Local Composition in Solvent + Polymer or Biopolymer Systems - The

J. Phys. Chem. B 2008, 112, 3005-3012

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Local Composition in Solvent + Polymer or Biopolymer Systems Ivan L. Shulgin and Eli Ruckenstein* Department of Chemical & Biological Engineering, State UniVersity of New York at Buffalo, Amherst, New York 14260 ReceiVed: August 13, 2007; In Final Form: NoVember 30, 2007

The focus of this paper is on the application of the Kirkwood-Buff (KB) fluctuation theory to the analysis of the local composition in systems composed of a low molecular weight solvent and a high molecular weight polymer or protein. A key quantity in the calculation of the local composition is the excess (or deficit) of any species i around a central molecule j in a binary mixture. A new expression derived by the authors (J. Phys. Chem. B 2006, 110, 12707) for the excess (deficit) is used in the present paper. First, the literature regarding the local composition in such systems is reviewed. It is shown that the frequently used Zimm cluster integral provides incorrect results because it is based on an incorrect expression for the excess (or deficit). In the present paper, our new expression is applied to solvent + macromolecule systems to predict the local composition around both a solvent and a macromolecule central molecule. Five systems (toluene + polystyrene, water + collagen, water + serum albumin, water + hydroxypropyl cellulose, and water + Pluronic P105) were selected for this purpose. The results revealed that for water + collagen and water + serum albumin mixtures, the solvent was in deficit around a central solvent molecule and that for the other three mixtures, the opposite was true. In contrast, the solvent was always in excess around the macromolecule for all mixtures investigated. In the dilute range of the solvent, the excesses are due mainly to the different solvent and macromolecule sizes. However, in the dilute range of the macromolecule, the intermolecular interactions between solvent and macromolecule are mainly responsible for the excess. The obtained results shed some light on protein hydration.

1. Introduction Systems composed of low molecular weight solvents and high molecular weight polymers or proteins, etc., are prone to various types of molecular clustering. We will consider a cluster as a micropart of a system in which the concentration differs from the bulk concentration. The first kind of clustering is the solvent clustering on high-weight polymers when the solvent as a vapor is adsorbed on the polymer.1-3 Examples are the clustering of benzene on rubber,1 toluene on polystyrene,1 water on cellulose,2 etc. Another kind of aggregation is the adsorption of water on a protein that leads to an excess (compared to the bulk) of the concentration of water in the vicinity of the protein surface.4-5 Finally, one more kind of clustering is the aggregation of polymer molecules in water or aqueous solutions. As an example, one should mention the self-assembled aggregation of polyether block copolymers in water and water + cosolvent mixtures under particular conditions (above a certain concentration (cmc) and temperature (cmt), which depend on the cosolvent type and amount).6 All the above-mentioned molecular aggregations in systems containing polymers, proteins, etc., are of industrial significance.1-6 In addition, the understanding of the molecular origin of such phenomena is of importance in the theoretical understanding of the above systems. The molecular clustering in systems composed of low molecular weight solvents and high molecular weight polymers, proteins, etc., has been investigated both experimentally and theoretically. The present paper is focused on the theoretical investigation of clustering on the basis of the fluctuation theory * Corresponding author. Phone: (716) 645-2911, ext. 2214. Fax: (716) 645-3822. E-mail: [email protected].

of Kirkwood and Buff (KB).7 In a previous paper,8 we applied the KB theory to the local composition in binary systems composed of two low molecular weight components. In the present paper, the systems are composed of low molecular weight solvents and high molecular weight polymers, proteins, etc. 2. Theoretical Background 2.1. The Zimm Cluster Integral. Zimm9 was the first to apply the KB theory to the solvent clustering in binary solvent (1) + polymer (protein) (2) mixtures. On the basis of the KB theory, he was the first to derive the following expression for the KB integral (KBI),

G11 ) kTkT +

c2V2 c12µ(c) 11

-

1 c1

(1)

where T is the absolute temperature; VR is the partial molar volume per molecule of species R; k is the Boltzmann constant; kT is the isothermal compressibility; cR is the bulk molar concentration of component R; µ(c) 11 ) (1/kT) (∂µ1/∂c1)T,P ) (∂ ln a1/∂c1)T,P, µ1being the chemical potential per molecule 1; P is the pressure; a1 is the activity of component 1; and Gij is the Kirkwood-Buff integral (KBI),7 which is defined as

