Investigation into the Diffusion of Water into HEMA-co-MOEP

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Biomacromolecules 2004, 5, 1194-1199

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Investigation into the Diffusion of Water into HEMA-co-MOEP Hydrogels Karina A. George,† Edeline Wentrup-Byrne,*,† David J. T. Hill,‡ and Andrew K. Whittaker§ Research Centre, School of Physical and Chemical Sciences, Queensland University of Technology, Brisbane, Australia 4001, and Polymer Materials and Radiation Group and Centre for Magnetic Resonance, University of Queensland, Brisbane, Australia Received November 18, 2003; Revised Manuscript Received March 8, 2004

Cross-linked homopolymers and copolymers of 2-hydroxyethyl methacrylate, HEMA, and ethylene glycol methacrylate phosphate, MOEP, have been synthesized, and the diffusion of water into these systems has been investigated. Only polymers with 0-20 mol % MOEP exhibited ideal swelling behavior as extensive fracturing occurred in the systems with greater than 20 mol % MOEP as the polymers began to swell during water sorption. Gravimetric studies were used in conjunction with magnetic resonance imaging of the diffusion front to elucidate the diffusion mechanism for these systems. In the case of the cross-linked HEMA homopolymer gels, the water transport mechanism was determined to be concentration-independent Fickian diffusion. However, as the fraction of MOEP in the network increased, the transport mechanism became increasingly exponentially concentration-dependent but remained Fickian until the polymer consisted of 30 mol % MOEP where the water transport could no longer been described by Fickian diffusion. Introduction Since the pioneering work of Wichterle and Lim, who demonstrated that poly(2-hydroxyethyl methacrylate) (PHEMA) hydrogels exhibited high biocompatibility and low thrombogenicity,1 hydrogel research has been focused primarily on the development of these materials for biomedical applications. PHEMA based hydrogels have now become one of the most commercially utilized biomedical hydrogels and are used widely in soft contact lenses and intraocular lenses. The success of PHEMA as a biomaterial has led to the investigation of novel copolymers to enhance particular properties,2-6 i.e., strength, imbibed water content, and stimuli-sensitivity, while retaining the biocompatibility and low biological response. The addition of co-monomers containing ionisable functional groups can result in hydrogels where the degree of intermolecular interactions between the chains depends on the pH of the surrounding fluid. Hence, a change in pH can cause either a decrease or increase in these interactions, leading to a pH dependence of the water content of the hydrogel.7 Many different polyelectrolyte hydrogel systems have been studied including those containing carboxyl, sulfono, and amino groups.7-10 To date, only limited research has been directed into systems containing pendent phosphate groups11,12 although the phosphate group is of biological importance in living systems. Incorporating * To whom correspondence should be addressed. Author address: Tissue BioRegeneration and Integration (TBRI) Program, Queensland University of Technology, 2 George St, GPO Box 2434, Brisbane, Queensland, 4001 Australia. Fax: +61 7 3864 1804. E-mail: [email protected]. † Queensland University of Technology. ‡ Polymer Materials and Radiation Group, University of Queensland. § Centre for Magnetic Resonance, University of Queensland.

this functionality into a hydrogel may not only improve its performance as a biomaterial but also open up an even wider range of potential applications. It is believed that the improved performance of hydrogels in biological environments is associated with the water imbibed within the system.13 Therefore, a thorough understanding of the water-polymer interactions of such systems is critical to the development of these materials for biomedical applications. The water transport mechanism into the polymer is particularly important for assessing the suitability of these materials as drug delivery systems, as the amount of drug released is dependent on the rate and transport mechanism of water diffusing into the polymer network. It is generally accepted that there are three models which describe the diverse range of responses of hydrophilic polymer networks in the presence of water. These models, proposed by Alfrey et al., are based on the relative rates of penetrant diffusion and polymer chain relaxation.14 1. Fickian diffusion (case I diffusion) occurs when the diffusion is significantly slower than the rate of relaxation of the polymer chains. 2. Case II diffusion occurs when the rate of penetrant diffusion is greater than the rate of relaxation of the polymer chains. 3. Anomalous diffusion (case III diffusion) accounts for behavior that lies between the extreme case I and case II models. Hence, the rate of penetrant diffusion is comparable to the polymer relaxation. The diffusion mechanism can be determined by a variety of techniques. The easiest and most utilized method is by gravimetric water sorption studies, where the mass of the water absorbed by the polymer, Mt, is monitored as a function of time, t. When Mt is normalized to the mass of water

