Ind. Eng. Chem. Res. 1994,33, 2288-2293
2288
Aqueous Two-Phase Systems. 1. Salt Partitioning H a r t o u n Hartounian,+ S t a n l e y I. Sandier,. and Eric
W.K a l e r
Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The partitioning of salts in aqueous two-phase systems formed from polyethylene glycol, dextran, and water produces an electrostatic potential difference between the phases which influences the partitioning of proteins and other biomolecules. In this paper we study the partitioning of, separately, NaC1, NaH2P04,and NaHS04 in such systems, as well as the resulting electrostatic potential differences. Polymer concentrations were measured by liquid chromatographyand ion concentrations by atomic absorption or inductively coupled emission spectroscopy. A combination of the UNIQUAC, Debye-Huckel, and Bransted-Guggenheim equations,after accounting for differences in the standard states, resulted in a thermodynamic model t h a t correlatesthe concentrations of all species (polymers, ions, and water) and allows calculation of the resulting electrostatic potential difference. At comparable concentrations, NaHS04 produces the largest electrostatic potential difference between the phases and NaCl the smallest. Introduction The partitioning behavior of biomoleculescan be altered and controlled by changing solution properties such as pH or ionic strength, or by changing the type of electrolyte. The role of electrolyte type and concentration is particularly important because the salt itself partitions unequally between the two aqueous polymer phases (Brooks et al., 1984; Bamberger et al., 1984; Haynes et al., 1989). This partitioning produces a difference in electrostatic potential between the coexisting phases, and this potential difference strongly drives the segregation of charged biomolecules such as proteins (Albertsson, 1986; Haynes et al., 1993). We have previously developed a thermodynamic model of polymer partitioning as a function of polymer molecular weight and temperature (Hartounian et al., 1993), and now extend it to model salt partitioning. The partitioning of various salts between two aqueous phases can be understood in terms of the molecular interactions in the mixture. The polymer-polymer, polymer-solvent, polymer-ion, and ion-ion interactions are functions of the polymer structures, the concentrations of the polymers and salt, the hydration of the polymer chain in aqueous solution, and the valence and the size of the ion. The equilibrium distributions of the polymers between the two phases depends on their structures and molecular weights. The choice of polymers determines the phase compositions and the nature of the polymerion short-range interactions. In addition, the distribution of ions between the two coexisting phases is determined by the ion-ion electrostatic interactions, which are a function of the size and valence of the ion. Different ions have different partitioning behaviors that, in general, can be related to their solubility in aqueous solutions of water soluble polymers (Zaslavskyet al., 1986,1987,1988,1991). This work describes ion partitioning in aqueous twophase systems. The partitioning of ions plays an important role in determining the partitioning of proteins as discussed in the following paper. We have investigated experimentally aqueous two-phase polymer systems composed of polyethylene glycol (PEG) and dextran (DEX) and measured the effects of ionic strength and salt type on the
* Author
to whom correspondence should be addressed.
