as a function of chain length and temperature and to interpret published partitioning data on pure alkanes and water. On the basis of these latter results, a contact free energy for the hydrophobic effect of 29 cal A-2 mol-1 is determined. The first strategy for handling disparate molecular sizes gives expressions for the numbers of contacts between differently sized globular particles by introducing weighting factors for particle numbers based on the number of contacts each particle can make with particles of the other type. The second strategy addresses the entropy of mixtures of differently sized globular particles by using a parameter in the partition function that gives the probability that a lattice with a given population of large globular particles can accept another large globular particle inserted at random. The third strategy evaluates chain configuration entropy by calculating the number of configurations available to a mixture of chain and globular particles without regard for intersegment chain connectivity, followed by correction for the fraction of configurations that are consistent with proper chain covalent bonding.

Introduction Geometric aspects of mixture components play an important role in determining the structure and properties of many types of complex fluids ranging from colloidal dispersions to molecular solutions. As one example of current interest, the biophysics community is currently concerned with the influence of solvent alkane chain length and solute size in oil-water partitioning experiments.1-12 An important issue is how to disentangle the geometric and energetic influences on oil-water partition coefficients so that contributions of the hydrophobic effect to the structure of biological molecules and amphiphilic aggregates can be properly parametrized. Oil-water partition experiments are used in numerous other applications including drug design, predicting partitioning into membranes and lipids, predicting bioavailability, and environmental impact.11 An improved understanding of the role of size and chain length is therefore broadly applicable. Lattice models13-15 are often used to describe thermodynamic properties of fluid mixtures. The form of these expressions is often physically informative and intuitive. Some of the conceptual and algebraic simplicity of the lattice approach is lost, however, when the system involves particles that occupy more than one cell. Multicell occupancy is necessary for describing large rigid or chain particles. A difficulty with multicell occupancy is that the configurations available to particles on the lattice can no longer be enumerated using a simple combinatorial expression, and one must devise methods to account for the coupling or interference among particles7 or

among bonds within multicell particles. Freed and coworkers16-22 and others23,24 have developed and applied a very powerful and highly successful lattice cluster theory (LCT) that gives the partition function as a series expansion. The lowest order term corresponds to a mean-field approximation. Higher order terms account for structural relationships among the chain segments and lattice residents. The purpose of the present work is to develop an alternate framework to treat geometry-related properties of mixtures using lattice statistical thermodynamic methods, emphasizing simple algebraic forms and avoiding computational approaches as much as possible. The approach introduces simple, physically informative terms to account for specific effects of having different particle sizes. Specifically, the approach strives to account for the effect of particle size differences on the number of contacts among particles of different types, the effect of multicell particles on the number of arrangements available to a mixture of a given composition, and the influence on chain molecule entropy of large globular particles. Analytical approximations introduced for this purpose are checked with simple computer simulations to evaluate their range of validity. The main application of the resulting approach is to develop expressions for the chemical potentials of short chain (alkane) and polymer molecules in various solvents and for globular solutes dissolved in short or long chain solvents. Therefore, the approach adopted here closely follows the early descriptions of lattice models of liquids,13-15 appropriately modified to account for effects of disparate molecular sizes.

10.1021/jp9845120 CCC: $18.00 © 1999 American Chemical Society Published on Web 06/30/1999

5870 J. Phys. Chem. B, Vol. 103, No. 28, 1999

Eads

Description of the Strategies Contact Number Weighting Strategy for Estimation of Interparticle Contact Numbers. Contact numbers are central to lattice models and were used, for example, by Guggenheim15 in some of the very early lattice treatments. The Freed lattice cluster theory also allows calculations of contact numbers for a variety of shapes.23,24 Contact numbers can be used to calculate the internal energy and, in this application, to account for the effect of globular solutes on the entropy of chain molecules. The goal of this section is to estimate the number of contacts in a lattice mixture containing particles having different sizes. The system considered here will consist of a lattice populated by nA small particles of type A, which occupy a single cell each; nB large globular particles of type B, which occupy more than one cell; and nE empty lattice cells. The nE cells may also be regarded as containing particles of type E, which occupy a single lattice cell. The total number of A and E cells will be denoted as nR ) nA + nE. The symbol R will denote a cell that contains either an A particle or an empty cell (or an E particle). Empty cells are used in lattice models to account for pressure, allowing much wider application of the resulting expressions,25-27 for example, allowing the development of equations of state. Unless otherwise noted, all particles (or chain segments) considered in this work will be globular. The term globular is here taken to refer to square particles on a two-dimensional lattice or cubic molecules on a three-dimensional lattice. The symbol mij is used to describe the total number of contacts between particles of types i and j, summed over the entire lattice. It is also useful to define limiting contact numbers, Zij. These are defined as the maximum number of particles of type j that can be packed around a particle of type i. These are illustrated for the limiting contact numbers ZAB and ZBA in Figure 1 for a two-dimensional square lattice. If all particle types occupy a single cell, then all Zij are given by the lattice contact number, Z. Consider first the number of contacts between R and B sites, mRB. In the limit of very dilute B, this quantity is given by mRB ) nBZBA (dilute B limit) since every B particle will be completely surrounded by R particles. In the limit of very dilute R sites, this is given by mRB ) nRZAB (dilute R limit) for a similar reason. For equally sized particles, the well-known expression for mRB is given by eq 1, where Z is the lattice contact number.

mRB )

nRnBZ (nRZ)(nBZ) ) nR + nB (nRZ) + (nBZ)

(1)

To obtain an expression that gives the correct limiting behavior for large B particles, and which reduces to eq 1 for the case of equally sized particles, one can replace each occurrence of nRZ with nRZAB and each occurrence of nBZ with nBZBA. In other words, rather than weighting all particle numbers with the lattice contact number Z, the weighting factor is replaced with the number of contacts each particle can make with particles of the other type. This procedure gives eq 2.

mRB )

ZABnRZBAnB nRnBF ) ZAB ZABnR + ZBAnB nR + nBF

(2)

This particle number weighting approach gives the correct limiting behavior while maintaining the same functional form as eq 1. It is intuitively appealing, and it gives close agreement with contact numbers determined from randomly arranged

Figure 1. Particle contact numbers. On a square lattice, the number of large globular B particles that can contact an A particle is 4, regardless of the size of the B particles. The number of single-cell A particles that can touch a large globular B particle is 4lB, where lB is the length of a B particle edge measured in lattice cell units. For a cubic lattice, the number of large globular B particles that can contact an A particle is 6, regardless of the size of the B particles. The number of single-cell A particles that can touch a large globular B particle is 6(lB)2, where lB is the length of a B particle edge measured in lattice cell units.