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Figure 1. Structures of (a) HEMA, (b) MOEP, and (c) EGDMA.

absorbed by the polymer at its equilibrium hydration level, M∞, the transport mechanism can be determined according to the equation Mt ) ktn M∞

(1)

where k and n are constant for a particular system. For a planar system, an exponent of 0.5 (n ) 0.5) is indicative of Fickian diffusion, whereas case III diffusion is characterized by an exponential between 0.5 and 1, with a limit of n ) 1 identifying case II transport.15 The work presented here is an investigation into the synthesis, characterization and water diffusion through methacrylate hydrogel systems containing pendent phosphate groups. The HEMA-co-MOEP copolymers (monomers shown in Figure 1) were synthesized by free radical polymerization in the bulk. Gamma radiolysis was used to initiate the reaction in an attempt to slow the polymerization and thus reduce the extent of thermal degradation occurring due to autoacceleration in the Tromsdoff region and avoid the use of potentially harmful chemical initiators. Experimental Section Polymer Synthesis. The monomers, HEMA and MOEP, and the cross-linking agent, ethylene glycol dimethacrylate, EGDMA, were all purchased from Sigma-Aldrich. MOEP (warmed in an oven to 40 °C) and EGDMA were purified by passing them through a dry alumina column, whereas HEMA was purified by distillation under a reduced atmosphere (3.0 mmHg). All reagents were stored under a nitrogen atmosphere for a maximum period of 4 days. The required proportions of HEMA and MOEP were accurately weighed and mixed with the EGDMA (0.5 wt %) and pipetted into Teflon tubes (internal diameter of 6.05 mm) which were stoppered at one end with a short glass cylinder. These tubes were purged with nitrogen after being placed under vacuum for 1 h. The polymerization was carried out in a 60Co AECL Gammacell 200 facility with a dose rate of 0.6 kGy/hr at room temperature for 47.5 h. After polymerization, the solid, glassy cylinders were removed from the moulds and subjected to a high vacuum at 50 °C for 6 days and finally at 80 °C for 24 h. Fourier transform-near-infrared spectroscopy (FT-NIR) was used

to confirm the absence of any monomer present in the cylinders.16 DSC. Modulated differential scanning calorimetry (MDSC) was carried out on a TA Instruments DSC Q 100 instrument to determine the glass transition temperature (Tg) of the copolymers. Approximately 5 mg of the vacuum-dried samples was placed inside sealed aluminum pans and first heated from 35 to 200 °C at a rate of 20 °C/min to obtain a constant thermal history. The reported Tg was then calculated from a second scan using the same parameters as the first, but with a modulation of 0.5 °C/80 s. Water Sorption Experiments. Following the post polymerization thermal treatment, the polymers were placed in distilled water which was maintained at 37 °C. Periodically, the cylinders were removed from the water bath, excess water was removed from the surfaces with a lint free tissue, and the cylinders were weighed. The amount of water absorbed was monitored gravimetrically for a period of 2-3 weeks. Magnetic Resonance Imaging (MRI). The cylindrical samples were immersed in distilled water at 37 °C. At predetermined times, they were removed and wrapped in polyethylene film, and NMR images were obtained using a Bruker AMX300 spectrometer. Images were acquired using a standard spin-echo, three-dimensional imaging sequence. The images were obtained by using 90° and 180° pulses of 14 and 28 µs with echo and repetition times of 8.4 ms and 2 s, respectively. The images consisted of 128 × 128 × 8 pixels, and all images have an in-plane resolution of 11.17 × 11.17 µm and a slice thickness of 3.75 mm in a field of view of 1.50 × 1.50 × 3.00 cm. The total acquisition time was approximately 24 min. Care was taken to confirm that full relaxation of the magnetization could occur during the 2 s repetition time and also that the T2 relaxation time of the water protons did not change appreciably across the images. Thus, the images provide an accurate picture of the relative water concentration across the cylinders. Theoretical water concentration profiles were calculated using an implicit finite difference method as described by Crank,17 with the diffusion coefficient being proportional to the exponential of the concentration. Other functional forms were considered including a concentration-independent diffusion coefficient and a diffusion coefficient that is proportional to concentration; however, visual comparison of experimental and calculated data showed poor agreement. Results and Discussion Glass Transition Temperatures. The systematic changes in the glass transition temperatures of the homopolymers and copolymers are shown in Figure 2. The solid line represents the predicted Tg values calculated by the Fox equation [2] with the assumption that there is ideal mixing of the monomers w1 w2 1 ) + Tg Tg1 Tg2

