Ind. Eng. Chem. Res. 1989,28, 1537-1542
1537
Polydispersivity Effects on the Behavior of Aqueous Two-Phase Two-Polymer Systems Choon-Hyoung Kang, Cheng-Kang Lee, and Stanley I. Sandler* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
Biomaterials may be separated by partitioning between two liquid phases formed when incompatible polymers such as dextran (DEX) and poly(ethy1ene glycol) (PEG) are dissolved in water. Since narrowly fractionated polymers are expensive, polymers with a wide dispersion in molecular weight are commonly used for such separations; however, this dispersion affects the liquid-liquid equilibrium. Here the phase equilibrium compositions of aqueous two-phase, two-polymer systems containing polydisperse dextran and several narrow poly(ethy1ene glycol) fractions were experimentally determined and found to be in agreement with the results of a completely predictive thermodynamic model including a recently proposed method to account for polymer polydispersivity. On the basis of our experiments and calculations, we find that (1)the phase equilibrium compositions are affected by polymer polymolecularity, with the smaller PEG molecules appearing in the DEX-rich phase and the smaller DEX molecules in the PEG-rich phase; and (2) the length of the tie lines, but not their location or slope, is affected by the degree of polydispersivity. In the study of the phase equilibrium of polymer solutions, polydisperse systems are usually treated as monodisperse systems with averaged distributed intensive properties such as average molecular weight. However, there are important differences; for example, the sharp phase boundaries found in monodisperse mixtures are not found in systems containing polydisperse species. Indeed, Tompa (1949) obtained better agreement when calculating phase diagrams for mixtures of one-solvent and twopolymer homologues differing only in molecular weight by treating the mixture as a ternary system rather than as a pseudobinary system with an averaged molecular weight. Koningsveld (1975) has extensively investigated phase separations in solutions containing polydisperse macromolecules by using the Flory-Huggins model and concluded that the polymolecularity of the macromolecules affects the phase diagrams to such an extent that it cannot be ignored in treatments that are intended to be quantitative. In dextran/poly(ethylene glycol) (PEG)/water systems, it was also experimentally observed that mixtures are more or less turbid outside the binodal curve and that the dextran in the bottom phase has a molecular weight distribution different from the dextran in the top phase (Albertsson, 1986). Usually, to account for the polydispersivity of a mixture, some number of pseudocomponents are chosen in a relatively arbitrary manner. A tuning procedure is then followed to fit the parameters to limited experimental data (Pedersen et al., 1985). Shishov et al. (1981) calculated the turbidity curve of polydisperse dextran/PEG/water systems using arbitrarily chosen pseudocomponents to represent polydisperse polymers. However, the inherent shortcoming in such a procedure is that it is nonunique, and trial-and-error calculations are needed to obtain a good fit of limited experimental data. Recent work has centered on the use of a continuous distribution to characterize the intensive properties of polydisperse species. The general conditions for phase equilibrium in terms of a continuous distribution have been formulated by Salacuse and Stell (1982), Gualtieri et al. (1982), Briano and Glandt (1983), and Kehlen and Ratzsch (1983). In their formulations, the equilibrium state, the state of minimum Gibbs energy, has to be located in the space of distribution functions, pressure, and tem-
* To whom correspondence should
be addressed.
