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Strain-Driven Swelling and Accompanying Stress Reduction in Polymer Gels under Biaxial Stretching Masayoshi Fujine,† Toshikazu Takigawa,† and Kenji Urayama*,‡ †

Department of Material Chemistry, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan Department of Macromolecular Science and Engineering, Kyoto Institute of Technology, Kyoto 606-8585, Japan

‡

S Supporting Information *

ABSTRACT: Strain-driven swelling and the accompanying stress reduction in polymer gels are investigated experimentally under general biaxial deformation, with strain being varied independently in two directions. The equilibrium degrees of strain-induced swelling and stress reduction depend markedly on both the magnitude and the type of the imposed deformation. When subjected to a constant elongation in one direction, the strain-induced swelling in biaxial stretching increases with an increase in the elongation in the other direction: When a gel is equibiaxially stretched by a factor of 2, the equilibrium volume is approximately 2.5 times greater than the initial volume. The degree of stress reduction in the direction of the smaller imposed strain is greater than that in the other direction: The stress in the constrained direction of planar extension is reduced by more than 40%. The time course of the stress reduction is governed by a diffusion process, i.e., the dynamics of swelling. The main features of the experimental results are satisfactorily explained by a classical theory for swelling of gels.

I. INTRODUCTION Polymer gels have attracted considerable attention from both academic and industrial viewpoints as unique soft solids that possess a particularly low elastic modulus. These gels are composed of flexible polymer networks and large amounts of solvent. A distinctive characteristic of gels is that they behave as thermodynamically semiopen systems that allows the transport of solvents through them. The degree of swelling (i.e., solvent content) is determined thermodynamically by the balance between the chemical potentials of the solvent inside and outside the gels. For instance, the degree of swelling of ionic gels is given by the gel volume that minimizes the total free energy, which includes contributions from network elasticity, isotropic mixing of a network and solvent, and Donnan-type potential. This feature makes the volume of gels responsive to a change in environmental parameters, such as temperature, solvent composition, and pH. The volume response to various types of external stimuli leads to potential applications of gels as soft actuators, sensors, and drug delivery systems.1−4 In addition to a change in the environmental parameters, externally imposed deformation (or stress) can induce a finite change in gel volume.5−15 The induced deformation significantly changes the elastic free energy of networks, resulting in a shift of the minimum of the total free energy to a new position, which is reflected as a finite change in gel volume. It is known that the gels swell further under a tensile strain and deswell under a compressive strain. This “strain-induced volume change” in the constrained state is observed as the dimensional changes in the unconstrained directions. When the imposed strain is kept constant, the external stress acting on the gels © XXXX American Chemical Society

decreases significantly with a change in gel volume. The time course of the change in gel volume and stress is governed by a diffusion process, and the characteristic time for this process depends primarily on the gel dimensions. The size-dependent characteristic time is a unique feature of force relaxation that results from the strain-induced volume change, which is substantially different from the force relaxation resulting from viscoelastic effects, featuring size-independent characteristic time. The characteristic time (τ) for typical polyacrylamide hydrogels with a thickness of 1 mm is a few hours; however, for fine gel fibers or thin gel films with a dimension of 1 μm, τ is on the order of milliseconds. Thus, the stress reduction phenomena, accompanied by a volume change under imposed deformation, can significantly affect the performance of gelbased actuators and sensors, with their effects becoming more pronounced in microfabricated and fiber gels that have particularly short characteristic times for swelling. It should be noted that even for a gel with fairly large dimensions, the effects of the strain-induced volume change on stress become significant when the strain rate is sufficiently slow to be comparable to the swelling rate.16,17 Gels undergo a complicated deformation process when employed as soft actuators or as industrial products. The magnitudes of the volume change and stress reduction are expected to depend on the type of deformation, although most of the previous reports5−7,13,14 have been limited to uniaxial Received: March 27, 2015 Revised: May 9, 2015