lattices, as shown below. The second form in eq 2 was used earlier28 and introduces the contact asymmetry parameter, F ) ZBA/ZAB. The contact asymmetry parameter is experimentally accessible in some cases and simplifies the notation. For evaluating the internal energy and other properties based on contact numbers, it will be necessary to determine mij for all combinations of i and j. For example, mAB can be determined from mRB by multiplying by the fraction of R sites that contain A particles

mAB )

nA m nR RB

(3)

To calculate mAA, consider a typical R site. The fraction XRB of its surface that contacts B particles is the ratio of the actual contact number, mRB, to the maximum contact number obtained from the limit of dilute R, nRZAB. This gives XRB ) mRB/nRZAB. The fraction of the surface of this R site that is left over for other R particles is thus XRR ) 1 - XRB. The number of R sites contacting another R site is therefore ZAAXRR. This times the fraction of R sites that contain A particles gives the average number of A particles around an R site. This times the number of A particles times 1/2 gives mAA. 2 2 1 nA 1 nA ZAA Z m mAA ) 2 nR AA 2 n 2 ZBB RB R

(4)

Lattice Mixtures of Particles and Chain Molecules

J. Phys. Chem. B, Vol. 103, No. 28, 1999 5871

Similar arguments give

1 1 ZBB mBB ) nBZBB m 2 2 ZBA RB

( ( (

(5)

) ) )

nEZAB 1 mEE ) nEZAA 2 nRZAB + nBZBA

(6)

mEB ) nBZBA

nEZAB nRZAB + nBZBA

(7)

mEA ) nAZAA

nEZAB nRZAB + nBZBA

(8)

In this model, the internal energy is determined by ascribing a nearest neighbor interaction energy, wij, for each type of contact between particles of type i and j. The total internal energy is just the sum of these pairwise interactions. It is assumed that nearest neighbor interactions with empty cells have zero energy. This gives

U ) mAAwAA + mBBwBB + mABwAB

()

1 1 nA Z n w + Z n w + mRBωRB U) 2 nR AA A AA 2 BB B BB

}

ZBB 1 nA 2ZAA wAA + w 2 nR ZAB ZBA BB

(11)

Equations 10 and 11 reduce to the symmetric regular solution equations13-15 if it is assumed that the particles have equal sizes and that E cells are absent. Note that the form of ωRB is composition dependent if empty cells or E particles are present and requires weighting of the contact energies according to contact number ratios. These expressions lead to activity coefficients having the form of the well-known van Laar expressions, although the adjustable parameters resulting from the present treatment have a different interpretation. Equation 11 is analogous to the interchange energy15 and the Flory χ parameter,29 given by wAB - 1/2(wAA + wBB). However, the use of contact numbers in eq 11 generalizes this expression to include effects of differing molecular sizes. Insertion Probability Strategy for Combinatorial Analysis for Globular Particles. Insertion probabilities play an important role in models of liquids because they are closely related to the chemical potential.30 Chain insertion probabilities can be used in both lattice and off-lattice cases31-33 to devise equations of state. In the present model, insertion probabilities are used to evaluate the number arrangements of large globular particles on a lattice. The following procedure, which closely follows a derivation of the Flory-Huggins entropy expression,14 will be used for this. First, the large B particles will be placed randomly on the lattice. The number of ways of arranging these B particles will be expressed in terms of a parameter, fi, which gives the fraction of cells in the lattice that can receive a B particle without overlap if i - 1 B particles are already present. fi is equivalent to the probability that a randomly selected location can accommodate a B particle, and is therefore an insertion probability. Once the B particles are situated, the small A and

1

NnLB

nB

nB

fi ∏NLfi ) n !∏ i)1

nB! i)1

(12)

B

The number of ways of arranging the nA A particles and nE E cells in the remaining nR lattice cells is

ΩR )

nR! nA!nE!

(13)

The total degeneracy is the product of ΩB and ΩR.

Ω)

(10)

where

{( )

ΩB )

(9)

Substituting formulas 3, 4, and 5 into eq 9 and collecting terms in mRB gives

ωRB ) wAB -

E cells will be arranged in the remaining volume. Methods for calculating fi will be described below. For this strategy it is convenient to define the location of a multicell globular (square on a 2D lattice, cube on a 3D lattice) particle as the location of the cell within the particle volume located closest to the lattice origin. This cell will be called the root cell of the particle. The number of ways, ωi, of putting the ith B particle into a lattice already containing i - 1 B particles is ωi ) NLfi, where NL is the number of cells in the lattice. The total number of ways of placing all nB B particles in the lattice is the product of these ωi for all particles, divided by nB! to account for the indistinguishability of the B particles.

nR!NnLB

nB

∏ fi

(14)

nA!nB!nE! i)1

If the particles have equal volumes of one lattice cell each, one can substitute fi ) (NL - i)/NL and the usual combinatorial formula for arranging three types of particles on a lattice results. The entropy, S, divided by Boltzmann’s constant, k, is given by the logarithm of the overall degeneracy (eq 14) to give eq 15.

S k

) nB ln(NL) + nR ln(nR) + nB - nA ln(nA) - nB ln(nB) nB

fi) ∏ i)1

nE ln(nE) + ln(

(15)

Chain Reconstruction Strategy for Combinatorial Analysis for Chain Molecules. Flory and Huggins independently developed simple expressions for the entropy of long chain molecules.34,35 Flory-Huggins theory is still valuable because it is physically informative and it gives an excellent qualitative description of many properties of long chain polymer systems. Derivation of the Flory-Huggins entropy depends on a mean field approximation that assumes, in effect, that chain segment insertion probabilities are uncorrelated. For short chain molecules such as alkanes, or for dilute polymer solutions, this mean field assumption is less valid. The theory gives no consideration to the sizes of globular particles that may be present, though it can be adapted to treat such particles as chain molecules. The chain reconstruction strategy gives a modification to the FloryHuggins entropy expression. Its main result is to provide a simple expression that considers the effect of large globular solutes on chain entropy, even for short or dilute chains. Consider the number of arrangements of nA chain molecules whose segments occupy a single lattice cell each and which consist of l segments per chain. The presence of nB globular solute molecules and nE empty cells or E-type particles will also be considered. To determine the partition function for this


Eads

mixture, the chain molecules will be conceptually broken into l cell-sized segments. These segments will be mixed with the nB B particles and nE lattice residents of type E. Initially, the number of configurations available to this system, without regard for covalent chain connectivity, will be evaluated. The resulting expression will then be corrected by multiplying by the fraction of those configurations that are consistent with having correct covalent chain connectivity. Determination of the uncorrected partition function follows the same procedure used above for the nonchain case, so the number of ways of arranging the B particles is given by eq 12. To determine the number of arrangements of the remaining lattice residents, it is assumed that the disconnected chain segments are distinguishable on the basis of their position in the chain, but not based on their specific chain of origin. This gives nR ) lnA + nE particles to arrange in the R sites. The number of arrangements for these nR objects in the volume not occupied by B particles is given by eq 16.