Gij )

∫0∞ (gij - 1)4πr2 dr

(2)

In the above equation, gij is the radial distribution function between species i and j, and r is the distance between the centers of molecules i and j. Equation 1 is equivalent to the usual

10.1021/jp076505u CCC: $40.75 © 2008 American Chemical Society Published on Web 02/19/2008

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expression employed10 to calculate the KBI, which involves the derivative of the chemical potential with respect to the mole fraction (see the Supporting Information for details). Because far from critical conditions the contribution of the compressibility term (kTkT) is very small,2,9-10 eq 1 acquires the form

G11 ≈

( )

c2V2 2

c1

∂c1 ∂ ln a1

-

P,T

1 c1

(

)

∂(a1/φ1) ∂a1

- V1

(4)

P,T

where φi ) ciVi is the volume fraction of component i in the mixture. Further, Zimm9 and Zimm and Lundberg1 introduced the notion of “cluster integral” G11/V1 to characterize the solvent clustering in the systems solvent (1)-polymer (protein) (2). They called G11/V1 a “cluster integral”, because1 “the quantity φ1G11/V1 is the mean number of type 1 molecules in excess of the mean concentration of type 1 molecules in the neighborhood of a given type 1 molecule; thus, it measures the clustering tendency of the type 1 molecules”. Using eqs 3 and 4, one can write the following expression for the cluster integral G11/V1:

G11/V1 ≈

( )

φ2 ∂c1 φ1c1 ∂ ln a1

P,T

-

(

)

∂(a1/φ1) 1 ) -φ2 φ1 ∂a1

- 1 (5)

P,T

The average number of solvent molecules in a cluster or the mean size of the cluster was considered to be given by2,11

(

)

∂(a1/φ1) φ1G11 + 1 ) -φ1φ2 υ1 ∂ ln a1

P,T

∆nij ) ciGij + ci(Vj - kTkT)

(3)

Equation 3 can be rewritten in the following frequently used form9

G11 ≈ - V1φ2

2.2. Expression for the Excess (or Deficit) in Solvent + Polymer (Protein) Systems. Recently, we demonstrated that the quantity ciGij does not represent the excess (or deficit) of molecules i around a central molecule j compared to the bulk. It was shown that the excess (or deficit) molecules i around a central molecule j is given by the expression35

In contrast to the traditional expression (∆Nij ) ciGij) for the excess, the new expression takes into account that owing to the central molecule j, there is a volume that is not accessible to molecules i. The difference between the above two expressions for the excess (or deficit) is particularly large for central molecules that possess large volumes.8,35 Far from critical conditions, one can neglect the compressibility term (kTkT) as compared to Vj. Consequently, one can write the following expression for the excess (or deficit) number of molecules i around a central molecule j:

∆nij ≈ ciGij + ciVj

( )

∆n11 ) (φ1G11/V1) + φ1

(9)

Therefore, φ1 should be added to the Zimm excess to get the true excess, ∆n11. Can the above equations provide information about the clustering? Let us consider the clustering of molecules 1 around a central molecule 1 in a volume, Vcorr, of radius R in which the concentration differs from that in the bulk. This volume Vcorr is usually called correlation volume. The total number, n11, of molecules 1 in this volume (which can be identified as the size of cluster) can be calculated using the expression36-37

n11 ) c1 (6)

(8)

The excess (deficit) of a solvent around a central solvent molecule φ1G11/V1 provided by the Zimm cluster integral is related to the true excess (or deficit) ∆n11via the expression

+ ∂ ln φ1 φ 2 ) φ2 ∂ ln a1

(7)

