I n d . Eng. C h e m . Res. 1988,27, 866-871
866
Pore Size Effects on Diffusion of Polystyrene in Dilute Solution Imtiaz A. Kathawalla and John L. Anderson* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Rates of diffusion of linear polystyrene (MW 1.69.3 X lo5) at dilute concentrations in tetrahydrofuran were measured as a function of pore size for thin membranes. The data for all molecular weights and pore sizes can be accurately correlated by a single relationship between DIDoand R,/R, where D and Do are the diffusion coefficients for the pore and bulk solutions, respectively, R, is the Stokes-Einstein radius of the polymer which is calculated from Do, and R is the pore radius. Our results indicate that the hindrance effect of the pores on this linear polymer is greater than for a rigid sphere of the same Stokes-Einstein radius. A theory for hindered diffusion that allows two length scales for the polymer, the radius of gyration for the partitioning effect of the pores and the Stokes-Einstein radius for the hindered mobility of the chains inside the pores, gives better agreement with the data than does the hard-sphere theory; however, this two-length-scale theory still predicts less hindrance than measured for R,/R < 0.1. Part of this discrepancy can be attributed to the noncircular geometry of the pores of our membranes. While our polymer could be considered “monodisperse” by normal standards, effects of the small degree of polydispersity might have been important when R,/R > 0.35 because in this range the hindrance was less than predicted by the theory. The diffusion of macromoleculesthrough pores of comparable size has been studied in the context of physiological transport across blood capillaries, mass-transfer limitations in microporous catalysts, and membrane separations. Pore size effects on the apparent diffusion coefficient can be quite large when A, the ratio of molecule-to-pore radius, is larger than 0.1; in fact, these effects are still measurable when X is as small as 0.01. Most theory and experiment have focused on macromoleculesthat are compact and approximately spherical in shape. In this paper, we report experimental results for the hindered diffusion of a linear polymer, polystyrene dissolved in tetrahydrofuran, and compare these results with a model for the diffusion of flexible-chain polymers through small pores. The diffusion coefficient ( D ) of macromolecules in porous media is defined by Fick’s law, with the concentration driving force based on the bulk solution concentration. For one-dimensional diffusion through a porous membrane having uniform, parallel pores, the flux of macromolecules ( N )is written as N = -DAC/l where N is based on the pore area, 1 is the membrane thickness, and AC is the polymer concentration difference between the two bulk solutions on either side of the membrane. A t infinite pore size, X 0, the diffusion coefficient equals the bulk solution value (Do). As X increases, D decreases because of two effects. First, the pore wall influences the distribution of macromolecules, and hence there is a partitioning effect described by a coefficient K , which is defined as the ratio of the mean concentration inside the pores divided by the bulk solution concentration at equilibrium conditions. For hard spherelhard wall interactions in a circular cylindrical pore, K equals ( 1 which is the fraction of the fluid volume inside the pore that is accessible to the center of each spherical molecule. van der Waals or electrostatic interactions between macromolecule and pore wall can significantly alter K from its entropic (steric) limit if these energies are comparable to the thermal energy kT. Calculations of K for purely steric macromolecule/ pore interactions have been performed for rigid particles of various shapes (Giddingset al., 1968) and random flight polymers (Casassa, 1967; Casassa and Ta-
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08S8-5885/88/2627-0866$01.50/0
gami, 1969). The second effect of the pore is a reduction of the mobility of a macromolecule, leading to a reduced intrapore diffusion coefficient (D,). The mobility of spherical macromolecules depends on position within the pore cross section, and D, is determined by averaging the mobility over the cross section using the single-molecule distribution function as a weighting factor. The diffusion coefficient defined in eq 1is the product of the partitioning and mobility effects:
D = KD,
(2)
Pappenheimer et al. (1951) considered a spherical molecule in a pore of circular cross section. Instead of averaging the mobility over the pore cross section, they substituted its value at the pore centerline when computing D,. The resulting expression, known as the “Renkin equation“ (Renkin, 1954), stands as the basic theoretical model for the hindered diffusion of “hard sphere” macromolecules:
DIDO = (1- X)*[1 - 2.1044X
+ 2.0888X3 - 0.948X5]
(3)
where X equals R,/R, R, is the sphere radius, and R is the pore radius. The first term is simply the steric partitioning fador discussed above. The centerline mobility correction, the term in square brackets, is only accurate as given in eq 3 for X 0.5; more precise calculations for larger values of X were published by Paine and Scherr (1975). Allowing for the dependence of the mobility on radial position within a circular pore, which can be done only for small X at the present time, one finds that DID, is slightly smaller than predicted from eq 3 (Anderson and Quinn, 1974; Brenner and Gaydos, 1977). The small number of experiments that have been performed with well-defined membranes and solutes have focused mainly on the range 0.4 > X > 0.05. The results for spherical macromolecules show reasonably good agreement with eq 3 (Beck and Shultz, 1972; Deen et al., 1981; Bohrer et al., 1984). A comprehensive review of theory and experiment related to hindered diffusion is provided by Deen (1987). A theoretical model for the hindered diffusion of linear, random coil polymers has been proposed by Davidson and Deen (1988). The partitioning effect is described by computing the entropic limitations imposed by the pore wall on the (assumed) random flight statistics of the polymer chains. The pore-wall effect on chain mobility, leading to 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 5 , 1988 867 the calculation of D,, is