Functionalized Microgel Swelling: Comparing Theory and Experiment

The variable ie is the effective degree of ionization of the polymer chains, ... of the average charge separation distance along the individual polyme...
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J. Phys. Chem. B 2007, 111, 11895-11906

11895

ARTICLES Functionalized Microgel Swelling: Comparing Theory and Experiment Todd Hoare* and Robert Pelton† Department of Chemical Engineering, McMaster UniVersity, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7 ReceiVed: March 25, 2007; In Final Form: July 25, 2007

A comprehensive gel swelling model accounting for the effects of added salt, counterion/polyelectrolyte charge condensation, inter-cross-link chain length distribution, polyelectrolyte chain stiffness, and direct chargecharge repulsion between fixed polymer network charges has been applied to predict water fraction profiles in -COOH-functionalized microgels based on poly(N-isopropylacrylamide). The model can successfully order the microgels according to their rheologically measured water fractions and explains key differences in observed microgel swelling according to the different functional group and cross-linker distributions in the microgels. The cross-linking efficiency is used as an adjustable variable to match the magnitude of the different model predictions with the experimental water contents from rheological measurements. The resulting cross-linking efficiency predictions are correlated with the ability of the different comonomers to facilitate chain transfer and/or radical termination in the polymerization environment. The model can capture the differing responses of the microgels in the presence of different salt concentrations and can account for the impact of many key physical parameters and heterogeneities in microgel swelling which the Flory-Huggins model cannot directly address.

Introduction Hydrogel systems which swell or deswell in response to specific environmental stimuli (i.e., pH, temperature, solute concentration, solvent composition, ionic strength, light, or electric field) are of technological interest as bioseparation matrices, bioreactors, superabsorbents, chemical sensors, and chemical transducers. Carboxylic-acid-functionalized microgels based on poly(N-isopropylacrylamide) (PNIPAM) may be of particular interest given their rapid, reversible responses to both temperature and pH changes. However, to better exploit the technological promise of responsive materials, an improved understanding of the impact of the underlying morphologies on the gel swelling responses to various stimuli is required. Indeed, the ability to predict changes in the microgel particle size (and thus the water content and the pore size) according to the local microstructure and chemical composition of the microgel would facilitate the design and synthesis of optimized microgels for specific applications. Significant effort has been invested in predicting the swelling behavior of hydrogel systems. The most widely used of these approaches is the Flory-Huggins thermodynamic theory,1 which treats gel swelling in terms of a series of osmotic pressure effects: the mixing of the polymer chains with the solvent, the elastic resistance of the cross-linked network to expansion or shrinking, and (for ionized gels) the Donnan equilibrium of mobile counterions within the gel network. Flory-Huggins theory has been remarkably successful in describing the swelling * To whom correspondence should be addressed: Phone: 1-905-5259140 ext. 27342. Fax: 1-905-528-5114. E-mail: [email protected]. † E-mail: [email protected].

of a wide range of bulk, polyelectrolyte hydrogels,2 according to the average composition of the gel. This success is facilitated by the fact that bulk hydrogels can be considered as largely homogeneous on the microscale, with any microscopic inhomogeneities averaged out over the macroscale dimensions of the gel. Microgels, however, exhibit systematic inhomogeneities within their nanoscale dimensions. Strong, systematic trends in radial cross-linking density3 and functional group density4,5 have been observed in PNIPAM-based microgel systems, trends which we have recently identified as being primarily related to the copolymerization kinetics of the different comonomers used to prepare the microgel.6 Given the critical importance of the cross-link density to the elastic contribution to gel swelling and the functional group density to the charge contribution to gel swelling, such nanoscale gradients may have very significant effects on the swelling responses of microgels. Current opinion is split on whether or not heterogeneities significantly impact gel swelling. Fernandez-Nieves et al. reported that swelling in poly(vinylpyridine) microgel particles can be adequately predicted by applying the Flory-Huggins gel swelling model to the bulk-phase composition7 but required a concentration-dependent Flory-Huggins parameter to successfully predict PNIPAM microgel swelling.8 Arleth et al.9 reported that classic Flory-Rehner theory can be used to fit swelling profiles in small (100 nm) microgel swelling using a modified form of the Flory-Rehner equation proposed by Hino and Prausnitz11; however, the water fraction predictions were significantly less accurate than those achieved with the smaller, more homogeneous particles9. Kokufuta et al.12