A

DOI: 10.1021/acs.macromol.5b00642 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules deformation. Uniaxial deformation is only a particular case among all physically possible deformations of gels.18 It was reported that a cross-linked rubber in marginal solvents showed a larger change in volume under equal biaxial extension than under uniaxial extension.19 However, to the author’s knowledge, there has been no corresponding study on anisotropic deformations, such as unequal biaxial stretching. Unequal biaxial deformation with strain being varied independently in two directions has been employed for the complete characterization of nonlinear elasticity in elastomers and gels because it covers a wide range of accessible deformation of elastomeric materials.20 Furthermore, the stress reduction caused by straininduced swelling under biaxial deformation has not been investigated yet, mainly because biaxial tensile measurements in solvents are not straightforward. The present study reports the volume change and stress reduction under various types of biaxial deformation for a polyacrylamide gel, which is one of the classical nonionic hydrogels, using a custom-built biaxial tester21 equipped with a water bath. We demonstrate that the degrees of volume change and stress reduction depend markedly on the type and magnitude of the imposed deformation. Further, we show that the degree of stress reduction differs considerably between the two directions when the imposed biaxial strains are unequal. We also compare the experimental results with predictions of the classical theory for swelling of gels. The results of this study provide an important basis for the fundamental understanding of swelling phenomena of gels and valuable information for the design of gel-based machines and devices that can be considered to be actuated in the presence of solvents.

Figure 1. Schematic diagrams of the gels under various states: (a) reference state (preparation state); (b) freely swollen state; (c) immediately after the imposition of a biaxial stretch of αx and αy on (b) (isovolumetric deformation); (d) equilibrium swollen state under a constant biaxial stretch of αx and αy; (e) immediately after the release of the imposed stretch from (d) (isovolumetric deformation). Note that the effective stretch governing the magnitude of tensile force is reduced from αi to αi/β∞ by the stretch-induced swelling, resulting in a reduction in tensile force.

σi (i = x, y, z) (i.e., force per cross section in the deformed state) is given by the derivative of ΔF with respect to λi:

II. THEORY The thermodynamics of strain-induced swelling and the accompanying stress reduction phenomena under general biaxial strain (Figure 1) can be described in the framework of the classical theory for swelling of gels. According to the Flory− Rehner model,22,23 the equilibrium swelling for nonionic gels is achieved through the balance between the resistive stress resulting from network elasticity and the expansive stress stemming from the osmotic pressure through the isotropic mixing of the network and solvent. The total change of free energy upon swelling (ΔF) is written as a sum of these two contributions (ΔFel and ΔFmix), using the classical rubber elasticity theory and the Flory−Huggins lattice model, respectively:

σi =

1 ⎛ ∂ΔF ⎞ 1 ⎛ ∂ΔF ⎞ ⎟ ⎜ ⎟= ⎜ l jlk ⎝ ∂li ⎠ V00λjλk ⎝ ∂λi ⎠

(2)

where lp (p = i, j, k) and V00 are the gel dimensions along the paxis and the gel volume in the reference state, respectively. Under a biaxial stretching of λx = λ0αx and λy = λ0αy at equilibrium, the stress in the z-direction must be zero σz /kBT = ns[ln(1 − ϕ) + ϕ + χϕ2] +

ncϕ0 4/3 ϕ001/3ϕαx 2αy 2

=0 (3)

where ns is the number of solvent molecules per unit volume, given by ns = Ns/(1 − ϕ)V, and nc is the number of network strands per unit volume, given by nc = Nc/V00. Equation 3 is also satisfied in the freely swollen state before stretching (αx = αy = 1 and ϕ = ϕ0):

ΔF /kBT = (ΔFmix + ΔFel)/kBT = Ns[ln(1 − ϕ) + χϕ] + Nc(λx 2 + λy 2 + λz 2 − 3)/2

⎛ ϕ ⎞1/3 ns[ln(1 − ϕ0) + ϕ0 + χϕ0 ] + nc⎜⎜ 0 ⎟⎟ = 0 ⎝ ϕ00 ⎠

(1)