(2)

Here w1 and w2 are the weight fractions of components 1

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Figure 2. Effect of composition on the glass transition temperature of poly(HEMA-co-MOEP).

and 2 in the polymer, Tg1 and Tg2 are the glass transition temperatures of the respective homopolymers, and Tg is the glass transition temperature of the copolymer. The agreement of the experimental Tg values with the calculated values implies that homogeneous networks are formed during the polymerization process and that the interactions between the monomer units, i.e., hydrogen bonding between the different monomers, does not significantly deviate from interactions experienced in the homopolymers. Behavior of Polymers when Immersed in Water. The behavior of the polymer cylinders when immersed in water was found to be dependent on the fraction of MOEP in the polymer. The polymers with 0-20 mol % MOEP did not fracture during swelling and displayed concentration-dependent Fickian water sorption. The water sorption properties of these hydrogels will be discussed in detail later. The polymers with 20 and 30 mol % MOEP retained its cylindrical shape during the swelling process although several fractures appeared perpendicular to the length of the cylinder (Figure 3a). This was attributed to a build up of stress between the glassy core and the hydrated swollen shell that surrounds the core and forms as water begins to diffuse into the network. Alfrey et. al. investigated the stresses associated with the sorption of solvents into cylindrically shaped polymeric networks by case II diffusion and reported that the largest tensile stress occurs between the glassy core and the swollen, flexible shell, parallel to the length of the cylinder (axial stress).14 It was observed that the fractures that resulted from this stress were orientated perpendicular to the cylinder length. Those copolymers with greater than 30 mol % MOEP exhibited catastrophic fracturing during swelling and all crumbled, resulting in the destruction of the cylindrical geometry (Figure 3b). The fact that these fractures were randomly orientated would indicate that stresses in all directions were responsible for this behavior. As the fractures formed, pieces of polymer peeled off from the surface of the cylinder. The size of these polymer pieces decreased as the amount of MOEP in the networks increased. It was also observed that the shell region of these polymers remained glassy throughout the duration of polymer fracturing, rendering the material less able to withstand the applied swelling

Figure 3. (a) Photograph showing the fractures in the 30 mol % MOEP hydrogel (top) and 20 mol % MOEP hydrogel (bottom). (b) Photograph of the crumbled cylinders with 40 mol % MOEP (far left), 60 mol % MOEP (left), 80 mol % MOEP (right), and 100 mol % MOEP (far right).

Figure 4. Sorption curves for the diffusion of water at 37 °C into poly(HEMA-co-MOEP) hydrogels.

stress. Thus, the incorporation of greater than 30 mol % MOEP leads to a network that is more rigid and less accommodating than those of polymers with lower MOEP contents and therefore the stress associated with swelling and water sorption is unable to be dissipated without extensive fracturing. Water Sorption Studies. The water uptake curves from a single run of each of the HEMA-co-MOEP hydrogels with 3-30 mol % MOEP and the HEMA homopolymer at 37 °C are shown in Figure 4. The fractional water content (St), kinetic parameters, and diffusion rates were calculated from these data. The value of St is defined as the ratio of the mass of absorbed water at equilibrium to mass of dry polymer

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Table 1. Summary of Values of n, D0, and A Determined from the Analysis of the MRI Images sample

n

D0 × 10-8/cm2s-1

A

PHEMA 3 mol % MOEP/HEMA 6 mol % MOEP/HEMA 10 mol % MOEP/HEMA 20 mol % MOEP/HEMA

0.48 0.54 0.57 0.53 0.54

17 ( 1 2.8 ( 0.3 3.9 ( 0.2 3.8 ( 0.5 3.5 ( 0.7

1.8 ( 0.2 1.9 ( 0.2 3.5 ( 0.1 3.4 ( 0.5

according to the equation St )

