Downstream Processing and Bioseparation - American Chemical

Chapter 2

Statistical Thermodynamics of Aqueous Two-Phase Systems 1

2,3

2

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Heriberto Cabezas, Jr. , Janis D. Evans , and David C. Szlag 1

Department of Chemical Engineering, University of Arizona, Tucson, AZ 85721 Center for Chemical Engineering, National Institute of Standards and Technology, Boulder, CO 80303

2

Hill's theory of solutions was used to model the phase diagrams of polymerpolymer aqueous two-phase systems. This theory expresses chemical potentials in terms of polymer-polymer osmotic virial coefficients. Scaling expressions for predicting these coefficients from the degree of polymerization and parameters were developed from Renormalization Group theory. For a two polymer system the parameters consist of two constants, b and b , and two exponents, υ and υ . Values for these parameters which are valid for all solutions were obtained from experiment. Phase diagrams at ambient conditions were predicted for three different systems consisting of aqueous mixtures of polyethylene glycol and dextran ranging in molecular weight from 3690 to 167,000. The predicted phase compositions are within 1-3% from experimental values which have uncertainties of about 1%. 1

2

2

3

I n the years since the pioneering studies of Albertsson and co-workers (1), aqueous two-phase extraction has gained wide acceptance as a method for the recovery and purification of proteins, enzymes, and other molecules and particles of biological origin. Interest i n these systems has been rekindled i n recent years (2) due to the rapid growth of the biotechnology i n d u s t r y i n an increasingly competitive e n v i r o n m e n t a n d to the identification of the cost of separation as the major component i n the price of bioproducts (3). This has created a need for a separation process which is gentle to sensitive biomolecules while offering high product recovery, high 3

Current address: Phillips Petroleum Company, P.O. Box 350, Borger, T X 79008 0097-6156/90/0419-0038$06.00/0 © 1990 American Chemical Society

In Downstream Processing and Bioseparation; Hamel, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.


2. CABEZASETAL.

Statistical Thermodynamics ofAqueous Two-Phase Systems 39

product purity and also economical operation and ease of scaling u p or d o w n . The aqueous two-phase extraction technique meets the above requirements and i n addition, allows us to bring to bear on the problem, the existing body of knowledge and experience for the operation and design of industrial liquid-liquid extractions (4, 5). The design and optimization of any liquid-liquid extraction process, including one involving aqueous two-phase systems, is predicated on the availability of a phase diagram for the system as a first step. Early progress in the prediction of phase diagrams was made by E d m o n d and Ogston (6), and additional theoretical advances have been accomplished recently by Prausnitz and coworkers (7) and by Sandler and coworkers (8) among others. A n overall review of this field has been given by Benge (9). Yet, at least two major problems remain. The simplest and most successful model, that of E d m o n d and Ogston is formulated for a solution under its o w n osmotic pressure and is not strictly applicable to the calculation of a phase diagram at constant temperature and pressure. The second problem is that there is no simple, quantitative way of accounting for the changes i n the phase diagram with polymer molecular weight. Our work addresses these two problems by adopting the solution theory of H i l l (10, 11) a n d by a d a p t i n g ideas f r o m the G r o u p Renormalization (12, 15) theory of polymer solutions to the prediction of the model parameters from the degree of polymerization of the phase forming polymers.

STATISTICAL MECHANICAL BASIS Our purpose i n this section is to derive a set of useful expressions for the chemical potentials starting with the principles of statistical mechanics. The expressions we shall obtain take the form of virial expansions similar to those of the Edmond and Ogston (6) but having a very different theoretical basis. O u r model parameters are isobaric-isothermal virial coefficients w h i c h are about an order of magnitude smaller than the osmotic virial coefficients i n the Edmond and Ogston model. We shall develop the theory neglecting the effect of polydispersity because we empirically d i d not find this to be very important at the level of accuracy commonly attainable i n experimental phase diagrams for these systems. To outline the fundamental basis of the m o d e l , we f o l l o w the notation of H i l l (10) and extend his derivation to a three component mixture. Component 1 is the solvent w h i c h i n our case is water, component 2 is a solute or polyethylene glycol, and component 3 is another solute or dextran. We base the theory on an isobaric-isothermal ensemble first introduced by Stockmayer (14). This choice of ensemble is the most appropriate because it yields expressions for the chemical potentials of the components w i t h temperature, pressure, and solute molality or mole fraction as the natural independent variables, and these are the independent variables normally used i n calculation, experiment, and industrial practice.