0888-5885/89/2628-1537$01.50/0
perature. The thermodynamics of continuous mixtures has been applied to various polydisperse systems. Based on the numerical method of Gauss-Laguerre quadrature, a systematic procedure to choose pseudocomponents to represent a continuous distribution in a mathematically correct and optimal way has been developed by Cotterman and co-workers (Cotterman et al., 1985, 1986) for the exponential distribution function of molecular weight or other properties and by Shibata et al. (1987) for more diverse distributions. An advantage of this procedure is that one can achieve the benefits of using a continuous description of a mixture by choosing pseudocomponents based on the appropriate Gaussian numerical quadrature formulas for the distributions that describe the system and then use the phase equilibrium algorithms now in use. In this study, we report experimental measurements for the liquid-liquid phase behavior for several specially prepared polydisperse dextran/PEG/water mixtures and compare them with the results of the phase equilibrium calculations with the pseudocomponents chosen based on the quadrature method. We have chosen the dextran/ PEG/water mixture since this system is of interest for the liquid-liquid extraction of proteins, cells, and other biomaterials. The phases that form are 80 w t 5% water or greater and nondenaturing to biologic materials that partition differently between the two aqueous phases (Albertsson, 1986). Further, because of the cost of highly fractionated dextran and PEG, it is likely that crude or polydisperse polymers will be used in commercial processes. Therefore, the goals of this study were to quantitatively determine (i) the effect of molecular polydispersivity on the liquid-liquid equilibrium (LLE) phase boundaries and (ii) the extent of fractionation of the phase-forming polymers. Finally, we also wanted to establish that both of these effects could be predicted with conventional solution thermodynamic models. Experiments
The dextran (A&, = 1.67 X lo5) used was obtained from Sigma Chemical Co. (St. Louis, MO). Narrow standard poly(ethy1eneglycol) fractions of number-average molecular weights 22000 and 5000 were purchased from Polyscience, Inc. (Warrington, PA), and a PEG fraction of number-average molecular weight 1000 was obtained from Fluka Co. (Rokonkoma, NY). The molecular weights of the polymers and their polydispersivity parameters are compiled in Table I. In this table, is the weight-av0 1989 American Chemical Society
1538 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 Table I. Molecular Weights and Polydispersivity Parameters of the Polymers Used in This Study component dextran T500 PEG 22000 5000 1000
MI$ 1.67 x 105 2.20 x 104 5.00 x 103 1.00 x 103
b = Mw/Mn 2.75 1.06 1.05 >1.10
erage molecular weight and I@,, is the number-average molecular weight. Four two-phase systems, each containing the same amount of dextran (4.2%w/w) and PEG (4.2%w/w) but
with different ratios of the three PEG fractions, were prepared from stock solutions of dextran (23.8% w/w), PEG 5000 (10% w/w), and PEG 1000 (40% w/w) and from solid PEG 22000. The total weight of each two-phase system was 1.5 g. System I was prepared by mixing deionized water with 0.263 g of the dextran solution, 0.125 g of the PEG 5000 solution, 0.048 g of the PEG 1000 solution, and 0.031 g of solid PEG 22000. System I1 was made by adding water to 0,263 g of the dextran solution, 0.125 g of the PEG 5000 solution, and 0.050 g of PEG 22000. Thus, in system I, the total 0.063 g of PEG consisted of 50% PEG 22000,20% PEG 5000, and 30% PEG
Table 11. Comparison of Experimental Equilibrium Compositions (in Weight Percent) with the Calculated Results for System I PEG exDtl calcd ( b = calcd ( b = 2.0)b calcd (6 = 2.75)'
feed bottom top KO bottom top K bottom top K bottom top K
DEX, 4.2 8.0 0.8 10 7.85 0.06 130 8.48 0.26 32.6 8.53 0.31 27.5
1000, 1.26 1.12 1.44 0.78 1.21 1.31 0.92 1.16 1.36 0.85 1.15 1.35 0.85
5000, 0.84 0.72 1.32 0.55 0.73 0.97 0.75 0.56 1.10 0.51 0.55 1.10 0.50
22000, 2.1 0.39 3.41 0.11 0.31 4.13 0.075 0.22 3.83 0.057 0.21 3.80 0.055
8d 0.536 0.468 0.520 0.527
nThe dextran was assumed to be monodisperse. *The dextran was assumed to be polydisperse with b = 2.0. 'The dextran was assumed to be polydisperse with b = 2.75. Q = weight fraction of the top phase. K = partition coefficient = concentration in the bottom phase/ concentration in the top phase.
Table 111. Comparison of Experimental Equilibrium Compositions (in Weight Percent) with the Calculated Results for System I1 PEG feed DEX, 4.2 5000, 0.84 22000, 3.36 Qd 0.60 0.395 0.628 exptl bottom 10.8 0.17 1.40 5.04 top Ke 0.43 0.078 63.5 10.5 0.65 0.16 0.602 calcd ( b = 1.0)" bottom 0.96 5.45 0.014 top K 750 0.68 0.029 11.08 0.44 0.16 0.631 calcd ( b = 2.0)b bottom 0.14 1.07 5.24 top 0.41 K 79.1 0.031 11.12 0.43 0.11 0.631 bottom calcd ( b = 2.75)c 1.07 5.24 0.13 top K 85.5 0.40 0.021 "he dextran was assumed to be monodisperse. *The dextran was assumed to be polydisperse with b = 2.0. 'The dextran was assumed to be polydisperse with b = 2.75. Q = weight fraction of the top phase. K = partition coefficient = concentration in the bottom phase/ concentration in the top phase.
