MAA Hydrogels

Article pubs.acs.org/Macromolecules

Aqueous Solute Partitioning and Mesh Size in HEMA/MAA Hydrogels Csaba Kotsmar,† Teresa Sells,† Nicole Taylor,† David E. Liu,† J. M. Prausnitz,† and C. J. Radke*,†,‡ †

Department of Chemical and Biomolecular Engineering, University of California, Berkeley, Berkeley, California 94720, United States Vision Science Group, University of California, Berkeley, Berkeley, California 94720, United States

‡

ABSTRACT: Using two-photon confocal microscopy, equilibrium partition coefficients, k, were measured for aqueous Na-fluorescein, fluorescently labeled dextrans with molecular masses ranging from 4 to 20 kDa, two fluorescently labeled proteins with opposite charges, anionic bovine serum albumin (BSA), and cationic avidin in anionic 70 wt % hydroxyethyl methacrylate (HEMA)/30 wt % methacrylic acid (MAA) gels saturated with aqueous phosphate buffer solution. Cross-linking density with ethylene glycol−dimethacrylate (EGDMA) ranged from 0 to 1 wt %. All partition coefficients, except for avidin, were considerably less than unity and diminished strongly with increasing Stokes− Einstein diameter of the free aqueous solute. The average mesh size of the wet gels, obtained from the zero-frequency oscillatory shear-storage gel modulus, ranged from 3.6 to 8.3 nm over the cross-link ratios studied. Except for Na-fluorescein, solute hydrodynamic diameters were larger than the smallest average gel mesh size. Yet, all solutes permeated the gels but with small partition coefficients less than about 0.001 for the largest diameter solutes in the small mesh size gels. To express deviation from ideal partitioning, we define an enhancement (or exclusion) factor, E ≡ k/(1 − φ), where φ is the polymer volume fraction in the gel and E is unity for point solutes. A hard-sphere excluded-volume Ogston mesh size distribution is adopted to predict a priori the measured enhancement factors as a function of average gel mesh size for those solutes that do not interact specifically with the anionic gel (i.e., for solutes with E < 1). Agreement between the extended Ogston distribution and experiment is qualitative for both enhancement factors and water content of the gels. The cationic protein, Fl-avidin, exhibits a large enhancement factor in the anionic gels due to strong specific interaction with the charged carboxylate groups of MAA. In this case, consideration must be given to both hardsphere size exclusion and specific complexation with the polymer strands.

1. INTRODUCTION Hydrogels are three-dimensional cross-linked polymeric networks that imbibe water and swell without dissolving.1−5 From a macroscopic viewpoint, hydrogels behave as a viscoelastic material,1,3−6 whereas from a microscopic viewpoint they are considered an aqueous polymer solution.3,7,8 Hydrogels are widely used in biomedical and pharmaceutical applications,1,9 as they can optimally mimic living tissue due to their soft consistency, high water content, and biocompatibility.1 These materials can be used to manufacture membranes, soft contact lenses,10−12 drug-delivery devices, and scaffolds for synthesizing artificial organs.1,13 A useful feature of hydrogels is their ability to uptake aqueous solutes and later deliver them in a controlled fashion. Accordingly, a critical characteristic of a hydrogel is the equilibrium partition coefficient, k, of a given solute defined by14−16 k ≡ Cgel /C bulk

bulk aqueous solute concentration, only in dilute solution where solutes do not interact with each other. For point solutes, the partition coefficient must reduce to the volume fraction of water in the gel, i.e., k = 1 − φ, where φ is volume fraction of polymer in the gel.14−22 The ratio k/(1 − φ) can be considered an enhancement factor E ≡ k/(1 − φ)

Solutes that bind specifically with the polymer chains exhibit enhancement factors significantly greater than unity. These may be thought of as adsorbed onto the polymer chains. Solutes that are partially rejected from the gel display enhancement factors less than unity, while E = 1 corresponds to ideal partitioning or to apparent ideal partitioning arising from compensation between size exclusion and specific adsorption. E = 0 reflects a solute that is too large to penetrate the water-filled spaces in the gel network. On the time scale of an absorption measurement, polymer chains in the cross-linked hydrogel matrix fluctuate rapidly. Thus, the water-filled spaces between polymer strands should not be considered as static “pores” as in a porous medium. Nevertheless, gel structure can be characterized by mesh size, ξ,

(1)

where Cbulk is the solute concentration in the external aqueous solution equilibrated with the gel and Cgel is the concentration of solute in the gel per unit volume of gel (i.e., per volume of water plus polymer). Equation 1 holds only for equilibrium partitioning where solute absorption in the gel is reversible. Further, the partition coefficient is constant, independent of © 2012 American Chemical Society

(2)

Received: September 3, 2012 Revised: October 12, 2012 Published: November 9, 2012 9177

dx.doi.org/10.1021/ma3018487 | Macromolecules 2012, 45, 9177−9187

Macromolecules

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on the equilibrium partitioning of fluorescein-labeled proteins BSA, ovalbumin, and lactalbumin were investigated in negatively charged SP-Sepharose gels at different ionic strengths.33 In all publications cited, little to no attention has been given the relationship between solute partitioning and independently measured gel mesh size. The goal of this study is to predict measured equilibrium partition coefficients of solutes in gels as a function of independently measured average mesh sizes where the solute and mesh sizes overlap. We focus on hydrogels reflective of soft-contact-lens materials that typically have relatively high polymer content, larger bulk moduli, and correspondingly smaller mesh sizes. Solute molecules applicable to soft contact lenses include salts, proteins, nutrients, polymers, and polymeric surfactants that can serve as wetting and comfort agents, water-soluble drugs, and vitamins. Once impregnated by solute, the lens can provide continuing performance improvement during wear.37,38 Solute partition coefficients determine the amount of material that can be loaded into a lens and later delivered. The sizes of polymeric wetting and comfort agents are typically comparable to the mesh size of the gels. In such cases, solute/matrix interactions are pronounced. We employ two-photon confocal microscopy to measure partition coefficients of sodium fluorescein, fluorescently labeled dextrans, fluorescently labeled ficolls, and two fluorescently labeled proteins with opposite charge: bovine serum albumin and avidin. The hydrogel is anionic consisting of 70 wt % HEMA/30 wt % MAA in phosphate buffer solution (pH = 7.4) with cross-link densities ranging from 0 to 1 wt %. Average gel mesh size is determined from oscillatory shear rheometry and Gaussian-chain rubber elastic theory. Solute sizes are determined from independent measurement of the bulk aqueous diffusion coefficient in a restricted diffusion cell and Stokes−Einstein theory. For those solutes that do not specifically interact with the anionic gel, prediction of solute partitioning is by the Ogston fiber-based mesh size distribution28 extended to account for hard-sphere excluded-volume interactions.

