Calculation of the Hydrodynamic Permeability of Gels and Gel-Filled

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Ind. Eng. Chem. Res. 2001, 40, 1694-1705

Calculation of the Hydrodynamic Permeability of Gels and Gel-Filled Microporous Membranes Alicja M. Mika* and Ronald F. Childs Department of Chemistry, McMaster University, Hamilton, Ontario, L8S 4M1 Canada

A new approach to the calculation of gel hydrodynamic permeability is presented in which the gel is modeled as an assemblage of spheres whose radius is a function of the gel concentration. Scaling methods coupled with the electrostatic persistence length are used to estimate the correlation length in equivalent semidilute solutions. The correlation length is taken to be equal to the sphere diameter in an assemblage, and the hydrodynamic permeability of the assemblage is calculated using the Brinkman and Happel models. It is shown that the calculated permeabilities are fully in accord with experimental data reported in the literature for membranes with incorporated poly(acrylamide) gels, as well as data reported in this work for membranes containing charged gels composed of poly(4-vinyl-N-benzylpyridinium salts). Introduction Mechanically strong and robust composite materials containing polymer gels confined in the pores of rigid microporous supports have recently been developed and tested as separation media (sorbents and membranes).1-6 The average pore diameter of the supports is typically in the micron range and much larger than the gel mesh dimension of the incorporated gel. As a result, molecular and viscous transport within the pores of the support as well as separation are determined by the properties of the incorporated gel.7 Depending on the nature of the gel, the basis of separation can include molecular sieving (size exclusion), Donnan exclusion of co-ions by charged gels, or specific interactions such as chelation. An important feature of these composite materials is that the microporous support provides mechanical strength. This means that the gel network can be optimized for a specific application without sacrificing its functionality or structural integrity. We have been exploring the properties and uses of membranes in which a polyelectrolyte gel is anchored within a polyolefin microporous support.2,3,5,6,8-13 Among other applications, such membranes function very effectively in ultra-low-pressure water softening, exhibiting high permeation rates (fluxes) with good bivalent ion rejections.6,12,13 We have examined the permeability of a series of membranes with different gel chemistries as a function of gel polymer volume fraction. The striking feature of this study was that a single relationship held for all of the gels we studied with the hydrodynamic permeability being simply dependent on the gel polymer volume fraction and independent of the gel chemistry or charge type.12 Hydrodynamic permeability, k, is a fundamental property of gels. It is defined from Darcy’s law as14

v)-

k ∇P η

(1)

where v is the superficial fluid velocity through the gel, η is the fluid viscosity, and P is the pressure. The * Corresponding author. Tel.: +1 905 5259140 ext. 23484. Fax: +1 905 5222509. E-mail: [email protected].

hydrodynamic permeability is closely related to the microstructure of the gel polymer network, with the square root of hydrodynamic permeability, xk (the hydrodynamic screening length), being regarded as a measure of the mean spacing between the gel polymer chains.15 The hydrodynamic screening length is used as a parameter in modeling molecular transport within gels.16,17 Measurement of the hydrodynamic permeability, k, of a gel is often very difficult because of its intrinsic nature and the ease of gel compression under the test conditions. For this reason, it would be particularly attractive if the permeability of a gel could be calculated from first principles. In the existing hydrodynamic theories of permeability, the polymeric gels are often modeled as networks of straight cylinders (fibers) of radius rf, and the Darcy permeability is expressed as a function of the volume fraction of chains (φ)15

k ) rf2f(φ)

(2)

where f(φ) depends on the spatial array of the chains. Several expressions for f(φ) have been obtained by various authors either from numerical solutions of the Stokes equations for regular and random arrays of the chains or from semiempirical fits of the experimental data (for a brief review see Kapur et al.15). Analysis of these models shows that, if rf is kept constant, none of the expressions for f(φ) fit the experimental data over a wide range of gel concentrations (vide infra). A much better fit of the experimental data can be achieved if rf is allowed to become a function of the gel concentration that increases with decreasing concentration. However, the need to allow the chains to “swell” with gel dilution raises the issue of whether the modeling of gel networks as arrays of straight cylinders of fixed radius is appropriate. The objective of this study was to develop an alternative model of gel structure that would allow quantitative predictions of the Darcy permeability of polymeric gels. The approach we have taken is to assume that confined gels can be treated as semidilute polymer solutions of the same polymer concentration (volume fraction). Scaling methods,18 in particular, Schaefer’s model of the

10.1021/ie000794q CCC: $20.00 © 2001 American Chemical Society Published on Web 03/03/2001