ΩR )

nR!

(16)

(nA!)lnE!

In determining the correction factor, C, for chain connectivity, the symbol p12 will be used to denote the probability that a segment from position 1 of a chain has at least one neighbor of a type suitable for segment 2. For bonds between all segment pairs i,j ) i + 1 other than 1 and 2, a slightly different parameter pij will be used. Expressions for p12 and pij will be derived below, though their numerical values are quite similar and it will often not be necessary to distinguish the two expressions. The probability that the first segment of the first randomly placed chain has a neighbor of type 2 is just p12. The probability that this neighbor does not belong to some other chain is unity. For the first segment of the second chain, the probability that it has a neighbor of type 2 is again p12. However, a correction is required to account for the probability that this neighbor does not belong to some other chain already connected. Since only chain 1 has been connected, this correction is 1 - 1/nA. For the first segment of the third chain, the corresponding probabilities are p12 and 1 - 2/nA. For all segments of type 1, the probability that they are all connected to segments of type 2 is nA

[(

)]

k-1

p12 1 ∏ k)1

nA

(17)

Similarly, for segment types 2 and 3, the probability that all pairs are properly connected is nA

[(

pij 1 ∏ k)1

)]

k-1 nA

(18)

and so on for all chain bonds. The overall correction factor is therefore given by eq 19.

C ) (p12) (pij) nA

( ) ( )

nA-1 nA(l-2)

) (p12)nA(pij)nA(l-2)

∏ k)0

nA! nnAA

1-

k

The partition function Ω is the product of ΩR, ΩΒ, and C. nB

Ω)

NnLBnnRRe (p12)nA(pij)nA(l-2) nnAAnnBBnnEE

fi ∏ i)1

(20)

In this expression, Stirling’s approximation was introduced to remove the factorials. Evaluation of eq 20 requires explicit expressions for fi, p12, and pij. The insertion probability fi is discussed below. To develop expressions for p12 and pij, begin by considering the expression for mAB (eq 3), which gives the total number of contacts between A segments and B particles. For a given mixture composition, the fraction XAB of the faces of a segment from position 1 on an A chain that contact B particles is given by the ratio of the actual contact number, mAB, to the maximum contact number mAB(max) ) nAZAB. The fraction of the surface of the A segment left over for R sites is XAR ) 1 - XAB. This times the fraction of the R site residents that are of type A, segment 2, gives the probability P12 that any given site on an A segment from position 1 contacts an A segment from position 2.

P1,2 )

(

)

nA mAB nA 1) lnA + nE nAZAB nR + FnB

(21)

Next, consider the probability that a given segment of type 1 has at least one neighbor of type 2. This is one minus the probability that it has no neighbors of type 2. The probability that one particular face does not have any neighbor of type 2 is 1 - P1,2 ) (Nc - nA)/Nc, where the parameter Nc ) nR + FnB is introduced to simplify the notation. This probability raised to the power ZAA gives the probability that none of the faces of a segment of type 1 has a neighbor of type 2. One minus this gives the desired formula for the probability that a segment of type 1 has at least one neighbor of type 2.

(

)

Nc - nA Nc

p12 ) 1 -

ZAA

(22)

For bonds i,j ) i + 1 other than that between segments 1 and 2, one must use the power of ZAA - 1 since it is known that at least one face of segment i is already occupied by segment i - 1.

pij ) 1 -

(

)

Nc - nA Nc

ZAA-1

(23)

By expanding eq 22 using binomial coefficients for the power ZAA, one can show that the limit for nA , Nc, which corresponds to the long chain or dilute cases, is given by

nA lim (p12) ) ZAA nA/Ncf0 Nc

(24)

If this long chain assumption is made, and if it is further assumed that the B particles occupy a single lattice cell, then eq 25 is obtained for the partition function.

l-1

Ω)

nA

(enA)nA(l-1)

nB

NNL L nnAAnnBBnnEE

( ) ZAA NLe

nA(l-1)

(25)

l-1

(19)

Equation 25 is identical to the Flory-Huggins combinatorial expression if the number of empty cells is set to zero.35



Figure 2. Total particle contact numbers. The heavy lines show the number of interparticle contacts of the indicated type determined from the lattice sampling algorithm. The broken lines show the predicted number of contacts of the indicated type based on eqs 3-5. The lattice sampling was averaged over 10 000 iterations on a cubic threedimensional lattice containing 4096 cells. The large B particles were cubes occupying 8 cells each.

Using the usual Boltzmann formula and eq 20, the entropy of a chain-containing mixture is given by eq 26.

S k

) nB ln(NL) + nR ln(nR) + nB - nA ln(nA) - nB ln(nB) nB

fi) + nA ln(p12) + nA(l - 2) ln(pij) ∏ i)1

nE ln(nE) + ln(

nA(l - 1) ln(nA) - nA(l - 1) (26) Numerical Consequences of the Assumptions Lattice Sampling Algorithm. As is common practice, a simple lattice sampling algorithm was used to determine true contact numbers mAA, mBB, and mAB for comparison with eqs 3-5. The same algorithm was used to explore the composition dependence of the insertion probability parameter fi. The algorithm uses random placement of B particles into a square (two-dimensional) or cubic (three-dimensional) lattice, one particle at a time. As each particle is successfully placed, the AA, BB, and AB contacts are tallied under the assumption that any cell not occupied by a B particle is occupied by an A particle. Following each particle insertion, the number of locations that can accept the root cell of the next B particle is evaluated. This divided by NL gives fi for the next particle. The local region surrounding each newly inserted B particle is searched, and running totals for the number of sites available and the contact numbers are adjusted. Periodic boundary conditions are used. The program can be run for multiple cycles (103-105) to obtain accurate average values in a few seconds to a few hours, depending on the lattice size, etc. Evaluation of Contact Numbers Using Numerical Lattice Sampling. Figure 2 compares contact numbers determined from numerical lattice sampling to the algebraic expressions given in eqs 3-5. Comparable results are obtained for two- or threedimensional lattices, and for various sizes of the B particles. In all cases the agreement is quite satisfactory, showing that the approximate formulas give good predictions of the contact numbers. Note that in Figure 2 the lattice sampling results do not extend above about 70% volume fraction of large B particles. Above this value, the probability of achieving arrangements of

