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J. Phys. Chem. B 2001, 105, 4572-4576
A Possible Mechanism for the Substrate Effect on Hydrogel Formation Jian Ping Gong, Akishige Kii, Jan Xu,† Yoshihiko Hattori, and Yoshihito Osada* DiVision of Biological Sciences, Graduate School of Science, Hokkaido UniVersity, Sapporo 060-0810, Japan ReceiVed: September 13, 2000; In Final Form: February 26, 2001
A possible mechanism for the formation of the substrate-induced interface during the gelation, which leads to heterogeneous polymerization, is proposed. According to this mechanism, a surface polymer deplete layer would be formed during the polymerization due to the high interfacial energy between the hydrophobic substrate and the aqueous polymerizing solution. When the polymer concentration attains the critical value at which an extensive entanglement occurs, and/or a chemical network forms, the thickness of the deplete layer increases dramatically to a macroscopic level to give an interface, due to the weak elasticity of the network. Theoretical model describing relations between the substrate surface tension, the increment in the surface tension of the solution upon polymerization, and the predicted position of the interface show a resonable agreement with the experimental data.
1. Introduction In the proceeding papers, we reported that the surface properties of hydrogels strongly depend on the surface nature of the substrates on which the gel is synthesized.1,2 To study the gel formation process, a novel method, electronic speckle pattern interferometry (ESPI), was first established to spatially and temporally monitor the entire polymerization and gelation between a pair of separated substrates: one is glass and the other is a hydrophobic plate.3 It was found that heterogeneous polymerization occurs on the hydrophobic surfaces. The substrate effect on the polymerization has been observed in a wide variety of hydrophilic vinyl-monomers, such as 2-acrylamide2-methyl-1-propanesulfonic acid (AMPS) and its sodium salt, the sodium salt of styrene sulfonate (NaSS), acrylic acid (AA), acrylamide (AAm), and N,N′-dimethyl acrylamide (DMAAm) in water.4 This kind of heterogeneous polymerization has been confirmed on various hydrophobic substrates, such as poly(tetrafluoroethylene) (Teflon), polyethylene (PE), polypropylene (PP), polystyrene(PS), and poly(vinyl chloride) (PVC) but does not occur on hydrophilic substrates, such as glass, sapphire, silicon, and mica. A correlation between the substrate effect and the surface tension of the hydrophobic surface was observed: The lower the surface tension of the substrate, the larger or bigger the heterogeneity formed in the hydrogel.4 Thus, the substrate effect might be associated with the interfacial energy change in the course of the polymerization as well as the network formation. By ESPI, it was found that the polymerization occurs homogeneously at the initial stage of reaction without any notable macroscopic effect of substrate. However, when the polymer concentration exceeded a critical value, for example, 0.1-0.2 M, an interface between the low viscous region facing the hydrophobic surface and the high viscous region facing the bulk suddenly appears parallel to the hydrophobic substrate in the vinicity of the substrate. The interface moves toward the bulk at first, and then retraces back to the hydrophobic surface, and finally disappears. The appearance of the interface accompanys the auto-accleration of the polymerization in the bulk region, which leads to the monomer diffusion from the surface region to the bulk region. The autoacceleration is due to the † On leave from the Institute of Chemistry, Chinese Academy of Sciences, China.