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polymer structure in semidilute solutions,19 coupled with the concept of electrostatic persistence length20,21 are used to estimate the correlation length in the solution. In the next step, the solution is modeled as an assemblage of spheres of diameter equal to the correlation length, and the hydrodynamic permeability of the assemblage is calculated using either the Brinkman22,23 or the Happel model.24 The result of this treatment is the estimation of the permeability of a gel itself. The permeability of gel-filled membranes can then be calculated from the permeability of the gel using corrections for the support porosity and pore tortuosity, as described by Kapur et al.15 In this paper, we show that the calculated permeabilities are fully in accord with experimental data reported in the literature for membranes with incorporated poly(acrylamide) gels, as well as data reported in this work for membranes containing charged gels composed of poly(4-vinyl-N-benzylpyridinium salts). A comparison with other gel models is also made. Theoretical Background Correlation Length in Semidilute Polymer Solutions. In a semidilute solution, the overall size of a polymer chain is not the significant length scale for most static and dynamic properties of the solution. The polymer chain network in a semidilute solution can be described quantitatively by a length scale ξp, the range of the monomer-monomer correlation function, which is proportional to the average distance between interchain contacts.25 In good solvents, ξp is the screening length, a concept first introduced by Edwards,26 which describes the distance beyond which there is no excludedvolume swelling. In theta solvents, ξp is the range of the pair correlation function, which is comparable to the mean distance between ternary contacts. It follows from the fundamental character of ξp that the solution properties such as the collective diffusion constant, the chain friction constant, and the sedimentation constant can be obtained if ξp is known.19 The length ξp decreases with the polymer concentration as the presence of other chains decreases the range of monomer pair correlations. The concentration dependence of ξp can be obtained from the relationship derived by Schaefer19 for semidilute solutions of semiflexible polymers. In this analysis, the degree of chain expansion due to excluded volume is quantified using the blob model.27 Three different solvent regimes, namely, good, marginal, and theta are used in the analysis of semidilute solutions. In the good solvent regime, short sequences of a polymer chain, called blobs, are presumed to be ideal, whereas long sequences are fully swollen because of the excluded-volume effects. The monomer interaction is a strong two-body effect. The good solvent regime is applicable only at low concentrations, typically for polymer volume fractions below 0.01. In the marginal regime, the chains are nearly ideal on all length scales, and the excluded-volume effects are screened and very weak. Weak two-body interactions control the thermodynamics. In the theta regime, the chains are ideal chains on all length scales and interact via three-body forces. As the polymer volume fractions in gels and gel-filled porous membranes are typically in the range of 0.04 e φ e 0.2, the marginal and theta regimes are applicable. The value of ξp throughout the marginal and theta

regime is given by the expression19

ξp )

n2a [n(1 - 2χ)φ + wa-6φ2]1/2

(3)

where n is the number of bonds in the persistence length and can be obtained from the characteristic ratio C∞, n ) C∞/6; a is the bond length; χ is the Flory-Huggins interaction parameter (reduced residual partial molar free energy); and w is the three-body excluded-volume parameter, which is assumed to equal 0 in the marginal regime and a6 in the theta regime. Correlation Length in Semidilute Solutions of Polyelectrolytes. Understanding of the conformations of polyelectrolyte chains in semidilute solutions, particularly in the absence of salt, is still unsatisfactory, and no definitive theory is yet available.28 It is generally accepted that, at infinite dilution and in the absence of small molecule electrolytes, a polyelectrolyte chain is nearly fully stretched because of the electrostatic repulsion of the charges along the chain. This feature is the basis of a theoretical approach developed by Lifson and Katchalsky,29 which is generally accepted as yielding correct scaling results.28 By applying scaling concepts to polyelectrolyte solutions, de Gennes et al.30 extended the theoretical description of polyelectrolyte solutions to higher concentrations and identified three different concentration regimes: (1) the dilute regime where polyion chains do not overlap and stay far from each other; (2) the dilute lattice regime where chains also do not overlap but where the electrostatic interactions between polyions are larger than thermal energies and the buildup of a three-dimensional periodic lattice is expected; and (3) the semidilute regime where chains overlap with each other, forming a hexagonal or cubic lattice of rigid rods or an isotropic transient network with a characteristic correlation length ξ, similar to that formed in semidilute solutions of neutral polymers. The isotropic structure is suggested to be the most probable structure. To form such a transient network, polyion chains require some flexibility and ability to bend. Odijk20 and Skolnick and Fixman21 showed that the concentration-dependent flexibility of polyelectrolyte chains can be explained by the counterion screening of the charges on the macromolecule. In this work, we examine two routes to the estimation of the correlation length ξ in semidilute polyelectrolyte solutions, namely, the scaling relations proposed by Odijk31 for wormlike chains32 and the Schaefer model discussed above. In the latter case, only the short-range electrostatic effect is incorporated into the model through the increased persistence length, while the electrostatic excluded-volume effect is neglected. The Odijk model is based on the existence of an electrostatic persistence length and the concept of local cylindrical symmetry.33 The electrostatic interactions along the polyions decrease local flexibility, and chains become rodlike along contour distances larger than the original Kuhn length. For a wormlike polyelectrolyte chain of contour length l with bare persistence length Lp bearing P elementary charges (qe) that are separated by a contour distance A ) lP-1, the electrostatic persistence length Le is given by the expression derived independently by Odijk20 and Skolnick and Fixman21

Le )

lB 4κ2A2

for A > lB

(4)

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where lB ) qe2/DkBT is the Bjerrum length (D is the dielectric constant of the solvent, and kBT is the Boltzmann factor) and κ is the Debye-Hu¨ckel parameter. For A< lB, Odijk adopted the counterion condensation theory in the form proposed by Manning34 and Oosawa35 and modified the Odijk and Skolnick-Fixman (OSF) relation (eq 4) to