estimated by using a Brinkmantype model (similar to the Debye and Bueche model) for a polymer chain centered on the pore axis and having a statistically averaged conformation (the so-called "preaveraging assumption"). Calculations from this theory show that Dp/Dofor a linear polymer is comparable to the value for a rigid sphere of identical Stokes-Einstein radius when X < 0.4, but K is smaller for the polymer; the net effect is that D / D o is smaller for the polymer because of the partitioning effect. It should be emphasized that this theory assumes only entropic effects of the pore wall and random flight statistics for the polymer chains; hence, it is strictly only valid for @solvents and negligible polymer/pore wall enthalpic interactions. Davidson and Deen (1988) suggest that the theory can be applied to linear polymers in good solvents if K is a general, unique function of R,/R, independent of the solvent, where R, is the radius of gyration of the polymer. The dependence of K on R,/R is assumed to be given by the model of Casassa (1967). Published results of hindered diffusion experiments with flexible polymers appear to be inconsistent. Cannell and Rondelez (1980) studied linear polystyrene in ethyl acetate, a moderately good solvent, and found DIDo to be significantly below the predictions of eq 3 when X is based on the Stokes-Einstein radius of the polymer. Guillot et al. (1985) reported data on the same system that seem to agree with eq 3, but this agreement could be due in part to adjustment of the pore density to fit the data. On the other hand, Deen et al. (1981) and Bohrer et al. (1984) measured the diffusion rates of dextrans as a function of pore size, using the same type of membrane as Cannell and Rondelez, and found that the hindrance was smaller for the polymer than for the equivalent hard sphere solute. Bohrer et al. (1988) have published data showing that the hindrance is greater for star-branched polyisoprene than for the linear polyisoprene of the same Stokes-Einstein radius. Our objective here was to measure the diffusion coefficient versus pore size for relatively monodisperse, linear polymers diffusing through membranes with a well-defined pore structure in order to answer the following questions: How does the hindered diffusion of a linear, coiled polymer in a good solvent compare to that for hard sphere molecules? Can the results over a broad range of molecular weight and pore size be correlated by a single relationship? Can a theory based on random flight statistics, such as that recently proposed by Davidson and Deen (1988), described the hindered diffusion of linear polymers in good solvents? Much of the justification for this work lies in the discrepancies within the literature on this subject as discussed in the preceding paragraph. Possible effects of polydispersity in molecular weight on the measured diffusion coefficient of a polymer fraction are estimated using our experimental results and a model for the molecular weight distribution. Experiments Linear polystyrene was purchased in five molecular weight fractions from the Pressure Chemical Company (Pittsburgh, PA). The solvent was HPLC grade tetrahydrofuran (THF) without a peroxide inhibitor. The polymer concentration was always smaller than 0.2[7]-l, where [v] is the intrinsic viscosity, so that interactions between different polymer molecules should not have been significant. The bulk solution diffusion coefficient of each polymer fraction was determined at 20 "C from quasielastic light scattering measurements in the laboratory of G.D. Patterson (Carnegie Mellon University, Brookhaven Instruments, Model 2030 digital correlator, Model 240
Table I. Polystyrene Diffusion Coefficients in B u l k Solution ( T H F ) at 20 O c a
Mwb 1.57 X 2.16 X 4.00 X 5.91 X 9.29 X
lo5 lo5 lo5 lo5 lo5
PIb 1.05 1.05 1.02 1.07 1.06
i07Dn.cm2/s light scatter- membrane inn' diffusiond 4.59 4.21 3.77 3.65 2.66 2.57 2.20 2.25 1.66 1.93
c,
mn/cm3 1.34 1.09 0.87 0.57 0.43
A In (Do)/AC,' cm3/mp 0.019 0.033 0.036 0.034 0.029
"PI = M,/M,,, C = 0.5CH0 where CHo is the high-side concentration of polymer in the diffusion cell. *As reported by the Pressure Chemical Company. 'Diffusion coefficient measured by quasi-elastic light scattering .$two polymer concentrations and interpolated at concentration C. Diffusion coefficient extrapolated to infinite pore size using the data in Figure 3. 'From light-scattering determinations.
goniometer). The determinations were made at two different polymer concentrations and the results were linearly extrapolated to infinite dilution. The concentration effect, A(ln Do)/AC,was small (less than 0.04 cm3/mg) even for the largest molecular weight. The light scattering values of Do are fit by the following expression:
(C
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0)
Do = 3.32
X
10-4M-0.554 cm2/s
(4)
The exponent on molecular weight agrees with literature data for this system (Mandema and Zeldenrust, 1977; McDonnell and Jamieson, 1977), but the prefactor is about 13% lower in our case. Credence is given to our results by the fact that we performed the experiments at different scattering angles; furthermore, our light-scattering results are in agreement with the membrane data extrapolated to X 0, as shown in Table I. The membranes were prepared from thin sheets of muscovite mica by the track-etch technique (Quinn et al., 1972), which creates uniform, parallel, capillary pores of rhomboidal cross section. Circular disks were cut from the mica sheets and a central region of 8 cm2 was irradiated to form tracks which were then etched with hydrofluoric acid to form pores. The number of pores per area (n)was determined from the time of irradiation and calibrations developed in our laboratory. The length of the pores (1) equaled the membrane thickness, which was found by weighing each disk prior to irradiation. Two methods were used to determine pore radius. The first involved measurement of the rate of diffusion of benzene in THF across the membrane. The diffusion cell is shown in Figure 1. Diffusion experiments were performed at least at two different stirring rates ( w ) , and the data, in the form of a mass-transfer coefficient for the benzene (kb),were extrapolated to infinite stirring rate to obtain the true membrane coefficient. Because the porosities of the membranes were low (