2

where kB is the Boltzmann constant, T is the absolute temperature, Ns is the number of solvent molecules, Nc is the number of elastically effective network strands, ϕ is the volume fraction of network, and χ is the Flory−Huggins solubility parameter. The deformation along each axis λi (i = x, y, z) is measured from a reference state, i.e., the preparation state of the network. A gel in the preparation state with a volume of V00 and ϕ = ϕ00 undergoes an isotropic expansion of λ0 (= ϕ001/3ϕ0−1/3) by free swelling. We consider, here, the biaxial deformation of αx and αy is exerted on a freely swollen rectangular gel in solvents with a network volume fraction of ϕ0 and a volume of V0 (Figure 1). The total deformation from the reference state is given by λi = λ0αi (i = x, y, z). The true stress

(4)

From eqs 3 and 4, the strain-induced volume change (V∞/V0 = ϕ0/ϕ) for the highly swollen gels with ϕ ≪ 1 and ϕ0 ≪ 1 is given by V∞/V0 = (αxαy)2/3

(5)

The infinitesimal elasticity theory gives the relation between the strains εi (i = x, y, z) under biaxial deformation with a Poisson’s ratio of μ:24 μ (εx + εy) εz = − 1−μ (6) B

DOI: 10.1021/acs.macromol.5b00642 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules A general expression of the volume change in finite biaxial deformation is obtained using eq 6 and the true strains εi = log αi (i = x, y, z) as V∞/V0 = αxαyαz = (αxαy)(1 − 2μ)/(1 − μ)

Equations 10−12 can also be obtained from eqs 5, 8a, and 9a, respectively, using αy = αx−μ, with the corresponding values: μ = 1/4 in eqs 5 and 8a, and μ = 1/2 for isovolumetric deformation in eq 9a. The theoretical predictions for the strain-induced volume change (V∞/V0) and accompanying force reduction [(σni0 − σni )/σni0] are compared with the corresponding experimental results obtained under various types of deformation in a later section of this paper.

(7)

From eqs 5 and 7, the equilibrium Poisson’s ratio is found to be μ = 1/4. This value is identical to that derived for uniaxial deformation using the same model17 or different theories,25,26 and the similar values (1/6 and 0.278) were derived using a slightly different model7 and a scaling approach.27 The experimental values of μ in uniaxial deformation of highly swollen nonionic gels were close to these theoretical values.5,7,13,14,17 The equilibrium stresses in the two stretching directions are obtained from eqs 2 and 3, but the expressions for nominal stresses (i.e., force per cross section before stretching) are more convenient for comparison with the experimental results. The corresponding nominal stresses (σni ; i = x, y) can be expressed as ⎞ ⎛ 1 σxn = σxαyαz = G0⎜⎜αx − 5/3 2/3 ⎟⎟ αx αy ⎠ ⎝

(8a)

⎛ ⎞ 1 σyn = σyαzαx = G0⎜⎜αy − 5/3 2/3 ⎟⎟ αy αx ⎠ ⎝

(8b)

III. EXPERIMENTAL SECTION 1. Preparation of Poly(acrylamide) (PAAm) Hydrogels. Poly(acrylamide) (PAAm) gels were prepared by the radical copolymerization of an acrylamide and N,N′-methylenebis(acrylamide) (cross-linker), using ammonium persulfate as initiator. The mixture, with a total concentration of 20 wt %, was dissolved in distilled water. The molar ratio [monomer]/[cross-linker] was 660. Gelation was performed at 5 °C for 24 h in a mold. The resultant gels were allowed to swell in distilled water until equilibrium was attained. Following washing out of unreacted materials, the network concentration in the fully swollen state was 5.6 wt %. These fully swollen hydrogels were used for the tensile measurements. The monomer and cross-linker concentrations were optimized so that the gels in the fully swollen state could be stretched biaxially to a significant extent, and the resultant forces could be sufficiently large to measure. 2. Biaxial Tensile Measurements in Water. A custom-built biaxial tester, equipped with a water bath, was used for the measurements. The details have been described elsewhere.21 The fully swollen hydrogel sheets, with dimensions of 65 × 65 × 4 mm, were stretched biaxially in water up to the desired strain with a crosshead speed of 1 mm/s. As the time required to reach the destination strain (less than 100 s) was considerably shorter than the characteristic time for the strain-induced swelling for the gel used in the experiment (ca. 3 × 104 s), no appreciable swelling occurred in the stretching process. The imposed strain was kept constant after reaching the desired value, and the tensile force along each axis was measured as a function of time until the force reached equilibrium. We used equibiaxial extension (αx = αy), planar extension (αy = 1), and biaxial extension (αx/αy = 3/2), where αi (i = x, y) is the principal ratio defined by the ratio of the dimensions in the undeformed and deformed states of the fully swollen gels. The gels were stretched to a state of αx = αy = 2.25 prior to all the measurements; this corresponded to the maximum stretch examined here. The imposed deformation was then released. This preconditioning procedure was carried out to avoid unfavorable strain-hysteresis effects. In principle, the equilibrium volume in the deformed states (V∞) can be determined directly from the equilibrium thickness of the biaxially stretched gel sheets in the water bath. However, this measurement was precluded by experimental difficulties; in particular, almost no difference in refractive index was observed between the highly swollen hydrogels and surrounding water. We, therefore used the following alternative method: The imposed strain was released, after equilibration of the forces at each deformation, and the gel sheets were then removed from the clamps. The thickness of the gel in the relaxed state (d∞r) was measured in air using a laser displacement sensor LT-9500 and LT-9010 M (Keyence). The strain-induced volume change (V∞/V0) was determined from β∞ (Figure 1e) as