∫oR g114πr2 dr

i, j ) 1, 2

(10)

which can be rewritten as

P,T

Hence, the average number of solvent molecules in a cluster was considered as the excess (deficit) (compared to the bulk) of solvent molecules plus one (the central molecule). The Zimm9 (or Zimm and Lundberg1) theory of solvent clustering became a popular tool for the estimation of solvent clustering in solvent (1)-polymer (protein) (2) systems.2,11-34 It has been used to estimate the solvent clustering in numerous systems.1,2,11-34 However, several authors noted some inconsistencies in the application of the above theory to the solvent clustering. For instance, Brown19 expressed doubts regarding the interpretation of φ1G11/V1 as the excess solvent molecules around a central solvent molecule. Klyuev and Grebennikov31 noted that the average number of solvent molecules in a cluster calculated with eq 6 can be less than one, even though the central molecule is already included in the cluster. The authors of the present paper recently found8,35 that φ1G11/V1 does not represent the excess (or deficit) of molecules 1 around a central molecule 1 compared to the bulk. The interpretation of φ1G11/V1 ) c1G11 as the excess (or deficit) was used in all publications based on the Zimm and Lundberg theory of solvent clustering and in all the papers that employed the KB theory of solutions (for more details, see refs 8 and 35).

n11 ) c1

∫0R (g11 - 1)4πr2 dr + c1 ∫0R 4πr2 dr

(11)

As soon as R becomes large enough for g11 to become unity, eq 11 can be rewritten as36

n11 ) c1

∫0∞ (g11 - 1)4πr2 dr + c1 ∫0R 4πr2 dr ) c14πR3 c1G11 + ) c1G11 + c1Vcorr (12) 3

The number of species 1 in the cluster is given by n11 + 1. Comparison of this number with the size of the Zimm cluster (eq 6) reveals that n11 + 1 is larger than φ1G11/υ1 + 1 by c1Vcorr. Thus, eq 12 provides a cluster size that includes all molecules 1 and not only the excess (or deficit). It also provides a clue as to why the average size of a cluster calculated using the expression (φ1G11/υ1 + 1) was often very small.11,19,26-27,31 However, the correlation volume is not usually known, even though it can be determined experimentally by small-angle X-ray scattering, small-angle neutron scattering (SANS), light-scattering (LS), etc. Although it is not yet possible to calculate the cluster size, one can, however, calculate the excesses, ∆nij,

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Figure 1. Cluster integral, G11/V1, versus solvent volume fraction (the solid line was calculated by us): (A) the toluene (1) + polystyrene (2) system (b, ref 1), (B) the water (1) + collagen (2) system (b, ref 12), (C) the water (1) + serum albumin (2) system, (D) the water (1) + hydroxypropyl cellulose (2) system, and (E) the water (1) + Pluronic P105 (2) system.

around central molecules, and these excesses will be considered as measures of the clustering. In the next section, various solvent-polymer (protein) mixtures will be examined. The excesses ∆nij will be used as measures of the clustering. Let us emphasize that our model implies that the systems considered behave like binary solutions. This behavior is debatable at low volume fractions of the solvent because then the macromolecules may acquire a gel-like structure imbibed with the solvent. 3. Excesses (or Deficits) in Various Solvent-Polymer (Protein) Mixtures 3.1. Toluene (1) + Polystyrene (2). The cluster integral G11/ V1 for this system was calculated by Zimm and Lundberg1 and

later by Lundberg.12 To the best of our knowledge, these papers contain the first calculation of the Kirkwood-Buff integrals (KBIs). The accuracy of the calculation of the KBIs mainly depends on the accuracy of the evaluation of the derivative µ(c) 11 ) (1/kT) (∂µ1/∂c1)T,P ) (∂ ln a1/∂c1)T,P (see eq 3). Our calculations were carried out using a molecular weigh of polystyrene of 247 800 and the experimental38 activity a1 at T ) 323.15 K. The activities were represented by the FloryHuggins equation,39 and the derivative (∂ ln a1/∂c1)T,P was calculated analytically. The molar volume, V, of toluene (1) + polystyrene (2) was calculated as V ) x1V01 + x2VS2 , where V01 is the molar volume of pure toluene, VS2 is the molar volume of polystyrene in the toluene (1) + polystyrene (2) mixture calculated from literature data,40 and xi is the molar fraction of component i. The results of the calculations of the cluster

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Figure 2. Excess of solvent molecules in the vicinity of a central solvent molecule: O, excess calculated with eq 8; and b, excess calculated with the Zimm expression (∆Nij ) ciGij). (A) The toluene (1) + polystyrene (2) system, (B) the water (1) + collagen (2) system, (C) the water (1) + serum albumin (2) system, (D) the water (1) + hydroxypropyl cellulose (2) system, and (E) the water (1) + Pluronic P105 (2) system.