Figure 3. Excluded volumes of large (B) particles and empty cells. The upper diagram shows the region that cannot be occupied by the root cell of a B particle if a B particle is located as shown. For a twodimensional lattice, the total excluded volume is VB,ex ) (2lB - 1)2. For a three-dimensional lattice, the total excluded volume is VB,ex ) (2lB - 1)3. The lower diagram shows the region that cannot be occupied by the root cell of a B particle if an empty cell is located as shown. The corresponding excluded volume due to an empty cell is VB ) lB2 for a two-dimensional lattice and VB ) lB3 for a three-dimensional lattice.

the particles that are sufficiently correlated to allow insertion of additional particles becomes too small to be achieved within reasonable amounts of computer time using the present algorithm. Algebraic Formulas for Insertion Probabilities. It is straightforward to calculate fi using the lattice sampling algorithm described above. Nonetheless, it is useful to have approximate algebraic expressions in keeping with the goal in the present work of avoiding the need for numerical computation. In deriving such approximate expressions, it is useful to consider the volumes and excluded volumes associated with the presence of B particles. As shown in Figure 3, large B particles on a D-dimensional lattice with an edge length of lB exclude a volume of VB,ex ) (2lB - 1)D from occupancy by the root cells of other B particles. Also, the presence of an empty cell implies that a volume equal to the volume of a B particle, VB ) lBD, is empty of B particle root cells. In analogy to treatments of dilute gases, at low B concentrations there is a negligible probability of overlap among excluded volumes, so one simple estimate of the insertion probability, valid only for sufficiently dilute B concentrations, is just the ratio of the available volume to the total volume.

fi )

NL - (i - 1)VB,ex NL

(27)

A somewhat different argument leads to a very simple expression that is valid at higher concentrations of B. Since


Eads

Figure 4. Insertion probability determined by various techniques. The lattice sampling run consisted of 10 000 iterations on a cubic threedimensional lattice containing 4096 cells. The large B particles were cubes occupying 8 cells each. See text for descriptions of the curves.

specifying that a specific cell is empty excludes lBD ) VB cells from hosting a B root cell, one can guarantee that a given region of the lattice is empty of B root cells by specifying that some small number, rex, of strategically selected cells within that volume contain empty space. In terms of rex, and the fraction of cells that are empty, the insertion probability is given by eq 28 if it is assumed that the positions of empty cells are uncorrelated.

fi )

(

)

NL - (i - 1)VB NL

rex

(28)

This expression is denoted the reserved volume expression for insertion probabilities. A simple estimate for the value of rex is obtained by asserting that the number of cells that must be reserved as empty is given by the ratio of the excluded volume of a B particle and the excluded volume of an empty cell with respect to B particles (the symbol rex was chosen to denote a ratio of excluded volumes). This gives eq 29 for rex.

rex )

(

)

2lB - 1 lB

D

(29)

Figure 4 compares the “correct” value of fi determined from lattice sampling to the predictions of the approximate equations derived above (eqs 27 and 28) and to the ideal expression that assumes equal molecular size. The ideal form gives too large an estimate of fi because it does not account at all for interference among molecules due to excluded volume. As expected, the dilute, additive excluded volume formula (eq 27) is correct in the limit of low B concentrations but rapidly diverges from the correct value at higher volume fractions. This form is very useful at low B concentrations because it gives correct results in an experimentally important concentration range often encountered, for example, in partitioning experiments. In contrast to the dilute B expression (eq 27), the reserved volume expression (eq 28) diverges from the correct curve only at relatively higher B volume fractions, and the divergence occurs to a lesser extent and in the opposite direction. This divergence is due to the mean field assumption concerning empty cells in the justification of eq 28. Figure 5 explores the dependence of the ratio [ln(fi)]/ [ln((NL - nBVB)/NL)], which gives the effective exponent or

Figure 5. Effective value of rex in eq 30 from lattice sampling. The thin line corresponds to a lattice sampling run consisting of 1000 iterations on a cubic three-dimensional lattice containing 4096 cells, with large B particles occupying 8 cells each. The heavy line corresponds to a lattice sampling run consisting of 1000 iterations on a cubic three-dimensional lattice containing 8000 cells, with large B particles occupying 27 cells each. The broken lines show the predicted value of the exponent determined via eq 29.

“true” value of rex. For the systems described in this figure, eq 29 gives the correct value for rex at low volume fractions, but the true value increases as the volume fraction of B increases. For smaller B particles, eq 29 gives a better estimate of the exponent over larger concentration ranges. This trend was observed for two-dimensional lattices as well (not shown). For particles differing in volume by a relatively small amount, this equation should be fairly accurate and thus provides a means of evaluating the entropy of mixing of globular molecules when they are not too different in size. Numerical Comparison of Chain Entropy Expression to Flory-Huggins Theory. This section explores differences in predictions made by the present model compared to the FloryHuggins theory with respect to chain length effects. Since taking the long chain or dilute limit for the formula for the probability of adjacent occupancy, p12, gives the Flory-Huggins partition function, within the present treatment the Flory-Huggins mean field approximation is equivalent to the assertion that p12 is given by eq 24. It is instructive to compare the numerical values of this probability using the limiting formula (eq 24) and the more complete expression (eq 22) resulting from the present work. These expressions are compared in Figure 6, where it has been assumed for simplicity that nB ) nE ) 0 and ZAA ) 6 (cubic lattice). Since this expression gives the probability that a given segment has at least one neighbor corresponding to the next sequential segment, the value cannot in principle exceed unity, and the probability should decrease monotonically as the chain length increases. Equation 22 correctly gives the expected behavior, while back-extrapolation of the Flory-Huggins long chain limit actually exceeds unit probability for chain lengths of 6 or less (this depends on the choice of Z). Especially for shorter alkane chain lengths, the Flory-Huggins limit significantly overestimates the probability of adjacent occupancy. For chain lengths above approximately 20, the two models are in substantial agreement. One consequence of this result is that compared to eq 22, the Flory-Huggins limit formula will overestimate the entropy for short chain fluids because it overestimates the probability that sequential segments may be adjacent when arranged


J. Phys. Chem. B, Vol. 103, No. 28, 1999 5875 Derivatives of the logarithm of the insertion probability nB nB fi) ) ∏i)1 ln(fi), are required for evaluating product, ln(∏i)1 entropic contributions to chemical potentials. One may use the explicit approximate expressions and evaluate the product and derivatives thereof directly. In the absence of any approximation for fi, one may still obtain the derivative of the logarithm of the product using the Widom method for the constant volume case. Since the logarithm of the product is the sum of the logarithms, one obtains for the difference between systems containing nB nB nB and nB - 1 particles ((∂/∂nB)[∑i)1 ln(fi)])V ) ∑i)1 ln(fi) nB-1 ∑i)1 ln(fi) ) ln(fnB). Chemical potentials are often described using the form

µB ) µ0B + kT ln(χBγB) Figure 6. Adjacent occupancy probability. The heavy line shows the occupancy probability vs chain length calculated using eq 22 (this work). The thin line shows the probability calculated using backextrapolation of the long chain limit (Flory-Huggins model). A cubic lattice with contact number Z ) 6 was used.