defined here as the distance between cross-links.1,17,23−27 The cross-linking process produces a distribution of polymer-strand molecular weights between linkages. A concomitant distribution of mesh sizes28,29 arises with an average mesh size, ⟨ξ⟩, that scales as the square root of the average molecular weight of chains between cross-links.2 Solute molecules larger than a limiting mesh size that permits percolation of the gel cannot penetrate. For that case, E = 0. Solute molecules smaller than this critical mesh size absorb into the gel but may occupy only a fraction of the mesh size space available because of size exclusion. If, in addition, this sized solute does not interact favorably with the polymer strands, then E < 1. Because smaller solutes occupy more and more of the water-filled space, as solute size decreases, E tends toward unity. E > 1 reflects favorable solute interaction with the polymer chains. Here, however, the solute may be large enough that it is rejected from the smaller mesh size domains and, thus, interacts with only a small fraction of the total polymer strands in the network. Quantifying mesh size is clearly important in determining how gel matrices absorb solutes and how such impregnated gels function in a variety of applications ranging from drug delivery to separation processes. In many applications, solute size is comparable with the average mesh size of a hydrogel.1,14−17,20,21,27 In such cases, excluded-volume interactions of the solutes with the gel matrix are crucial. Because of the variety of applications, a large effort has been expended on obtaining partition coefficients in hydrogels, usually by back-extraction.11,20,21 Published work falls in two classes: small solutes, such as salts and small sugars, where E usually is close to unity, and larger solutes, such as polymers, surfactants, and proteins, whose sizes are comparable to the mesh size.1,14−17,20,21,27 In this second class, the enhancement factor can range from quite small to large values depending on whether or not there is strong attractive interaction with the gel polymer. Most investigated systems fall into the first class. In the second class, almost all studies consider high-water-content gels where φ is less than 0.1. Most work has been done on uncharged hydrogels.2,16,20,21 Equilibrium partition coefficients of fluorescein-labeled ficolls in pure agarose and agarose−dextran composite gels were measured as a function of gel composition and ficoll size.20 BSA and ficolls with different sizes were studied in agarose gels by Lazzarra and Deen.21 Tong and Anderson16 measured equilibrium partition coefficients of two proteins and two linear polymers as a function of polymer content in polyacrylamide (PA) gels. Far less is known about charged networks despite their widespread application.15,30−33 Russell and Carta15 measured partition coefficients of fluorescently labeled dextrans in anionic and cationic polyacrylamide gels as well as those for negatively charged proteins in anionic gels. They determined two characteristics of hydrogels from these measurements: polymer fiber radius and mesh size. This same group studied partitioning and transport of myoglobin in cationic, acrylamide-based hydrogels.31 Chromatography has also been used to investigate interactions of hydrogel networks and solute molecules. Multiple research groups have established that oppositely charged macromolecules are favorably adsorbed in gels at low ionic strengths.30,34−36 Uncharged macromolecules or macromolecules with the same charge as that of the gel, however, are nearly completely excluded when mesh size is close to that of these macromolecules.36 The effects of electrostatic interactions

2. EXPERIMENTAL SECTION 2.1. Chemicals. Sigma-Aldrich (St. Louis, MO) provided all monomers and chemicals used in synthesizing hydrogels: 2hydroxyethyl methacrylate (HEMA, Cat. No. 128635-500G), methacrylic acid (MAA, Cat. No. 155721-500G), ethylene glycol dimethacrylate (EGDMA, Cat. No. 335681-100 ML), thermo-initiator for free-radical polymerization, 4,4′-azobis(4-cyanovaleric acid) (decomposition temperature 69 °C, 11590-100G), and Sigmacote (SL2-100 ML) to hydrophobize the glass surfaces prior to polymerization. Sodium phosphate dibasic heptahydrate (Na2HPO4·7H2O, SX0715-1) and sodium phosphate monobasic monohydrate (NaH2PO4·H2O, SX0710-1), purchased from EMD Chemicals Inc. (Darmstadt, Germany), and sodium chloride (NaCl, S271-3), purchased from Fisher Scientific (Fair Lawn, NJ), were dissolved in distilled/deionized (DI) water to prepare a pH = 7.4 phosphate buffer saline solution (PBS: 0.15 M NaCl, 0.017 M Na2HPO4·7H2O, and 0.003 M NaH2PO4·H2O) used in all solution preparations. Fluorescein sodium salt (Na-fluorescein, F-6377) was purchased from SigmaAldrich (St. Louis, MO). Fluorescein isothiocyanate dextrans (FITCdextran4, MW 4000 g/mol; FITC-dextran10, MW 10 000 g/mol; FITC-dextran20, MW 20 000 g/mol) and fluorescein isothiocyanate ficoll (branched polysaccharide, MW ∼ 70 000 g/mol, FITC-Ficoll70) were purchased from TdBCons (Uppsala, Sweden). The supplied labeled ficoll demonstrated fast diffusion into the gels indicative of a broad molecular-weight distribution. The sample was accordingly purified by gel permeation chromatography on a GE Healthcare 9178