Mt Mp

(3)

where Mt is the mass of absorbed water at time t and Mp is the mass of the dry polymer. Figure 4 shows that the maximum equilibrium water content of 59% occurs in the 6 and 10 mol % MOEP hydrogels. Miyata et al. also studied a series of HEMA-co-MOEP hydrogels which were polymerized in a solution of methanol.11,12 They reported that a maximum equilibrium water content of 740% occurred when there was 6 mol % MOEP present in the network.12 It should be noted that the large deviation between the water contents of the hydrogels from the two studies can be attributed to the different polymerization techniques employed. The hydrogels, in the present investigation, were polymerized in the bulk, whereas the hydrogels previously studied were polymerized in solution and are therefore more porous. In the previous study by Miyata et al. the rational for maximum equilibrium water uptake to occur in the hydrogel with 6 mol % MOEP was due to the phosphate groups acting as both a hydrophilic group and a cross-linking agent.12 At low MOEP content, the distance between the phosphate groups limits the interactions between these groups, and therefore, they behave as independent hydrophilic groups. The hydrogel is then capable of increased water uptake. However, when the concentration of the phosphate groups is greater, they are in closer proximity to each other, and hydrogen bonding between the groups can occur. This, in turn, increases the cross-linking density in the network making it more rigid, and consequently the amount of water imbibed by the system decreases. The exponent, n, from eq 1 was also calculated for the various networks, and these values are shown in Table 1. From the calculated n values the transport mechanism was found to occur principally by Fickian diffusion. It should be noted that eq 1 is based only on the first term of eq 4 which describes Fickian diffusion

()

St Mt 4 Dt ) ) 1/2 2 M∞ S∞ π r

1/2

-

()

Dt 1 Dt r2 3π1/2 r2

3/2

+ ...

(4)

where S∞ is the fractional water content at equilibrium, D is the diffusion coefficient, t is immersion time, and r is the radius of the dry polymer cylinder. Therefore, eq 1 provides a simple but approximate estimate of transport mechanism. Sorption Rate. As the MOEP content increases, there is a concomitant increase in the rate of water sorption into the network. This is attributed to the incorporation of the polar

Figure 5. Effect of phosphate concentration in the polymer on the initial sorption rate at 37 °C.

phosphate groups, which increase the hydrophilicity of the network and provide a greater “driving force” for the water sorption process. Figure 5 shows that the rate of water sorption that occurs during the initial linear region of the sorption curve systematically increases with the concentration of phosphate groups in the network. The sorption rates were determined as the mass of absorbed water by one gram of polymer per unit time, and errors were estimated according to error in the slope of the respective sorption curves. For the polymers with 0-20 mol % MOEP, the increase in the sorption rate is approximately linear with the MOEP content. Overshoot in the Sorption Curve. All of the sorption curves exhibit a maximum in the water uptake which gradually decreases to an equilibrium value at longer times. Previously, this has been attributed to the slower rate of the relaxation of the polymer chains compared to the rate of swelling with water.2 Comparison of the magnitude of the overshoots reveals that, as the MOEP content increases, the overshoot becomes more pronounced implying that the increase in physical cross-linking causes the polymer chains to become less mobile. MRI Investigation of the Water Distribution. Despite the fact that both the n values and the shape of the sorption curves suggest that the ingress of water is governed by Fickian diffusion, a comparison with theoretical Fickian sorption curves calculated according to eq 4 shows small systematic deviations from the experimental data. An imaging technique such as MRI makes it possible to visualize the diffusion front and hence the distribution of water through the hydrogel during the sorption process. This, in turn, should lead to a better insight into the diffusion process. Gravimetric studies are based upon the measurement of macroscopic changes, which are the result of diffusion at the molecular level. The determination of the diffusion coefficient of a polymer-penetrant system by gravimetric studies involves the continual monitoring of the mass change so a sorption curve can be constructed. Only one diffusion coefficient can be calculated from any sorption curve giving the average diffusion coefficient for the period of the sorption process. MRI allows direct information concerning the distribution of water throughout the polymer to be determined. From a single image taken at a particular time, the instantaneous diffusion coefficient can be calculated. This allows a greater insight into the sorption process, for

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the boundary between the swollen and glassy regions becomes more abrupt. The result is that the interface between the glassy core and swollen outer layer becomes sharper, and consequently, the stresses in this region are magnified. These observations support our conclusions concerning the occurrence of stress related fractures in the polymers with high MOEP contents (>20 mol % MOEP). Determination of the Diffusion Mechanism and Coefficient. Theoretical sorption curves have been calculated based on an exponentially concentration-dependent Fickian diffusion coefficient, where Figure 6. T2 weighted MRI images showing the intensity of water protons across the diameter of the cylinder containing 3 mol % MOEP/ HEMA hydrogel after (a) 3 h 30 min, (b) 12 h, (c) 24 h, and (d) 43 h immersion in water.