(30)

where µ0B is the chemical potential in some standard state, χB is the mole fraction of B in the solution, and γB is the activity coefficient. Ideal solutions are those that conform to this expression if γB is set to unity. On the basis of statistical mechanical arguments, Ben-Naim36 advocates the following form for the chemical potential.

µB ) µ/B + kT ln(FB)

(31)

In this expression, µ/B is called the pseudochemical potential, and FB is the number density (molar concentration) of B particles in the solution. One advantage of eq 31 over eq 30 is that, for a fluid that obeys classical statistical thermodynamics, the pseudochemical potential represents appropriately averaged interactions of a B particle at a fixed position with all other particles in the system. In partitioning experiments, the solute B concentration is often very low, and interactions among B particles can be ignored. In this limit, one obtains eq 32 for the dilute limit pseudochemical potential. Figure 7. Chain length dependence of entropy per lattice cell. The heavy line shows the entropy calculated using eq 26 (this work) for a lattice containing only chain molecules. The thin line shows the corresponding curve determined from the Flory-Huggins long-chain model.

randomly on the lattice. This effect should be especially important for molecules such as alkanes. Figure 7 shows the entropy per unit volume for neat A chains vs chain length for the present theory. Qualitatively, one would expect the entropy per unit volume to increase monotonically from zero for a chain length of one to some limiting value as the chain length increases and end effects become unimportant. This is the behavior exhibited by the present theory. Chemical Potentials, Partition Coefficients, and Concentration Units The chemical potential of component i, µi, is the derivative of the free energy with respect to the number of molecules of that substance. In calculating derivatives with the present model, one may adopt the perspective of Widom30 that the chemical potential is determined by the change in free energy due to the insertion of a single particle. For example, in the case of constant volume, insertion of a B particle will result in the elimination of VB empty cells, so that (∂nE/∂nB)V ) -VB. In this work it will be assumed that additive volume is a good approximation to constant pressure. In this case, this derivative is zero since the system expands by VB upon B insertion.

µ/B0 ) lim (µB - kT ln(FB)) FBf0

(32)

The partition coefficient Koil/wat in density units is given by eq 33 if the solute is dilute in both solvents. The subscripts oil and wat in the term Koil/wat are meant to be suggestive of oil and water, though they may represent any solvents.

kT ln(Koil/wat) ) kT ln

( )

FB,oil 0 0 ) µ/B,wat - µ/B,oil FB,wat

(33)

In some cases the solute may not be dilute in both phases, for example, when one is considering the partitioning of an alkane between water and its own pure liquid. In this case, one uses

kT ln(Koil/wat) ) µ/B,wat - µ/B,oil

(34)

Interpretation of Partitioning Data Partition Coefficients for Globular Solutes between Globular Solvents Having Different Sizes. Equation 35 gives the additive volume (approximately constant pressure) chemical potential of dilute B particles in A solvent, µB, calculated using the dilute B approximation (eq 27) of fi.


µB )

( ) ( ) ( ) [ ] ( ( ) ( ∂U ∂nB

+P

P

∂V ∂nB

-T

P

∂S ∂nB

Eads

()

) µ/B + kT ln

P

nB NL

2 nR ) 1/2ZBBwBB + ZBAωRB + PVB + nR + nBF VB NL - VB,exnB nB kT ln + ln VB,ex NL NL - VB,exnB

))

(35)

In this expression, P is the pressure and V is the volume. The dilute pseudochemical potential is given by eq 36.

µ/0B ) 1/2ZBBwBB + ZBAωRB + PVB

(36)

This shows that for a nonchain solvent A, the dilute pseudochemical potential has no contribution arising from configurational entropy. This result holds regardless of the functional form used for fi, because for systems in the limit of dilute B, fi ) 1. Next, consider the partition coefficient of B particles between two globular solvents, denoted oil and wat (but not necessarily corresponding to oil and water) in which both solvents are viewed as consisting of particles that occupy one lattice cell each. The partition coefficient will be defined as the ratio of concentrations of B particles in the two solvents, where the concentration units are in particles per unit volume. As mentioned above, this choice of units is advocated by BenNaim36 because it gives a meaningful measure of the difference in solvation energies for the solute in the two solvents. Note that in all the preceding expressions the unit of volume is tacitly chosen as the volume of a lattice cell. Since the two types of solvent particles may have different sizes with respect to each other, the cell sizes in the two lattices may be different. One must scale the number of lattice cells by the cell volume in each lattice. Therefore, using density units the partition coefficient Koil/wat is given by eq 37.

[ ] ) [ )] [ ( ( ( (

) )

nB NLVcell oil ln(Koil/wat) ) ln nB NLVcell wat

nB NL oil (Vcell)wat ) ln + ln ) nB (Vcell)oil NL wat

]

ln(Kχoil/wat)

(

)

(

)

NL - VB,exnB nB VB µB,ent ) kT ln + kT ln + VB,ex NL NL - VB,exnB

[ ]

+ ln

(Vcell)wat (Vcell)oil

(37)

Equation 37 introduces the definition Kχoil/wat ) (nB/NL)oil/ (nB/NL)wat. For dilute B, this is equivalent to the ratio of mole fractions of B in the two solvents. Using eqs 33, 36, and 37, one obtains eq 38 for Kχoil/wat in eq 37.

ln(Kχoil/wat) )

It is useful to broaden the scope of the internal energy term in these expressions to include effects that may arise in specific solvents such as water. In these cases, solute-induced orientational restrictions on water molecules and other phenomena not entirely ascribable to contact energies may be operating. Such interactions will be lumped with the internal energy and given the symbol gB and referred to as “contact free energies”7 when partition experiments are considered. This terminology will be adopted in the remainder of this work. Xenon Solvation in Alkanes. Experimental data show an important role of molecular size in partitioning experiments. A dependence of solute chemical potential on solvent alkane chain length was observed experimentally by De Young and Dill,1 who noted that the alkane-water partition coefficient of benzene depended on the alkane chain length. A series of papers2-12 appeared that discussed the best way to handle these size effects. At issue in these publications is the contribution of size-related entropy effects. In particular, use of volume fraction units, based on the Flory-Huggins expression, to account for solvent chain length effects or molecular size effects in general gives a parametrization of the hydrophobic effect that differs substantially from the generally accepted value based on mole fraction units. A comprehensive review11 of this literature has recently appeared. An earlier study of the gas-liquid partitioning of xenon in alkanes as a function of solvent chain length and temperature also indicates a strong dependence of the chemical potential (of xenon in this case) on the alkane chain length.37 This study spans the range of liquid alkanes from 6 to 20 carbon atoms and a range of temperatures from 10 to 50 °C. These data are of very high quality and will be considered in detail to evaluate the ability of the present model to account for chain length effects on the chemical potentials of solutes. Ben-Naim converted the data in the original paper to his preferred density units.38 This section examines these data in terms of the present model. To examine the influence of short chain entropy on the chemical potential of dilute globular solutes, one can combine the dilute B expression for fi (eq 27) with eq 26 for the entropy of a mixture of nA chain A molecules, nB globular B solute particles, and nE empty cells. From this, one obtains the contribution of the entropy to the chemical potential of dilute B particles at constant pressure (additive volume) shown in eq 39.