E-mail:
[email protected]. t Depo Tech Corporation, 11025 N. Torrey Pines Rd., Suite 100,La Jolla, CA 92037. 0888-5885/94/2633-2288$04.50/0
distributions of the salts and polymers between the two coexisting phases. We have also developed a thermodynamically-based correlation of the observed ion partitioning in the aqueous two-phase systems. The approach we use yields a simpler and more tractable free energy expression, and a simpler partitioning model, than the more complete model of Haynes et al. (1993). However, it may not be accurate at very high salt concentrations. Experimental Section PEG 8000 and DEX 500were obtained from the Sigma Chemical Company, and sodium chloride and monobasic sodium salts of phosphate and sulfate from the Fisher Scientific Company. Stock solutions of PEG 8000, DEX 500, and several salts were made by weight, and their concentrations confirmed by refractive index, polarimetry, and atomic absorption measurements, respectively. Twophase systems were prepared gravimetrically in centrifuge tubes from the stock solutions and deionized water. After mixing the samples were centrifuged for at least 10 min. When complete phase separation was achieved, a sample of the top phase was withdrawn using a pipet and a sample of the bottom phase obtained through a syringe piercing the bottom of the tube. Samples were volumetrically diluted and the polymer concentrations analyzed using a liquid chromatography system with a differential refractometer detector (Waters Model 410) and multisolvent delivery system (Waters Model 600) with four gel permeation columns in series (two ultrahydrogel linear, one ultrahydrogel 250, and one ultrahydrogel 120). Data were collected using a chromatography data station (Waters Maxima 820 on an IBMA T computer). In all cases the mobile phase was 0.1 N sodium nitrate solution at a flow rate of 1mL/min. Sodium ion concentrations in solution were determined using a Perkin Elmer 220 atomic absorption spectrometer by the method described by Kopp and McKee (1983), and confirmed with inductively coupled emission spectroscopy (JY 70 plus Spectroanalyzer, Instruments SA/Jobin-Yvon Company). The measured pH’s were 5.4 for the phosphate system and 7.0 when either chloride or sulfate was used. A t these pH’s, the salts chiefly dissociate to &?Pod- and HS04-; calculations using the activity-based equilibrium constants show that the concentrations of the other ions are negligible ( K a , ~ 2 p = ~ , 6.2 X lo4, Ka,~p0,2-= 1.7 X 10-13; Ka,~so4= 1.2 X 1P2(Masterton et al., 1985)). 0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 2289 Thermodynamics of Water/Polymer/Salt Solutions The thermodynamic properties of aqueous solutions of polymers and ions depend on long-range electrostatic interactions between the ions and short-range nonionic and ionic interactions with the uncharged polymers. In this view, there are three contributions to the excess Gibbs free energy, GEX, of the solution: (1) ion-ion interactions, described by an extended Debye-Huckel expression, (2) water-polymer interactions, described by the UNIQUAC model (Abrams and Prausnitz, 1975),and (3) ion-polymer interactions, described by a Brornsted-Guggenheim expression. With this expression for GEX the distributions of the polymers, solvent, and ions are then computed by minimizing the total Gibbs free energy. Each component of the free energy and the method of calculation are described in the following sections.
range interactions between ions and the short-range interactions of the ions with each other and with the polymer (Newman, 1973; Haynes et al., 1989). Thus the excess Gibbs free energy due to electrostatic, ion-ion, and ion-polymer short-range interactions is
-GEX* RT
2
- - c t d I d B a d I ) E ; n j + ~t & J~Z O@ 3
i j m i(4) m j
J
The first term in eq 4 is due to long-range electrostatic interactions, and the second term is the contribution from short-range ion-ion and polymer-ion interactions (Newman, 1973). The total Gibbs free energy is G = GEX' GId, where the ideal Gibbs free energy is
+
Ion-Ion and Polymer-Ion Interactions The nonideality of electrolyte solutions is due to the strong long-range interactions of ionic species in solution. The influence of these interactions on the Gibbs free energy is accounted for by the Debye-Huckel expression for the excess Gibbs free energy of the solution:
where G E X * ~ is- ~the excess Gibbs free energy (in the unsymmetric convention, see below), a is the DebyeHuckel constant, B is the Debye-Huckel parameter, a is the average value of the radii of pairs of hydrated ions, I is the ionic strength, xw and Mw are the mole fraction and molecular weight of the solvent, and
(Newman, 1973) n, and pWoare, respectively, the number of moles and standard statechemical potential of the water, and Xje is the standard state activity of the ion. Local Polymer-Polymer-Water Interactions The thermodynamics of aqueous polymer solutions has been investigated extensively (Flory, 1942;Huggins, 1942; Edmond and Ogston, 1968; Cabezas et al., 1990; King et al., 1988),and local composition models such as UNIQUAC have been used to account for the nonidealities of polymer solutions (Kang and Sandler, 1987). The UNIQUAC expression for the excess free energy for polymers in water is