Figure 7. Normalized profiles of the water concentration within the 6 and 20 mol % MOEP/HEMA hydrogels after 18 h immersion in water where C0 is the concentration of water at the cylinder surface. The overlaid curve is that generated for a concentration-independent Fickian diffusion mechanism after 18 h where D ) 8 × 10-8 cm2/s.

example, as the dependence of the diffusion coefficient on the water content of the polymer can be examined. Figure 6 shows MRI images from a cross-section of the hydrogel with 3 mol % MOEP after being immersed in water over a 43 hour period. The intensity of the signal in these plots represents the concentration of water in the sample. This plot shows that at short sorption times the concentration of water is greatest at the surface. The concentration gradually decreases toward the center of the cylinder where the polymer is in a glasslike state. Figure 7 shows the water profiles for the 6 and 20 mol % MOEP hydrogels after 18 h immersion in water. For comparison purposes, a calculated concentration-independent Fickian profile has been included in this figure. Previously we have shown that the water concentration profiles during diffusion of water into poly(HEMA) and copolymers of HEMA with hydrophobic monomers are of this Fickian form.2,3 A comparison of the experimental profiles with the calculated Fickian profile reveals that the incorporation of phosphate groups into the system causes the transport mechanism to deviate from true Fickian diffusion. This is reflected in the change in shape of the water profile from concave for concentration-independent Fickian diffusion to convex for water diffusing into the MOEP-containing systems. As the concentration of MOEP in the network increases, the experimental profile becomes more convex and

D ) D0eA(C/C0)

(5)

and D0 is the diffusion coefficient at the limit of zero penetrant concentration, A is a proportionality constant, and C/C0 is the concentration of water relative to the concentration at the surface. Previously, Crank has considered in some detail the case of concentration-dependent diffusion coefficients.17 Using the A and D0 values which best describe the water distribution through each polymer during the sorption process, mass uptake curves can be generated which agree closely with the experimental sorption curves up to the point where the glassy core begins to disappear. The experimental water distribution profile and the mass uptake sorption curve each overlayed with the calculated curves for the 6 mol % MOEP hydrogel are shown in Figure 8. The excellent fit of the calculated data to the experimental data is clearly demonstrated in these figures. The calculated curve in Figure 8a was obtained by substituting eq 5 into eq 6 C - C1 C0 - C1

)1-

2

∞

1 J0(aRn)

∑ r n)1 R

n

J1(rRn)

exp(-DRn2t)

(6)

where C is the water concentration, C1 is the initial water concentration in the polymer, C0 is the water concentration at the surface of the polymer. J0(x) and J1(x) are the Bessel functions of zero and first order, respectively, and a is defined as O < a < r. The Rn values are the positive roots of the Bessel functions of the first kind. The calculated curve in Figure 8b was obtained by substituting eq 5 into eq 7 Mt M∞

)

St S∞

∞

)1-

∑2 n-1

4

r Rn

2

exp(-DRn2t)

(7)

where the Rn values are the positive roots of the Bessel function of the first kind and of zero order. It was observed that once the glassy core disappears there is an abrupt change in the dimensions of the cylinder. Hence, it can be concluded that the diffusion kinetics change at this point and the concentration-dependent Fickian model for water ingress no longer adequately describes the system. Table 1 shows the values of D0 and A prior to the disappearance of the glassy core. As mentioned above, Crank has previously considered in great detail the analysis of the swelling of polymers in the