Table IV. Comparison of Experimental Equilibrium Compositions (in Weight Percent) with Calculated Results I11 PEG feed DEX. 4.2 1000. 0.63 5000. 0.63 22000. 2.94 exptl bottom 10.2 0.51 0.43 top 0.313 0.69 0.97 4.57 Ke 33 0.74 0.44 calcd ( b = bottom 9.57 0.59 0.51 0.21 top 0.03 0.65 0.72 5.03 K 319 0.91 0.71 0.042 calcd ( b = 2.0)* bottom 10.05 0.56 0.36 0.16 top 0.18 0.67 0.81 4.82 K 55.8 0.84 0.44 0.033 calcd ( b = 2.75)c bottom 10.07 0.56 0.36 0.15 top 0.15 0.68 0.81 4.83 K 67.1 0.82 0.44 0.031
for System
Qd 0.567 0.565 0.594 0.594
The dextran was assumed to be monodisperse. The dextran was assumed to be polydisperse with b = 2.0. The dextran was assumed to be polydisperse with b = 2.75. d Q = weight fraction of the top phase. ' K = partition coefficient = concentration in the bottom phase/concentration in the top phase.
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1539 Table V. Comparison of Experimental Equilibrium Compositions (in Weight Percent) with the Calculated Results for System IV PEG exptl
feed bottom top
DEX, 4.2 11.83
bottom top
10.84 0.004 2710 11.29 0.13 86.8 11.32 0.11 102.8
5000,0.63 0.37 1.0 0.37 0.48 0.72 0.67 0.32 0.80 0.40 0.32 0.80 0.40
Ke calcd ( b = l.O)O
K calcd ( b = 2.0)*
bottom top
K calcd ( b = 2.75)c
bottom top
K
22000,3.57
Qd 0.663
5.43 0.15 5.70 0.026 0.12 5.52 0.022 0.11 5.52 0.020
0.602 0.639 0.637
"The dextran was assumed to be monodisperse. bThe dextran was assumed to be polydisperse with b = 2.0 'The dextran was assumed to be polydisperse with b = 2.75. d Q = weight - fraction of the top phase. e K = partition coefficient = concentration in the bottom phasejconcenhation in the top phase.
1000 on a weight basis, and the PEG in system I1 consisted of 80% PEG 22000 and 20% PEG 5000. Systems I11 and IV were prepared in a similar way. The weight percentages of the components in each of the systems are listed in Tables 11-V. The concentration of each PEG fraction in the twophase systems was determined by gel permeation chromatography (GPC). A Waters HPLC system (Millipore, Bedford, MA) was used for this purpose, and four columns, two Ultrahydrogel linear, one Ultrahydrogel 120, and one Ultrahydrogel 250, were used in the GPC analysis. The column temperature was kept constant at 45 "C, and deionized water was used as a solvent at a flow rate of 0.5 mL/min. A refractive index detector (Waters Model RI 410) was used to detect the PEG concentration after calibration curves had been obtained using standard solutions of PEG 22000,5000, and 1000 of various dilutions. The experimental two-phase systems were well mixed and centrifuged to obtain two clear phases. Samples taken from both phases were quantitatively diluted with four volumes of deionized water before analysis to ensure linear detector response. Dextran concentrations were determined by standard polariometric techniques. The results appear in Tables 11-V.