n
The Debye-Huckel theory is valid only for dilute electrolyte solutions and for ions of approximately equal size (so an average diameter can be chosen), and many corrections and extensions have been proposed (Pitzer, 1973, 1977; Robinson and Stokes, 1959; Waisman and Lebowitz 1970,1972; Blum, 1975). In particular, predictions of ion activities are improved when a BrcanstedGuggenheim type term is added (Guggenheim,1935,1966): (3)
GEi
where is the excess Gibbs free energy (again in the unsymmetric convention), mi is the molality of the ion i, and & is the osmotic virial coefficient for the ion pair i-j. A similar equation has been used with the Debye-Hackel expression to calculate the thermodynamic properties of mixtures of polymers, ions, and water (Haynes et al., 1989); in that work polymer-polymer interactions were also described by a virial expansion. More recently, the mean spherical approximation for the primitive model of electrolyte solutions (Waisman and Lebowitz, 1970;Blum, 1975;Blum and Horye, 1977) has been used (Haynes et al., 1993). This model is better suited for ions of very different size and higher electrolyte concentrations than the extended Debye-Htickel model we use, but represents an undue complication for the solutions we consider. We use the Guggenheim extension of the Debye-Huckel free energy expression here to account for both the long-
(5)
with
and (7)
where wi is the weight percent of component i, r[ and qi' are the volume and surface parameters of the polymer i per unit mass, z = 10 is the coordination number, and Mi is the molecular weight of component i. Here the sum is over each of the polymer species and water. Also, T i j = exp(-aij/T) where aij is the interaction parameter for the pair of components i and j. The contributions of the polymers and water to GEX were calculated using the UNIQUAC interaction parameters obtained by Kang and Sandler (1987). That work was only concerned with phase equilibrium calculations of aqueous two-phase systems free of salt at constant temperature. The effect of
2290 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994
temperature on the equilibrium distributions of the polymers is reported elsewhere (Hartounian et al., 1993). McMillan-Mayer and Lewis-Randall Standard States We have written the total Gibbs free energy as a sum of terms resulting from the various interactions between the components in solution. However, care must be taken since the conventions used for the standard states and the activity coefficients in these equations are not the same. UNIQUAC uses a symmetric convention in which yi 1 as xi 1,while activity coefficients of ions in electrolyte solution theories are based on an unsymmetric convention in which yi* 1 as mi -+ 0. For phase equilibrium calculations of mixtures of ionic and nonionic species the excess Gibbs free energy relations must be internally consistent. The relation between the symmetric (GEX) and unsymmetric (GEX*)excess Gibbs free energy is -+
+
-
where yi” is the infinite dilution activity coefficient. For the calculations reported here, the UNIQUAC symmetric excess Gibbs free energy relation was converted to the unsymmetric convention. Further, the extended Debye-Hiickel electrostatic expression is based on McMillan-Mayer theory (McMillan and Mayer, 1945),in which the independent variables are the molar concentrations of the solute, temperature, and the chemical potential of the solvent, and in which the Helmholtz free energy is the most convenient thermodynamic function. On the other hand, solution thermodynamic models based on Lewis-Randall theory (such as UNIQUAC) use the molal concentrations of the solutes, temperature, and pressure as the independent variables, and the Gibbs free energy is the main thermodynamic function. The conversion of the McMillan-Mayer convention to the Lewis-Randall convention for electrolyte solutions was carried out by Cardoso and O’Connell (1987). They obtained activity coefficient expressions for components in a multisalt/multisolvent mixture, and they found no difference between the Lewis-Randall and McMillanMayer expressions for a solution with a single solvent. Using Friedman’s approach (Friedman, 1962, 1972a,b), we have obtained a rigorous relationship between the thermodynamic properties calculated in McMillan-Mayer and Lewis-Randall theories (Hartounian, 1993). The result is that the excess Helmholtz free energy of a state of molality m and osmotic pressure II is related to the excess Gibbs free energy of the system via
-AEX cRT
GEX
+
-K T
where AEX and G E X are the excess Helmholtz and Gibbs free energies, c is the ion number density, and Q is the “correction factor”. Q varies from 0.113 to 0.205 for 1:l electrolytes at 25 OC at 6 m ionic strength (Vaslow, 1972). For ion concentrations up to 1 m, Q is negligibly small (Vaslow,1972;Kabiri-Badr, 1990),so this term is neglected in the phase equilibrium calculations presented here. The correction factor Q may be more important for solutes which have large partial molar volumes, such as for proteins. This problem has been discussed by Haynes et al. (1993).