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We begin w i t h the canonical partition function for a three component system w h i c h is given by Equation 1 and where the independent variables are temperature, volume and mole numbers. ΟίΝ^Ν^,ν,Τ) = Xe

Î i / k T

(1)


i

The summation i n Equation 1 is taken over all of the energy states of the ensemble. F r o m a series of transformations of Equation 1 we obtain a new partition function (Γ) whose independent variables are temperature, pressure, solvent mole number, and the chemical potentials of the solutes (components 2 and 3). These transformations consist of first creating a partition function w i t h pressure rather than volume as an independent variable, and then using this result to create yet another partition function i n w h i c h we have switched independent variables f r o m solute mole numbers to solute chemical potentials. These operations are analogous to the Legendre transforms commonly employed i n thermodynamics. Γ ( Ν ! , Ρ , Τ , μ 2 , μ 3 ) = f^Vi/VT

A

N i

where:

=Ie-PV/kTQ

( N l

=

,

N 2 /

^ ^ i=2,3 Ni>0 N

3 /

e

N

^ T

eâ/kT^ (2)

V,T)

v

Q = canonical partition function The right hand side of Equation 2 includes a power series i n the solute activities, a2 and a , of components 2 and 3, respectively. Expanding the series a n d taking the logarithm of both sides of the equation yields Equation 3. 3

ΙηΓ = 1 η Δ [ 1 + χ χ X i-2.3 NiÔ 0

N i

af]

where: βΐ = Δ , β Α / Κ Ι 7 Ν ι Δ

Δ

Ο

= ΔΝ

1

=

0

0


(3)

2. CABEZASETAL.

Statistical Thermodynamics of Aqueous Two-Phase Systems 41 Δ = ΔΝι = 1 0

Expansion of the logarithm about unit activity and collection of terms of like power i n solute activity yields Equation 4 where the index j refers to the series expansion of the logarithm. Since solute activity approaches unity only as the solute molalities approach zero, this expansion is strictly valid only for dilute solutions.

îjHl -ΙηΓ-ΙιιΔο + Ni £ Σ Μ ' > «I Downloaded by UNIV OF CINCINNATI on February 18, 2015 | http://pubs.acs.org Publication Date: January 24, 1990 | doi: 10.1021/bk-1990-0419.ch002

Τ

P

From the Gibbs-Duhem equation and Equation 4 we obtain Equations 5 which express molality (n\2,1x13) as a power series i n activity. i^TT

L a , = m =lJe,aaP)ai 2

kTdlna

2

j*i

(5a)

kT31na

3

jî

(5b)

where: n\2 = N2/N1 Π13 =

N3/N1

The inverse of Equations 5 give activity as a power series i n molality. Taking this inverse and collecting terms i n like powers of the molalities up to first order we obtain Equations 6 which give the solute chemical potential as a power series i n solute molality w i t h osmotic virial coefficients that are functions of temperature and pressure. μ2 (Τ, P,m2,m3) - μ5 (Τ, Ρ, Ο, Ο) = kT In a2 = - kT · [In m + 2

2C22

m2 + 2C23irt3 + ...]

μ (Τ Ρ π\2 ηΛ3)-μ§(Τ Ρ 0 0) = kTlna3= -kT*[lnm3 + 2C23m2 + 3

/

/

/

/

where:

/

/

Q; = C«

μ? =

2C33TO+ .··]

(T,P)

L i m μι (Τ,Ρ,ητ^ητο) m2->0 rrbÔ


(6a) (6b)

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Finally from the Gibbs-Duhem equation and Equations 6 w e obtain Equation 7 which gives the solvent (water) chemical potential i n terms of solute molalities and the aforementioned coefficients.

μ ι ί Τ , Ρ , ι η ^ ι τ ^ - μ ? ( Τ, Ρ, Ο, Ο) == - kT ( m + ma + C22 ml + 2C23 m


2

2

+C33 mi + ...)

(7)

Equations 6 a n d 7 are the fundamental expressions g i v i n g the chemical potentials as functions of solute molalities and H i l l osmotic virial coefficients. W e can obtain an expression for the osmotic pressure b y considering the pure solvent i n equilibrium w i t h a solution under its o w n osmotic pressure. This is expressed by Equation 8. μ (Τ, Ρ, m , m ) = μι (Τ, Ρ - π, Ο, Ο) τ

2

3

( 8 )

Next we integrate the pressure derivative of the pure solvent chemical potential (at zero solute molality).