enhanced monomer consumption in that region and results in the inhomogeneous network structure formation in a range of several millimeters in thickness in the hydrogel. The specific features characterizing the effect of the hydrophobic substrate on the gel formation are summerized as follows. (1) The interface appears at a certain “critical time” (or critical concentration) leading to the heterogeneous polymerization. (2) The appearance of the interface on the surface occurs almost simultaneously with the auto-acceleration in the bulk region. (3) The interface moves toward the bulk during the reaction at first, retraces back to the hydrophobic surface, and then disappears when the polymerization is complete. The maximum distance of the interface from the hydrophobic surface attains a range of several millimeters. (4) A correlation between the surface tension of the substrate and the maximum distance of the interface to the hydrophobic substrate is observed. The lower the surface tensions of the substrates, the further the interface travels from the hydrophobic surface, and the stronger the heterogeneity of the hydrogel formed. (5) The interface appears even in the absence of a crosslinking agent if the polymerization is carried out under a sufficiently high monomer concentration. (6) The interface is substantially suppressed when the polymerization is carried out in ethanol. No interface appears when the hydrophilic monomer is copolymerized with hydrophobic monomers in ethanol. (7) The interface appears in various hydrophilic monomer systems in water, but does not appear for the cross-linking reaction of prepolymers in water. The appearance of the interface might be associated with the high interfacial energy between the hydrophobic substrate and the hydrophilic polymer and/or gel during the polymerization process. The interface in a range of millimeters from the hydrophobic surface is, however, too thick to be explained in terms of the deplete layer of polymer solution. According to the thermodynamic principle, the thickness of a polymer deplete layer on a solid surface should not exceed an order of the length of a polymer radius for a polymer solution, which could not be observed in a macroscopic level. In this paper, a possible mechanism for the formation of the substrate-induced interface has been proposed and theoretical
10.1021/jp003244e CCC: $20.00 © 2001 American Chemical Society Published on Web 05/01/2001
Substrate Effect on Hydrogel Formation
J. Phys. Chem. B, Vol. 105, No. 20, 2001 4573
relations describing the position of the interface have been derived as a function of the substrate surface tension and of the increment in the surface tension of the solution upon polymerization. The theoretical results are compared with the experimental data and the validity of the essential feature of the model is discussed. 2. Experimental Section Surface Tension Measurement. Solution. The surface tension measurements of monomer and polymer solutions were performed at 20 °C using the drop weight method.5 The droplet lifetime of all samples is 5-10 s. The data were calculated using the correction factors of Harkins and Brown.6 Gel. The surface tension of the gels was determined by using the method developed by Andrade et al.7 The contact angle measurement was performed on the equilibrately swollen hydrogel by the sessile drop-goniometer method using laboratory made equipment. Air and n-hexane were used as probes. The volumes of the liquid drops were controlled between 3 and 5 µL in order to keep a partial sphere. To obtain the equilibrium contact angles, the measurements were carried out 30 min after the delivery of the probe fluids onto the gel surface. The surface tension of the gel was calculated by combining the harmonic mean equation with Young’s equation for water-water immiscible fluids-gel system.7 Polymerization. The in situ measurement of the polymerization was the same as described in previous works.3,4 The polymerization was carried out with an aqueous solution of prescribed 2-acrylamide-2-methyl-1-propanesulfonic acid (AMPS) concentration (0.2, 0.75, and 1.0 M), 0.003 M 2-oxoglutaric acid as the ultraviolet initiator, and with and without 0.05 M N,N′-methylenebisacrylamide (MBAA) as the cross-linking agent. The solution was poured into the reaction cell and purged with nitrogen for 40 min. The cell was a cuvette with 200 × 20 mm in its cross-section area and with 5 mm in the light path. The internal surfaces at two sides of the cell were a Teflon plate and the glass wall of the cuvette, respectively. The distance between the Teflon surface and the glass wall was 16 mm. Other polymerization conditions as well as the methods for the in situ monitoring of the polymerization using ESPI were the same as those described in previous papers.3,4 3. Modeling If the appearance of the interface in the vicinity of the hydrophobic substrate surface could be associated with the increased interfacial tension between the hydrophobic substrate and the polymer solution/network formed during the polymerization, we need discuss how the interfacial tension increases during the polymerization. The surface tension of the polymerizing solution, γL, as the first approximation, can be expressed as
γL ) (1 - φ0)γW + (φ0 - φp)γm + φpγp
(1)
Here, γW, γm, and γp are the surface tension of water, monomer, and polymer/or gel, respectively. φ0 is the initial volume fraction of the monomer and φp is the volume fraction of the polymer at a certain reaction time, respectively. We have neglected the contributions of the cross-linking agent and the UV initiator to the surface tension in order to simplify the problem. The surface tension of the polymerizing solution can be written in a form with two terms