1 [(Z ω ) - (ZBAωRB)oil] kT BA RB wat

(38)

This expression emphasizes that by using mole fraction units in expressing partition coefficients for globular solvents, one automatically cancels any influence of different lattice cell volumes in the two solvents. This has the effect of eliminating solvent size contributions, leaving only the internal energy term.

[ ( )

( )]

1 - p12 1 - pij nA2 ZAA + (ZAA - 1)(l - 2) kTF p12 pij Nc(Nc - nA)

(39) The dilute B pseudochemical potential is given by eq 40.

(

)

ZAA 1 - p12 l p12

µ/B0 ) g0B + kTF

(40)

In this expression, g0B gives the contact free energy for very dilute solute in pure chain solvent. It is also assumed that the number of empty cells is small, that the unit of volume is given by the volume of the lattice cell, and that the difference between p12 and pij is negligible, so that the symbol p12 replaces both. Equation 40 shows that the chemical potential of B solutes is raised due to the effect of B on the chain entropy. In particular, the presence of B particles reduces the probability that randomly



Figure 8. Contribution of chain entropy to the chemical potential of B particles. The heavy line shows the prediction of eq 40 (this work). The broken line shows the prediction of the Flory-Huggins model (eq 41) extrapolated to short chains.

arranged A chain segments will be positioned consistent with having correct covalent connectivity. The net result is that the presence of B particles decreases the chain entropy. In the Flory-Huggins limit (long chain, single cell B particles), the corresponding expression is given by eq 41.

(l -l 1)

0 µ/B,FH ) g0B + kT

(41)

Figure 8 compares the size-related entropic contribution to the additive volume chemical potential predicted by this work (eq 40) compared to the short-chain extrapolation of the FloryHuggins theory (eq 41). Considering the case for F ) 1 shown in Figure 8, the present theory predicts that the contribution of chain entropy to the chemical potential is significantly less than the prediction of the Flory-Huggins theory for the small chain lengths typical of alkanes. A second consequence of the present theory is that the contact asymmetry parameter F scales the effect of chain length. Recall that F is the ratio of limiting contact numbers, F ) ZBA/ZAB, so that as the solute B particle size is increased, the corresponding increase in F reflects an increase in the number of contacts B particles can make with A segments. Hence, B particles have higher chemical potential if they are larger because they interfere more with the A chains. This is in contrast with nonchain A solvents, where the size of B particles generates no special entropic contribution to the dilute chemical potential or partition coefficient because there is no chain entropy to influence (eq 36). Based on eq 40, it is possible to express the gas-alkane partition coefficient Kgas/alk as shown in eq 42.

kT ln(Kgas/alk) ) ∆g0B + kTF

(

)

ZAA 1 - p12 l p12

(42)

The term ∆g0B represents the contact free energy. The second term in eq 42 arises from the influence of globular B solute particles on chain entropy. Note that the unknown parameters in eq 42 can be determined by least-squares fitting to the published partitioning data on the xenon-alkane system.37,38 The adjustable parameters include ∆g0B, F, and ZAA. Since it is not known at the outset how many methylene groups correspond to a lattice cell, a fourth adjustable parameter is introduced to relate the length of the chain in lattice cells, l, to the number of

Figure 9. Fit of model to alkane-gas xenon partitioning. The data are from ref 28. The lines through the data are calculated using eq 42 and the parameters listed in Table 1.

TABLE 1: Parameters from Fit to Xenon-Alkane Partitioning Dataa T

∆g0B

F

10 20 30 40 50

-1.186 -1.128 -1.095 -1.057 -1.014

1.315 1.221 1.202 1.178 1.132

a Data were fit to eq 42. Values of Z AA and a were fixed at 12 and 1, respectively.

carbon atoms in the alkane chain, n. The expression used is l ) an, so 1/a is the number of carbon atoms per lattice cell. Figure 9 shows the resulting fit to the published values of RT ln(Kgas/alk), with Kgas/alk expressed in density units.28 The data are fit to within better than 0.5% of their values. Chan and Dill11 fit these data using the Flory-Huggins model at a single temperature and showed that it gives an acceptable fit as well, though the apparent fitted volume of the xenon atom is anomalously large. The Flory-Huggins model treats the solute as a chain molecule as well. Table 1 gives the results of the fits according to the present model. When all parameters are allowed to vary, the values of ZAA and a are found to be highly interdependent. As long as the ratio of ZAA/a remains about 12, good fits to the data can be obtained. For example, values of ZAA ) 12 and a ) 1, or ZAA ) 6 and a ) 0.5 fit the data nearly as well and give closely similar values for the other fitted parameters. The parameters in this table correspond to fixed values of ZAA ) 12 and a ) 1, corresponding to close-packed spheres. Contact energies and contact asymmetry parameters are much better defined. According to Table 1, near 20 °C the contact energy is about -1.1 kcal/mol, and the xenon atoms can contact about 1.2 times as many methylene groups compared to the number of xenon atoms that can fit around a methylene. It is also interesting that the parameters are only slightly dependent on temperature. All of the parameters produced by fitting the present model to the xenon partitioning data appear physically reasonable. Partitioning of Pure Alkanes into Gas and Water Phases. The entropic contribution to the chemical potential of chain molecules will be considered in order to account for partitioning experiments involving pure alkanes and their vapors or involving pure alkanes and alkanes in water. If it is assumed that no large globular B particles are present, and if it is further assumed that the difference between p12 and pij is negligible, then eq 26


Eads

can be simplified to give eq 43 for the entropy of a mixture of chain molecules and E-containing cells.

S ) nR ln(nR) - nA ln(nA) - nE ln(nE) + k nA(l - 1)[ln(p12) - ln(nA) - 1] (43) The constant pressure (additive volume) derivative is given by eq 44.

