Article pubs.acs.org/Macromolecules
Electromechanical Equilibrium Properties of Poly(acrylic acid/ acrylamide) Hydrogels Katsiaryna Prudnikova†,‡,§ and Marcel Utz*,†,‡,⊥ †
Center for Microsystems for the Life Sciences, ‡Department of Mechanical and Aerospace Engineering, and ⊥Department of Chemistry, University of Virginia, Charlottesville, Virginia 22904, United States ABSTRACT: Thermomechanical properties of poly(acrylic acid-co-acrylamide) hydrogels have been measured for a range of gels while systematically varying the acrylamide/acrylic acid ratio. The gels have been equilibrated with a buffer solution at constant pH and salinity. The gels were characterized in terms of their equilibrium swelling ratio, elastic modulus, and electrochemical potential. The results are in quantitative agreement with the predictions from a recently published thermodynamic field theory.
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INTRODUCTION Polyelectrolyte hydrogels (PHG) consist of cross-linked macromolecules that carry dissociable electric charges along the polymer backbone. In a polar solvent such as water, the ionizable groups dissociate, providing a powerful entropic driving force that leads to substantial swelling. Because of this unique structure, dry PHG can absorb many times their own weight in water. They are therefore widely used as superabsorbers in diapers and cleaning products. In the swollen state, PHG exhibit rubber elasticity, i.e., solid-like behavior, in spite of water contents in excess of 90%. The equilibrium swelling ratio is sensitive to small changes in outside conditions, such as temperature, pH, and salinity of the surrounding solution. Consequently, PHGs have been studied widely as chemically, thermally, and electrically controlled actuators.1−9 Several authors have provided theoretical approaches toward understanding the electrical and mechanical behavior of PHG.2,8,10−15 Much of this work has focused on the kinetics of the electrically induced swelling processes either analytically2,9 or by finite element simulation.13 Rubinstein et al. have developed a scaling theory of polyelectrolyte gels, and applied it to predict gel elastic modulus as a function of solvent salinity.11 De Gennes has sketched a theory based on nonequilibrium thermodynamics coupling solvent flux and electrical fields,12 which has recently been shown to quantitatively predict the bulk streaming potentials observed in PHG exposed to gradients in solvent pressure.16 On the other hand, a comprehensive field theory of the thermodynamic equilibrium state of PHG in the presence of solvent, electrical fields, and mechanical stress has only been reported recently by Hong et al.17,18 It differs from scalingtheory approaches in that it attempts to make absolute predictions on the gel properties, rather than merely their scaling. Like other theoretical approaches, it is based on a combination of Donnan equilibrium with the Flory−Rehner theory of polymer gels.19,20 This approach is widely used to rationalize the equilibrium swelling behavior of PHG.21−26 The © 2012 American Chemical Society
particular merit of the theory by Hong et al. lies in its rigorous incorporation of the effects of electric fields and mechanical stress and strain tensors. In this contribution, we compare the results of an experimental study on poly(acrylic-co-acrylamide) gels with its predictions. The composition of the gels (acrylic acid versus acrylamide) as well as their cross-link density was varied, and the equilibrium swelling degree, electrical (Donnan) potential, and the elastic modulus were measured. As discussed in the remainder of this paper, the agreement between the theory and the experimental results is excellent for the swelling degree and modulus across the entire range of compositions and cross-link densities studied. At the same time, while the scaling of the Donnan potential with gel composition is captured correctly, its magnitude is underestimated consistently by about 20%.