integrals G11/V1 are presented in Figure 1A. The other KBIs (G12 and G22) have been calculated from G11 using the expressions35 V1∆n11 ) -V2∆n21 and V1∆n12 ) -V2∆n22 with eq 8 for ∆nij. The obtained KBIs were used to calculate the excesses (or deficits) from eq 8, and the Zimm excesses (or deficits) (∆Nij ) ciGij). The results of these calculations are plotted in Figures 2A and 3A. They demonstrate that toluene is in excess around both (toluene and polystyrene) central molecules. The excesses (or deficits) from eq 8 and the Zimm excesses (or deficits) (∆Nij ) ciGij) will be compared in the Discussion and Conclusion Section of the article. 3.2. Water (1) + Collagen (2). The cluster integral G11/V1 for this system was calculated by Zimm and Lundberg1 and

Lundberg.12 Bull’s data at T ) 298.15 K have been used for the activity of water in this system.4 Starkweather11 has determined the activity of water in the concentration range 0 eφ1 e 0.2 and noted that it depends on the volume fraction as a1 ) 12φ12, the expression that was used in our calculations. The molecular weight and the partial specific volume of collagen were taken from ref 41. The results of the calculations are presented in Figures 1B, 2B, and 3B. In contrast to the toluene + polystyrene mixture, the solvent (water) is in deficit around a central water molecule but in excess around a protein molecule. 3.3. Water (1) + Serum Albumin (2). Again, Bull’s data at T ) 298.15 K have been used for the activity of water.4 Starkweather11 has determined the activity of water in the

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J. Phys. Chem. B, Vol. 112, No. 10, 2008 3009

Figure 3. Excess of solvent molecules in the vicinity of a central solute molecule: O, excess calculated with eq 8; b, excess calculated with the Zimm expression (∆Nij ) ciGij). (A) The toluene (1) + polystyrene (2) system, (B) the water (1) + collagen (2) system, (C) the water (1) + serum albumin (2) system, (D) the water (1) + hydroxypropyl cellulose (2) system, and (E) the water (1) + Pluronic P105 (2) system.

TABLE 1: Results of Calculations Regarding the Excess (or Deficits) in the Systems Investigated system toluene (1) + polystyrene (2) water (1) + collagen (2) water (1) + serum albumin (2) water (1) + hydroxypropyl cellulose (2) water (1) + Pluronic P105 (2)

composition range

∆n11 > 0

∆n21 > 0

0 e φ1 e 1 0 e φ1 e 0.2 0 e φ1 e 0.2 0 e φ1 e 0.2

yes no no yes

yes yes yes yes

0 e φ1 e 1

yes

yes

concentration range 0 eφ1 e 0.2 and noted that it depends on the volume fraction as a1 ) 29φ12, the dependence that was used in our calculations. The molecular weight and the partial specific volume of serum albumin were taken from ref 41. The

results are plotted in Figures 1C, 2C, and 3C. The results obtained for the excess (or deficit) of water molecules around both central molecules (water and serum albumin) are comparable to those obtained for the water + collagen mixture. 3.4. Water (1) + Hydroxypropyl Cellulose (2). There are several papers11,14,28 in which the KB theory of solutions was applied to water + cellulose (or cellulose derivatives) mixtures. The water + hydroxypropyl cellulose mixture was selected in this paper because there are accurate data regarding the activity of water.42 The data in the range 0 eφ1 e 0.2 have been represented by the expression

ln(a1/φ1) ) t1φ2 + t2φ22

(13)

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Figure 4. The contribution to ∆n12 due to different volumes of solvent and solute: (A) toluene + polystyrene, (B) water + collagen, (C) water + serum albumin, (D) water + hydroxypropyl cellulose, and (E) water + Pluronic P105. O, ∆n12 calculated for a real system using eq 8 (see Figure 3) and b, ∆n12 calculated for an ideal system using eq 14.