γL ) γL0 + φp(γp - γm)
(2)
where the first term
γL0 ) γW - φ0(γW - γm)
(3)
does not change during the polymerization and the second term would increase upon the polymerization if polymer or gel has a higher surface tension than monomer. The interfacial tension between a polymer solution and a solid surface itself has been the object of many studies. For simplicity, we use the geometric mean rule proposed by Girifalco and Good8 to express the interfacial tension 1/2
γSL ) γS + γL - 2Φγs
1/2
γL
(4)
where γSL is the interfacial tension between the polymer solution and the solid surface and γS is the surface tension of the substrate. The parameter Φ is a constant depending on the molecular structure, and it is unity if molecular diameters obey a geometric mean law. As the value of Φ would not qualitatively affect our following discussion, we set Φ ) 1. So that, upon polymerization,
dγSL 1/2 1/2 ) (γp - γm)(1 - γS /γL ) dφp
(5)
The obtained relation indicates that the interfacial tension increases upon polymerization if γp > γm and γL > γS are satisfied simultaneously. If γL changes slowly with φp, the total interfacial free energy change at a polymer volume fraction of φp is 1/2
1/2
∆γSL ≈ φp(γp - γm)(1 - γS /γL )
(6)
According to thermodynamic principles, a polymer deplete layer with a thickness of ξ would form at the interface to eliminate the higher interfacial tension and minimize the Gibbs free energy of the system. ξ is determined by the following relation9
∆γSL = Πξ
(7)
here Π is the osmotic pressure of the polymer solution. The above equation means that the increased interfacial energy equals the work done against the osmotic pressure, as a result of the entropy penalty. For a semidilute polymer solution, ξ is the correlation length of the solution, at most several hundred angstroms and does not appear on the macroscopic level. This explains why we could neither observe any macroscopic interface formation nor spatial distribution in kinetics at the initial stage of polymerization. However, when the polymerization proceeded further, the macromolecules began to form entanglements with each other or even to form a permanent cross-linked network by the presence of the cross-linking agent. At this stage, the entangled polymer, or the chemically crosslinked network, would be expelled from the surface by a distance d, which, we suppose, is determined by
∆γSL = Gd
(8)
where G is the compressive elastic modulus of the entangled or the cross-linked polymer in solution. Since the value of G of a weakly entangled or weakly cross-linked network is much lower than that of the osmotic pressure of a polymer solution, we can expect a very large value of d, probably macroscopically
4574 J. Phys. Chem. B, Vol. 105, No. 20, 2001
Gong et al.
Figure 1. Surface tensions of aqueous solutions of monomer, polymer, and hydrogels. (a) AMPS, (b) AA. (]) monomer, (0) polymer, (O) gel. Temp: 20 °C.
detectable as observed in the preceding papers. Combining eq 6 and 8, we have 1/2
d ≈ φp(γp - γm)(1 - γ1/2 S /γL )/G
(9)
4. Discussion (1) Estimating the Order of d. Value of ∆γSL. As has been predicted in eq 9, the formation of the interface should occur only in a system where the monomer solution has a lower surface tension than that of the polymer solution or the gels. The surface tension of a hydrogel has not been well studied. A few reports on the measuring of surface tension exit.7,10-12 According to Andrade et al., the surface tension of a hydrogel is neither sensitive to the water content (or polymer concentration) nor to the structure of the gel. It almost has the same value as that of water (γw ) 73 mN/m at 20 °C) even though the surface tension of a monomer solution or a polymer solution strongly depends on the concentration.6 We have measured the concentration dependencies of the surface tension of the AMPS (monomer) solution, the PAMPS solution, and the PAMPS gel in equilibrate swollen state according to the method proposed by Andrade et al.7 The results are shown in Figure 1a. As shown in the figure, the surface tension of the AMPS monomer solution decreases with the increase in the concentration, which may result in a preferential adsorption of the molecules at the hydrophobic surface. In contrast with that, the PAMPS solution increases the surface tension with the increase in the concentration, which results in a negative interfacial adsorption. The result for the PAMPS solution is quite unique since the surface tension of a polymer solution usually decreases in a similar manner with its monomer solution. The surface tension of the PAMPS gel, as Andrade et al. indicated,7 is not sensitively dependent on the concentration but has a value lower than water. From Figure 1a, the difference in the surface tension between the polymer solution and the monomer solution is about 10mN/m in the discussing concentration (c.a. 0.2 M); namely, φp(γp - γm) ≈ 10 mN/m. Using γS ) 23.9 mN/m at 20 °C for the Teflon substrate13 and γL0 ≈ 60 mN/m from Figure 1, ∆γSL is estimated to be 4 mN/m from eq 6. To confirm that the substrate effect is a general feature whenever a monomer that has a lower surface tension than its
Figure 2. Maximum distance of the interface to the hydrophobic substrate surface, dmax, as a function of the square root of the substrate surface tension, γ1/2 s . (O) AMPS, gel (0) DMAAm. The straight lines are the least-squares fits of the experimental data that are replotted from Figure 9 of ref 4. According to eq 9, the intercepts of the lines at the horizontal axis give the values of γ1/2 L .