Le )

1 for A e lB 4lBκ2

(5)

The bare persistence length Lp and the electrostatic persistence length Le are additive, giving the total persistence length Lt ) Lp + Le. The derivation of eqs 4 and 5 is based on the assumption that the polyelectrolytes are near the rod limit so that excluded-volume effects can be neglected and the electrostatic interactions can be treated as a perturbation of the interactions associated with the bare persistence length.20 This assumption imposes a validity condition on eqs 4 and 5, namely, that κLt . 1. In case of κLt > 1, these equations become a reasonable first-order approximation. This approximation becomes very poor if κLt < 1.31 To obtain scaling relations for salt-free polyelectrolyte solutions, Odijk31 assumed that no interchain stiffening occurs, i.e., only charges on one chain increase the persistence length, and that only the uncondensed counterions screen the polyion. Under these assumptions, A g lB, and the Debye-Hu¨ckel parameter is given by

κ2 ) 4πAc

(6)

where c is the concentration of charge-bearing chain segments per unit volume of the solution (c ≡ NP/V, where N is the number of polyelectrolyte chains in the volume V). The correlation length ξ scales with the concentration c as

(

ξ = Lp +

1 16πlBAc

)

-1/4

(4πAc)1/8(Ac)-3/4

(7)

Equation 7 is valid only if ξ . Le.31 Hydrodynamic Permeability. At high polymer concentrations (in the semidilute regime), there is an obvious analogy between the sedimentation of polymer chains in a solvent and the permeation of solvent through a porous plug of polymer material.36 The scaling theory applied to sedimentation treats a semidilute polymer solution as a continuum formed by entangled chains that can be divided into spheres (blobs) of diameter equal to the correlation length ξ. According to Sun,37 there is some evidence to believe that the solvent is forced in an orderly fashion around the blobs but cannot penetrate their interior. If this is assumed to be a valid model for the viscous flow of solvent through a semidilute polymer solution (gel), the hydrodynamic permeability can be calculated by treating the solution (gel) as a porous plug made of impenetrable spheres (blobs) of diameter equal to the correlation length. There are several hydrodynamic theories aimed at obtaining a relationship between the fractional void volume of a porous plug and its permeability or relative settling velocity. One of the oldest is that of Brinkman22,23 for the flow of fluids through swarms of particles (hard spheres) of radius R occupying the volume fraction φ in a spherical cloud. The equation for

the hydrodynamic permeability of such a suspension is a modification of Darcy’s empirical relationship and is given by

k)

1/2 R2 4 8 3+ -3 -3 18 φ φ

[

(

) ]

(8)

Because of its empirical basis, the equation is not regarded as a rigorous solution to the problem, and it yields zero when φ ) 2/3. Nevertheless, its simplicity is attractive, and the polymer concentration range in our system is far below this value. In the mathematical treatment developed by Happel,14,24 the porous plug is modeled as an assemblage of hard spheres of radius R. It is assumed that such an assemblage can be considered as a system of identical unit cells each containing a sphere surrounded by a spherical fluid envelope of radius Re. The fluid volume contained in the envelope is sufficient to make the fractional void volume in the cell identical to that in the entire assemblage. This means that

( ) R Re

3

)φ)1-

(9)

where is the void volume in the assemblage. The outside surface of each cell is assumed to be frictionless, so the disturbance due to each particle is confined to the fluid cell associated with the particle. Solving the Stokes equation with the boundary conditions defined by the cell model leads to the following relationship for the permeability:

k)

[

]

9 1/3 9 5/3 2 2R2 3 - ( /2)φ + ( /2)φ - 3φ 9φ 3 + 2φ5/3

(10)

Happel also applied the cell model to arrays of cylinders instead of spheres.38 By analogy to spherical assemblages, such arrays of cylinders can be modeled as sets of identical unit cells consisting of two concentric circular cylinders: a fiber as the inner cylinder and a fluid envelope as the outer cylinder. By considering two cases of flow, namely, parallel to the axes of the cylinders and perpendicular to the axes, he derived two expressions for the hydrodynamic permeability

for the parallel flow k)

[

( )]

re 1 4rf2re2 - rf4 - 3re4 + 4re4 ln 2 rf 8re

(11a)

for the perpendicular flow k)

[ ( ) ( )]

4 4 re2 re 1 r e - rf ln 4 rf 2 r 4+r4 e f

(11b)

where re ) rfφ-1/2 is the radius of the fluid envelope. The permeability for random orientation of cylinders, where the cylinders are not parallel to each other, is obtained by taking two-thirds of that for perpendicular flow plus one-third of that for parallel flow at equal void volumes. Experimental Section The porous substrate used was a poly(propylene) microfiltration membrane (3M Company) produced by