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THEORY Following Hong et al.17,18 the free energy of a gel is separated into contributions due to the entropy elasticity of the polymer chains Wel, the free energy of mixing of the polymer Wmix and free ions Wion with the solvent, and the free energy due to the electrical displacement polarization Wpol. Using the Flory− Huggins expression for Wmix, the ideal mixing law for Wion, and the Flory−Rehner model for the elasticity of the polymer chains, Hong et al. obtained for the stress field
σij =
NkT 1 (FiK FjK − δij) + det F ε ⎛ ⎞ 1 ⎜D D − DmDmδij⎟ − Πδij ⎝ i j ⎠ 2
(1)
where N is the concentration of polymer chains (per dry volume of polymer), F is the deformation gradient, ε the Received: November 10, 2011 Revised: December 19, 2011 Published: January 13, 2012 1041
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electrical permittivity of the gel, D the electrical displacement field, and Π represents the osmotic pressure
Π=
(see Appendix for derivation). In the present work, the elastic properties of the samples were measured by a uniaxial compression experiment. This yields Young’s modulus Y, which is obtained from the Lamé constants as
vsCs kT ⎡ C+ + C − − 2vsc0 − log ⎢ 1 + vsCs vs ⎣ Cs ⎤ χ 1 ⎥ − − 1 + vsCs (1 + vsCs)2 ⎥⎦
Y= (2)
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with the molecular volume of the solvent vs (18 cm3/mol for water), the solvent concentration (per unit volume of dry polymer) Cs, the concentration of positive and negative free ions in the gel C±, and the salt concentration in the solvent c0. Inside the gel, away from the Debye layer at the gel/solution interface, both the stress and the electric displacement field vanish. Assuming isotropic swelling Fij = δijλ0, on finds from (eq 1) for the equilibrium osmotic pressure
(4)
The concentrations of positive and negative free ions in the gel are given by the Boltzmann distribution
∓ eϕ C± = c0ϕsCs exp (5) kT where c0 is the salt concentration in the solution, and v± are the molar volumes of positive and negative ions, respectively, ϕ denotes the Donnan potential within the gel, and the ions are assumed to have a valency of 1. Finally, due to charge neutrality deep inside the gel, we have C0 + C − = C+ (6) where C0 denotes the concentration of fixed negative charges inside the gel. As detailed in ref 17, eqs 2−6 represent a nonlinear system of six independent equations, which can be solved numerically to yield the six unknowns λ03, ϕ, C±, Π, and Cs, given the input parameters χ, N, vs, v±, c0, and C0. In the present work, gels with varying composition and cross-link density were synthesized, and their swelling ratio λ03, the Donnan potential ϕ, and the elastic modulus were measured. While λ03 and ϕ emerge directly from solving the above equations, the elastic modulus can be obtained from eq 1 by assuming the strain to be of the form
Fik = λ 0(δik + uik)
G=
⎛C + C 1 − 2χ N⎞ NkT ⎟⎟ , Λ = kT ⎜⎜ − 3 + + − λ0 λ0 ⎠ vs λ 06 ⎝ λ0
EXPERIMENTAL SECTION
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RESULTS AND DISCUSSION Parts A and B of Figure 1 show the experimental results for the swelling degree and Young’s modulus of the gels, respectively (closed symbols). The least-squares fit of the theory laid out above to the data is represented by solid lines; the bands indicate the 95% confidence regions of fit, as discussed in more detail below. The swelling degree (Figure 1A) increases with the acrylic acid content, and decreases with the density of crosslinks. Pure acrylamide gels (xAA = 0) exhibit swelling ratios between 3 and 10, whereas acrylic acid gels (xAA = 1) reach values of up to 35 at the lowest cross-link density.27 These swelling degrees are consistent with values reported by other groups; e.g.,Vasheghani-Farahani et al. have obtained swelling degrees between 5 and 10 for a gel very similar to our sample with intermediate cross-link density at xAA = 0.1 around neutral pH.28
(7)
where uik = ∂ui/∂xk ≪ 1 represents the infinitesimal strain applied to the swollen network. Inserting this into (eq 1), assuming the electrical displacement to vanish, one finds to first order in the infinitesimal strains
u + uki σik = 2G ik + δik Λujj 2 with the Lamé constants
(10)