μ ι

(Τ,Ρ,0,0)-μι(Τ,Ρ-π,0,0)

= | V i (Τ, P) d P = V i π

L«

(9)

Inserting Equation 7 i n the left hand side of Equation 8 and then i n Equation 9 a n d assuming the solvent to be incompressible, w e obtain Equation 10 for the osmotic pressure (π).

kT

= m + ma + C22 m2 + 2C23 m ma + C33 m? + ... 2

2

(10)

SCALING LAW EXPRESSIONS In this section we "semi-empirically" adapt some scaling ideas from the G r o u p Renormalization theory (12, 15) of polymer solutions to obtain expressions for the osmotic virial coefficients of Equations 6 and 7 i n terms of the degree of polymerization. In the f o l l o w i n g discussion we w i l l occasionally omit the indices on the osmotic virial coefficients for the sake of simplicity.


2.

CABEZAS ET AL.


It is w e l l k n o w n that the osmotic pressure of a solution of one polymer can be scaled w i t h a single dimensionless variable "S" w h i c h is proportional to polymer concentration at least for the case of mixtures with good solvents i n the dilute to semidilute regime (12, 17). This implies that the osmotic compressibility factor (π/cRT) can be expressed as some function of "S" only as shown i n Equation 11.

-f^=l+F(S)


ckT

(11)

From the Group Renormalization theory of polymer solutions (12) we k n o w that "S" is proportional to "b" w h i c h depends on the nature of the polymer, the polymer concentration "c", and the degree of polymerization " N " raised to the power of an exponent "3υ" as shown i n Equation 12. S = b

C N

3»

( 1 2 )

U s i n g Equation 12 we expand F(S) i n a Taylor series about infinite dilution (c = 0) and insert the result, truncated to first order i n polymer concentration (c), into Equation 11 to obtain Equation 13.

ckT

= 1 + Be + ....

where:

B= bN

, (13) v

3 u

A t this point we note that Equation 13 is the M c M i l l a n - M a y e r (16) expansion for the osmotic compressibility factor w h i c h is fundamentally different f r o m the analogous expansion that was obtained f r o m the formalism of H i l l (Equation 10). We also identify Β as a M c M i l l a n - M a y e r osmotic virial coefficient. W e have shown that there is a scaling relation for Β of the form given in Equation 13. However, we have not shown that an analogous relation exists for the H i l l osmotic virial coefficients (C). W e start the proof with the exact relation between Β and C shown i n Equation 14. C = - L ( B - V° + l - K i R T ) Vi 2


(14)

44 where:

DOWNSTREAM

γ° Ki

^

=

=

e


polymer partial molar volume at infinite dilution

the isothermal compressibility of pure water

To make further progress we need the well k n o w n empirical fact that the polymer partial molar v o l u m e scales linearly w i t h the degree of polymerization (N) as given by Equation 15. V =V°mN

(15)


ö

where:

v°m

=

effective monomer partial molar volume

Next we propose that there is a relation for C of the form given by Equation 16. C =bN " 3

( 1 6 )

Finally, we insert Equations 15 and 16 into Equation 14, substitute for Β in terms of the scaling expression of Equation 13, and solve for b to obtain Equation 17. b

_ b Vi

V° V i Ν "" 3

, K i RT 1

2Ν

3 υ

(17)

Since the Renormalization Group approach is strictly applicable only for very long polymers, we take the limit of Equation 17 as Ν becomes very large to obtain Equation 18. b = — Vi

N-> -

(18)

This shows that at least for the case of large polymers, there exists a scaling expression for C of the f o r m of E q u a t i o n 16 where the proportionality constant b is given by Equation 18. One should be aware of the fact that i n obtaining this result we have tacitly assumed that the exponent υ i n Equation 13 which is defined i n a M c M i l l a n - M a y e r ensemble is the same as that i n Equation 16 which is defined i n an isobaric-isothermal



2. C A B E Z A S E T A K


ensemble. W e justify this assumption on the fact that the two exponents are numerically indistinguishable w h e n evaluated f r o m experimental osmotic virial coefficients. We also feel this to be appropriate for the level of approximation i n our approach. The scaling expression of Equation 16 is applicable to a solution of one polymer i n water. It therefore gives us relationships for C22 and for C33 which represent the interaction of either polymer 2 or 3 with itself but not C23 which represents the interactions of polymers 2 and 3. Furthermore, the results from Group Renormalization theory do not give as straightforward guidance on the functional f o r m of C23 as they do for the other two coefficients (13). W e have, therefore, developed on empirical grounds a relationship for C23 which still embodies the fundamental scaling concepts from theory (see Equation 19c). The expressions for the three virial coefficients are given by Equations 19. (19a)