a thermally induced phase separation process.39 The substrate had a bubble point pore diameter of 0.57 µm, a porosity of 84.5 vol %, and a thickness of 110 µm. The polymer used was poly(4-vinylpyridine) (P4VP), MW (1.5-2.0) × 105 (Polysciences, Inc.). R,R′-Dichloro-pxylene (DCX) (Aldrich) was used as a cross-linker and benzyl bromide (Aldrich) as a quaternization agent. All reagents were used without further purification. The gel-filled membranes were made by in situ crosslinking of poly(4-vinylpyridine) with R,R′-dichloro-pxylene in N,N-dimethylformamide (DMF). The membranes were manufactured according to the following procedure. A weighed sample of poly(propylene) substrate was placed on a glass plate, and a solution of poly(4-vinylpyridine) and cross-linker (5-10 mol % with respect to P4VP) was applied to the sample. After the excess solution was removed by gentle application of a Teflon roller, a Teflon gasket was placed around the sample, and another glass plate was put on top of it. The gasket thickness was such that a gap of a few millimeters was left between the sample and the top plate. A few drops of DMF were placed around the membrane sample to saturate the space with the solvent vapor and prevent the membrane from drying. After 3-5 days were allowed for the cross-linking reaction to occur at room temperature, the sample was removed from the glass plate and immersed in a quaternization solution containing 5 vol % benzyl bromide in DMF to convert the remaining pyridine into pyridinium sites. Some P4VP-filled membranes were not subjected to this treatment. The reactions were carried out at room temperature for 2-3 days. The gel-filled membranes were washed with several portions of methanol, followed by a thorough wash with deionized water. The membranes were conditioned in 0.1 M HCl, washed with deionized water, and stored in dilute HCl (pH 2-3). The mass of incorporated gel was determined from the difference between the dry mass of a pore-filled membrane sample (dried in a vacuum at room temperature to a constant mass) and that of the substrate. The gel concentration (volume fraction), φ, was calculated from

φ ) (mm,dry - ms)v2/VssMu

(12)

where mm,dry is the mass of a pore-filled sample (in the dry state), ms is the mass of the substrate in the sample, v2 is the partial molar volume of the gel polymer, Vs is the substrate volume in the sample, s is the substrate porosity, and Mu is the molar mass of the polymer repeating unit. The value of v2 ) 180 ( 2.5 cm3/mol used was determined from the density measurements.40,41 In case of any change in the volume of the pore-filled sample compared to the original volume of the substrate sample, eq 12 was modified to

φ ) (mm,dry - ms)v2/[Vm - (ms/dPP)]Mu

(13)

where Vm is the volume of the swollen sample and dPP is the density of poly(propylene) (0.91 g/cm3). The water flux through the membranes was measured using a dead-end cell with a membrane active area of 38.50 cm2. The cell was fitted with a thermocouple and a pressure gauge. Pressurized nitrogen was used to force water through the membrane. The permeate was collected over a measured period of time and weighed (accuracy ) (0.0001 g). The temperature and pressure

of the water on the pressurized side of the membrane were measured with an accuracy of (0.5 °C and (5 kPa, respectively. The flux at 25 °C (in kg/m2s) was calculated from the mass of permeate divided by the time and the membrane active area. Each measurement was repeated two or more times with a reproducibility of (5%. The empirical permeabilities, km, of the membranes were determined from the flux-pressure measurements and calculated according to the equation

km ) -

Qdη Am∆P

(14)

where Q is the volume flow across the membrane (m3/ s), d is the membrane thickness (m), η is the water viscosity (0.00089 Pa s), Am is the membrane area (m2), and ∆P is the transmembrane pressure (Pa). The permeabilities of the membranes were derived from the slopes of the straight lines obtained by plotting Qη/Am as a function of -∆P/d. The Darcy permeability of the pore-filling gel, k, was obtained by correcting the membrane permeability, km, for the substrate porosity s and pore tortuosity τ through the so-called tortuosity factor s/τ15

k ) km(s/τ)-1

(15)

The tortuosity τ was estimated as τ ) Ks/K* ) 2.5, where Ks ) 5 is the Kozeny constant (for porosity ≈ 2/3) and K* ) 2 is the Kozeny constant for parallel capillary pores of circular cross section.15 Results and Discussion In testing the ability of the model outlined above to predict the permeability of pore-filled membranes, we examined two separate cases. The first involves a neutral gel where charge effects are absent. In the second case, the permeability of a charged polyelectrolyte gel was examined. Neutral Poly(acrylamide) Gels. Poly(acrylamide) is a water-soluble polymer, biocompatible with various biological environments.42 The hydrodynamic permeability of poly(acrylamide) (PA) gels stabilized within a microporous poly(vinylidene fluoride) membrane was studied by Kapur et al.15 as a function of polymer concentration, φ, for 0.04 e φ e 0.16. They found that the experimental values of k can be approximated by the empirical relationship k ) 4.35 × 10-22φ-3.35 (m2). In this section, we examine how predictions of the model compare to these experimental data. The calculation of ξp from eq 3 requires that the values of C∞ and χ be known. Although the characteristic ratio is readily available for a large number of polymers, values of the Flory-Huggins interaction parameter, particularly for water as a solvent, are less available. The reported values of the interaction parameter bear substantial uncertainty, as they depend on the assumptions made when the parameter is calculated from thermodynamic measurements. In the original Flory-Huggins lattice theory,43 the interaction parameter χ is considered to be concentration-independent and inversely proportional to temperature. It is known from experimental data that this is an oversimplification and that the interaction parameter is a complicated function of both concentration and temperature. Theories, such as that proposed by Yamakawa44 to account for the effect of the properties of pure