( )

()

( )

nA p12 ∂ S + (l - 1) ln + ) -l ln ∂nA Pk nR e nEnA(l - 1) 1 - p12 ZAA (44) nR(nR - nA) p12

(

)

The entropic contribution to the pseudochemical potential, µ/A,ent, is therefore given by eq 45.

() ( )

µ/A,ent ) µA,ent - kT ln kT(l - 1) ln

nA ) nR

(

Figure 10. Chain length dependence of the entropic contribution to alkane vapor-alkane liquid partitioning. The curve was generated using the last term of eq 48 (in units of RT).

)

nAe nEnA(l - 1) 1 - p12 - kTZAA (45) nRp12 nR(nR - nA) p12

The pseudochemical potential of an alkane in itself, µ/alk A,ent, is given by the limit of eq 45 in which nE/nR approaches zero. This gives eq 46.

( )

µ/alk A,ent ) kT(l - 1) ln

e lp12

(46)

For the case of a dilute alkane in a nonchain solvent (denoted wat), the pseudochemical potential, µ/wat A,ent, is given by the limit of eq 45 in which nA/nR approaches zero. Using the dilute limit for p12 (eq 24), this gives eq 47

( )

µ/wat A,ent ) kT(l - 1) ln

e ZAA

(47)

The alkane/water (or alkane/vapor if one treats the vapor phase as a lattice also) partition coefficient Kalk/wat is given by eq 48.

( )

/ - µ/alk kT ln(Kalk/wat) ) µ/wat A A ) ∆gA + kT(l - 1) ln

lp12 ZAA

(48) In this expression, µ/wat and µ/alk are the pseudochemical A A potentials of alkane A chains in water and alkane solvent, respectively, p12 refers to the neighbor probability (eq 22) for the neat alkane liquid, ∆g/A is the contact free energy including all contributions to the partition coefficient that do not arise from chain entropy, and equal lattice cell volumes for both solvents are assumed. Figure 10 shows the numerical value of the second term of eq 48 as a function of chain length for a lattice with a contact number ZAA ) 6. For the range of liquid alkanes, this makes a fairly constant contribution of about -2.2 kT to the partition coefficient. Changing the value of ZAA or scaling the number of methylene groups per lattice cell changes the magnitude of this term, but it remains fairly flat over the range of liquid alkanes for reasonable values of these parameters. Figure 11 gives a fit of eq 48 to published partitioning data39 for pure alkanes and their vapors as a function of alkane chain

Figure 11. Fit of model to alkane vapor-alkane liquid partitioning. The data from ref 39 were fit to eq 48 as described in the text. The contact energy is proportional to the molecular surface area. The chain entropy reduces the tendency of the alkanes to reside in the liquid state because more independent conformations are available in the gas phase.

length. As with the xenon partitioning data, the number of methylene groups per lattice cell was scaled using the expression l ) an. The contact energy was expressed as a function of accessible surface area of the alkanes, calculated using the expression area ) 121.68 + 31.04n, where the area is in Å2 and n is the number of carbons in the alkane chain. This expression was parametrized using published solvent-accessible surface areas.6 The best fit parameters to these data are ∆g/A ) 23.11 cal Å-2 mol-1, ZAA ) 13.65, and a ) 1.67. The average difference between experimental and fit energies is 0.04 kcal mol-1. Figure 12 shows a fit of eq 48 to published partitioning data for pure alkanes with water. The data in the units used here are from the work of Ben-Naim36 based on the original work of McAuliffe.40 Though the original data extend to shorter alkanes, only those alkanes that exist as liquids at room temperature were considered in this analysis. The best fit to these data gives ZAA ) 12.5, a ) 0.96, and ∆g/A ) 29.3 cal Å-2 mol-1. As was found above for liquid/vapor partitioning, the chain entropy contribution tends to force the alkanes out of the neat liquid phase in order to increase the conformational entropy by removing interference caused by the presence of other chains. The value of ∆g/A determined from this fit is the contact


Figure 12. Fit of model to alkane partitioning between water and pure alkane liquid. The data for alkanes that are liquid at 298 K were taken from refs 36 and 40 and fit to eq 48.

contribution to the hydrophobic effect. Chan and Dill11 suggested, on the basis of the use of globular cyclohexane as a solvent, that the contact contribution to the hydrophobic effect is approximately 34 cal Å-2 mol-1. The value extracted from alkane/water partitioning data using the present short-chain model is in much better agreement with this value compared to the value of 47 cal Å-2 mol-1 based on using the FloryHuggins expression.2,3 Summary Three strategies for treating thermodynamic and structural properties of lattice mixtures of differently sized particles and chain molecules have been described and illustrated. The contact number weighting strategy gives expressions for the numbers of contacts between differently sized particles, leading to expressions for the internal energy. The key to this strategy is to weight the particle numbers by the number of contacts these particles can make when surrounded by particles of the other type. The approach gives the correct limiting behavior for extreme concentration ratios and for equally sized particles, and the predictions agree well with the results of lattice sampling calculations. The entropy of size-asymmetric lattice mixtures can be handled by using the insertion probability strategy, which uses the insertion probability parameter fi. Given this parameter, a combinatorial expression for the number of system configurations can be derived. This strategy gives several advantages. Since the insertion probability parameter appears as a multiplicative factor in the combinatorial expression, it yields an additive contribution to the entropy. The resulting expression is exact (assuming fi is known) and offers algebraic and conceptual simplicity compared to other recent exact lattice treatments of rigid lattice objects.19,20,23 The insertion probability also has a well-defined value for dilute systems, which allows quantitative treatment of experimentally important partitioning experiments. A very simple computational lattice sampling approach can be used to quickly obtain accurate values of this parameter up to fairly high volume fractions of the large particles. Approximate algebraic expressions were presented for estimating fi on the basis of simple physical considerations. The reserved volume approach appears particularly promising for giving accurate values, especially if the component sizes are not too different. Observed deviations between the approximation and the simulations may provide insight into future improvements.