10.1021/jp072360f CCC: $37.00 © 2007 American Chemical Society Published on Web 09/26/2007

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observed significant differences in the swelling responses of “homogeneous” poly(NIPAM-co-AA) gels prepared by acrylic acid monomer copolymerization with those of gels prepared by prepolymerizing the acrylic acid macromonomer (effectively imposing inhomogeneities within the gel microstructure), which cannot be understood by bulk Flory-Huggins theory alone. Simple Donnan equilibrium also cannot explain the dramatic swelling differences observed by Jones and Lyon13 depending on whether acrylic acid residues were localized in the shell or in the core of AA-NIPAM microgels. Finally, our recent work in preparing microgels with the same bulk functional monomer and cross-linker content but different functional group distributions4 resulted in microgels with sharply different responses to ionization, a result which would not be predicted based on bulk Flory-Huggins modeling. In our recent publications,6 a terminal copolymerization kinetics model was applied to predict the compositions of 20 concentric “shells” within a series of functionalized microgels, permitting the prediction of functional group distributions in poly(NIPAM)-based microgels prepared via copolymerization of methacrylic acid (MAA), acrylic acid (AA), vinylacetic acid (VAA), fumaric acid (FA), and maleic acid (MA). In this paper, we apply the local compositional knowledge acquired from this kinetic model (i.e., the compositional distribution in a fully dehydrated microgel) with both classical Flory-Huggins theory as well as a more complex theory accounting for ion condensation, background electrolyte, chain stiffness, and direct chargecharge repulsion between the fixed charges on the polymer network to predict swollen-state morphologies in microgels. In particular, we aim to better understand the effect of microgel morphologies and compositional heterogeneities on microgel swelling and identify the predictive approaches required to accurately forecast the effects of microgel heterogeneities.

Nc is the effective number of polymer chains in the gel volume, χ is the Flory-Huggins polymer-solvent interaction parameter, and f is the average number of ionic groups present between cross-linking points. The reference volume Vo can be expressed in terms of the volume fractions according to the scaling relationship (V/Vo) ) (φo/φ) ) (R/Ro)3, where R is the microgel radius in the swollen state and Ro is the microgel radius in the reference state. To use this model for predictive purposes, the reference state is defined according to the heuristic given by Wu et al., equating φo to the volume fraction of polymer within the microgel particles under the polymerization conditions.10 The kinetic model results can be used to estimate the local values of Nx, Nc, and f according to the predicted compositions of each shell. Nx can be expressed as

Theory

In any predictive gel swelling model, the accurate prediction of the Flory-Huggins parameter χ is a particular challenge. To achieve χ estimates for heterogeneous microgels at temperatures below the lower critical solution temperature, the group contribution approach developed by Hansen15 is applied. The solubility parameter of a polymer is predicted based on the group contributions of each of the monomer building blocks of the overall polymer structure16 and weighting those contributions based on their mole fraction in the polymer being considered, according to the additivity relationship

In Flory-Huggins theory, gel swelling is determined by the sum of three different contributions, the elasticity of the gel network Πelas (typically restricting swelling), the mixing of polymer chains with the solvent Πm (typically driving swelling), and (for ionic gels) a Donnan-type potential due to the translational entropy of the counterions ΠD (also driving swelling).1,14 These contributions can be expressed mathematically using eqs 1-3. The equilibrium swelling state φ/φo at a specific composition can be calculated by setting the sum of these osmotic pressures to zero

Πelas )

[( ) ( ) ]

kTNc 1 φ φ Vo 2 φ o φo

1/3

)

[( ) ( ) ]

kTNAφo 1 φ φ VsNx 2 φo φo kTNA (φ + ln(1 - φ) + χφ2) Πm ) Vs ΠD )