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components on the interaction parameter, are difficult to apply because most of the characteristic properties of pure components required by these theories are unknown. The most common alternative approach is to treat χ as an empirical function and use a suitable model, such as the one proposed by Qian et al.,45 to fit the experimental data. In this work, a typical value of χ ) 0.4 was initially assumed. The sensitivity of the calculated gel permeability to changes in this parameter was also examined. The reported46 characteristic ratio of PA, C∞ ) 11.14, gives n ) 1.86, which, together with χ ) 0.4, defines the crossover from the good regime to the marginal regime at φ∼ ) 0.0075 and the crossover to the theta regime at φ+ ) 0.2 (in Schaefer,19 eqs 13 and 20, respectively). The calculation of ξp as a function of polymer volume fraction, φ, for 0.04 e φ e 0.2, was performed using eq 3 for a ) 1.54 Å and w ) 0 (marginal regime). The calculated values of ξp in this concentration range change from 43.6 Å (φ ) 0.04) to 19.5 Å (φ ) 0.20). Substituted as 1/2ξp ) R into the Brinkman equation (eq 8) and the Happel equation (eq 10), they give the Darcy permeability as a function of the gel concentration. The results are plotted in Figure 1, together with the experimental data of Kapur et al.15 As expected, the permeability calculated with the Brinkman equation is 25-30% higher than that calculated with the Happel equation (Figure 1a). As can be seen in Figure 1b, the relative differences between the calculated and experimental values of gel permeability, δ ) (kcalculated - kexperimental)/kexperimental, are small. Both equations predict the permeability of the gel quite well. The largest differences are found close to the borders of the concentration range tested; at the low concentrations, the permeability is underestimated, while it is overestimated at the higher concentrations. Overall, the differences are smaller with the Happel model. For comparison, Figure 1c shows differences between PA gel permeabilities measured by Kapur et al.15 and by Tokita.47 The differences are expressed as σ ) (kKapur - kTokita)/[1/2(kKapur + kTokita)]. As can be seen by comparing Figure 1b with Figure 1c, the absolute values of σ are substantially larger than the relative differences between the calculated and experimental results, δ. The effect of solvent quality (χ value) on the calculated gel permeability is profound, as shown in Figure 2, which contains permeability data calculated using Happel’s equation with ξp values obtained for three different values of χ, namely, 0.45, 0.40, and 0.35. The overall effect is an increase in permeability as χ increases, i.e., the solvent becomes poorer (Figure 2a). Such an effect of solvent quality has been reported in the literature15,36 and found in the present work (vide infra). For χ ) 0.45, there is a crossover from the marginal to the theta regime at φ+ ≈ 0.1. The unrealistic “jump” in ξp, and consequently in the permeability as a function of φ, has been smoothed by a polynomial approximation of the data calculated at this χ value. The relative differences between the calculated and experimental permeabilities decrease as the solvent quality improves and become less than or equal to (50% at χ ) 0.35 (Figure 2b). Charged Poly(4-vinyl-N-benzyl pyridinium) Gels. A series of membranes with different polymer volume fractions was prepared by cross-linking poly(4-vinylpyridine) in solutions of different polymer concentrations. The Darcy permeability of the membranes was deter-

Figure 1. Darcy permeability k of poly(acrylamide) gels as a function of the polymer volume fraction φ: (a) k determined experimentally (Kapur et al.)15 and k calculated from the equations of Brinkman (eq 8) and Happel (eq 10) by substituting the sphere radius R for 1/2ξp calculated from Schaefer’s equation (eq 3); (b) relative differences δ ) (kcalculated - kexperimental)/kexperimental between the calculated and experimental values of k; (c) relative differences σ ) (kKapur - kTokita)/[1/2(kKapur + kTokita)] between the gel permeabilities measured by Kapur et al.15 and Tokita.47

mined by measuring the volume flux Q/Am as a function of the applied pressure, ∆P. Sample data from these measurements are presented in Figure 3, which shows the repeated flux measurements for two gel-filled membranes with different polymer volume fractions (Figure 3a) and the average flux values obtained for two separate samples with the same polymer volume fraction (Figure 3b). For all membranes tested, the standard deviation in the volume flux was less than 5% of the mean value for each applied pressure. The values of km


Figure 2. Darcy permeability k of poly(acrylamide) gels as a function of the polymer volume fraction φ: (a) k determined experimentally (Kapur et al.)15 and k calculated from Happel’s equation (eq 10) by substituting the sphere radius R for 1/2ξp calculated from Schaefer’s equation (eq 3) for different values of the Flory-Huggins parameter χ; (b) relative differences δ ) (kcalculated - kexperimental)/kexperimental between the calculated and experimental values of permeability k.

reported here were obtained by extrapolations to zero pressure. At the highest pressures applied (300-400 kPa), the decline of km compared to the value at the lowest pressure was less than 5%. The correlation coefficients in the linear fits of the experimental data were >0.99, and the standard error in estimation of km was less than 2.5% for all membranes. The introduction of electrostatic charge into a gel is expected to increase the rigidity of the chains in the polymer network and, consequently, the hydrodynamic permeability. Indeed, the positively charged poly(vinylpyridinium) (P4VP+) gels stabilized in porous poly(propylene) membranes show substantially higher permeabilities than the PA gels discussed above (Figure 4). The measured permeability of the P4VP+ gels follows the empirical relationship

k ) 6.10 × 10-22φ-3.55 (m2)

(16)

with a correlation coefficient R2 ) 0.9899 and a standard error in the estimation of the slope of less than 5%. Both the numerical factor and the exponent are larger than those found for the PA gels.