where G0 is the shear modulus in the fully swollen state before stretching, given by G0 = nckBTϕ01/3ϕ00−1/3. The initial stresses immediately after biaxial stretching without volume change (σni0; i = x, y) are obtained from ΔFel as ⎞ ⎛ 1 σxn0 = G0⎜⎜αx − 3 2 ⎟⎟ αx αy ⎠ ⎝

(9a)

⎞ ⎛ 1 σyn0 = G0⎜⎜αy − 3 2 ⎟⎟ αy αx ⎠ ⎝

(9b)

Equations 8 and 9 show that the strain-induced swelling causes a reduction in tensile forces from σni0 to σni (i = x, y). The straininduced swelling will lead to a decrease in the effective strain governing the magnitude of the external force. This is simply because, immediately upon the release of the imposed biaxial stretch, the gel dimensions (Figure 1e) are greater than those in the state before stretching due to the strain-induced swelling (Figure 1b). Notably, eqs 8 and 9 also indicate that the degree of force reduction, represented by (σni0 − σni )/σni0, not only depends on the magnitude of the imposed deformation but also differs between the two stretching directions. The corresponding theoretical results for uniaxial deformation were derived in earlier studies:17 V∞/V0 = αx1/2

σxn

⎛ 1 ⎞ = G0⎜⎜αx − 3/2 ⎟⎟ αx ⎠ ⎝

⎛ 1 ⎞ σxn0 = G0⎜αx − 2 ⎟ αx ⎠ ⎝

V∞/V0 = β∞3 = (d∞r /d0)3

(10)

(13)

where d0 is the gel thickness before stretching. Recovery to the initial volume occurred slowly after the release of the imposed strain and required about 1 day. As the measurement of d∞r was complete within 5 min after the strain release, the volume recovery process had no substantial effect on d∞r. The time required for the dimensional change from αiλ0 to β∞λ0 by isovolumetric deformation was less than 1 min. Although the clamped parts in the gels were not subjected to the imposed biaxial deformation, they did not have any significant

(11)

(12) C

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Macromolecules influence on the values of V∞/V0 because the volume of the clamped parts was less than 10% of the total volume. After the measurement of d∞r, the gels were left in water to undergo volume recovery to the initial state. The gels showed perfect recovery of volume after 1 day, which allowed us to use the same specimens for measurements under various deformations. All measurements were conducted at 25 °C. Uniaxial tensile experiments were conducted in the same way as biaxial experiments using a rectangular gel, with dimensions of 65 × 5 × 4 mm.

IV. RESULTS AND DISCUSSION Figure 2 shows the strain-induced volume change (V∞/V0) of the poly(acrylamide) (PAAm) hydrogels as a function of αx

Figure 3. Photographs of a polyacrylamide hydrogel sheet (a) in the fully swollen state before stretching, (b) immediately upon the release of an imposed biaxial stretch of αx = αy = 1.6 after equilibration of strain-induced swelling, and (c) after the re-equilibration of swelling in water in the strain-free state for 1 day. The volume of the gel becomes approximately twice under the imposed biaxial stretch, and it reverts to the initial value sufficiently long after the release of the imposed strain. The arrows indicate the location of the edges of the transparent gel sheet. Figure 2. Degree of strain-induced volume change as a function of αx in various types of deformation for a polyacrylamide hydrogel. The solid lines represent the theoretical predictions calculated using eq 5.