()

()

kTNAφo f φ kTNc f φ ) VsNx φo Vo φ o

1/3

(1)

(2)

(3)

Here, φ is the volume fraction of the polymer in the swollen gel, φo is the volume fraction of the polymer in the reference state, k is Boltzmann’s constant, T is temperature, Vo is the volume of the gel in the reference state, Vs is the molar volume of the solvent (water), NA is Avogadro’s constant, Nx is the average number of monomer units between cross-linking points,

Nx )

n1 + (1 - x)n2 + n3 xn2

(4)

where n1 is the number of moles of NIPAM within the shell, n2 is the total number of moles of cross-linker within the shell, n3 is the number of moles of functional monomer within the shell, and x is the cross-linking efficiency (i.e., the percentage of crosslinker monomer in which both vinyl groups react). On this basis, the number of subchains in the shell volume Nc can be defined as

Nc ) 2NAxn2

(5)

and the number of ionic groups between cross-linking points f is defined as

f)

δ2poly )

n3 n2

(6)

∑ fiδmi ) ∑ fi (δ2d + δ2p +δ2h) ) 2 ∑ Fdi ∑ Fpi2 ∑ fi V + 2

[( ) ( ) ( )] m

Vm

+

∑ Ehi

(7)

Vm

Here, δpoly is the overall polymer solubility parameter, fi is the mole fraction of component i in the polymer considered, δmi is the solubility parameter of monomer unit i, δd is the dispersion force contribution to solubility, δp is the polarity contribution to solubility, δh is the hydrogen-bonding contribution to solubility, Vm is the molar volume of the unit being considered (in this case, the individual monomer residues), ΣFdi is the sum of the dispersion force group contributions, ΣFpi is the sum of the polarity group contributions, and ΣEhi is the sum of the hydrogen-bonding group contributions. The effect of -COOH ionization on the solubility parameter is estimated according to the approach of Barra et al.17 using their group contribution values for the sodium ion (the counterion in our work) in the additivity relationship given in eq 7. While reasonable compara-

Functionalized Microgel Swelling

J. Phys. Chem. B, Vol. 111, No. 41, 2007 11897

tive values of the solubility constant are generally produced via this approach, it must be noted that the absolute predicted values may deviate ∼10-20% from the actual values. However, good agreement is observed between the δpoly values calculated using the ref 16 approach and those calculated using independent group contribution theories such as those developed by Fedors18 and Stefanis et al.19 For a NIPAM homopolymer, ref 16 predicts δpoly ) 22.2 J1/2 cm-3/2, while Fedors predicts δpoly ) 24.5 J1/2 cm-3/2 and Stefanis et al. predict δpoly ) 22.1 J1/2 cm-3/2. The Flory-Huggins constant χ may subsequently be estimated by summing the enthalpic and entropic contributions using eq 8, a residual free-energy function

Vs(δw - δpoly) + 0.34 RT 2

χ ) χH + χ s )

(8)

ΠD )

{

[(

) ]}

ieFV2 RT ieFV2 - 2fcoionccoionMo 1 + Mo fcoionccoionMo

1/2

-1

(10)

Here, δw is the solubility parameter of water (the solvent), and R is the gas constant. The enthalpic contribution is an extension of the regular solution theory of Hildebrand et al.,20 while the entropic contribution is a semiempirical constant from the work of Bristow and Watson21 applied to gels by Dong and Hoffman.22 The basic Flory-Huggins gel swelling theory presented in eqs 1-3 is, however, inherently limited in its application given the many simplifying assumptions made in the derivation of the model.23 Of particular interest, the potential impacts of added mobile electrolyte, counterion condensation to the fixed network charges, inter-cross-link chain length distributions, chain stiffness, and direct charge-charge repulsion between the fixed network charges cannot be addressed using the basic FloryHuggins model but may have large impacts on the swelling behavior of hydrogels. The latter two considerations are particularly interesting in the case of heterogeneous functionalized microgels given the highly inhomogeneous cross-linker distribution6 and strong apparent effects of charge-charge repulsion4 that we have observed. Although the Donnan term (eq 3) can be modified to account for some of the complicating electronic factors,24 an improved model is required to comprehensively examine the importance of all of the listed variables on swelling. In response, we have combined the polyelectrolyte gel swelling theory of Hasa et al.25 with the direct polyelectrolyte repulsion term derived by Katchalsky and Michaeli26 in order to predict microgel swelling. In this analysis, the total osmotic pressure Π of a polyelectrolyte gel can be written as the sum of four components (eq 9)