Equilibrium Calculations As is evident from Table I, the molecular weight of the dextran is distributed over a wide range, whereas the molecular weights of the PEG fractions are narrowly distributed. To describe the molecular weight distribution of the dextran, the Lansing-Kraemer (LK) or logarithmic-normal distribution was used (Granath, 1958):
The polydispersivity parameter, b, for this distribution is
b
9
Mw/M,, = exp(2/p4)
(2)
Thus, the parameters ,@ and Mo in eq 1can be determined given Mnand Mwor one of the average molecular weights and the parameter b. Due to their narrow distributions, each of the PEG fractions considered here was assumed to be of a single molecular weight. In the quadrature method, pseudocomponents for the polydisperse species in the feed are obtained with the Gaussian quadrature formula appropriate to the molecular weight distribution. The overall system is then considered to consist of a finite number of pseudocomponents for
dextran and each of the monodisperse species (each PEG fraction and water). To choose the pseudocomponents to represent dextran, whose molecular weight was assumed to be distributed according to the L-K distribution, we recognize that phase equilibrium calculations are linear operations on the molecular weight distribution. Therefore, the problem is analogous to choosing the optimum quadrature points for numerical integration. Given the form of the L-K distribution, the optimum numerical integration method is Gauss-Hermite quadrature (see, for example, Shibata et al. (1987)), which gives the following expression for replacing an integral with a summation
where G(x) is any function. Comparing eq 1and 3, we see that x = (1/p) In (M/Mo) and that n is the number of quadrature points (pseudocomponents) used. Therefore, the optimal quadrature points for the evaluation of the integral or the mathematically optimal pseudocomponents, zk, are obtained from the zeros of the Hermite polynomials (Stroud and Secrest, 1966)
and the weight factors in the quadrature are n!,l/zZkn+l wk
=
(5)
[Hn+l(zk)1
The values of the locations of the zeros and weight factors can be found in Abramowitz and Stegun (1972). Thus, the molecular weight of the kth pseudocomponent is calculated from Mk = M oexp(@zk),where zk is the kth zero of the Hermite polynomial, and the weight fraction of this pseudocomponent is obtained from n wk
= (total weight fraction of dextran)(Wk/
wk)
(6)
k=1
The only unspecified quantity in the calculation is n, the number of pseudocomponents that are used to represent the continuous distribution. Usually n = 3 or 4 is satisfactory. Once the continuous species is represented by a finite number of pseudocomponents chosen as above, the LLE phase composition of the multicomponent system is com-
1540 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 Table VI. Physical Properties of the Components Used in the Equilibrium Calculations (Kang and Sandler. 1988b) component 1o2r‘ 102q ’ dextran 2.717 1.96 PEG 3.87 2.57 water 5.11 7.78
puted by solving the following equivalence condition for the chemical potentials (7)
Apd = A p d f Ab..f 11 = Ap..ff 11
(8)
and the following summation conditions for each component in the coexisting phases = i
&ff = CXiF= 1 1
(9)
i
Here the subscript 0 refers to the solvent, ij indicates pseudocomponent j of species i, and the superscript F denotes the feed composition. For our calculations, we have used A p j = RT In aj (10) where the activity of species j , a’, is represented by the mass fraction -based UNIQUAb model (Abrams and Prausnitz, 1975). In this model, 0; M;Oj’ In a; = In 4;’ + 5Mjq/ In - + M,l; - -Clk’Xk 4if xi
+
where = volume fraction of species j = XIrlf/xXkrkf
(12)
experiment when dextran is assumed to be polydisperse (larger value of the b parameter) than when dextran was taken to be a monodisperse component. In the polydisperse calculations, three pseudocomponents were chosen for dextran according to the Gauss-Hermite quadrature method described above. As previously shown, three pseudocomponents can represent the extent of polydispersivity for the species we consider here described by the L-K distribution (Kang and Sandler, 1988b). From both the experimental and calculated results, we see that there is a polymer fractionation effect, as would be expected from the work of Flory (1953), with the larger molecule weight PEG fractions tending to remain predominantly in the top PEG-rich phase, while the smaller molecular weight PEG fractions are more uniformly distributed between the top phase and the bottom dextran-rich phase. This is most evident by examining the very sensitive distribution coefficient, K , defined as K = weight percent of polymer in the bottom phase/ weight percent of polymer in the top phase (15) In each of the cases we see that the polydisperse dextran is strongly partitioned into the bottom phase, and the largest molecular weight poly(ethy1eneglycol) (PEG 22000) is strongly partitioned ( K