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ATKA Explorer system (Piscataway, NJ) equipped with a UV/vis detector set at 280 nm. 20 mg of FITC-Ficoll70 was dissolved in 4 mL of PBS solution and injected onto a Suphadex 75 10/300 column (Code No. 17-5174-01; ID No. 0716135) from GE Healthcare (Piscataway, NJ). The column was washed with identical PBS solution at 0.5 mL/min flow rate. The leading section of the main peak, corresponding to the higher-molecular-weight species, was collected to narrow the molecular-weight distribution. As outlined below, the bulk diffusion coefficient of this fraction in PBS solution was determined giving a hydrodynamic diameter of 18.5 nm. Fluorescein-labeled bovine serum (BSA, Alexa Fluor488 conjugate Fl-BSA, A13100) and avidin fluorescein conjugate (Fl-avidin, A821) were obtained from Invitrogen (Eugene, OR) and used as received. All experiments were performed at ambient temperature. 2.2. Gel Synthesis. 70 wt % HEMA/30 wt % MAA hydrogels were synthesized by simultaneous copolymerization and cross-linking of monomers with EGDMA as the cross-linking agent at 0−1 wt % in aqueous solution.2,39 In a typical reaction, HEMA and MAA (measured gravimetrically), EGDMA (measured volumetrically using a micropipet), and 0.5 wt % (of the total monomer content) 4,4′azobis(4-cyanovaleric acid) initiator (measured gravimetrically) were mixed. Then 30 wt % (of the total monomer content) DI water was added to the mixture. The reaction mixture was magnetically stirred until the initiator was fully dissolved. Nitrogen gas was then bubbled through the reaction mixture for 30 min to remove dissolved oxygen, followed by minimal degassing under vacuum at 40 kPa for 45 min with less than 2% change in HEMA/MAA composition. The degassed reaction mixture was slowly poured between two flat glass plates previously hydrophobized with Sigmacote and separated by a 1.5 mm spacer. Free-radical thermally initiated polymerizations took place in an oven whose temperature was raised from 70 to 100 °C over a 60 min period and then maintained at 100 °C for 60 min. The oven was then cooled to 50 °C and kept at this temperature for 24 h. The gel sheet was separated from the glass plates and placed in DI water to soak for 7 days to remove all unreacted monomers. DI water was refreshed daily during the soaking period. Hydrogel films (6 mm × 6 mm) were then cut and placed into scintillation vials filled with PBS for no less than 7 days to allow complete swelling. The buffer solution was changed at least once daily to ensure equilibration with the surrounding PBS solution. Since the carboxylic groups of the MAA monomer units dissociate above about pH 5.5, the gel network swollen in PBS is negatively charged. However, because the ionic strength of the PBS sets a Debye length of about 0.5 nm, network charges are effectively screened. The thickness of the swollen gel samples used for thermal gravimetric analysis (TGA), rheological, and two-photon confocal microscopy experiments was between 2 and 2.7 mm depending on the hydrogel cross-link density. For the rheological measurements, swollen hydrogel samples were hole-punched into 8-mm-diameter discs. For the two-photon confocal microscopy experiments, swollen hydrogel samples were cut into 4 mm × 4 mm squares, whereas thermal gravimetric analysis (TGA) experiments utilized 1 mm × 2 mm swollen rectangles. 2.3. Water Content. Water content was determined by thermal gravimetric analysis (TGA, Model 2950, TA Instruments, New Castle, DE). Equilibrium-swollen hydrogel were placed into open platinum pans, heated at 5 °C/min from 30 to 80 °C, and kept there for 100 min under a nitrogen atmosphere. In the last 15 min, sample mass remained constant, indicating total water evaporation from the polymer matrix. At least three repeat TGA measurements were performed for each gel sample. Figure 1 reports the water content of our gels as a function of crosslink density. Reproducibility of the measurements was within 1%. Hydration of the charged carboxylic groups in the MAA at pH 7.4 swells the HEMA/MAA gels to about 80% water content. As expected, increasing the density of cross-links decreases the water content, almost linearly. The range of water contents, however, is not large. Reproducibility of the measurements was within 1%. 2.4. Linear Shear Rheology. The average mesh size of each gel was determined from the zero-frequency shear storage modulus,

Figure 1. Water volume fraction, 1 − φ, versus cross-link density (wt % EGDMA) for 70 wt % HEMA/30 wt % MAA hydrogels in aqueous PBS solution measured by thermal gravimetric analysis. G′(0).40 Oscillatory shear rheometry in the linear viscoelastic region (LVE) was performed on a Physica MCR301 Rheometer (Anton-Paar, Ashland, VA). The rheometer was adapted by replacing the bottom plate with a solvent bath to maintain aqueous-solution saturation of the hydrogels. The top 8-mm-diameter circular plate was sandblasted, and the bottom solvent bath was roughened by hand sanding, each to prevent sample slippage during oscillation. All samples were first subjected to an amplitude sweep at 10 Hz to establish the linear viscoelastic range (LVE) where both the storage and loss modulus, G′ and G″, respectively, are independent of strain. In assessing the LVE range and in subsequent moduli measurements, the applied normal force must be high enough to ensure full contact between the gel sample and the measuring plate of the rheometer. Figure 2 graphs the shear storage modulus at zero frequency as a function of applied normal force at 0.02% strain and 0.02 Hz for the

Figure 2. Zero-frequency shear storage modulus G′(0) versus normal force FN for the 70 wt % HEMA/30 wt % MAA hydrogel in aqueous PBS solution with differing EGDMA cross-link densities expressed in wt %. Strain is 0.02%, and frequency is 0.02 Hz. measured range of cross-link density. At low normal force, G′(0) values decline almost linearly. In this region, however, results are not reproducible. At an applied normal force near 1 N (depending on the cross-link density), the gel and the measuring plate are apparently in full contact, and G′(0) asymptotes to a reproducible constant value that we identify as the storage modulus. Separate compression experiments to higher normal forces do not expel measurable water from the gel, indicating that at 1 N applied force the gel remains fully water saturated. Figure 3 shows the two moduli as a function of frequency for the HEMA/MAA gel at 0.25 wt % cross-link density. For frequencies less than 0.1 Hz, both moduli are safely within the zero-frequency limit. We adopt 0.02 Hz to specify zero-frequency values. G′(0) in Figure 3 is about 0.25 MPa. With a Poisson ratio of 1/2, the corresponding Young’s dilatational modulus is 3G′(0) or 0.75 MPa, commensurate with results reported for commercial soft contact lenses.41 The 70% 9179


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π 2D d [ln(A(t ) − A∞)] = − 2 0 dt L

(4)

where L is the total liquid column height in the diffusion cell (1.5 cm), D0 is the desired solute diffusion coefficient, and A∞ is the final absorbance reflecting a uniform solute concentration over the whole diffusion cell. The latter value was determined by mixing (sloshing) the cell solution and measuring the final absorbance. Figure 4 illustrates a typical plot of ln[A(t) − A∞] versus time for fluorescein-labeled dextran (4000 g/mol). The linearity demanded in

Figure 3. Shear storage modulus G′ (squares) and shear loss modulus G″ (circles) as a function of frequency for the 70 wt % HEMA/30 wt % MAA hydrogel in aqueous PBS solution with 0.25 wt % EGDMA cross-link density. Strain is 0.02%, and applied normal force is 1 N.