( ) ( ) ( ) (

of polymer in the microgel under the synthetic conditions at which the cross-links were formed (T ) 70 °C) is a better reference state for PNIPAM-based hydrogels given that this condition is the closest objective representation of the zerostrain point in the network (the definition of the reference state). Use of a collapsed-gel reference state also permits direct comparisons between our model and the basic Flory-Huggins model presented in eqs 1-3. The Donnan term ΠD for an electrolyte solution is defined by the equation25

δ∆Fm Π ) Πm + ΠD + Πelas + Πelec ) δV T,n δ∆FD δ∆Felas δ∆Felas δV T,n δV T,n δV

)

T,n

(9)

Here, Πm is the solvent-polymer mixing contribution (favoring swelling), ΠD is the Donnan equilibrium contribution (favoring swelling), Πelas is the network elasticity contribution (favoring deswelling), and Πelec is a term accounting for electrostatic interactions between the fixed charges within the polyelectrolyte (repulsive, driving swelling) and salt counterions and the polyelectrolyte (attractive, driving deswelling). The mixing contribution Πm is defined by the Flory-Huggins expression given eq 2, adapted in this case by renaming φ as V2, the volume fraction of polymer in the swollen gel. Hasa et al. 25 used the dry state as their reference condition (φo ) 1) for model evaluations. However, the average volume fraction

Here, Mo is the monomer molecular weight, and F is the density of the dry polyelectrolyte, both calculated as average properties based on the kinetic model-predicted compositions of each volumetric shells considered. The variable ie is the effective degree of ionization of the polymer chains, given in our microgel system by the expression

[

]

n3 Φ ie ) RΦp ) (Di) n1 + n2 + n3 p

(11)

where Di is the degree of ionization of the carboxylic-acidcontaining monomers in the polymer, R is the overall degree of ionization (i.e., the fraction of total monomers within the polymer chain which are ionized), and Φp is the osmotic coefficient. The osmotic coefficient, defined as the ratio of ions which are actually “free” of the polyelectrolyte and can give rise to the Donnan potential, is estimated using the correlation given by Alexandrowicz and Katchalsky27 (Figure 8 in ref 27) for vinylic polyelectrolytes. This corrected cylindrical chain model also accounts for the lack of full extension in real chains with lower degrees of ionization and/or inherent stiffness due to steric or electronic effects. The application of this correlation to our microgel system requires the calculation of the average charge separation distance along the individual polymer chains dCC, defined according to our kinetic model compositional results as

(

dCC ) bo

)

n1 + n2 n3

(12)

Here, bo is the effective length of a monomer unit, equal to the x-axis projection of two carbon-carbon single bond lengths (taken as 2.55 Å as per Katchalsky and Michaeli’s average estimate for vinylic polymers26). The concentration of free Cl- ions within the gel phase (ccoion) is calculated according to the Donnan equilibrium between the anionic-functionalized microgel containing xCOOH polymerbound ions within a total gel volume Vmicrogel and the suspension medium containing xNaCl moles of salt in a total volume of Vsuspension. The moles of chloride coion xCl inside of the gel phase can be written based on the neutrality condition in both phases (gel and bulk)28,29

xCl(xCOOH + xCl) V2microgel

)

(xNaCl - xCl)2 V2suspension

(13)

The xCl can be solved analytically using the quadratic formula and converted to the Cl- ion concentration ccoion by dividing