The details of the gel synthesis have been published elsewhere,27 and are only summarized briefly here. Acrylic acid (AA) and acrylamide (AM) were used as comonomers. N,N′-methylenebis(acrylamide) (nBisA) and 2,2-dimethyl-2-phenylacetophenone (DMPA) were used as received as cross-linker and photoinitiator, respectively. The polymerization was carried out in a potassium phtalate/methyl alchohol buffer solution pH 4.00 (Fisher) with a small amount of added dimethyl sulfoxide (DMSO); the resulting gels were then equilibrated in a potassium acid phtalate/sodium hydroxide buffer solution at pH 5.00. The salt concentration of both buffer solutions was c0 = 0.05 M. All copolymerizations were performed at an overall monomer concentration of 3.8 M, while the mol fraction of AA xAA was varied between 0 and 1. In this way, three sequences of copolymers were created using different initial concentrations of nBisA = 22 mM, 43 mM, and 86 mM. The photoinitiator DMPA concentration was equal to 6.78 mM and was kept constant during all copolymerizations. Samples were polymerized in a square mold made of two microscope slides (Fisher) separated with PDMS spacers of 2.2 mm thickness. Polymerization was initiated by exposure to an UV lamp (λ = 365 nm) at room temperature for 10 min. The samples were then carefully removed from the mold and equilibrated in a buffer solution at pH 5.00 for 7 days, at which point the sample mass reached a constant value. Sample mass was monitored as a function of time on an analytical balance after gentle drying with paper towels. The elastic modulus of the same gels was measured by a quasi-static uniaxial compression measurement using a TA Instruments QA 800 dynamic mechanical analyzer. A sample of 15 mm diameter was exposed to a preload of 1 mN, which was then gradually increased to 7 N at a rate of 2 N/min. Sample thickness ranged from 2.5 mm to 3 mm, depending on the swelling degree. Experimental error was estimated from running three independent samples per data point. The Donnan potential in each sample was measured by gentle indentation of the sample surface with a Ag/AgCl electrode, as described in detail in ref 27. A reference Ag/AgCl electrode was kept in solution, while the surface of the gel sample was gently pressed with the working Ag/AgCl electrode. The resulting potential difference was recorded with a GAMRY Reference 600 Potentiostat.
Π 0 = NkT(λ 0−1 − λ 0−3) (3) Since the polymer molecules and the solvent can be regarded as incompressible, and the contribution of the free ions to the total volume of the system is negligible, the equilibrium swelling ratio λ03 satisfies
λ 03 = 1 + vsCs
⎤ Nvs λ 05 NkT ⎡ ⎢3 − ⎥ λ 0 ⎢⎣ 1 − 2χ + (C+ + C −)vs λ 03 ⎥⎦
(8)
(9) 1042
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termination. In order to fit the experimental data, the chain density was therefore taken to be proportional to the concentration of cross-linker: N = ηcCnBisA, where the crosslinker efficiency ηc was used as a fit parameter. Several authors have explored the efficiency of cross-linkers in the formation of polymer networks;30,32,33 typical values for ηc range from 1% to 50%. At pH 5, acrylic acid groups in the copolymers dissociate partially, whereas the acrylamide groups do not. Therefore, changing the ratio of AA vs AM monomers in the gel provides control over the concentration of fixed charges in the gel. However, this is complicated by the fact that the pKa value, and therefore the degree of acrylic acid dissociation at fixed pH, varies with the concentration of acrylic acid.34 In the present work, this effect was captured by assuming the concentration of fixed negative charges C0 to grow with the acrylic acid fraction xAA as αx C0 = AA exp( −βxAA) vm (11) where α and β were fit constants, and vm denotes the molecular volume of the monomers (which is almost the same for acrylic acid and acrylamide). The four adjustable parameters are listed in Table 1, along with their physically meaningful range, and the best fit value Table 1. Adjustable Model Parameters with Their Meaningful Physical Range
Figure 1. (A) Swelling degree of poly(acrylic acid-co-acrylamide) gels prepared with cross-linker concentrations of 22 mM (green diamonds), 43 mM (red squares), and 86 mM (blue circles) in equilibrium with buffer solution at pH = 5. (B) Elastic moduli of the same samples. Solid lines represent the best fit of the model to the experimental data (cf. text); the bands indicate 95% confidence intervals of fit.