Table 1. UNIQUAC Interaction Parameters, a~ (K)from Kang and Sandler (1987) PEG 8OOO DEX 500 water
PEG 8OOO 0 2894.50 -130.45
DEX 500 -68.16 0
water -243.33 -273.92
219.11
0
Parameter Estimation The distribution of polymers, ions, and water between the two phases at equilibrium depends on the values of the interaction parameters in the excess Gibbs free energy expressions (eqs 3 and 4). Experiments show (see below) that the low concentrations of salts (0.05 M) of interest to us here do not significantly affect the distribution of the polymers between the coexisting phases, so we will assume that the salt does not affect the polymer-polymer and polymer-solvent interactions in solution. Thus in the UNIQUAC excess Gibbs free energy relation the interaction parameters used are those determined previously by correlating experimental liquid-liquid equilibrium (LLE) data for aqueous polymer solutions (Table 1, from Kang and Sandler, 1987). Liquid-liquid equilibrium data measured at different polymer and salt feed compositionswere used to determine the ion-ion and ion-polymer second virial coefficients by minimizing the objective function 2
F
n
F , ~ ( ww$P)’ $p=l i=l
where wpi is the weight percent of component i in phase p, and n is the number of components. Using a set of
initial guesses for the values of the osmotic virial coefficients Bij, the phase equilibrium compositions of the polymers, water, and ions were calculated by minimizing the total Gibbs free energy of the system
Here GTod is the molar Gibbs free energy of the twophase system, nT is the total number of moles in the twophase system, Gi” is the standard state molar Gibbs free energy of component i, nip is the mole number of component i in phase p, np is the total number of moles in phase p, and Gp”” and G y are the excess and ideal molar Gibbs free energies of phase p , respectively. To calculate the phase equilibrium for each feed composition, the 10 unknown concentrations (five components in two phases) were reduced to four unknowns using the mass balance and electroneutrality constraints. Then the total Gibbs free energy of the system, eq 11,was computed using the measured phase compositions as the initial guesses and an estimated set of interaction parameters. The calculation was continued by varying the phase compositions until the global minimum for the total Gibbs free energy was obtained using the Nelder-Mead optimization algorithm (Nelder and Mead, 1965). Then the objective function in eq 10 was calculated and compared with a preset tolerance (l0-lo). If the objective function was greater than this value, a new set of osmotic virial coefficients was chosen and the calculation repeated until a global minimum of the objective function was obtained. The second virial coefficients obtained in this way are presented in Table 2.
--
Table 2. Osmotic Virial Coefficients, & (kg/mol), for Polymers and Ions
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2291
NaCl Na -2.01 -1.17 0.00 -1.87
PEG 8OOO' DEX 500 Na c1
c1 -2.96 -0.54 -1.87 0.00
24
I
NaHzPO4 Na -2.45 -0.89 0.00 -0.26
PEG 8000 DEX 500 Na
HzPOi -9.10 -3.71 -0.26 0.00 NaHSOi
Na -3.06 -0.53 0.00 -0.11
PEG 8OOO DEX 500 Na HSO4
HSOi -12.87 -6.16 -0.11 0.00
12
0 0
NaCl
hl
NaHS04
0.9
1.2
1.5
1.8
Figure 2. Variation of tie-lie length with total concentration of sodium chloride. The lines are guides to the eye. 1
NaH2P04 0
0.6
Salt Concentration, M
-NO Salt
10
0.3
0.95
8
F3 g 6
0.9 0.85
00
0.8
t
\ 0.75
NaH2P04
0
0.7
0
5
10
15
20
25
Dextran 500, WT% Figure 1. Measured LLE for aqueous mixtures of PEG8000 and DEX500 with three different salts at a feed concentration of 0.05 M.