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by the total gel volume in the suspension. Estimates of ccoion are made under the dilute conditions used for dynamic light scattering (15 ppm microgel solution) such that the model results and the experimental swelling results can be compared directly. However, less than a 3% difference in ccoion is observed when the gel concentration is increased from 15 ppm to 1 wt %, making comparisons with the rheological data also relevant. The Donnan equilibrium-predicted concentrations are also similar to those measured experimentally under similar conditions by Hasa and Ilavsky.30 The activity coefficient fcoion is set equal to unity, reasonable given that Nagasawa et al.31 showed that an anionically functionalized linear polymer exhibited fCl- > 0.75 at all degrees of ionization. Parameter sensitivity analysis on the final model also showed that fcoion has a 90% of the bulk functional group content. The outermost shells of MAA-NIPAM and AA-NIPAM, meanwhile, are predicted by the gel swelling models shown in Figures 2 and 3 to contain a higher polymer chain fraction and lower

Functionalized Microgel Swelling functional group density. This is consistent with the mobility modeling result which implied the presence a thinner ionpermeable shell containing fewer charges and a lower void fraction. The fundamental understanding of microgel swelling achieved within this model framework can also be used to explain two unusual swelling behaviors observed with the different functionalized microgels. First, FA-NIPAM swells significantly less than VAA-NIPAM upon ionization despite their identical overall charge content and highly similar radial functional group distributions. At 25 °C, FA-NIPAM swells 3-fold by volume upon ionization (i.e., changing the pH from 3.5 to 10), while VAA-NIPAM exhibits a 7-fold increase in microgel volume when ionized. This difference can be rationalized based on the effective degree of ionization term incorporated into the model. The average distance between consecutive -COO- groups in FA-NIPAM and MA-NIPAM is fixed by the diacid monomer structure at three carbon-carbon bond lengths (corresponding to an osmotic coefficient Φp ∼ 0.25), while the monoacid comonomers are, on average, separated by many more carbons (>25 in all cases, corresponding to Φp > 0.75). Consequently, the effective degree of ionization ie of the chain segments is lower in FA-NIPAM than that in VAA-NIPAM at all values of R. Since the Donnan term ΠDon has a direct dependence on ie (eq 10) and the charge-charge repulsion term Πrep varies with i2e (eq 19), the osmotic pressure driving microgel swelling is significantly lower in FA-NIPAM compared to that in VAANIPAM; correspondingly, less ionization-driven swelling is observed. The significant differences in the observed swelling behaviors of the two diacid-functionalized microgels MA-NIPAM and FANIPAM can also be rationalized. Structurally, MA and FA are extremely similar; both are diacids which react significantly slower than NIPAM in a free-radical environment and localize predominantly at the shell of the microgel. Thus, although FA does has a limited capacity for functional monomer block formation, the radial and chain distributions of functional groups in these two microgels are predicted to be very similar via kinetic modeling. However, upon ionization, MA-NIPAM increases in volume by a factor of 2.2, while FA-NIPAM increases in volume by a factor of 3.1. This difference can be attributed directly to the significantly higher (approximately double) cross-linker concentration in the outermost shells of MA-NIPAM compared with that of FA-NIPAM. As noted earlier, this near-surface cross-linker enrichment was identified via the kinetic model in cases where the high molar excesses of functional monomer were required to achieve a targeted -COOH loading. Because a FA excess of 70% and a MA excess of 780% are required to achieve the targeted 6.5 mol % functional monomer content in the final microgels, the cross-linker is significantly more surface localized in MA-NIPAM. Given that this higher degree of crosslinking is present in the same near-surface radial location as the majority of the ionizable functional groups, electrostaticsdriven swelling is significantly reduced by a locally higher crosslink content in MA-NIPAM. In particular, the charge-charge repulsion contribution which becomes a large driver of swelling in shells with high -COOH contents is dramatically reduced due to the N2x dependence of eq 20. Thus, while the radial distribution of -COOH groups in MA-NIPAM and FA-NIPAM is similar, the increased cross-link concentration in the outermost shell results in a lower degree of ionization-driven swelling in MA-NIPAM. Both of the above examples illustrate how kinetic and thermodynamic models of microgel swelling can be applied to