Figure 5. This indicates that the stress reduction in Figure 4 is primarily attributed to the strain-driven swelling whereas the viscoelastic contribution is negligibly small. An increase in volume from V0 to V∞ causes a reduction in effective elongation governing the magnitude of stress from αi to αi/ β∞ in the i-direction, which can be understood from comparison of Figures 1b and 1e. Figure 6 shows semilogarithmic plots of the normalized reduction of nominal stress versus time for the data of the planar extension in Figure 4a. The data points for σxn and σyn overlap over the entire time domain. This confirms that the stresses along the two different axes in unequal biaxial stretching obey the same dynamics. The data points fall on a single straight line over a wide t region, except for the shorttime region t < 5 × 103 s. This indicates that the swelling dynamics obey a multiexponential function, with the longest relaxation term having the greatest amplitude, as is the case of free swelling of gels:28,29

under various deformations. All types of stretching deformation drive further swelling (V∞/V0 > 1). In each deformation, the magnitude of volume change increases with an increase in αx. The volume change V∞/V0 in biaxial stretching increases with an increase in αy when compared at the same value of αx. The largest effect is with equibiaxial extension, followed, in order, by unequal biaxial extension (αx/αy = 3/2), planar extension (αy = 1), and uniaxial extension. Equibiaxial stretching leads to a large increase in volume. For instance, in the case of αx = 2.25, V∞ is approximately 3 times the value of V0. It should be noted that the strain-induced volume change is reversible; i.e., the volume slowly decreases to the initial value after release of the imposed strain (Figure 3). The solid lines in Figure 2 correspond to the theoretical prediction (eq 5) for each deformation. The theory satisfactorily describes the αx dependence of the volume change under various types of deformation. Figures 4a−c show the nominal stress (σni : i = x, y) along each axis as a function of time (t) after the imposition of the three types of biaxial deformation, i.e., equibiaxial stretching (αx = αy = 1.3), planar extension (αx = 1.8 and αy = 1), and unequal biaxial stretching (αx = 1.65 and αy = 1.1), respectively. The nominal stress is defined as the force per cross section before stretching. In each deformation, the stresses slowly decay and reach the equilibrium values in the region of t > 8 × 104 s. Several tens of percent of the total stress reduction occurs in the time region of t < 1 × 103 s. In the corresponding time scale, the gels stretched without surrounding water (i.e., in air) showed no appreciable reduction in stress, as can be seen in

σ n(t ) − σ n(∞) = σ n(0) − σ n(∞)

m

⎛ t⎞ ⎛ t⎞ ⎟ ≈ A1 exp⎜ − ⎟ (t > 5 × 103 s) ⎝ τ1 ⎠ ⎝ τi ⎠

∑ Ai exp⎜− i=1

The inverse of the slope of the corresponding straight line gives the longest characteristic time, τ1 ≈ 2.7 × 104 s. The diffusion coefficient of networks (D) is found to be D ≈ 5.7 × 10−10 m2/ s using the relationship D ≈ d∞2/τ1, where d∞ is the equilibrium thickness of the deformed gels. This value of D is of the same order of magnitude as has been reported for free swelling of several poly(acrylamide) hydrogels.28,29 Figure 7 illustrates the magnitude of the total reduction in nominal stress Δσni (Δσni = σni0 − σni (∞)) divided by the initial value (σni0) as a function of αx for various deformations. The relative stress-reduction Δσni /σni0 decreases with an increase in αx for each deformation, while Δσni /σni0 in biaxial stretching is D

DOI: 10.1021/acs.macromol.5b00642 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 5. Nominal stress as a function of time after the imposition of a biaxial stretch of αx = αy = 2 for a polyacrylamide hydrogel in air. No appreciable change in stress is observed.