where t1 and t2 are adjustable parameters (t1 ) 2.925 and t2 ) -0.397). Equation 13 was used to calculate the derivative µ(c) 11 , and the partial specific volume of hydroxypropyl cellulose was taken as that of the hydroxyethyl cellulose, which is provided in ref 43. The results are plotted in Figures 1D, 2D, and 3D. 3.5. Water (1) + Pluronic P105 (2). Pluronic P105 is6 a polyether block copolymer (EO37PO58EO37, where EO and PO denote ethylene oxide and propylene oxide segments, respectively). At elevated temperatures6 (above 40-60 °C), this block copolymer self-assembles in water and in water + cosolvent as micelles. At low temperatures and concentrations, the block copolymer molecules are present in solution as independent polymer chains.6 We carried out the calculation at the “low”

temperature of 24 °C because activity data were available only at that temperature.44 The molecular weight and the partial specific volume of Pluronic P105 were also taken from ref 44. The activity in the water + Pluronic P105 mixture was represented by the Flory-Huggins equation44 and the derivative (∂ ln a1/∂c1)T,P was calculated analytically. The calculated excesses (or deficits) around both central molecules, water and Pluronic P105, are plotted in Figures 1E, 2E, and 3E. 4. Discussion and Conclusion In this paper, the KB theory of solutions was applied to binary mixtures containing a low molecular weight solvent and a high molecular weight polymer, protein, etc.

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We used our new expression35 for the excess (or deficit) to examine the local composition in binary solvent (1) + polymer (protein) (2) mixtures. Several mixtures ((I) toluene (1) + polystyrene (2), (II) water (1) + collagen (2), (III) water (1) + serum albumin (2), (IV) water (1) + hydroxypropyl cellulose (2), and (V) water (1) + Pluronic P105 (2)) were considered. The excess (or deficit) of solvent molecules around a central solvent molecule (∆n11) and around a central solute molecule (∆n12) were calculated (see Figures 2 and 3), and the results are summarized in Table 1. In addition, the calculated excesses (or deficits) were compared with the Zimm excesses (or deficits) ∆Nij ) ciGij in Figures 2 and 3. The comparison shows that ∆n11 and ∆N11 are different in magnitude but have the same sign. Such a result is expected, because for the systems considered here, |G11| . V1, and for this reason, ∆n11 and ∆N11 have comparable values. In contrast, the comparison between ∆n12 and ∆N12 reveals striking difference between the two excesses (deficits). Whereas for all systems considered ∆n12 > 0 and large, ∆N12 e 0 and is smaller in absolute value than ∆n12. It is worth noting that ∆N12 e 0 contradicts the experimental results,41 which reveal that the proteins preferentially uptake water in aqueous solutions. Let us emphasize the physical significance of the obtained results regarding the excesses (deficits). ∆n11 > 0 means preferential hydration (or solvation in the case of toluene + polystyrene mixture) of the solvent molecules, and ∆n12 > 0 means that the polymer (protein) molecules are preferentially hydrated (or solvated for the toluene + polystyrene mixture). The signs and magnitudes of ∆n11 and ∆n12 depend on two factors that are of “enthalpic” and “entropic” nature. The former is due to differences in the intermolecular interactions: (1) solvent-solvent and solvent-solute for ∆n11 and (2) solutesolute and solvent-solute for ∆n12. As suggested by Kauzmann (as quoted by Timasheff),45 the latter is due to the different sizes of the solute and solvent molecules. According to Kauzmann, the smaller molecules can more easily penetrate in the vicinity of a central molecule, and for this reason, the vicinity of a central molecule is enriched in the smaller molecules. One can see from Table 1 that ∆n11 > 0 for three mixtures, but not for water + collagen and water + serum albumin mixtures. The very strong H-bonding46 between two water molecules and the very small size of the water molecule suggest that ∆n11 > 0. The opposite inequalities (∆n11 < 0 in water + collagen and water + serum albumin mixtures) can be attributed to the much stronger H-bonding of the water molecules to some functional groups of the protein (collagen and serum albumin) than to the water molecules. In contrast to ∆n11, ∆n12 > 0 for all mixtures investigated, and hence, the polymers or proteins are preferentially hydrated (solvated in the case of toluene + polystyrene mixture). Let us examine separately the contributions to ∆n12 provided by the “entropic” and “enthalpic” factors. The contribution to ∆n12 provided by the different sizes of the solvent and polymer (protein) molecules will be evaluated from the excess in an ideal mixture of components possessing the same volumes as the analyzed mixture. In this case,35 one can write the following expression for ∆n12,