solvated polymer or gel is polymerized in water, the concentration dependence of the surface tension for AA solution, PAA solution, and PAA gel were also measured and the results are shown in Figure 1b. As shown in Figure 1b, the PAA solution and PAA gel have much higher surface tensions in comparison with that of AA monomer solution, and the surface tension of PAA gel also shows a constant value in the investigated concentration range. From Figure 1b, we can get φp(γp - γm) ≈ 10mN/m. ∆γSL is estimated to be 4mN/m which is almost the same as that of AMPS system. Therefore, although the solvated PAMPS solution increases its surface tension with the increase in the concentration, the phenomenon may be explained in a general manner. G at Entanglement Concentration. According to the model, the macroscopic surface polymer deplete layer to give an interface appears above the polymer entanglement concentration, where the solution osmotic pressure decreases and where the elastic modulus of the entangled network or gel becomes important. Now we estimate the magnitude of the elastic modulus G at the entanglement concentration. According to the scaling theory, the concentration of the polyelectrolyte solution at which the entanglement occurs is14
Ce ≈ n(B/b3)6/5 (2ACs)3/5N-4/5
(12)
Here, N is the degree of polymerization, B is a value that depends on the solvent quality, and A is the number of monomers between charges; b is the monomer length, and n is a constant having a value of 5 < n < 10. By putting A ) 1, B ) 1, b ) 3 × 10-10 m, and Cs ) 0.85 M, N ) 103, we obtain Ce ≈ 0.15-0.3 M for 5 < n < 10. This entanglement concentration is well in agreement with the concentration of 0.1-0.2 M at which the interface appeared and the autoacceleration began, as shown in Figure 6 of ref 4. The elastic modulus of the entangled polymer solution is given as follows if the salt concentration Cs is much higher than that of the polymer C.14
G ≈ kTb-3n-2(Cb3)3/2 B-3/2(1 + 2ACs/C)-3/4
(13)
Here, k is the Boltzmann constant and T is the absolute temperature. At the entanglement concentration, G ≈ 14.8 Pa - 3.7 Pa. If we choose the value of G )10 Pa, and ∆γSL ) 5 mN/m, we obtain d ) 0.5 mm from eq 9, indicating that the position of the interface is far enough to be observed macro-
Substrate Effect on Hydrogel Formation
Figure 3. Time profiles of the change in the interface position for the polymerization of AMPS with various monomer concentration in the presence (a) and absence (b) of the cross-linking agent. AMPS: (]) 0.2 M, (0) 0.75 M, (O) 1.0 M. MBAA: 0.05 M. UV initiator: 0.003 M. Substrates: Teflon and glass.
Figure 4. Schematic illustration of the interface formation of an entangled polymer solution or chemically cross-linked gel on the hydrophobic surface.
scopically from the hydrophobic surface. This result suggests that the substrate effect might appear at the concentration where the entanglement or cross-linking starts. (2) Effect of Substrate Surface Tension. According to eq 9, the interface position, d, would linearly change with the square root of substrate surface tension γ1/2 s . dmax, the largest distance between the interface to the substrate surface observed experimentally, is shown in Figure 2 as a function of γ1/2 for the s polymerization of AMPS and dimethylacrylamide (DMAAm). The linearity of AMPS is poor but that of DMAAm is reasonably good. According to eq 9, the interception of the straight 1/2 line at the horizontal axis gives out the value of γL , and γL is found as 57.7 mN/m for PAMPS and 48.5 mN/m for PDMAAm, which are lower than the experimental values in Figure 1.