parameter Flory interaction cross-linker efficiency dissociation degree dissociation exponent
χ ηc α β
physical range
fitted value
0.45−0.55 0−0.3 0.1−0.5 0−2
0.505 ± 0.01 0.05 ± 0.005 0.32 ± 0.05 1.14 ± 0.05
obtained from the data shown in Figure 1 . A normalized overall mean square deviation of fit was computed as
Three separate samples of each composition were made; the error bar shown in Figure 1A represents ±3 standard deviations. Error bars for the other data points are of similar or smaller size, and have been omitted from the figure for clarity. The Young’s moduli of the gels, as obtained from uniaxial compression, are shown in Figure 1B. The samples with the lowest cross-link density exhibit the lowest modulus, as expected. For a given cross-link density, the modulus appears to be almost constant as the acrylic acid content is increased. This may seem surprising, since the swelling degree of the gels is a very strong function of acrylic acid content. The data in Figure 1 were used to fit the model, with the Flory interaction parameter χ, the density of cross-linked chains N and the density of dissociated charges on the polymer backbone C0 as adjustable parameters. The remaining quantities appearing in the model, the solvent salt concentration c0, and the molar volumes of the solvent and salt, are well-known. Even for the free parameters, there are quite narrow physical bounds. χ is known to be in the vicinity of 0.5 for aqueous acrylate gels.29−31 The experiments described here were designed to control the density of chains and the polymer charge density through the composition of the gels. The concentration of chains N is proportional to the concentration of cross-linker. However, not every cross-linker molecule that is incorporated into the polymer network leads to an active branching point; many are lost due to the formation of short loops or chain
3
σ2 =
M
⎛ (λ 3 − λ′ 3 )2 kj kj
∑ ∑ ⎜⎜
k=1 j=1 ⎝
λkj 3
+
(Ykj − Y ′kj )2 ⎞⎟ ⎟ Ykj ⎠
(12)
where k runs over the three different cross-link densities and j over the acrylic acid concentrations. Primed quantities are calculated from the fit parameters by numerically solving the system of equations (eq 1−eq 6), whereas their unprimed counterparts λ3 and Y denote the experimental values. The fitted parameter values given in Table 1 were obtained by minimizing σ2. The error bars represent estimated 95% confidence contours based on the 95% quantile of the F distribution with p = 4 fit parameters and n = 46 independent data points.35 As shown in parts A and B of Figure 1, the overall agreement of the theory with the experimental swelling and modulus data is remarkably good. At the lowest cross-link density, however, the fit overestimates the saturation of the swelling degree with acrylic acid content; the experimental data seems to exhibit a finite slope as xAA = 1 is approached, whereas the fit clearly saturates. Also, the swelling degree of pure acrylamide gels (xAA = 0) is overestimated somewhat. The experimental data suggest a finite slope of the swelling degree at this point, which is not represented by the model. The moduli of the gels with the lower two cross-link densities are very well represented by the 1043
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fitting procedure, albeit at the expense of somewhat worse fit of the swelling and modulus data. It has been noted previously that the Donnan potential data fall onto a single curve if they are plotted against the actual concentration (spatial density) of ionizable groups in the gel.27 In fact, in simplified terms, the Donnan potential can be seen as a potential difference that arises due to the charge imbalance at the Debye layer at the gel surface, and should therefore be proportional to the number density of (dissociated) negative charges on the polymer backbone in the swollen gel. Using (eq 11) and the values obtained from the fit in Table 1, Figure 3 plots the Donnan potential ϕ versus the spatial
model, even including a slight downward trend as the acrylic acid content increases. The most highly cross-linked gels, however, exhibit a modulus that increases from 350 to 450 kPa as the acrylic acid content increases from 0 to 0.3, and then stays level. The model, by contrast, fits the value at xAA = 0, but shows a much smaller increase, which is followed by a slight decline. It therefore underestimates the moduli of the most highly cross-linked gels at large acrylic acid content by about 25%. The fact that this deviation surfaces at the most swollen gels at the highest cross-link density suggests that it may be due to finite extensibility of the polymer chains, which is neglected in the Flory−Rehner approach, which underlies the model. In addition to the swelling degree and the elastic modulus, the Donnan potential of the gels has been measured using the approach shown in Figure 2A; Figure 2 B shows the results.
Figure 3. Donnan potential data plotted as a function of true spatial density of dissociated charges on the polymer backbone C0/λ03, determined from eq 11 and the parameters given in Table 1.
concentration of negative backbone charges C0/λ03. Indeed, the points do seem to fall onto a straight line, even though there is some scatter, including two outlying data points around C0/λ03 = 0.08 mol/L. The model predictions for the three cross-link densities (solid lines in Figure 3) almost collapse on a single curve, which very closely approximates a straight line. However, they are consistently below the experimental data by a few millivolts. Nonetheless, the fact that both the experimental and model data essentially fall onto a straight line provides an independent consistency check and suggests that eq 11, combined with the values of Table 1, correctly captures the dissociation behavior.
Figure 2. (A) Setup for measuring the Donnan potential; (B) Donnan Potential values (closed symbols), and predicted values from the model (solid lines), using the best fit parameters given in Table 1.