C22 = biN?*

C33 =

(19b)

C23 = b [ N ^ N 2

3

3 u j

]2

The scaling expressions for C 2 2 and C33 are given i n terms of a proportionality constant b i which depends on temperature and pressure, the degrees of polymerization N 2 and N 3 , and the scaling exponents V2 and v for polymers 2 and 3 respectively. Theory (12) indicates that for model linear polymers the value of the exponent should be universal while the value of the proportionality constant should vary w i t h the chemical nature of the polymer. In fact, the value (17) of the universal exponent is set at 0.59. W e empirically found that a value of 0.60 w h i c h is very close to the theoretical value was adequate to correlate the molecular weight or Ν dependence of the virial coefficients of the polyethylene glycols since these are close to being linear polymers. H o w e v e r , this same value proved completely inadequate for the dextrans w h i c h are branched polymers. For the dextrans we set the value of the exponent at 0.6948 which is still not an unreasonable number. A s implied by Equations 19, we also found that a single proportionality constant w i t h a value of 0.9012 was adequate to correlate the self interaction virial coefficients of both polymers. Since the proportionality constant represents the interaction of two monomers of the same polymer i n the presence of water, it w o u l d seem that monomers of polyethylene glycols a n d monomers of dextran have s i m i l a r self 3


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BIOSEPARATION


interactions possibly indicating that both interactions are dominated b y the hydroxyl group. The scaling expression for C23 is an empirical geometric mean type rule involving the same exponent values used i n the relations for C22 and C33 but with a different value for the proportionality constant. W e found that a value of 1.1841 for b2 was adequate to reproduce phase diagrams. W e were not able to find any relation between b i and b2 possibly indicating that the interactions between a monomer of polyethylene glycol and a monomer of dextran are very different from the self interactions mentioned before.

ESTIMATION OF MODEL PARAMETERS There are four final model parameters after the scaling laws are inserted i n Equations 6 and 7. These are Ό2, v$, b i , and b2- There are many methods that one could use to obtain values for these parameters. However, we shall be concerned with only three of them. First, if experimental data are available on the osmotic pressure of the two-polymer system versus the molality of the polymers, one could simply insert Equations 19 into Equation 10 a n d n u m e r i c a l l y fit the four parameters. Alternatively, if there are data on the osmotic pressure of only one of the polymers i n water one could obtain values for C22 or C33 but not for C23. Although this method is the least ambiguous, we d i d not follow it because the necessary data were not available. Second, if M c M i l l a n - M a y e r (16) osmotic virial coefficients (Bip are available from light scattering or some other experiment for several molecular weights of the same polymer, one could use Equation 20, V1C22 = B 2 - V 2 + J-KiRT 2

(20)

which relates the virial coefficients from the H i l l theory to the M c M i l l a n Mayer osmotic virial coefficients to calculate C22/ C 3 3 , or C23 for several molecular weights. One could then fit the four parameters previously mentioned above to the Qj's using Equations 19. U s i n g this method with the osmotic virial coefficients of Prausnitz and co-workers (7) we calculated values of C22 for polyethylene glycols 8000 and 3350 and then fitted b i and V2 using Equation 19a. We tried the same procedure for the dextrans but found that the value of V2 so obtained c o u l d not represent the phase diagrams accurately. T h i r d , if experimental phase diagrams are available one could use tie lines and the phase equilibrium relations of Equations 21 to solve for three of the four parameters.

μ|(Τ, Ρ, m j , m j ) = μ?(Τ, Ρ, m ? , m f )


(21a)

2. CABEZASETAL.


μ£(Τ, Ρ, m j , m j ) = \i\ (Τ, P, m f , m f )

( 2


μ£(Τ, Ρ, m j , m j ) = μ? (Τ, Ρ, m f , m f )

( 2

ib)

i ) c

A modification of this procedure was used by us to estimate values for b2 and υ3. We simply adopted the previously calculated values for b i and i>2, used Equations 21 with Equations 6, 7 and 19 to obtain Equations 22 which can be solved for b2 and v^, and applied Equations 22 to one tie line from each of the experimental phase diagrams of Figures 1, 2, and 3 (King, R. S.; Blanch, H . W . ; Prausnitz, J. M . , University of California at Berkeley, unpublished data presented at the A C S Meeting i n A n a h e i m , C A , 1986). The numerical solution of equations 22 for b2 and b y applying Equations 19b and c gave us several slightly different values for the two parameters and thus we adopted the arithmetic average.