J. Phys. Chem. B, Vol. 111, No. 41, 2007 11905 explain unusual observed microgel swelling phenomena not explained by conventional qualitative observations and colloid analysis techniques. Conclusions Shell-by-shell swelling calculations based on monomer concentration gradients determined via a microgel copolymerization kinetics model can generate water fraction profiles as a function of radial distance for -COOH-functionalized microgels A comprehensive model of microgel swelling incorporating the effects of salt, counterion/polyelectrolyte charge condensation, cross-link length distribution, chain stiffness, and direct charge-charge repulsion can successfully predict the relative water contents of a series of -COOH-functionalized microgels according to their local composition profiles. The cross-linking efficiency can be used as an adjustable variable to match the model predictions with the experimental water contents from rheological measurements. The resulting cross-linking efficiency predictions correspond to the probability of chain transfer in each of the functionalized monomer systems. Model predictions can be used to understand the different responses of the microgels as the salt concentration is increased and the apparent anomalies in the experimental results (i.e., the difference in swelling observed between MA-NIPAM and FANIPAM). Acknowledgment. The Natural Sciences and Engineering Research Council of Canada (NSERC) is acknowledged for funding of this work. Supporting Information Available: Model parameter sensitivity analyses for s, n, x, Nx, χ, and cion, variable definitions and units for all parameters used in the model, and a copy of the MathCad model used to perform the thermodynamic swelling calculations are provided. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Flory, P. J. Principles of Polymer Chemistry; Cornell Press: Ithaca, NY, 1953. (2) (a) Hooper, H. H.; Baker, J. P.; Blanch, H. W.; Prausnitz, J. M. Macromolecules 1990, 23, 1096-1104. (b) Ricka, J.; Tanaka, T. Macromolecules 1984, 17, 2916-2921. (3) (a) Varga, I.; Gilanyi, T.; Meszaros, R.; Filipcsei, G.; Zrinyi, M. J. Phys. Chem. B 2001, 105, 9071-9076. (b) Berndt, I.; Pedersen, J. S.; Lindner, P.; Richtering, W. Langmuir 2006, 22, 459-468. (4) Hoare, T.; Pelton, R. Macromolecules 2004, 37, 2544-2550. (5) (a) Zhou, S.; Chu, B. J. Phys. Chem. B 1998, 102, 1364-1371. (b) Kratz, K.; Hellweg, T.; Eimer, W. Colloids Surf., A 2000, 170, 137149. (6) (a) Hoare, T.; McLean, D. J. Phys. Chem. B 2006, 110, 2032720336. (b) Hoare, T.; McLean, D. Macromol. Theory Simul. 2006, 15, 619632. (7) Fernandez-Nieves, A.; Fernandez-Barbero, A.; Vincent, B.; de las Nieves, F. J. Macromolecules 2000, 33, 2114-2118. (8) Fernandez-Barbero, A.; Fernandez-Nieves, A.; Grillo, I.; LopezCabarcos, E. Phys ReV. E 2002, 66, 051803. (9) Arleth, L.; Xia, X.; Hjelm, R. P.; Wu, J.; Hu, Z. J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 849-860. (10) Wu, J. Z.; Huang, G.; Hu, Z. B. Macromolecules 2003, 36, 440448. (11) Hino, T.; Prausnitz, J. M. J. Appl. Polym. Sci. 1996, 62, 16351640. (12) Kokufuta, E.; Wang, B.; Yoshida, R.; Khokhlov, A. R.; Hirata, M. Macromolecules 1998, 31, 6878-6884. (13) Jones, C. D.; Lyon, L. A. Macromolecules 2003, 36, 1988-1993. (14) Shibayama, M.; Tanaka, T. AdV. Polym. Sci. 1993, 109, 1-62. (15) (a) Hansen, C. M. J. Paint Technol. 1967, 39, 104-117. (b) Hansen, C. M. J. Paint Technol. 1967, 39, 511-514.

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