HEMA/30% MAA gel in PBS is purely elastic with little to no viscous component. To determine the average mesh size of the hydrogels, rubber elastic theory was adopted to describe the measured storage moduli:42−44 G′(0) = RTCchϕ1/3, where Cch is the molar chain density per dry volume of polymer. Peppas et al.2 specify the gel average mesh size as ⟨ξ⟩ = lc−c(2CnMch/Mr)1/2φ−1/3, where lc−c is the length of a covalent carbon−carbon bond in the backbone (0.154 nm), Cn is the Flory characteristic ratio or rigidity factor,42,44 which for HEMA−MAA gels equals 6.9,2 Mr is the molecular weight of a repeat unit, 112.7 g/mol for the 70% HEMA/30% MAA copolymer, and Mch is the average molecular weight of a polymer chain between two cross-links. Because Mch = ρ2/Cch where ρ2 is the density of the dry polymer (1070 kg/m3 for 70% HEMA/30% MAA), the average gel mesh size is available from measured zero-frequency storage modulus

⟨ξ⟩ = lc − c

2Cnρ2 RT M rG′(0)

Figure 4. Plot of ln(A − A∞) versus time from UV/vis absorbance for aqueous FITC-dex4 in the restrictive diffusion cell. The indicated slope gauges the bulk diffusion coefficient in aqueous PBS solution. Here, A is the optical absorbance at time t. eq 4 does not emerge until after about 3 days. Scatter in the data at very long times arises from the small differences in absorbance as equilibrium is approached. The slope of the data in the linear region gives the solute diffusion coefficient following eq 4. Table 1 reports the

Table 1. Properties of Solutes

φ−1/6 (3)

Equation 3 applies strictly to uncharged gels. Ilavsky et al.45−47 establish that for the sufficient electrostatic screening, as in our aqueous PBS solution, polymer strands in a polyelectrolyte gel are well approximated as random Gaussian chains. Therefore, eq 3 is valid for our gels. 2.5. Solute Size. To ascertain the hydrodynamic radii of the chosen solutes, we determine bulk dilute diffusion coefficients in a restricted diffusion cell48,49 using UV/vis absorption. From the measured diffusion coefficient, the hydrodynamic radius of the aqueous solute is ascertained by the Stokes−Einstein equation.18,19,50 The experimental setup is similar to that described by Stewart and Newman.49 A 4 mm wide UV quartz cuvette (path length 10 mm) was filled with aqueous solute in PBS to a level of 0.75 cm. Less-dense PBS solution (pH 7.4) was gently placed atop the more dense solute solution. Absorption measurements relative to a PBS-solution-filled reference cell were taken at 491 nm with an Ocean Optics spectrometer (Model ADC-1000, Dunedin, FL) using the UV/vis DH-2000 light source focused 1.5 mm above the bottom of the cuvette. Ocean Optics OOIBase32 Platinum software recorded the decay of absorbance as solute diffused into the aqueous solvent. Because of the small diffusion coefficients of the solutes studied, experiments lasted several days. Fluorescein-tagged solutes bleach upon prolonged exposure to light. To minimize this effect, the experimental setup included an automated shutter programmed to open only during a measurement interval via a custom BAS script file. Typical solute concentrations were in the range between 10−5 and 10−6 M where light absorbance, A, is linear with concentration. Newman and Chapman48 established that during the later period of solute mixing in the diffusion cell, the slope of absorptivity versus time is a constant

a

solute

MW [g/mol]

D0 × 107 [cm2/s]

2aS [nm] measda

2aS [nm] lit.

Na-fluorescein FITC-dextran4 FITC-dextran10

376 4 000 10 000

65.9 14.3 7.99

0.74 3.4 6.1

Fl-avidin Fl-BSA

68 000 66 500

6.9 6.14

7.05 7.96

FITC-dextran20 FITC-Ficoll70

20 000 70 000

5.27 2.63

0.7672 3.117 4.717 5.415 ∼673 7.2674 7.8614 7.221,15 6.717

9.3 18.5

Calculated from the Stokes−Einstein equation.

solute molecular weights, measured diffusion coefficients and their calculated Stokes−Einstein hydrodynamic diameters, 2aS, compared to available literature values. Agreement between literature and measurement is good. Table 1 demonstrates that the two proteins studied are more compact in solution compared to the linear polymers of smaller or comparable molecular weight. 2.6. Solute Partition Coefficients. Partition coefficients of the fluorescently labeled solutes in the hydrogels were determined directly using two-photon confocal microscopy.51−54 Advantages of twophoton fluorescence attenuation are excitation only in a very small volume (∼0.1 μm3) and minimal photobleaching, thereby permitting concentration-profile assessment. For our dilute solute concentrations, between 10−5 and 5 × 10−4 M, fluorescence intensity measured both in solution and in the gel was proportional to dye concentration.15 Thus, provided there is no interference to fluorescence excitation or emission by the gel structure, the partition coefficient is simply the ratio of dye 9180


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intensities in the gel and in the solution. Two-photon fluorescence attenuation also permits interrogation of spatial equilibration throughout the gel and of reversibility toward desorption. We used a Carl Zeiss (Jena, Germany) 510 LSM META NLO AxioImager confocal microscope equipped with a Spectra-Physics (Santa Clara, CA) MaiTai HP DeepSee laser for two-photon imaging at 780 nm. Fluorescence emission was collected with a Plan-Neofluar 10×/0.30 NA objective (Carl Zeiss GmbH) using a 500−550 nm emission filter. Prior to the partition coefficient measurement, all gels were soaked in the pertinent aqueous solute solution under magnetic stirring for at least 48 h at 400 rpm. Subsequently, a 1 mm thick layer of the aqueous solution in a small Petri dish was placed on the microscope platform and scanned in the vertical (z) direction with 2 μm steps to a depth of at least 500 μm. Afterward, a corresponding solute-equilibrated 4 mm × 4 mm, 2−2.7 mm thick gel sheet was placed on a microscope slide (VWR Micro Slides, 48300-047, VWR International, West Chester, PA), covered (Microscope Cover Glass, 12-541-B, Fisher Scientific, Fair Lawn, NJ), and placed on the microscope for scanning in the z direction at the same laser power and detector setting as those during scanning of the bulk-aqueous solute solution. During each experiment, background fluorescence intensity was recorded and subtracted from the solution and gel signals. Penetration depth of the two-photon technique in the gel matrix was ascertained as about 1 mm since, as demonstrated in Figure 5, intensities at deeper regions declined from that relative to the dye concentration at smaller penetration depths.

placing them into solute-free PBS solution under magnetic stirring. Figure 6 pictures fluorescent micrographs for desorption of Na-

Figure 6. Fluorescence micrographs upon desorption of Nafluorescein from a 70 wt % HEMA/30 wt % MAA hydrogel with 0.25 wt % cross-link density at different desorption times: 20 min, 40 min, and 1 h, imaged with two-photon confocal microscopy. fluorescein from the 70 wt % HEMA/30 wt % MAA hydrogel with 0.25 wt % cross-linking at different desorption times. At zero time, the gel is completely saturated with solute. As time elapses, the fluorescent intensity decreases from the center toward the two surfaces of the gel sheet as solute desorbs and diffuses toward the surrounding solution. After sufficient time, total desorption from the gel is observed, although complete desorption can require many days depending on the solute. The exception is Fl-avidin, whose desorption occurs only near the gel surface even after several weeks. Save possibly for Flavidin, the observations of uniform concentration profiles after equilibration, complete solute desorption into pure solvent, and measured partition coefficient that are independent of solute concentration confirm that for the solutes studied, eq 1 applies, and the partition coefficients are constant in dilute solution.