Ο = (mj - m f ) + (mj - m f ) + C + 2C 0=

2 3

[mj m j - m f m f ] + C

l n ^ i + 2C £

2 2

[(mj) - (mf ) ] + 2

2 2

3 3

2

[(mj) - (mf ) ]

(mj - m f ) + 2 C

2 3

( m j - m f ) 4- 2 C

2 3

2

2

(mj - m f )

m

Ο = In 5 î l + 2 C mf

3 3

(mj - m f )

(

2 ) 2 a

(22b)

(22c)

where C22, C33, and C23 are given by Equations 19.

PHASE DIAGRAM CALCULATIONS W e have calculated phase diagrams at ambient conditions for three different polyethylene glycol-dextran systems using our model and have compared the results to the experimental phase diagrams of K i n g et aL (King, R. S.; Blanch, H . W . ; Prausnitz, J. M . , University of California at Berkeley, unpublished data presented at the A C S Meeting i n Anaheim, C A , 1986). These calculations are illustrated i n Figures 1 to 3. We covered a wide range of polymer molecular weights i n order to observe the polymer molecular weight dependence of the virial coefficients. Thus, the calculated system represented i n Figure 1 consists of polyethylene glycol 3350 ( M W = 3690) and dextran T-500 ( M W = 167,000); that i n Figure 2 consists of polyethylene glycol 3350 ( M W = 3690) and dextran T-70 ( M W = 37,000); and

American Chemical Society Library 1155 15th St., N.W. In Downstream Processing and0Λ. Bioseparation; Washington, 20038 Hamel, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1990.

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48

20

10

30

Wt % DEXTRAN

Figure 2. Phase diagram for an aqueous mixture of Polyethylene Glycol 3350 ( M W = 3690) and Dextran T-70 ( M W = 37,000) at 25°C Ο Experiment. Δ Model.



2.

CABEZASETAL.

Statistical Thermodynamics of Aqueous Two-Phase Systems 49

Figure 3. Phase diagram for an aqueous mixture of Polyethylene Glycol 8000 ( M W = 8920) and Dextran T-500 ( M W = 167,000) at 25°C. Ο Experiment. Δ Model.



50

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that i n Figure 3 consists of polyethylene glycol 8000 ( M W = 8920) and dextran T-500 ( M W = 167,000). The tie lines i n Figures 1, 2, and 3 were calculated from the phase equilibrium relations represented by Equations 21 which upon substitution of the chemical potential models become Equations 22. There are three equations and four unknown molalities. For each tie line we therefore set a value for one of the molalities, which i n our case was that of dextran i n the bottom phase, and simultaneously solved Equations 22 for the remaining three molalities. The numerical algorithm used was the same one that Edmond and Ogston (6, 9) used for their model. The virial coefficients used in Equations 22 for all the calculations were predicted from the scaling expressions of Equations 19. Comparing the predicted to the experimental tie lines i n Figures 1 to 3 indicates that the agreement (1-3 wt%) between the two is, overall, satisfactory. This is particularly true if one realizes that the experimental uncertainty i n the determination of the phase compositions is of the order of 1.0 w t . % , especially near the critical point. A n additional source of discrepancy between theory and experiment may be a small difference between the average molecular weights of the polymers actually used i n the experiments and the molecular weights that we assumed i n our model calculations.

CONCLUSIONS The present work applies the formalism of H i l l and the modern theory of polymer solutions to the development of a new and very simple model for the prediction of phase diagrams i n aqueous polymer-polymer systems. The fundamental model expressions are rigorous for constant temperature and pressure calculations. The scaling expressions allow us to predict the osmotic virial coefficients i n the H i l l equations from the number average molecular weight of the polymers and model parameters. The model involves four parameters: b i , b2, V2, and which appear i n the scaling expressions. Calculated phase diagrams are most sensitive to b2 because it determines the value of C23 w h i c h represents the interactions between unlike polymers w h i c h dominate phase separation. To establish the validity of our ideas, we have applied our model to the calculation of phase diagrams i n aqueous polyethylene glycol-dextran systems and shown that it can predict phase behavior accurately for a wide range of polymer molecular weights and composition. W e hope to extend this approach to other polymer systems and to include the effect of polydispersity i n a forthcoming paper (Cabezas, H . , Jr.; Evans, J.D.; Szlag, D . C . Fluid Phase Equilibria, i n press).