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tion. Since the diffusion coefficient is approximately proportional to the number of these holes per unit volume, it should increase in an exponential manner with penetrant concentration. Conclusions The incorporation of phosphate pendant groups into crosslinked PHEMA hydrogels significantly alters the mechanism of water transport into the polymer matrix. The transport mechanism of water through the polymers varies from concentration-independent Fickian diffusion for a PHEMA hydrogel to concentration-dependent Fickian diffusion, for polymers with 3-20 mol % MOEP. The change in transport mechanism is attributed to the presence of the phosphate group which acts both as a hydrophilic group and, at high concentrations, a physical cross-linker which makes the networks more rigid. The brittle nature of hydrogels with 40-100 mol % MOEP was attributed to the formation of cross-links between phosphate groups. These cross-links resulted in a stiff network that could not efficiently dissipate the stress created at the glass/rubber interface of the hydrating polymer. For biomedical applications, the 3-20 mol % MOEP polymers appear to be promising materials, and more work is being conducted on these systems to investigate their bioactivity. Figure 8. (a) Normalized water concentration profile for the 6 mol % MOEP/HEMA copolymer after 12 h immersion. The calculated water profiles are for an exponentially concentration-dependent diffusion mechanism with the values of D0 and A as shown in the figure. (b) Normalized sorption curve for the 6 mol % MOEP/HEMA hydrogel with the theoretical sorption curve calculated for an exponential concentration-dependent diffusion mechanism where D0 ) 3.9 × 10-8 cm2/s and A ) 1.9.

case of a concentration-dependent diffusion coefficient. In the current study, many different forms of concentration dependence were considered, and it was found that a satisfactory fit to the experimental data could only be obtained by invoking an exponential dependence of D on concentration. This behavior can be explained by reference to Eyring’s “hole theory of diffusion”.18 According to Eyring’s theory, as a result of thermal fluctuations, there are a number of holes present in all materials. Diffusion takes place when a molecule leaves its current position and migrates into one of these holes. In order for a hole to be formed, a number of van der Waals bonds must be broken. This gives rise to a site of higher energy. It follows that the amount of energy required to form a hole increases with the size of the hole. Thus, according to Boltzmann’s law, the concentration of holes decreases exponentially with their size. Using the assumption that the polymer-penetrant bonds are weaker than the polymer-polymer bonds, then the energy required to form a hole of a certain size decreases linearly with increasing penetrant concentration. Consequently, the number of holes big enough to permit diffusion should increase exponentially with increasing penetrant concentra-

Acknowledgment. Sean McElwain, School of Mathematics, QUT is thanked for his interesting and stimulating conversations concerning the mathematics of diffusion. K.A.G. thanks the Faculty of Science, QUT for their financial support. References and Notes (1) Wichterle, O.; Lim, D. Nature 1960, 185, 117-118. (2) Ghi, P.; Hill, D. J. T.; Whittaker, A. K. J. Polym. Sci. Polym. Phys. 2000, 38, 1939-1946. (3) Hill, D. J. T.; Moss, N. G.; Pomery, P. J.; Whittaker, A. K Polymer 1999, 41, 1287-1296. (4) Sun, Y.; Hunag, J.; Lin, F.; Lai, J. Biomaterials 1997, 18, 527-533. (5) Refojo, M. F.; Yasuda, H. J. Appl. Polym. Sci. 1965, 9, 2425-2435. (6) Peniche, C.; Cohen, M. E.; Vazquez, B.; San Roman, J. Polymer 1997, 38, 5977-5982. (7) Bae, Y. H.; Kwon, I. C. In Biorelated Polymers and Gels; Okano, T., Ed.; Academic Press: San Diego, CA, 1998; pp 93-134. (8) Galaev, O. Y.; Mattiasson, B. Tibtech 1999, 17, 335-340. (9) Hoffman, A. S. MRS Bull. 1991, 16, 42-46. (10) Qiu, Y.; Park, K. AdV. Drug DeliVery ReV. 2002, 54, 321-339. (11) Miyata, T.; Nakamae, K.; Hoffman, A. S.; Kanzaki, Y. Macromol. Chem. Phys. 1994, 195, 1111-1120. (12) Nakamae, K.; Miyata, T.; Hoffman, A. S. Makromol. Chem. 1992, 193, 983-990. (13) Bruck, S. D. J. Biomed. Mater. Res. 1973, 7, 387-404. (14) Alfrey, T.; Gurnee, E. F.; Lloyd, W. G. J. Polym. Sci., Part C 1966, 12, 249-261. (15) Peppas, N., A.; Brannon-Peppas, L. J. Food Eng. 1994, 22, 189210. (16) Ghi, P.; Hill, D. J. T.; Whittaker, A. K. J. Polym. Sci. Polym. Chem. 1999, 37, 3730-3737. (17) Crank, J. The Mathematics of Diffusion, 2nd ed.; Clarendon Press: Oxford, U.K., 1975; Chapter 7. (18) Prager, S.; Long, F. A. J. Am. Chem. Soc. 1951, 73, 4072-4075.

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