J. Phys. Chem. B, Vol. 105, No. 20, 2001 4575 Equation 9 explains also why any suppression of polymerization does not occur at the glass surface. For a glass surface, the surface tension of the substrate is larger than the polymerizing solution, i.e., γs g γL, and therefore, an extra adsorption layer of polymer might exit at the surface. Thus, the interfacial energy does not increase with polymerization, and no interface should form. (3) Effect of Solvent. Since the organic solvent has a lower surface tension than that of water, this will lead to a lower value of γL of the mixing solution, as shown by eq 1. Therefore, the substrate effect should be suppressed or even disappear when the surface tension of the solution is as low as that of the hydrophobic substrate since γL ≈ γs leads to ∆γSL ≈ 0. This is in good agreement with the fact that the distance between the Teflon surface and the interface is decreased when the PAA is prepared in ethanol (Figure 8, ref 4). No appearance of the interface in the presence of a few percent of hydrophobic monomers, such as styrene and 2,2,2-trifluorenethyl acrylate (TFEA)4 can be explained by the same reasoning. (4) Effect of Monomer Concentration. According to the proposed model, the interface is formed above a certain polymer concentration at which entanglement occurs, and no interface should be observed if the polymerization is carried out in a dilute solution. Therefore, the effect of monomer concentration is studied by changing the AMPS concentration from 0.1 to 1.0 M, keeping the amount of MBAA and initiator constant. The interface appears when the monomer concentration is not less than 0.2 M and no interface is observed at 0.1 M monomer concentration. This coincides well with the previous experimental fact that the interface is formed at a polymer conversion of 10-20% when monomer concentration is 1 M. Figure 3a shows the time profiles of the interface position for various monomer concentration. We could not find a clear correlation between the position of the interface and the monomer concentration. However, the time profiles at a higher concentrations (0.75 M, 1.0 M) are quite different than those at a lower concentration (0.2 M). The interface moves toward the bulk at first, and then, returns back to the Teflon surface at a high concentration, whereas the interface never returns at a low concentration. (5) Effect of Cross Linkage. Since the elastic modulus G increases gradually with the progress of the polymerization, due to increased physical entanglement and the chemical cross linking, the position of the interface should change with the cross-linking reaction. It should be noted that entanglement concentration for the occurrence of the interface and the autoacceleration is much lower than that of the percolational sol-gel transition where the molecular weight of the chemically cross-linked network goes to infinity. Above the threshold, the elastic modulus increases exponentially with the relative distance to the threshold15
( )
Gs ) G0
p - pc pc
t
(14)
Here p is the probability for one chemical function to have reacted and pc is that at the sol-gel transition point. The exponent t = 2.7 in Rouse model. Gerard Hild et al. showed that in the early stages of the polymerization of vinyl monomer in the presence of a small amount of divinyl monomer, most of the bifunctional monomer yields pendant double bonds.16 The reactivity of these pendant unsaturations is far lower than that of the monomers involved, consequently the pendant double bonds are still numerously left, even at the sol-gel transition
4576 J. Phys. Chem. B, Vol. 105, No. 20, 2001 point. As these double bonds are slowly consumed in the final stages of the reaction (when not much monomer is left over), the network becomes tighter; and its modulus increases slowly with reaction time. The slow increase in the conversion rate of polymerization at the later stage of the reaction corresponds to the slow increase in the (p - pc)/pc, which brings about the increase in the modulus of the gel. The increase in G would allow the interface to move back to the hydrophobic substrate surface, as indicated by eq 9. This explains why we observe that d decreases with the acceleration until it is finished. At the end of the reaction, the elastic modulus of the gel reaches a value as high as G ≈ 103 Pa, therefore d ≈ 5 µm. This explains why the light line showing the position of the interface moves toward the hydrophobic surface and finally stops or disappears on the surface. In the case of a polyelectrolyte gel, such as PAMPS, the consumption of monomer weakens the screening of electrostatic repulsion of the polyelectrolyte chain and also gives an increased G. Equation 9 also suggests that a neutral gel should show a larger distance of the interface than that of a polyelectrolyte gel even if they have the same value of ∆γSL. This is because a neutral gel usually has a smaller elastic modulus G than a charged gel due to its lower osmotic pressure. This explains why a stronger interface is observed in AA system (Figure 8, ref 4), in which PAA dissociates less than a few percent of carboxyl, despite that ∆γSL of the AA system is the same as that of AMPS system. The significant substrate effect in DMAAm, as shown in Figures 8 and 9 of ref 4 can be explained by the same reasoning. According to the proposed mechanism, the substrate effect should also occur even in the absence of the cross-linking agent if the polymer exceeds the entanglement concentration. However, the return of