Results and Discussion As shown in Figure 1,the tneasured polymer-water LLE for mixtures of PEG 8000, DEX 500, and water a t a salt feed concentration of 0.05 M do not depend on the type of salt. The binodal curve calculated using the UNIQUAC model with previously determined parameters also matches all of the data adequately (Figure 11, so the assumption that low concentrations of salt do not affect the partitioning of polymers is justified. The tie lines also do not change appreciably as salt is added up to 1.5 M (asshown in Figure 2). On the other hand, the partitioning of salt between the two phases does depend strongly on both salt type and tie-line length (Figure 3). All of the salts (NaC1, NaH2Po4, and NaHS04) preferentially partition into the bottom (DEX-rich) phase. Monobasic sodium sulfate partitions to the bottom phase more strongly than does the monobasic sodium phosphate or sodium chloride. The partitioning of the salts studied in this work increases with increasing tie-line length. The thermodynamic model correlates these experimental results well (Figure 3).
NaCl
e
-Model
0.65
0
5
10
15
20
25
30
Tie Line Length, w/w%
Figure 3. Partition coefficients, defined as the ratio of molality of the ion in the top phase to molality of the ion in the bottom phase, for various salts as a function of tie-line length. The solid lines are model fits.
The electrostatic potential difference between the two phases for different salts can be calculated from our thermodynamic model for aqueous mixtures of polymers and salt. The electrostatic potential difference between the two phases, A$, for a 1:l electrolyte is (Albertsson, 1986) Aq=-ln RT
(Y%T"P) Top Bot
2R Y- Y+ where yi is the activity coefficient of ion i. From a feed containing 50 mM salt (either NaC1, NaHzPO4, or NaHSod) and for a given tie-line length, the value of the electrostatic potential difference between the coexisting < A $ N ~ S O(Figure ~ phases increases as A$N,CI < Arhrld~go~ 4). This trend reflects the partitioning behavior of these salts, i.e., monobasic sodium sulfate partitions to the
2292 Ind. Eng. Chem. Res., Vol. 33, No. 10, 1994 4
r
L
/
t
0
10
15
20
I
I
/
5
I
25
30
Tie-Line Length, w/w%
Figure 4. Calculated electrochemical potential difference between the two phases for various salts as a function of tie-line length.
bottom phase more strongly than does monobasic sodium phosphate or sodium chloride. The UNIQUAC solution thermodynamic model describes the aqueous polymer solution accurately. The interaction parameters used in the UNIQUAC model are fixed and the calculations are completely predictive; there was no correlation of experimental data. Compared to the osmotic virial expansion model (Haynes et al., 1989) which fails to describe accurately aqueous polymer solutions with concentrations above 20 wt % ,the UNIQUAC model accurately predicts the equilibrium compositions of phases made with different molecular weights of PEG and DEX at any feed composition. Also, we have previously shown that the UNIQUAC model leads to accurate phase behavior predictions of polymer solutions as a function of polymer molecular weight, polymer polydispersity, and temperature. Analysis of the osmotic second virial coefficientsin Table 2 provides clues to the partitioning behavior of ions in aqueous two-phase systems. In general, the value and the sign of an osmotic virial coefficient reflects the type of interaction between two components in solution. A negative osmotic virial coefficient corresponds to an attractive two-body interaction, e.g., an attractive electrostatic or hydrophobic interaction, while a positive osmotic virial coefficient corresponds to repulsive interactions between two components, e.g., repulsive electrostatic or excluded volume effects (King et al., 1988). All of the polymer-ion osmotic virial coefficients are negative (Table 2), so there are attractive forces between the ions and the polymers. The magnitude of these attractive forces depends on the size of the ion, the polarizability of the polymer chain by the ionic field, and the polarizability of the ion itself. Comparison of the absolute values of the cation osmotic virial coefficients with PEG in Table 2 shows &f&Na < pgz2p04 < &%%.: The same pattern holds for the PEG-anion coefficients. In principle, the osmotic second virial coefficients for PEG-Na should be the same in all cases since they are independent of concentration and account for binary interactions in solution. That the calculated values are somewhat different may be due to three-body interactions in solution.