3. RESULTS 3.1. Gel Mesh Size. Figure 7 shows G′(0) measured on gels with varying cross-link densities. The zero-frequency storage

Figure 5. Fluorescence intensity versus position in a 70 wt % HEMA/ 30 wt % MAA hydrogel with 0.1 wt % cross-link density and equilibrated with Na-fluorescein. Figure 5 shows uniform fluorescent intensity in the first 1.3 mm depth from the upper surface of the gel sample. Reversing the gel sheet shows identical penetration behavior. Therefore, we collect solute intensities averaged only within the penetration depth. No less than three gel samples were measured; their intensities were averaged. Maximum error was less than 10% among the three samples. Measured partition coefficients were independent of solute concentration over the investigated concentration range.55 Figure 5 also confirms that gel equilibration is established within the 48 h of soaking because the solute concentration is uniform (within the penetration depth). For shorter equilibration times of a day or less, a profile with descending intensity is observed. Confirmation of equilibrium is not readily possible by other methods for determining partition coefficients, notably that of back-extraction. The constant-intensity region demonstrated in Figure 5 further confirms that the fluorescence-emission spectrum of our solute dyes is unaffected by interaction with the gel structure in agreement with Tong and Anderson for polyacrylamide gels.16 Fl-avidin, however, strongly attaches to the anionic-gel polymer chains, giving rise to adsorptive slowing of penetration into the gel. In this case, 48 h is not long enough for equilibration of the entire gel.56 Accordingly, we utilize the fluorescence intensity at the gel surface (within 50 μm from the surface) to calculate the partition coefficient. To establish that solute partitioning is reversible, desorption experiments were performed with solute-saturated gel samples by

Figure 7. Shear storage modulus G′(0) measured with oscillatory shear rheometry on 70 wt % HEMA/30 wt % MAA hydrogels for varying cross-link densities (wt % EGDMA). 0.02% strain and 0.02 Hz. Typical error bars are shown.

modulus increases by a factor of 4 over the range of cross-link densities studied. Even without added chemical cross-links, G′(0) is 0.2 MPa. Most likely, EGDMA impurity in the supplied HEMA monomer (HEMA Product Specification, Sigma-Aldrich, St. Louis, MO) initiated this cross-linking. Physical or trapped entanglements and even self-chemical cross-linking between polymer chains are alternate explanations. We performed time−temperature superposition measurements44,57,58 on the 0 wt % cross-linked gel to determine 9181


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whether physical cross-links are present in the structure (data not shown). No physical cross-linking was evident from these results. As shown by Caykara et al.59 for p(HEMA/MAA) hydrogels, self-cross-linking is also possible via ester-bond formation between the hydroxyl groups of HEMA and the carboxylic groups of the MAA copolymer. If chain density Cch varies linearly with the degree of cross-linking, G′(0) should increase linearly with cross-link density. Figure 7 confirms a linear relationship, except for the highest cross-link densities where trapped entanglements may contribute.60 Figure 8 graphs average mesh size measured from the zerofrequency storage modulus and eq 3 as a function of polymer

Figure 9. Measured enhancement factors of Na-fluorescein (filled squares), FITC-dextran4 (open diamonds), FITC-dextran10 (solid triangles), Fl-BSA (open squares), and FITC-dextran20 (filled circles) versus measured average mesh size of 70 wt % HEMA/30 wt % MAA hydrogels with varying cross-link densities. Typical error bars are shown. Lines correspond to extended Ogston theory with af = 2 nm. Solid lines correspond to filled symbols, and dashed lines correspond to open symbols.

Figure 8. Measured average mesh size (filled squares) as a function of polymer volume fraction expressed as (π/φ)1/2 exp(φ) erfc(φ1/2) for the 70% HEMA/30% MAA hydrogels of varying cross-link densities. Error bars are about the size of the data points. The abscissa expression and the straight lines arise from extended Ogston theory for two polymer-strand radii: aS = 1.5 and 2 nm.

volume fraction for 70 wt % HEMA/30 wt % MAA gels. The abscissa format and the straight lines in this figure follow from extended Ogston theory described below. Average mesh size increases from 3.6 to 8.3 nm while the polymer volume fraction increases minimally from 0.14 to 0.23. 3.2. Solute Partitioning. Figure 9 shows the measured partition coefficients (open and closed symbols) divided by water volume fraction (i.e., the enhancement factor) on a logarithmic scale against the measured average mesh size. Equilibrium partition coefficients of Na-fluorescein, FITCdextrans, and anionic Fl-BSA are reported in the 70% HEMA/ 30% MAA gels with 0.05, 0.1, 0.25, 0.5, and 1 wt % cross-link densities. Lines in this figure correspond to extended Ogston theory with no adjustable parameters. All partition coefficients correspond to enhancement factors much smaller than unity with larger solutes exhibiting progressively smaller enhancement factors. Enhancement factors rise with increasing mesh size, as expected. Likewise, Figure 10 reports measured partition coefficients as a function of polymer volume fraction for the solutes in Figure 9 but now in the classic manner on a semilogarithmic scale.16,31 Again lines on this figure are drawn according to extended Ogston theory for hard-sphere solutes described below with no adjustable parameters (i.e., the polymer-strand radius was established from the results in Figure 8). Measured partition coefficients rise exponentially with increasing water content. Our measured partition coefficients agree with those of Russell and Carta for FITC-dex10 in anionic polyacrylamide gel15 and with those reported by Lazzara and Deen for BSA in agarose