the interface back to the hydrophobic surface should be less significant since there is no chemical cross-linking process to increase the modulus G. A comparable study with and without the cross-linking agent, MBAA, was carried out and the time profiles of the interface position in the course of polymerization in the absence of MBAA for various monomer concentrations is shown in Figure 3b. As shown in Figure 3, in the presence of a cross-linking agent, the interface first moves toward the bulk, and then returns back to the hydrophobic surface for the increased monomer concentration (0.75 and 1.0 M). In the absence of the cross-linking agent, the interface appears simultaneously with the auto-acceleration, which is the same as that with the cross-linking agent, but it hardly returns to the hydrophobic surface with the progress of the reaction even for a high monomer concentration (0.75 M). This fact suggests that the elastic modulus of the linear polymer does not substantially increase with the progress of the polymerization to push the interface back to the hydrophobic surface, which supports the relation given by eq 9. 5. A Possible Mechanism Based on the above results, the mechanism for the appearance of the interface concurrent with the inhomogeneous gelation at the hydrophobic surface could be described as follows (Figure 4): At the initial stage of reaction, the polymerization proceeds homogeneously. The increase in the interfacial energy due to polymerization leads to the formation of the surface deplete layer of polymer with a thickness of several hundred angstroms, as a result of the balance between the increased interfacial energy and the work done against the osmotic pressure of the solution. When the polymer concentration increases high enough to make the entanglement, or the chemically cross-linked network, the
Gong et al. force required to repel the networked polymer chains from the hydrophobic substrate exceeds the elastic modulus of the network, which is much smaller than the osmotic pressure of a semidilute solution. Thus, the distance being repelled increases to a value of macroscopic order, and we observe an interface in the region near the hydrophobic substrate (Figure 4a). Therefore, the interface is the boundary between a low polymer concentration region where the polymer chains are not in contact with each other (solution region) and a high polymer concentration region where the polymer chains are entangled or chemically cross linked (gel region). With the progress of the polymerization, the elastic modulus of the gel region increases and gradually pushes the interface back to the hydrophobic surface (Figure 4b). Once the interface is formed, the polymerization in the high concentration region is accelerated due to the suppression of radical termination between growing chains by the so-called “gel effect”. The quicker consumption of the monomer in this region in turn results in a gradient in the monomer concentration, and leads to the diffusion of the monomer to this region from the low concentration region near the hydrophobic surface. This gives rise to the spatial inhomogeneous distribution of polymer networks. However, the proposed mechanism is in contradiction with the experimental phenomena in that no substrate effect occurs for the cross-linking reaction of an aqueous polymer solution. Also, no interface appears on the Teflon surface immersed in a concentrated polymer solution even for a prolonged time. A diffusion-coupled reaction model might be necessary in understanding the substrate effect on hydrogel formation. Acknowledgment. This research was supported by Grantin-Aid for the Specially Promoted Research Project “Construction of Biomimetic Moving System Using Polymer Gels” from the Ministry of Education, Science, Culture and Sports, Japan. The authors sincerely thank Mr. T. Ohta of the glassware shop and Mr. Y. Hirata of the machine shop, of the Electronic Research Institute, Hokkaido University for their help in making the special reaction cells for us. References and Notes (1) Narita, T.; Hirai, A.; Xu, J.; Gong, J. P.; Osada, Y. Biomacromolecules 2000, 1, 162. (2) Gong, J. P.; Kagata, G.; Kurokawa, T.; Narita, T.; Osada, Y.; Nishimura G.; and Kinjo, M. J. Am. Chem. Soc. 2001, in press. (3) Zhang, X. M.; Xu, J.; Okawa, K.; Katsuyama, Y.; Gong, J. P.; Osada, Y.; Chen, K. S. J. Phys. Chem. B 1999, 103, 2888. (4) Kii, A.; Xu, J.; Gong, J.; Osada, Y.; Zhang, X. J. Phys. Chem. B 2001, 105, 4565. (5) Matijevic, E., Ed. Surface and Colloid; Wiley-Inter Science: New York, 1969; Vol. 1, p 135. (6) Harkins, W. D.; Brown, F. E. J. Am. Chem. Soc. 1919, 41, 499. (7) Andrade, J. D.; Ma, S. M.; King, R. N.; Gregonis, D. E. J. Colloid Interface Sci. 1979, 72, 488. (8) Girifalco, L. A. and Good, R. J. J. Phys. Chem. 1957, 61, 904. (9) de Gennes, P. G. Scaling Concept in Polymer Physics; Cornell University Press: Ithaca, NY; 1979. (10) Szabo, D.; Akiyoshi, S.; Matsunaga, T.; Gong, J. P.; Osada, Y.; Zrinyi, M. J. Chem. Phys. 2000, 113, 8253. (11) Nakamura, T; Hattori, M.; Kawasaki, H.; Miyamoto, K.; Tokita, M.; Komai, T. J. Phys. ReV. E 1996, 54, 1663. (12) Good, R. J.; Kotsidas, E. D. J. Adhesion 1979, 10, 17. (13) Billmeyer Jr., F. W. Textbook of Polymer Science; John Wiley & Sons, Inc.: New York, 1962. (14) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Macromolecules 1995, 28, 1859. (15) Physical Properties of Polymer Gels; Addad, J. P. C., John Wiley & Sons: New York, 1996. (16) Hild, G.; Rempp, P. Pure Appl. Chem. 1981, 53, 1541.