The magnitude of the absolute values of the osmotic virial coefficients for the DEX-anion are in the following order B D E X - ~ < i BDEX-H~PO,< BDEX-HSO,,so the affinity of the anion for the DEX-rich phase increases in the order C1- < HzPO4- < HS04-. This trend agrees with the experimentally observed salt partitioning: the monobasic sodium sulfate ion partitions more strongly into the bottom phase than do the monobasic sodium phosphate and sodium chloride ions. In the ion partitioning model of Haynes et al. (1989) the polymer interactions with K+ and C1- are set to zero, and the ion-polymer osmotic virial coefficients are calculated relative to these species. The absolute values of P E G anion and DEX-anion virial coefficients calculated in this way show the same trend as observed here when the anion is changed from C1- to HS04-. However, Haynes et al. assumed constant values for the PEGcation and DEXcation virial coefficients regardless of the salt used. The above discussion shows that the effective cation-polymer virial coefficients do depend on the anion present. Haynes et al. (1993) recognized this as well, and obtained saltspecific cation-polymer virial coefficients from low angle light scattering data. At very low salt concentrations (0.05 M), the solution behavior of ions in aqueous polymer solutions is well described by the Debye-Huckel Br~nsted-Guggenheim expression used here. However, at high salt concentrations, other effects, such as ion hydration and ion pairing may be important, and it may be necessary to account for finite ion sizes (Blum, 1975). Summary We have shown experimentally that the partition coefficients of the salts NaC1, NaHzP04, and NaHS04 decrease from unity as the tie-line length increases in aqueous two-phase systems. The partitioning to the bottom phase of monobasic sodiumsulfate is stronger than the partitioning of monobasic sodium phosphate or sodium chloride. A combination of the UNIQUAC, DebyeHuckel, and Br~nsted-Guggenheim models in a selfconsistent Gibbs free energy expression provides a good correlation of all of the experimental data. The Gibbs free energy minimization procedure is an efficient way to calculate liquid-liquid equilibria for aqueous mixtures of polymers in the presence of low concentrations of salt. Acknowledgment The work was supported by the DuPont Company and the Delaware Research Partnership. Nomenclature a = average value of the radii of pairs of hydrated ions A E X = excess Helmholtz free energy aij = interaction parameter for the pair
i
of components i and
E = Debye-Huckel parameter c = ion number density F = objective function, Faraday constant GEX = excess Gibbs free energy GEX' = excess Gibbs free energy, unsymmetric convention GEX = excess Gibbs free energies of phase p = ideal Gibbs free energy G r d = ideal molar Gibbs free energies of phase p Gio = standard state molar Gibbs free energy of component i GToM = molar Gibbs free energy of the two-phase system I = ionic strength
dd
Ind. Eng. Chem. Res., Vol. 33, No. 10,1994 2293