J. Phys. Chem. B, Vol. 103, No. 28, 1999 5879 The chain reconstruction strategy allows evaluation of the entropy of chain molecules under conditions for which the usual Flory-Huggins mean field approximation was not intended. In particular, the present model works equally well for short chains, dilute chains, and long chains, while effects of the presence of globular solutes can be easily handled. The key to this strategy is to enumerate the lattice configurations without regard for chain connectivity and then to correct the resulting combinatorial expression for the fraction of configurations that are expected to be correctly covalently connected. Several examples were described to illustrate and exploit the strategies. For globular solutes in globular solvents, the theory predicts that solute particle size should not influence the partition coefficients through entropic effects, though particle size does influence the internal energy of solvation. As summarized by eq 38, the model predicts an effect of solvent size on partition coefficients expressed in density units for systems involving only globular molecules. This arises from differences in the number of lattice cells per unit volume in differently sized solvents. When partition coefficients are expressed in mole fraction units for such systems, the effect of lattice cell size is automatically factored out, and the partition coefficient becomes a measure of the contact free energies. The model is therefore consistent with the successful use of mole fraction units in classical partitioning experiments involving globular molecules. However, it should be emphasized that the result of such an analysis is a contact free energy of transfer, whereas using density units gives a difference of solvation free energies (pseudochemical potentials). For polymeric and short chain (alkane) solvents, the model predicts an effect of chain length on the chemical potential of globular solutes. This is summarized by eq 42 for globular solutes in alkanes. The origin of the effect is that the presence of globular solutes reduces the number of ways that the chain segments can be arranged, consistent with having proper covalent connectivity. Larger globular solute particles have a proportionally larger effect. The Flory-Huggins model predicts a similar effect, but the ability of the present model to handle short chains and globular solutes of variable size allows a more detailed accounting of published experimental data. For example, the model gives high-quality fits of xenon partitioning between alkane solvents and the gas phase. All the resulting parameters are physically reasonable, the temperature dependence of the data is well described and dominated by the chain entropy, and a good estimate of the contact energy for xenon in alkanes is determined. Application of the strategies to partitioning data on alkanes between pure alkane and water or vapor was also carried out. The fitted value for the hydrophobic contact free energy, 29 cal Å-2 mol-1, is significantly different from the value of 47 cal Å-2 mol-1 obtained using the Flory-Huggins approach, and in closer agreement with the value of 34 cal Å-2 mol-1 based on partitioning of globular solutes between cyclohexane and water.11 According to the model developed here as summarized by eq 48, the contribution of chain entropy to the partition coefficient is only slowly dependent on chain length for chains long enough to form liquids at room temperature. The physical origin of this entropy gain is that by leaving the neat chain solvent, the alkane molecule’s conformation is no longer coupled to the other chains in the solvent. The lattice model presented in this work provides a simple set of expressions for interpreting and separating different contributions to solvation free energy differences determined from partitioning experiments. Though by design it is not a

5880 J. Phys. Chem. B, Vol. 103, No. 28, 1999 rigorous solution to the lattice combinatorial problem, it leads to very simple expressions whose terms correspond to specific effects of particle size, and which appear to be sufficiently accurate to describe experimental results in terms of these physically meaningful parameters. References and Notes (1) De Young, L. R.; Dill, K. A. J. Phys. Chem 1990, 94, 801. (2) Sharp, K. A.; Nicholls, A.; Fine, R. F.; Honig, B. Science 1991, 252, 106. (3) Sharp, K. A.; Nicholls, A.; Friedman, R.; Honig, B. Biochemistry 1991, 30, 9686. (4) Holtzer, A. Biopolymers 1992, 32, 711. (5) Tun˜o´n, I.; Silla, E.; Pascual-Ahuir, J. L. J. Phys. Chem. 1994, 98, 377. (6) Giesen, D. J.; Cramer, C. J.; Turhlar, D. G. J. Phys. Chem. 1994, 98, 4141. (7) Chan, H. S.; Dill, K. A. J. Chem. Phys. 1994, 101, 7007. (8) Kumar, S. K.; Szleifer, I.; Sharp, K.; Rossky, P. J.; Friedman, R.; Honig, B. J. Phys. Chem. 1995, 99, 8382. (9) Sharp, K. A.; Kumar, S. Rossky, P. J.; Friedman, R. A.; Honig, B. J. Phys. Chem. 1996, 100, 14166. (10) Ben-Naim, A.; Mazo, R. J. Phys. Chem. B 1997, 101, 11221. (11) Chan, H. S.; Dill, K. A. Annu. ReV. Biophys. Biomol. Struct. 1997, 26, 425. (12) Shimizu, S.; Ikeguchi, M.; Nakamura, S. Shimizu, K. J. Chem. Phys. 1999, 110, 2971. (13) Prigogine, I. The Molecular Theory of Solutions; Interscience: New York, 1957. (14) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover: New York, (1986 reprint).

Eads (15) Guggenheim, E. A. Mixtures; Oxford University Press: London, 1952. (16) Bawendi, M. G.; Freed, K. F. J. Chem. Phys. 1988, 88, 2741. (17) Freed, K. F.; Bawendi, M. G. J. Phys. Chem. 1989, 93, 2194. (18) Dudowicz, J.; Frfeed, K. F.; Madden, W. G. Macromolecles 1990, 23, 4803. (19) Li, W.; Freed, K. F.; Nemirovsky, A. M. J. Chem. Phys. 1993, 98, 8469. (20) Li, W.; Freed, K. F. J. Chem. Phys. 1995, 103, 5693. (21) Foreman, K. W.; Freed, K. F.; Ngola, I. M. J. Chem. Phys. 1997, 107, 4688. (22) Freed, K. F.; Dudowicz, J. Macromolecules 1998, 31, 6681. (23) Krukowski, A. E.; Chan, H. S.; Dill, K. A. J. Chem. Phys. 1995, 103, 10675. (24) Baker, D.; Chan, H. S.; Dill, K. A. J. Chem. Phys. 1993, 98, 9951. (25) Sanchez, I. C.; Lacombe, R. H. Nature 1974, 252, 381. (26) Sanchez, I. C.; Lacombe, R. H. J. Phys. Chem. 1976, 80, 2352. (27) Lacombe, R. H.; Sanchez, I. C. J. Phys. Chem. 1976, 80, 2568. (28) Eads, C. D.; Robosky, L. C. Langmuir 1999, 15, 2661. (29) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (30) Widom, B. J. Chem. Phys. 1963, 39, 2808. (31) Dickman, R.; Hall, C. K. J. Chem. Phys. 1986, 85, 4108. (32) Honnell, K. G.; Hall, C. K. J. Chem. Phys. 1989, 90, 1841. (33) Dickman, R.; Hall, C. K. J. Chem. Phys. 1986, 85, 3023. (34) Huggins, M. L. J. Phys. Chem. 1942, 46, 151. (35) Flory, P. J. J. Chem. Phys. 1942, 10, 51. (36) Ben-Naim, A. SolVation Thermodynamics; Plenum Press: New York, 1987. (37) Pollack, G. L.; Himm, J. F. J. Chem. Phys. 1982, 77, 3221. (38) Ben-Naim, A.; Marcus, Y. J. Chem. Phys. 1984, 80, 4438. (39) Ben-Naim, A.; Marcus, Y. J. Chem. Phys. 1984, 81, 2016. (40) McAuliffe, C. J. Phys. Chem. 1966, 70, 1267.

Strategies for Evaluating Thermodynamic Properties of Lattice

Recommend Documents