Two routes have been taken to calculate the correlation length in the P4VP+ gels, both based on the assumptions of the Manning-Oosawa counterion condensation theory and the OSF theory of electrostatic persistence length. In the first approach, the Odijk expressions for Le (eq 4 or 5), κ (eq 6), and ξ in salt-free polyelectrolyte solutions (eq 7) are applied. In an alternative approach, the electrostatic interactions in the gel are accounted for by increasing the number of bonds in the persistence length from n ) C∞/6 to p ) n + ne, where ne ) (0.83)2Le/3a is the number of bonds in the electrostatic persistence length. The constant 0.83 is specific to a tetrahedrally bonded chain.19 The Schaefer relationship (eq 3) is then used, together with the OSF expression for the electrostatic persistence length (eq 5) and the Debye-Hu¨ckel parameter in the absence of salt (eq 6), to calculate the correlation length in the P4VP+ gels. The characteristic ratio C∞ ) 11.2 for poly(4-vinylpyridine)46 was used to calculate the rigidity index n, together with the C-C single bond length a ) 1.54 Å and the monomer length am ) 2.52 Å (tetrahedral bonding). The calculations were performed for the concentration range 0.04 e φ e 0.2 where, following the Schaefer criteria, marginal regime conditions existed. The bare persistence length Lp ) 12.5 Å was estimated from the characteristic ratio assuming tetrahedral bonding. The Debye-Hu¨ckel parameter κ and, subsequently, Le were calculated as a function of the gel concentration assuming validity of eq 5. The parameter κLt changed from 1.35 to 2.07 in the concentration range considered. This implies that eq 5 gives a good approximation of the electrostatic persistence length in the P4VP+ system. The correlation length ξ was calculated from eq 7 with the numerical prefactor taken equal to 1. The correlation length ξ calculated according to Odijk and Schaefer is presented, along with the electrostatic persistence length, as a function of gel concentration φ in Figure 5. The electrostatic persistence length Le is approximately 1 order of magnitude smaller than ξ, thus indicating the applicability of eq 5 to the P4VP+ system. There are small differences between ξ values obtained by these two routes, which increase toward the lower concentration border. Overall, ξ calculated using the Schaefer expression is larger. Both the Brinkman equation (eq 8) and the Happel equation (eq 10) were again used to calculate the Darcy permeability (Figure 6). The electroviscous effect, which is negligible at very high and very low salt concentrations,48 was disregarded in these calculations. The permeability calculated with the Brinkman equation and the ξ values obtained from the modified Schaefer expression overestimates the true value by 50-100% over the entire concentration range. The Odijk relationship for ξ coupled with the Brinkman equation underestimates the permeability at the lower gel concentrations and overestimates it at φ > 0.1. The deviations from the experimental values are fairly small, (50%. The Happel equation for permeability brings both estimations closer to the real values. Particularly, the Schaefer relationship for ξ leads to very close predictions of permeability at the lower gel concentrations (φ e 0.08). The discrepancies between the Schaefer-Happel model calculations and the experimental results increase with increasing gel concentration but stay below +50% over the entire tested concentration range (Figure 6d).

1700


Figure 3. Sample Qη/Am data as a function of -∆P/d for P4VP+ gel-filled membranes: (a) measurements for two membranes with different polymer volume fractions, φ, in the pores; (b) mean values for two samples with the same polymer volume fraction, φ ) 0.080 ( 0.001. Permeability km is the slope of the straight lines. Correlation coefficients: (a) R2 ) 0.9965 for φ ) 0.072 and R2 ) 0.9914 for φ ) 0.142, (b) R2 ) 0.9902.

Figure 4. Darcy permeability k of poly(4-vinyl-N-benzylpyridinium chloride) (P4VP+) and poly(acrylamide) gels (Kapur et al.15) as a function of the polymer volume fraction φ. Correlation coefficient for empirical fit of P4VP+ data: R2 ) 0.9899.

Figure 5. The correlation length, ξ, calculated from the modified Schaefer equation (eq 3) and the Odijk equation (eq 7), and the electrostatic persistence length, Le, calculated from the OSF model (eq 5), as functions of the polymer volume fraction, φ, in poly(4vinyl-N-benzylpyridinium chloride).