Figure 6. Semilogarithmic plots of the normalized nominal-stress change versus time for the data in Figure 4b. The longest characteristic time (τ1) is evaluated to be 2.7 × 104 s from the inverse of the slope of the straight line.

reduction in effective strain on stress in the direction of the smallest imposed strain. The lines in the figure correspond to the theoretical predictions calculated using eqs 8, 9, 11, and 12. The main features observed in the experiments are well reproduced by the theory, although the theory tends to underestimate Δσn/σn0 in the x-direction under planar extension. The classical model employed here satisfactorily describes the experimental results for the degrees of strain-induced swelling and the accompanying force reduction. This good agreement, however, seems somewhat unexpected because our previous study21 showed that the neo-Hookean elastic energy, which is identical with Fel used here, failed to describe the nonlinear stress−strain relations in biaxial deformation for fully swollen polyacrylamide hydrogels. A more complicated function of Fel was required to describe the biaxial stress−strain data.21 The same tendency is also observed for the biaxial data of the gel used here, which is shown in the Supporting Information. It is also known that the ϕ- and ϕ0-dependences of the initial modulus for gels in good solvents deviate appreciably from the predictions of the classical

Figure 4. Nominal stresses in the two directions as a function of time after the imposition of various types of biaxial deformation for a polyacrylamide hydrogel in water.

greater than that in uniaxial stretching. In unequal biaxial stretching, Δσn/σni in the y-direction (i.e., the axis of the smaller strain) is larger than that in the x-direction. In particular, Δσn in the constrained (y-) direction for planar extension becomes greater than 40%. The strain-induced swelling occurs isotropically, resulting in the most pronounced influence of the E

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need to be considered when actuating microfabricated gels or fiber gels with particularly short characteristic time for swelling because they will markedly influence the mechanical performance of soft actuators and sensors driven in solvents.

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ASSOCIATED CONTENT

S Supporting Information *

Figure S1. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.macromol.5b00642.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (K.U.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Mr. Ryosuke Mishima at Kyoto Institute of Technology for the support of the experiments.

Figure 7. Degree of total reduction of nominal stress in each direction [Δσni : Δσni = σni0 − σni (∞)] as a function of αx for a polyacrylamide hydrogel in water. In the figure, Δσni is reduced by the initial nominal stress at each deformation (σni0). The lines represent the theoretical predictions calculated from eqs 8, 9, 11, and 12.

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REFERENCES

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model, and the excluded volume effect needs to be considered for interpretation.30 In fact, the polyacrylamide hydrogels show the relation G0 ∼ ϕ0−2.3 in the fully swollen state,31 which is in agreement with the theoretical prediction for gels in good solvents where the excluded volume effect is considered. The successful fit of the classical theory employed here results at least partly because V∞ and σn(∞) in Figures 2 and 7 are relative values compared to the initial ones rather than the absolute values. It is noteworthy that the simple classical theory successfully catches the main features of the observations, but a theory with more developed forms for Fel and Fmix will be required for the quantitative description of the experimental results on the basis of the absolute values.

V. CONCLUSION The degrees of strain-driven volume change (V∞/V0) and the accompanying stress reduction (Δσn/σn0) depend markedly on both the magnitude and the type of imposed deformation. In all types of stretching deformation, V∞/V0 increases and Δσn/σn0 decreases as the magnitude of the strain increases. When compared under the same stretch in one direction, V∞/V0 in biaxial deformation increases with an increase in the stretch in the other direction. In the case of equibiaxial deformation with αx = αy = 2.25, the gel volume increases to approximately 3 times the initial value. The induced volume change leads to a considerable reduction in tensile force as a result of a decrease in the effective strain governing the magnitude of stress. The time course for the force reduction is governed by the dynamics of swelling. In unequal biaxial stretching, Δσn/σn0 values are not identical in the two directions, and Δσn/σn0 in the smaller strain direction is considerably greater than that in the other direction: In the case of planar extension, Δσn/σn0 values in the constrained direction are greater than 40%. A classical model for swelling of gels satisfactorily describes the main features of the experimental results regarding the dependence of V∞/V0 and Δσn/σn0 on the magnitude and type of imposed deformation. The strain-driven volume change and accompanying stress reduction observed here are important factors that F

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