2. CABEZAS ET AL.


ACKNOWLEDGMENTS The authors are grateful to the N a t i o n a l Institute of Standards and Technology of the U n i t e d States of A m e r i c a for financial and material support.

LEGEND OF SYMBOLS


aj B, Bjj

-

b, b ^ b2 c QCij

-

Γ

_ _ -

Ajsjj θ ji μμ υ, V[ π

-

k Ki m, mj Ν Nj Ρ Q R S Τ V Vi Vi

Activity of component i , dimensionless. McMillan-Mayer osmotic virial coefficient for components i a n d j , L/Mole. scaling law proportionality constants, functions of temperature and pressure, dimensionless. molar concentration of solute or polymer i , M o l / L . H i l l osmotic virial coefficient for components i and j , dimensionless. energy of system i n quantum level i , J. Boltzmann's constant, J/molecule -°K isothermal compressibility of pure solvent or water, bar " · N[/N\ dimensionless molality of polymer i . degree of polymerization moles of component i . system pressure, bar. canonical partition function. gas constant. scaled dimensionless polymer concentration. system temperature, °K system volume, L . molar specific volume of pure solvent or water, L / M o l . partial molar volume of solute i at infinite dilution, L / M o l . proportionality factor in Equation 3, dimensionless. partition function for Stockmayers's isobaric-isothermal ensemble. isobaric-isothermal partition function for N i Moles, proportionality factor in Equation 4, dimensionless. chemical potential of component i , J/Molecule. scaling exponents of component i , dimensionless. osmotic pressure, bar. 1

f

Superscripts Τ Β ο

-

top phase. bottom phase, infinite dilution.


52

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Subscripts 1 2 3

-

components. solvent or water. component 2 or polyethylene glycol, component 3 or dextran.

LITERATURE CITED Downloaded by UNIV OF CINCINNATI on February 18, 2015 | http://pubs.acs.org Publication Date: January 24, 1990 | doi: 10.1021/bk-1990-0419.ch002

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Albertsson, P. A. Partition of Cell Particles and Macromolecules; 3rd Ed., John Wiley & Sons: New York, NY, 1986. Walter, H.; Brooks, D. E.; Fisher, D. Partitioning in Aqueous Two-Phase Systems: Theory, Methods, Uses, and Applications to Biotechnology; Academic Press: New York, NY, 1985. Godfrey, P. B.; Kohll, Ε. Α.; Gelbein, A. P. Proc. Econ. Intl. 1985, 5, 12. Treybal, R. E. LiquidExtraction;2nd Ed., McGraw-Hill: New York, NY, 1963. Kula, M. -R.; Kroner, Κ. H.; Hustedt, Η. In Purification of Enzymes by Liquid-Liquid Extraction; Fiechter, Α., Ed.; Advances in Biochemical Engineering No. 24; SpringerVerlag: New York, NY, 1982;p73. Edmond, E.; Ogston, A. G. Biochem.J.1968, 109, 569. King, R. S.; Blanch, H. W.; Prausnitz, J. M. AIChEJ.1988, 34, 1585. Kang, C. H.; Sandler, S. I. Fluid Phase Equilibria 1987, 38, 245. Benge, G. G.; Master Thesis, Virginia Polytechnic Institute and State University, Blacksburg, 1986. Hill, T. L. J. Am. Chem. Soc. 1957, 79, 4885. Hill, T. L. J. Chem. Phys. 1959, 30, 93. Schafer, L. Macromolecules 1982, 15, 652. Schafer,L.;Kappeler, C. J. de Phys. 1985, 46, 1853. Stockmayer, W. H. J. Chem. Phys. 1950, 18, 58. Oono, Y. In Statistical Physics of Polymer Solutions; Prigogine, I.; Rice, S. Α., Eds.; Advances in Chemical Physics No. 61; John Wiley & Sons: New York, NY, 1985; 301. McMillan, W. G.; Mayer, J. Ε. J. Chem. Phys. 1945, 13, 276. Des Cloizeaux, J.; Jannink, G. Les Polymères en Solution: leur Modélisation et leur Structure; Les Editions de Physique: France, 1987. Le Gillou, J. C.; Zinn-Justin, J. Phys. Rev. Lett. 1977, 39, 95.

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