∆nid 12 )

x1x2V2(V2 - V1) (x1V1 + x2V2)2

(14)

where xi is the bulk mole fraction of component i. ∆n12 and ∆nid 12 are compared in Figure 4. One can see that the contribution of ∆nid 12 is important in the dilute range of the

TABLE 2: Values of 100∆nid 12/∆n12 (%) for φ1 ) 0.2 system toluene (1) + polystyrene (2) water (1) + collagen (2) water (1) + serum albumin (2) water (1) + hydroxypropyl cellulose (2) water (1) + Pluronic P105 (2)

100∆nid 12/∆n12 (%) 88.0 88.0 88.0 88.1 88.2

solvent (0 eφ1 e 0.2). The values of 100∆nid 12/∆n12 for φ1 ) 0.2 are listed in Table 2. Figures 4A and E show that the “enthalpic” contribution to ∆n12 becomes dominant at high φ1 (φ1 g 0.4 - 0.5). Protein hydration has been the focus of attention for more than a century.47 Many theories have been suggested to explain protein hydration.41,48 These theories can be subdivided into three groups:48 (1) those based on the physical adsorption of water vapor on the protein surface, (2) those based on the stoichiometric binding of water molecules to specific functional groups of the protein, and (3) models that have considered the water-protein system as a simple aqueous solution. Our results regarding the excess number of water molecules in the vicinity of a protein molecule indicate that at low humidity (φ1 e 0.20.3), the excess of water in the vicinity of a protein molecule is due mainly to the difference in the sizes of the water and protein molecules. Pauling5 connected the hydration of proteins to the adsorption of water molecules on the polar groups of the former. He assumed that each polar group adsorbs one water molecule. Later, Kuntz and Kauzmann41 criticized this approach because it provided only one-fourth of the hydration level found experimentally. Our simple analysis indicates that the difference in the sizes of water and protein molecules may constitute an important factor in protein hydration. For water (1) + serum albumin (2) mixture, Figure 4C provides at φ1 ) 0.2, ∆n12 ≈ 540 and ∆nid 12 ≈ 430. Therefore, almost the entire increase in the hydration number compared to the bulk is due to the difference in the sizes of water and protein molecules and a smaller fraction (110 molecules) is probably due to the binding of the water molecules to the specific functional groups of serum albumin. Supporting Information Available: Derivation of the expressions for the Kirkwood-Buff integrals. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Zimm, B. H.; Lundberg, J. L. J. Phys. Chem. 1956, 60, 425-428. (2) Starkweather, H. W. Polym. Lett. 1963, 1, 133-138. (3) Water in Polymers; Rowland, S. P., Ed.; ACS Symposium Series 127, American Chemical Society: Washington, DC, 1980. (4) Bull, H. B. J. Am. Chem. Soc. 1944, 66, 1499-1507. (5) Pauling, L. J. Am. Chem. Soc. 1945, 67, 555-557. (6) Alexandridis, P.; Yang, L. Macromolecules 2000, 33, 5574-5587. (7) Kirkwood, J. G.; Buff, F. P. J. Chem. Phys. 1951, 19, 774-777. (8) Shulgin, I. L.; Ruckenstein, E. Phys. Chem. Chem. Phys. 2008, 10, 1097-1105. (9) Zimm, B. H. J. Chem. Phys. 1953, 21, 934-935. (10) Matteoli, E.; Lepori, L. J. Chem. Phys. 1984, 80, 2856-2863. (11) Starkweather, H. W. Macromolecules 1975, 8, 476-479. (12) Lundberg, J. L. J. Macromol. Sci. Phys. 1969, B3, 693-710. (13) Williams, J. L.; Hopfenberg, H. B.; Stannett, V. J. Macromol. Sci. Phys. 1969, B3, 711-725. (14) Orofino, T. A.; Hopfenberg, H. B.; Stannett, V. J. Macromol. Sci. Phys. 1969, B3, 777-788. (15) Lundberg, J. L. J. Pure Appl. Chem. 1972, 31, 261-281. (16) Starkweather, H. W. Clustering of Solvents in Adsorbed Polymers; In Structure Solubility Relationship in Polymers; Harris, F. W., R. B. Seymour, R. B., Eds.; Academic Press: New York, 1977.