Figure 10. Measured enhancement factors of Na-fluorescein (filled squares), FITC-dextran4 (open diamonds), FITC-dextran10 (solid triangles), Fl-BSA (open squares), and FITC-dextran20 (filled circles) versus polymer volume fraction of 70 wt % HEMA/30 wt % MAA hydrogels with varying cross-link densities. Lines correspond to extended Ogston theory with af = 2 nm. Solid lines correspond to filled symbols, and dashed lines correspond to open symbols.

gel.21 However, our results are larger than those for BSA in a polyacrylamide gel.16 Note in Figures 9 and 10 that FITC-dextran20, with a hydrodynamic diameter of 9.3 nm, partitions into the 3.6 nm average mesh size gel. Clearly, large, interconnected liquid-filled voids must be present in the gel to permit this partitioning. For the largest solute studied, FITC-Ficoll70 with a hydrodynamic diameter of 18.5 nm, however, no partitioning was detected. Apparently, regions of liquid space of this size in the gel are minimal and/or are not physically accessible. As shown in Table 2, the partition coefficient of positively charged Fl-avidin was 23.9 in the 0.05 wt % cross-link density gel with the largest ⟨ξ⟩ = 8.3 nm. We note that Fl-avidin in the 8.3 nm mesh-size HEMA/MAA gel takes weeks to equilibrate. 9182


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Table 2. Role of Solute Charge on Partition Coefficient charge 2aS [nm] ka

Fl-BSA

FITC-dextran10

Fl-avidin

anionic 7.9 0.04

neutral 6.1 0.05

cationic 7.1 23.9

a

70 wt % HEMA/30 wt % MAA hydrogel of 0.05 wt % cross-link density.

Desorption is even slower. For this reason, we only pursued partitioning measurements in the least cross-linked gel. The molecular weight and Stokes diameter for Fl-avidin are about the same as those for Fl-BSA (see Table 1) whose comparative enhancement factor in the 0.05 wt % cross-linked gel is about 0.05. Because these two proteins are about equal in size, the large value of 23.9 for the partition coefficient of Fl-avidin is remarkable. Fl-avidin is a counterion for the anionic gel. Because of strong electrostatic screening in our PBS solvent, however, nonspecific Donnan uptake61 cannot explain such a high partition coefficient for Fl-avidin.

Figure 11. Ogston mesh size distribution from eq 5 at φ = 0.2 for three polymer-strand radii. Distributions are not normalized.

al.65 for agarose gels. We establish the polymer-chain radius from the data in Figure 8, although the range of water contents studied is limited. Thus, af is no longer an adjustable parameter when predicting measured partition coefficients. The Ogston distribution in eq 5 can be extended to account for solute excluded volume by replacing the liquid-space radius by r by r + aS, where aS is the solute hydrodynamic radius. Let g(r;af,aS) denote this extended distribution. The hard-sphere solute enhancement factor then follows as

4. THEORY 4.1. Subunity Enhancement Factors. Figures 9 and 10 demonstrate that partition coefficients for the listed solutes are small, as low as 0.001. Significant screening of the gel network charges by the PBS solvent eliminates the effect of electrostatic repulsion. Further, the dextran solutes are uncharged. Thus, the results in Figures 9 and 10 suggest that the studied solutes are partially excluded from the gels because of their size. Comparison of Table 1 and Figure 8 reveals that the hydrodynamic radii of all studied solutes, except sodium fluorescein, are near to or greater than the average mesh size of the HEMA/MAA gels. Nevertheless, these large solutes do penetrate the gels reversibly and reproducibly, even for the smallest 3.6 nm average mesh size gel and the largest 9.3 nm diameter solute. This result is explained by a mesh size distribution exhibiting liquid-filled voids that are large enough to allow solute access. Following others,14−16,20−22,33,62−64 we employ the theoretical mesh size distribution of Ogston for a random assembly of infinitely long fibers28 a f g0(r ; a f ) = 2φ(1 + r /a f ) exp[−φ(1 + r /a f )2 ]

∞

E HS ≡

π exp(φ) erfc φ

S

∞

∫0 g0(r ; a f ) dr

(7)

The meaning of EHS is the volume of liquid available to the solute in the gel divided by the total volume of space (i.e., total liquid volume in the gel). The numerator on the right of eq 7 accounts for complete rejection of solute in spaces of mesh sizes less than 2aS but also for the reduction of volume available to the solute (i.e., excluded volume) in water-filled spaces of mesh sizes greater than 2aS. Evaluation of the integrals in eq 7 gives the desired result E HS = exp{−4φ[(aS/a f )(1 + aS/a f )]}

(5)

(8)

Equation 8 pertains strictly to finite-size solute rejection from a gel where the only interaction between the solute and the polymer matrix arises from excluded volume. Also implicit in eq 8 is that all mesh sizes percolate the gel. For a point solute, eq 8 correctly reduces to the ideal partitioning limit of k = 1 − φ. As shown by the lines in Figure 9, eq 8, in conjunction with eq 6, predicts hard-sphere solute partitioning as a function of gel average mesh size. Likewise, lines in Figure 10 give the predicted hard-sphere solute enhancement factors as a function of polymer volume fraction from extended Ogston theory in eq 8. 4.2. Superunity Enhancement Factors. A large positive enhancement factor emerges for the aqueous counterion protein, FI-avidin. The likely explanation is specific adsorption of the cationic protein onto the anionic polymer chains. However, because avidin is also large compared to the gel mesh sizes, size exclusion remains with a hard-sphere enhancement factor similar to that of BSA. Hence, avidin is not expected to adsorb onto polymer chains incorporated in spaces of mesh sizes less than 2aS = 8 nm. To account for both size exclusion

where g0(r;af) dr is the volume fraction of water-filled spaces with radii between r and r + dr and af is the fiber radius, taken here as the polymer-strand radius including bound water. In the Ogston distribution, ξ = 2r. Figure 11 displays this distribution for φ = 0.2 and for three polymer-chain radii. Larger strand radii lead to broader distributions. As written, the distribution in eq 5 is not normalized. The average mesh size follows from eq 5 by definition ⟨ξ⟩ = af

∫a g (r ; a f , aS) dr

φ (6)

Thus, the Ogston distribution provides a self-consistent relationship between average mesh size and water content. Equation 6 justifies the form of the abscissa adopted in Figure 8. Two polymer-stand radii are considered in Figure 8 with a nominal value of af = 2 nm, giving reasonable agreement with experiment. This value is larger than physically expected but is within the range cited by Lazzara and Deen21 and Johnson et 9183