As emphasized by Schaefer,19 unrealistically sharp transitions between different solution regimes predicted by the blob model are, in reality, broad and smooth changes in statistical properties. In the case discussed here, the transition from the marginal to the theta regime is predicted to take place at φ ≈ 0.2. A direct

crossover from dilute to semidilute theta behavior without an intervening transition through the marginal or good solvent regime is possible in polymer-solvent systems at the theta temperature. When it is assumed that the theta regime governs the chain statistics in the system studied, the Schaefer model for ξ combined with


Figure 6. Darcy permeability k of P4VP+ gels as a function of the polymer volume fraction, φ: (a) k determined experimentally and calculated from the Brinkman equation (eq 8) using ξ values calculated from the Schaefer equation (eq 3) and the Odijk equation (eq 7); (b) k determined experimentally and calculated from the Happel equation (eq 10) using ξ values calculated from the Schaefer equation (eq 3) and the Odijk equation (eq 7); (c) relative differences δ ) (kcalculated - kexperimental)/kexperimental between the experimental permeabilities and those calculated from the Brinkman equation; (d) relative differences δ ) (kcalculated - kexperimental)/kexperimental between the experimental permeabilities and those calculated from the Happel equation.

the Happel equation for k gives a nearly perfect prediction of the hydrodynamic permeability. The results of these calculations are presented in Figure 7. The line representing the calculated gel permeability fits the experimental points very well (Figure 7a). The relative differences δ between the calculated and experimental values of the gel permeability shown in Figure 7b do not exceed 7%. This excellent fit suggests that the system might be close to the theta conditions. Lowering the charge density in a polyelectrolyte gel should affect ξ (and, consequently, the permeability) in two opposing ways, namely, decreasing ξ through lower chain rigidity and increasing it through reduced solvent quality. The latter effect should be particularly strong for polyelectrolytes with hydrophobic backbones.49 In this work, two membranes filled with P4VP+ gels with only 20% of the pyridine nitrogen atoms quaternized were examined. The permeability of these gels was found to be similar to or higher than that of the fully quaternized gels of the same concentration, indicating that the effect of reduced charge is outweighed by the increased gel hydrophobicity (Figure 8). The SchaeferHappel model calculations for the partly quaternized

gels and χ ) 0.4 underestimate the experimental data by about 60%. A very good fit is obtained, however, for χ ) 0.47, i.e., for a poorer solvent. The calculations were carried out assuming the conditions of theta regime. Effect of Cross-Linking. In the modeling presented above, we have assumed that the macroscopic rigidity of the gels introduced by cross-links can be neglected. Silberberg50 postulated that such assumption is valid only when the correlation length (blob size) is much smaller than the average chain length between crosslinks; otherwise, the structural heterogeneity imposed by the cross-links will dominate the gel behavior. From a hydrodynamic point of view, heterogeneity of the polymer segment distribution in a polymer network should result in essentially nondraining and freely draining regions. The majority of the gel polymer should be located in the nondraining regions. The flux through the draining regions can be very high because of the low effective concentration of the gel polymer in these regions. As a result, fluid can move faster through a heterogeneous gel even though only part of its volume is accessible.

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Figure 8. Darcy permeability, k, of the fully and partially quaternized P4VP+ gels as a function of the polymer volume fraction φ. Permeability of the partially quaternized gels was calculated for two values of the Flory-Huggins parameter χ.

Figure 7. Darcy permeability k of P4VP+ gels as a function of the polymer volume fraction, φ: (a) k determined experimentally and calculated from the Happel equation (eq 10) for the ξ values calculated from the Schaefer equation (eq 3) in the theta regime; (b) relative differences δ ) (kcalculated - kexperimental)/kexperimental between the calculated and experimental permeabilities k.

Silberberg50 estimated that, for poly(acrylamide) gels, the conditions under which gels and semidilute solutions of equivalent concentrations show similar behavior would only arise at high polymer concentrations (φ > 0.15) and low degrees of cross-linking. He showed that the introduction of cross-linking increases the permeability of low-concentration gels by nearly 2 orders of magnitude as compared to the permeability of non-crosslinked semidilute solutions calculated from sedimentation data. (It should be noticed that there is a very large discrepancy between the hydrodynamic permeabilities of PA gels reported by Silberberg50 and those reported by Kapur et al.15 or Tokita.47 The values given by Silberberg are approximately 2 orders of magnitude larger than those reported by Kapur et al.) For the majority of the gels tested in this work, including the poly(acrylamide) gels of Kapur et al.,15 the average correlation length calculated from the semidilute solution theory is close to or larger than the average chain length between cross-links. However, despite this fact, the calculated permeability fits the experimental data well over the tested concentration range of 0.04 e φ e 0.20 and the nominal degrees of gel cross-linking

of 5-10 mol % for the P4VP gels and 6 wt % for the PA gels. It is well-known51,52 that large-scale concentration heterogeneities do not appear during gelation but during swelling of a gel in a good solvent as it reaches equilibrium. The polymer concentration in the gel as prepared is expected to be as uniform as it is in the original solution used to for the gel preparation. This is particularly the case for gels produced by cross-linking of polymer solutions, such as the poly(4-vinylpyridine) gels discussed here. Because pore-confined gels are not allowed to swell to any substantial degree, they are expected to remain uniform unless some drastic conditions are introduced during the preparation phase. Thus, the structure of a polymeric gel network formed and retained in submicron confinement should be very similar to the statistical transient network of a semidilute solution with the same polymer segment concentration. This represents a major difference between confined and bulk swollen gels. To further test the effect of cross-linking on the properties of confined gels, we prepared a series of membranes with similar gel polymer concentrations, φ ) 0.085 ( 0.005, but different degrees of cross-linking. The hydrodynamic permeability of the membranes and the pressure-driven rejection of KCl from 10 mM solution under a pressure of 300 kPa were measured. The results are summarized in Table 1. Doubling the crosslinking from 0.05 to 0.10 mol of R,R′-dichloro-p-xylene per mole of the repeating unit had no effect on the membrane permeability or the salt rejection, demonstrating that essentially the same porous network of charged channels existed in both these gels. The permeability of these membranes can be predicted from the model. On further increases in cross-linking to 0.15 and 0.20, this picture changed; the permeability increased,