Measurements were repeated for three independent samples per data point; the error bars represent ± σ. Because of the negatively charged polymer backbone, the Donnan potential values are negative; i.e., the electrical potential inside the gel is less than in the surrounding solution. As expected, the values obtained are zero within experimental error for pure acrylamide gels, and then increase in magnitude with increasing acrylic acid content. More highly cross-linked samples exhibit higher Donnan potential. The values increase rapidly at low acrylic acid content, and then start to saturate around xAA = 0.5. The solid lines in Figure 2 show the predicted Donnan potential ϕ, using the fit parameter values obtained from the swelling and modulus data (Table 1). The predicted values show the same scaling and trends as the experimental data: gradual increase with xAA and saturation around xAA = 0.5. The scaling with cross-link density is also similar to the experimental data. However, the predicted values underestimate the measured data by a factor of 1.5 to 2, particularly at high acrylic acid concentrations. These discrepancies notwithstanding, the agreement in overall behavior and scaling is remarkable. It should be noted that an even better agreement could have been obtained by incorporating the Donnan data into the data
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CONCLUSIONS A thermodynamic field theory for the electromechanical behavior of polyelectrolyte gels has been applied to the case of acrylamide/acrylic acid copolymer gels. Samples have been made at different cross-link densities, and with monomer compositions ranging from pure acrylamide to pure acrylic acid. The swelling degree and elastic moduli of the gel have been used to determine four free parameters of the model, for which physically plausible values were obtained. The fit of the swelling data is excellent, and the moduli are well represented, except at the highest cross-link densities in combination with high acrylic acid content. The model, in combination with the fitting parameters obtained in this manner, predicts the correct scaling of Donnan potentials with gel composition but underestimates the experimental values systematically by about 30%. Nonetheless, the model comes very close to quantitatively capturing the electromechanical behavior of the gels over a wide range of cross-link densities and gel compositions, in spite 1044
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of being based on very simple treatments of solvent−polymer interactions (Flory−Huggins), polymer network entropy (Flory−Rehner), and osmotic pressure (ideal solution theory).
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APPENDIX: ELASTIC CONSTANTS The linear elastic properties for a swollen gel can be derived from (eq 1). We assume that the gel is in equilibrium with a solvent, and has swollen to an isotropic state with Fij = δijλ0. An additional (small) displacement ui produces the strain Fij′ =
∂xi ∂ui ∂u ∂x + = Fij + i k ∂Xj ∂Xj ∂xk ∂Xj
(13)
Since Fij is isotropic, we find
Fij′ = λ 0(δij + uij)
(14)
where uij = ∂ui/∂xj is the infinitesimal strain. The stress given in (eq 1) depends on the strain through the determinant det F′, the explicit expression FikFjk, and through the osmotic pressure Π. In the following, the first order variations of these terms in the infinitesimal strain uij are derived. Making use of the identity det(I + A) = exp(tr log(I + A)), and expanding the matrix logarithm and exponential to first order in uij, we find
1/det(Fij′) ≈ λ 0−3(1 − ukk)
(15)
Equation (eq 4) expresses the swelling ratio λ03 in terms of the solvent concentration Cs. In the presence of an additional infinitesimal strain, (eq 4) becomes
λ 03(1 + ukk) = 1 + vsCs
(16)
Using this to substitute Cs in (eq 2) and expanding to first order in ukk, one finds
Π ≈ Π0 −
1 − 2χ ⎤ kT ⎡ vs(C+ + C −) ⎢ ⎥ukk + vs ⎢⎣ λ 03 λ 60 ⎥⎦
(17)
with the equilibrium osmotic pressure Π0 given by eq 3. By definition, the stress vanishes for uij = 0. Together with this, the expressions in eqs 14, eq 15, and eq 17 can be introduced into eq 1. Retaining only terms of first order, we obtain the Lamé formulation of isotropic linear elasticity (eq 8), with the Lamé constants given by eq 9.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Present Address §
Department of Materials Science and Engineering, Drexel University, Philadelphia PA.
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ACKNOWLEDGMENTS This work has been supported by the US National Science Foundation under Grant Number DMR 0647790. We gratefully acknowledge helpful discussions with Prof. Zhigang Suo, School of Engineering and Applied Science, Harvard University.
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REFERENCES
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