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ignoring a corresponding spatial distribution of those sizes is the assumption that all mesh sizes are accessible to the solutes. For the larger mesh size spaces, sample percolation may not be present in spite of polymer-strand thermal fluctuations. Finally, because our range of studied volume fractions is small, theory may not be accurately assessed. We consider all solutes as effective hard spheres characterized by their hydrodynamic radii. Flexible linear polymers, however, may alter their shape upon entering a gel network. Our measured solute diffusion coefficients in the HEMA/MAA gel scale with the bulk diffusion coefficients suggesting little configuration alteration.56 Further, Figures 9 and 10 and Table 1 indicate that the measured partition coefficients follow the expected size trend of the measured hydrodynamic radii. This observation also speaks against major solute shape change in the HEMA/MAA gels. Also FITC-Ficoll70, the largest of the solutes with a hydrodynamic diameter of 18.5 nm, does not enter the gels within the precision of our measurement. Figure 11 shows that this finding is consistent with the Ogston mesh size distribution. Average mesh sizes reported here are based on measured zero-frequency storage moduli interpreted by elastic rubber theory of random coils. For highly cross-linked gels, randomchain statistics may be suspect. Although our mesh sizes are physically reasonable, gel mesh sizes depend on the particular experimental method chosen, such as electrophoretic mobilities, 67 photon-correlation spectroscopy,68 chromatography,30,31,33 TEM of freeze-etched hydrogels,69 DSC,70 or mixed-solute exclusion.71 The counterion protein, avidin, exhibits a very large partition coefficient compared to the co-ion protein, BSA, each of about the same hydrodynamic size (see Tables 1 and 2). Thus, based on hard-sphere size alone, avidin should be partially rejected from the 70 wt % HEMA/30 wt % MAA gel with an enhancement factor of about 0.03. The measured enhancement factor, however, is ∼25. Application of eq 9 then gives a value of K = 5500 for Henry’s adsorption constant, indicating strong attractive interaction with the polymer matrix. This interaction is not due to a favorable Donnan electrostatic field in the gel because the ionic strength of the aqueous PBS solution effectively screens the chain charges. Nevertheless, screening does not preclude specific binding to charged sites along the polymer chain.

and specific adsorption, we assume that the solutes are dilute and follow Henry’s law for equilibrium adsorption on the polymer chains. Let n be the moles of adsorbed solute per polymer volume in the gel and let K be Henry’s adsorption constant (dimensionless) so that n = KEHSCbulk. The product EHSCbulk corresponds to the spatially averaged concentration of solute in the liquid volume of the gel. Accordingly, we express the overall enhancement factor including both size exclusion and specific adsorption as ⎞ ⎛ φ k K⎟ = E HS⎜1 + 1−φ 1−φ ⎠ ⎝

(9)

where EHS is the hard-sphere or size-exclusion enhancement factor, such as given by extended Ogston theory. Several points are salient. With no specific adsorption, the overall enhancement factor reduces to that of size exclusion only. For large solutes (relative to the average mesh size) greater-than-unity enhancement factors mean that the amount of specific adsorption exceeds the amount of rejection due to size exclusion. It is also possible to have small overall enhancement factors near unity, either above or below, that arise from a combination of positive specific adsorption and negative adsorption due to size rejection.

5. DISCUSSION Comparison of measured average mesh sizes and partition coefficients to extended Ogston theory shows only qualitative agreement in Figures 8−10. For the solutes in Figures 9 and 10, the partition coefficients are overpredicted for the smaller solutes (e.g., for Na-fluorescein) and underpredicted for the largest solutes (BSA and FITC-dextran20). Disagreement is especially apparent in the smallest 3.6 nm average mesh size gel (0.23 polymer volume fraction). Further, the apparent size of the polymer-strand radius even with possible attached bound water appears large for a HEMA/MAA gel. A number of reasons for the lack of agreement are clear. The Ogston mesh size distribution is well understood to apply only at low polymer content because fiber overlap is not accounted for.14−16,20−22,33,62−64 Our focus, however, is on larger solutes in lower water content gels. To apply the Ogston distribution to larger polymer content gels, Bosma and Wesselingh66 provided a simple correction for fiber overlap. Their expression for the resulting mesh size distribution is identical to that in eq 5 with φ replaced by ln[(1 − φ)−1]. Given this new distribution and following the development outlined in the Theory section, we find that the gel average mesh size in eq 6 and the expression for the hard-sphere enhancement factor in eq 8 also remain identical but again with φ replaced by ln[(1 − φ)−1]. Comparison of the so-extended Bosma−Wesselingh distribution to the data in Figures 8−10, however, provides no significant improvement over extended Ogston theory in eqs 6 and 8. This observation, in addition to the necessity for a large polymer-chain radius, apparently arises from an overestimate of excluded volume in g(r;af,aS), the extended Ogston liquid-space distribution. Although eq 5 is a consistent application of the Ogston distribution, overestimation of excluded volume requires a larger strand radius to coincide theory and data. Figures 9 and 10 also reveal that the Ogston mesh size distribution is too narrow as the measured partition coefficients in the smallest average mesh size gels are much larger than predicted. Implicit in employing a mesh size distribution while

6. CONCLUSIONS We measured hydrogel equilibrium partition coefficients of solutes where the solute diameter is comparable to or larger than the independently measured average mesh size of the gel. In such cases, solute/matrix interactions play a critical role. The hydrogels were 70 wt % HEMA/30 wt % MAA synthesized with cross-link densities ranging from 0 to 1 wt % and swollen in pH 7.4 PBS solution to a water content of about 80 wt % due to dissociation of the carboxylic groups on the MAA monomer units along the polymer chains. These compositions mimic soft-contact-lens materials with relatively high polymer contents and correspondingly small mesh sizes. Gel average mesh sizes were determined independently from oscillatory shear rheometry and Gaussian-chain rubber elastic theory. The average mesh size decreased from 8.3 to 3.6 nm while the corresponding polymer volume fraction increased from 0.14 to 0.23. Two-photon confocal microscopy quantified the partition coefficients of sodium fluorescein, fluorescently labeled dextrans, and two fluorescently labeled proteins with opposite 9184