Ka,i = activity-based equilibrium constants Mi = molecular weight of component i mi = molality of ion i n = number of components nip = mole number of component i in phase p np = total number of moles in phase p nT = total number of moles in the two-phase system n, = number of moles of water Q = correction factor, eq 9 q; = surface parameters of polymer i per unit mass r; = volume parameters of polymer i per unit mass wi = weight percent of component i wpi = weight percent of component i in phase p x = mole fraction z = coordination number a = Debye-Huckel constant & = osmotic virial coefficient of ion pair i-j A\k = electrostatic potential difference between two phases yj = activity coefficient of ion i yf = infinite dilution activity coefficient of ion i Xi” = standard state activity of ion j pwo= standard state chemical potential of water II = osmotic pressure Literature Cited Abrams, D. S.; Prausnitz, J. M. StatisticalThermodynamicofLiquid Mixtures: A New Expression for the ExcessGibbs Energy of Partly or Completely Miscible Systems. AIChE J. 1975,21,116. Albertsson, P.-A. Partition of Cell Particles and Macromoelcules, 3rd ed.; Wiley-Interscience: New York, 1986. Bamberger, S.; Seaman, G. V. F.; Brown, J. A.; Brooks, D. E. The Partition of Sodium Phosphate and Sodium Chloride in Aqueous Dextran Poly(ethy1ene glycol) Two-Phase Systems. J. Colloid Interface Sci. 1984,99,187. Blum, L. Mean Spherical Model for Asymmetric Electrolytes. I Method of Solution. J. Mol. Phys. 1975,30,1529. Blum, L.; Holye, J. S. Mean Spherical Model for Asymmetric Electrolytes. 11 Thermodynamic Properties. J.Phys. Chem. 1977, 81, 1311. Brooks, D. E.; Sharp, K. A.; Bamberger, S.;Tamblyn, C. H.; Seaman, G. V. F.; Walter, H. Electrostatic and Electrokinetic Potentials in Two Polymer Aqueous Phase Systems. J. Colloid Interface Sci. 1984,102,1. Cabezas,H., Jr.; Evans, J. D.; Szlag,D. C. Statistical Thermodynamics of Aqueous Two-Phase Systems. ACS Symp. Ser. 1990,419,38. Cardoso, M. J. E. De M.; O’Connell, J. P. Activity Coefficients in Mixed Solvent Electrolyte Solutions. Fluid Phase Equilib. 1987, 33,315. Edmond, E.; Ogston, A. G. An Approach to the Study of Phase Separation in Ternary Aqueous Systems. Biochem. J. 1968,109, 569. Flory, P. J. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1942,IO, 51. Friedman, H. L. Ionic Solution Theory; Interscience Publishers: New York, 1962. Friedman, H. L. Lewis-Randall to McMillan-Mayer Conversion for the Thermodynamic Excess Functions of Solutions. Part I. Partial Free Energy Coefficients. J. Solution Chem. 1972a,1, 387. Friedman, H. L. Lewis-Randall to McMillan-Mayer Conversion for the Thermodynamic Excess Functions of Solutions. Part 11.Excess Energy and Volume. J. Solution Chem. 1972b,1, 413. Guggenheim, E. A. The Specific Thermodynamic Properties of Aqueous Solutions of Strong Electrolytes. Philos. Mag. Ser. 7 1935, 19,588. Guggenheim, E. A. Mixtures of 1:l Electrolytes. Trans. Faraday SOC.1966,62,3446. Hartounian, H. Protein Partitioning in the Aqueous Two-Phase Systems of Polymers and Salt. Ph.D. Thesis, University of Delaware, 1993.