Ind. Eng. Chem. Res., Vol. 40, No. 7, 2001 1703 Table 1. Effect of Cross-Linking on Permeability and Selectivity of P4VP+-Filled Polypropylene Microporous Membranes degree of cross-linking (mol/mol)

km × 1018 (m2)

KCl rejection (%)

0.05 0.10 0.15 0.20

1.40 1.40 2.47 3.03

49.9 50.8 45.8 36.8

and the salt rejection dropped. It is worth of noting, however, that the increase in permeability is relatively small (about 100% for the highest degree of crosslinking) compared to about 2 orders of magnitude increase described by Silberberg50 for PA gels when the cross-linking was changed from 0 to 8%. It was noticed in this work that bulk P4VP gels formed using the same solutions as were used to prepare membranes were not clear at these high degrees of cross-linking and that solvent syneresis was observed. The syneresis could also be observed on the surface of the membranes. This clearly indicates that cross-linking in excess of 10 mol % caused microphase separation and the formation of micropores in the gels. The decrease of salt rejection also supports this explanation and is in a qualitative agreement with the space-charge model53-55 for salt rejection by charged capillaries. For such gels, the model calculations discussed above are invalid. Comparison with Other Models. As mentioned before, alternative models treat gels as regular or random networks of straight cylinders (fibers) of radius rf. Contrary to our model, in which the basic dimension defining the gel structure (correlation length/sphere diameter) scales with the polymer concentration, there is no obvious rationale for any change in the radius of fibers as their concentration in the network increases. To examine the ability of other models to predict the hydrodynamic permeability of gel networks, we calculated the hydrodynamic permeability of random arrays of cylinders as a function of gel polymer volume fraction, φ, by treating the fiber radius as a fitting parameter and using the Happel equations (eqs 11). The results calculated for two radii, 0.85 and 1.4 nm, are presented in Figure 9, together with the empirical relationship found for poly(4-vinyl-N-benzylpyridinium) gels. It is evident from the graph that the model fails to predict the permeability of the gels. The slope of the calculated parallel lines is very different from that of the empirical relationship, and the difference between the calculated and experimental data can be nearly 2 orders of magnitude over the examined concentration range. Because k ) f(φ) for PAA gels has nearly the same slope as that for P4VP+ gels, the fiber model also fails to predict the permeability of these gels. Other equations developed to calculate the hydrodynamic permeability of cylindrical arrays, e.g., the Drummond and Tahir relationship,56 give results very similar to those obtained from the Happel equations. Conclusions The existing models of gels as regular or random networks of straight cylinders of fixed radius have a limited applicability, as the hydrodynamic equations for viscous flow through such networks fail to predict the hydrodynamic permeability of gels over a wide range of gel concentrations. The model developed here, in

Figure 9. Darcy permeability, k, of P4VP+ gels as a function of the polymer volume fraction φ: empirical fit of experimental data (solid line) and calculated for two fiber radii (dashed lines) from Happel’s equations (eqs 11).

which gel networks are treated as an assemblage of spheres of concentration-dependent radius, provides an instrumental advance in the calculation of hydrodynamic permeability. The most important feature is that the sphere radius, the basic parameter in the calculations of permeability, can be estimated from the fundamental properties of the gel polymer chains and thermodynamics of the gel polymer-solvent systems. The data required for this estimation are readily available for a large number of polymers. The Flory-Huggins parameter, χ, which was used in this work as a fitting parameter, can be obtained from measurements of vapor-liquid equilibria (VLE) or the activities of the solvents in the given systems. It is shown that the sphere radius can be obtained from the pair correlation length in equivalent semidilute solutions, which can, in turn, be calculated using wellestablished scaling methods. The hydrodynamic (Darcy) permeabilities of homogeneous charged and neutral polymeric gels can be predicted with good accuracy over a wide range of polymer concentrations using the estimated sphere radius and the Happel cell model for viscous flow through assemblages of spheres. The differences between calculated and empirical permeabilities are smaller than the deviations found between experimental data from different sources. The substrate porosity and the pore tortuosity are additional parameters required in the calculations when the model is applied to gel-filled microporous membranes. The quantitative description of gel structure developed in this work opens the door to subsequent modeling and calculations of molecular transport such as diffusion and electrodiffusion of neutral and charged species in gels. In principle, the separation properties of gel-filled

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Received for review August 31, 2000 Revised manuscript received January 22, 2001 Accepted January 24, 2001 IE000794Q

Calculation of the Hydrodynamic Permeability of Gels and Gel-Filled

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