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(3) D’Errico, G.; De Lellis, M.; Mangiapia, G.; Tedeschi, A.; Ortona, O.; Fusco, S.; Borzacchiello, A.; Ambrosio, L. Structural and mechanical properties of UV-photo-cross-linked poly(N-vinyl-2pyrrolidone) hydrogels. Biomacromolecules 2008, 9, 231−240. (4) Meyvis, T. K. L.; Stubbe, B. G.; Van Steenbergen, M. J.; Hennink, W. E.; De Smedt, S. C.; Demeester, J. A comparison between the use of dynamic mechanical analysis and oscillatory shear rheometry for the characterisation of hydrogels. Int. J. Pharm. 2002, 244, 163−168. (5) Anseth, K. S.; Bowman, C. N.; Brannon-Peppas, L. Mechanical properties of hydrogels and their experimental determination. Biomoterials 1996, 17, 1647−1657. (6) Baker, B. A.; Murff, R. L.; Milam, V. T. Tailoring the mechanical properties of polyacrylamide-based hydrogels. Polymer 2010, 51, 2207−2214. (7) Fornasiero, F.; Krull, F.; Prausnitz, J. M.; Radke, C. J. Steady-state diffusion of water through soft-contact-lens materials. Biomaterials 2005, 25, 5704−5716. (8) Wijmans, J. G.; Baker, R. W. The solution-diffusion model-A review. J. Membr. Sci. 1995, 107, 1−21. (9) Ratner, B. D.; Hoffman, A. S. Synthetic hydrogels for biomedical applications. In Hydrogels for Medical and Related Applications; Adrade, J., Ed.; American Chemical Society: Washington, DC, 1976; Vol. 31, pp 1−36. (10) Castillo, E. J.; Koenig, J. L.; Anderson, J. M. Protein adsorption on hydrogels: II. Reversible and irreversible interactions between lysozyme and soft contact lens surfaces. Biomaterials 1985, 6 (5), 338− 345. (11) Guan, L.; González-Jiménez, M. E.; Walowski, C.; Boushehri, A.; Prausnitz, J. M.; Radke, C. J. Permeability and partition coefficient of aqueous sodium chloride in soft contact lenses. J. Appl. Polym. Sci. 2011, 122, 1457−1471. (12) Luensmann, D.; Zhang, F.; Subbaraman, L.; Sheardown, H.; Jones, L. Localization of lysozyme sorption to conventional and silicone hydrogel contact lenses using confocal microscopy. Curr. Eye Res. 2009, 34, 683−697. (13) Peppas, N. A.; Langer, R. New challenges in biomaterials. Science 1994, 263, 1715−1720. (14) Pluen, A.; Netti, P. A.; Jain, R. K.; Berk, D. A. Diffusion of macromolecules in agarose gels: Comparison of linear and globular configurations. Biophys. J. 1999, 77, 542−552. (15) Russell, S. M.; Carta, G. Mesh size of charged polyacrylamide hydrogels from partitioning measurements. Ind. Eng. Chem. Res. 2005, 44, 8213−8217. (16) Tong, J.; Anderson, J. L. Partitioning and diffusion of proteins and linear polymers in polyacrylamide gels. Biophys. J. 1996, 70, 1505− 1513. (17) Shalviri, A.; Liu, Q.; Abdekhodaie, M. J.; Wu, X. Y. Novel modified starch−xanthan gum hydrogels for controlled drug delivery: Synthesis and characterization. Carbohydr. Polym. 2010, 79, 898−907. (18) Waters, D. J.; Frank, C. W. Hindered diffusion of oligosaccharides in high strength poly(ethylene glycol)/poly(acrylic acid) interpenetrating network hydrogels: Hydrodynamic vs. obstruction models. Polymer 2009, 50, 6331−6339. (19) Amsden, B. An obstruction-scaling model for diffusion in homogeneous hydrogels. Macromolecules 1999, 32, 874−879. (20) Kosto, K. B.; Panuganti, S.; Deen, W. M. Equilibrium partitioning of Ficoll in composite hydrogels. J. Colloid Interface Sci. 2004, 277, 404−409. (21) Lazzara, M. J.; Deen, W. M. Effects of concentration on the partitioning of macromolecule mixtures in agarose gels. J. Colloid Interface Sci. 2004, 272, 288−297. (22) White, J. A.; Deen, W. M. Equilibrium partitioning of flexible macromolecules in fibrous membranes and gels. Macromolecules 2000, 33, 8504−8511. (23) deGennes, P. G. Dynamics of entangled polymer solutions. II. Inclusion of hydrodynamic interactions. Macromolecules 1976, 9, 594− 598. (24) deGennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.

charge: anionic bovine serum albumin and cationic avidin. Solute sizes were determined from independent measurement of the bulk aqueous diffusion coefficients in a restricted diffusion cell and Stokes−Einstein theory. For the large hydrophilic solutes studied here, measured partition coefficients exhibit considerable size exclusion in concert with the size trend of the solutes’ measured bulk-solution hydrodynamic radii. To describe solute partitioning for solutes that do not specifically interact with the anionic gel, an extended Ogston mesh size distribution is proposed to account for excludedvolume interactions. Sizes of all studied solutes, except sodium fluorescein, are near to or greater than the average mesh size of the gels. Nevertheless, large solutes up to 9.3 nm in diameter do penetrate even the smallest 3.6 nm mesh size gels, however, with very low distribution coefficients. Large-solute penetration is attributed to a mesh size distribution that exhibits some liquid-filled voids large enough to allow solute access. The counterion protein, avidin, exhibits a very large partition coefficient compared to that of the co-ion protein, BSA, although both have nearly the same hydrodynamic size. In the 0.05 wt % cross-link density gel with the largest average mesh size of 8.3 nm, the enhancement factor of BSA is about 0.03, whereas that of avidin is ∼25. This large number is due to specific adsorption of the cationic protein on the anionic polymer chains. An extended Ogston mesh size distribution, when applied with no adjustable parameters, shows qualitative agreement with experimental partition coefficients for solutes that do not specifically interact with the polymer strands. The Ogston mesh size distribution is well understood to apply only at low polymer content because fiber overlap is not taken into account. Adoption of the overlap-corrected Bosma/Wesselingh mesh size distribution provides little improvement. Our results also reveal that the extended Ogston mesh size distribution is too narrow as the measured partition coefficients in the smallest average mesh size gels are much larger than those predicted. Because the polymer volume fraction range of the studied hydrogels is small, however, extended Ogston theory may not be accurately assessed. To attain quantitative prediction of partition coefficients especially for large solutes in lower water content gels, there is a clear need for improved understanding of mesh size and spatial distributions.

■

AUTHOR INFORMATION

Corresponding Author

*Tel +1 510 642 5204; Fax +1 510 642 4778; e-mail radke@ berkeley.edu. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Alcon Laboratories for financial support. We also thank Joshua Tan and Kevin Yeh for performing the restricteddiffusion measurements and Jerome Fox for help in gel chromatography.

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REFERENCES

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