3012 J. Phys. Chem. B, Vol. 112, No. 10, 2008 (17) Coelho, U.; Miltz, J. J.; Gilbert, S. G. Macromolecules 1979, 12, 284-287. (18) Starkweather, H. W. In Water in Polymers; ACS Symposium Series 127; American Chemical Society: Washington, DC, 1980; p 433. (19) Brown, G. L. In Water in Polymers; ACS Symposium Series 127; American Chemical Society: Washington, DC, 1980; p 441. (20) Saeki, S.; Holste, J. C.; Bonner, D. C. J. Polym. Sci. Polym. Phys. Ed. 1982, 20, 805-814. (21) Misra, A.; David, D. J.; Snelgrove, J. A.; Matie, G. J. Appl. Polym. Sci. 1986, 31, 2387-2398. (22) Saeki, S.; Tsubokawa, M.; Yamaguchi, T. Macromolecules 1987, 20, 2930-2934. (23) Saeki, S.; Tsubokawa, M.; Yamaguchi, T. Macromolecules 1988, 21, 2210-2213. (24) Horta, A. Macromolecules 1992, 25, 5651-5654. (25) Horta, A. Macromolecules 1992, 25, 5655-5658. (26) Vancso, G. J.; Tan, Z. Can. J. Chem. 1995, 73, 1855-1861. (27) Tan, Z.; Vancso, G. J. Macromolecules 1997, 30, 4665-4673. (28) Beck, M. I.; Tomka, I. J. Macromol. Sci. Phys. 1997, B36, 19-39. (29) Benczedi, D.; Tomka, I.; Escher, F. Macromolecules 1998, 31, 3055-3061. (30) Benczedi, D.; Tomka, I.; Escher, F. Macromolecules 1998, 31, 3062-3074. (31) Klyuev, L. E.; Grebennikov, S. F. Zh. Fiz. Khim. 1999, 73, 17001702. (32) Sabzi, F.; Boushehri, A. Eur. Polym. J. 2005, 41, 974-983. (33) Sabzi, F.; Boushehri, A. Eur. Polym. J. 2005, 41, 2067-2087. (34) Kilburn, D.; Claude, J.; Schweizer, T.; Alam, A.; Ubbink, J. Biomacromolecules 2005, 6, 864-879.

Shulgin and Ruckenstein (35) Shulgin, I. L.; Ruckenstein, E. J. Phys. Chem. B 2006, 110, 1270712713. (36) Ben-Naim, A. Statistical Thermodynamics for Chemists and Biochemists; Plenum Press: New York, 1992. (37) Hill, T. L. Statistical Mechanics: Principles and Selected Applications; McGraw-Hill: New York, 1956. (38) Wong, H. C.; Campbell, S. C.; Bhethanabotla, V. R. Fluid Phase Equilib. 1997, 139, 371-389. (39) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomes de Azevedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; PrenticeHall: Englewood Cliffs, NJ, 1986. (40) Boyer, R. F.; Spencer, R. S. J. Polym. Sci. 1948, 3, 97-127. (41) Kuntz, I. D.; Kauzmann, W. AdV. Protein Chem. 1974, 28, 239345. (42) Aspler, J. S.; Gray, D. G. Macromolecules 1979, 12, 562-566. (43) Andersson, G. “The Partial Specific Volume of Polymers at High Pressures”; In The Physics and Chemistry of High Pressures; Papers read at the symposium held at Olympia, London, June 27-29, 1962; Society of Chemical Industry: London, 1963. (44) Gu, Z.; Alexandridis, P. Macromolecules 2004, 37, 912-924. (45) Timasheff, S. N.; Xie, G. Biophys. Chem. 2003, 105, 421-448. (46) Eisenberg, D.; Kauzmann, W. The Structure and Properties of Water, Oxford University Press: Oxford, 1969. (47) Timasheff, S. N. Biochemistry 2002, 41, 13473-13482. (48) Shamblin, S. L.; Hancock, B. C.; Zografi, G. Eur. J. Pharm. Biopharm. 1998, 45, 239-247.