Haynes, C. A.; Blanch, H. W.; Prausnitz, J. M. Separation of Protein Mixtures by Extraction: Thermodynamic Properties of Aqueous Two-Phase Polymer Systems Containing Salts and Proteins. Fluid Phase Equilib. 1989,53,463. Haynes, C. A.; Benitez, F. J.; Blanch, H. W.;Prausnitz, J. M. Application of Integral Equation Theory to Aqueous Two-Phase Partitioning Systems. AIChE J. 1993,39, 1539. Huggins, M. L. Some Properties of Solutions of Long Chain Compounds. J. Phys. Chem. 1942,46,151. Kabiri-Badr, M. Thermodynamics of Salt-Polymer Aqueous TwoPhase Systems: Theory and Experiment; The University of Arizona: Tucson, 1990. Kang, C. H.; Sandler, S. I. Phase Behavior of Aqueous Two-Phase Systems. Fluid Phase Equilib. 1987,38,245. King, R. S.; Blanch, H. W.; Prausnitz, J. M. Molecular Thermodynamics of Aqueous Two-Phase Systems for Bioseparations. AIChE J. 1988,34,1585. Kopp, J. F.; McKee, G. D. Methods for Chemical Analysis of Water and Wastes; National Technical Information Service, U.S.Department of Commerce-EPA-600/4-79-020 Springfield, VA, 1983. Masterton, L. W.; Slowinski, E. J.; Stanitski, C. L. Chemical Principles, 6th ed.; Saunders CollegePublishing: New York, 1985. McMillan, W. G.; Mayer, J. E. The Statistical Mechanics of Multicomponents Solutions. J. Chem. Phys. 1945,13,276. Newman, J. Electrochemical Systems; Prentice Hall: Englewood Cliffs, NJ, 1973. Nelder, J. A.; Mead, R. A Simple Method for Function Minimization. Comput. J. 1965,7,307. Pitzer, K. S.Thermodynamics of Electrolytes. I: Theoretical Basis and General Equations. J.Phys. Chem. 1973,77,268. Pitzer, K. S. Electrolyte Theory- Improvements Since Debye and Htickel. Acc. Chem. Res. 1977,10,371. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1959. Vaslow, F. Water and Aqueous Solutions: Structure, Thermodynamics, and Transport Processes; John Wiley & Sons: New York, 1972. Waisman, E.; Lebowitz, J. L. Exact Solution of an Integral Equaton for the Structure of a Primitive Model of Electrolytes. J. Chem. Phys. 1970,52,4307. Waisman, E.; Lebowitz,J. L. Mean Spherical ModelIntegral Equation for Charged Hard Spheres. I Method of Solution. J.Chem. Phys. 1972,56,3086. Zaslavsky, B. Y.; Bagirov, T. 0.; Borovskaya, A. A.; Gasanova, G. Z.; Gulaeva, N. D.; Levin, V. Y.; Masimov, A. A.; Mahmudov, A. U.; Mestechkina, N. M.; Miheeva, L. M.; Osipov, N. N.; Rogozhin, S. V. Aqueous Biphasic Systems Formed by Nonionic Polymers: Effects of Inorganic Salts on Phase Separation. Colloid Polym. Sci. 1986,264,1066. Zaslavsky, B. Y.; Miheeva, L. M.; Gasanova, G. Z.; Mahmudov, A. U. Influence of Inorganic Electrolytes on Partitioning of NonIonic Solutes in an Aqueous Dextran-Poly(ethy1ene glycol) Biphasic System. J. Chromatogr. 1987,392,95. Zaslavsky, B. Y.; Miheeva, L. M.; Aleschko-Ozhevskii, Y. U.; Mahmudov, A. U.; Bagirov, T. 0.;Garaev, E. S. Distribution of Inorganic SaltsBetween the CoexistingPhases of Aqueous Polymer Two-Phase Systems. J. Chromatogr. 1988,439,267. Zaslavsky, B. Y.; Borovskaya, A. A,; Gulaeva, N. D.; Miheeva, L. M. Influence of Ionic and Polymer Composition on the Properties of the Phases of Aqueous Two-Phase Systems Formed by Non-Ionic Polymers. J. Chem. SOC.,Faraday Trans. 1991,87,141. Receioed for reuiew September 23, 1993 Revised manuscript received June 14,1994 Accepted June 28,1994. Abstract published in Advance ACS Abstracts, September
1, 1994.
