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Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Distinctive Characteristics of Internal Fracture in Tough Double Network Hydrogels Revealed by Various Modes of Stretching Thanh-Tam Mai,† Takahiro Matsuda,‡ Tasuku Nakajima,§,∥ Jian Ping Gong,*,§,∥ and Kenji Urayama*,† †

Department of Macromolecular Science & Engineering, Kyoto Institute of Technology, Sakyo-ku, Kyoto 606-8585, Japan Graduate School of Life Science and §Faculty of Advanced Life Science, Hokkaido University, N21W11, Kita-ku, Sapporo, Hokkaido 001-0021, Japan ∥ Soft Matter GI-CoRE, Hokkaido University, Sapporo, Hokkaido 001-0021, Japan Downloaded via UNIV OF SOUTH DAKOTA on July 10, 2018 at 02:18:08 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: The cyclic stretching measurements in various geometries including uniaxial, planar, unequal, and equal biaxial extension reveal the distinctive features of the internal fracture in the double network (DN) hydrogels with high toughness, which are composed of the rigid and brittle first network and the soft and ductile second network. The initial modulus, residual strain after unloading, dissipated energy (D), dissipation factor (Δ; the ratio of D to input strain energy), and the ultimate elongation of network strands (λi,m*) in each loading−unloading cycle are evaluated as a function of the imposed maximum elongation in the i-direction (λi,m, i = x, y) in each cycle. The modulus reduction and Δ depend on the stretching mode when compared at the same λi,m, but each of them exhibits a universal relation independently of the stretching mode when the corresponding magnitude of the deformation tensor (mm; mm = (I1,m2 − 2I2,m)1/2 where I1,m = λx,m2 + λy,m2 + λz,m2 and I2,m = λx,m2λy,m2 + λy,m2λz,m2 + λz,m2λx,m2) is used as a variable. This is in contrast to that Δ of the filler-reinforced elastomers, which undergo apparently similar mechanical hysteresis, which shows the corresponding universal relation using I1,m as a variable. The difference in governing variable indicates that the influence of the cross-effect of strains (λiλj; i,j = x,y,z and i ≠ j) on Δ is pronounced in the DN gels whereas it is minimal in the filled elastomers. Characteristically, λi,m* is close to λi,m in every type of deformation, indicating that in the end of the loading most of the chains with lower extensibility than λi,m undergo fracture whereas most of the long chains with higher extensibility than λi,m remain intact. The elongation λi,m* has no appreciable cross-effect of strains in contrast to the modulus reduction as well as Δ.

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INTRODUCTION

The classical types of DN gels were synthesized by a twostep free radical polymerization:11 The first step synthesizes a covalently cross-linked polyelectrolyte (first network), and subsequently, the second network of neutral polymer is made in the presence of the first network. The concentration of the second network is several tens of times larger than that of the first network, and the first network is tightly while the second network is loosely cross-linked.13 Recently, several novel methods including molecular stent,22 one-pot,23 extrusion three-dimensional printing technique,24,25 and free shapeable methods26 have been reported to fabricate the tough and functional DN gels. Nevertheless, the classical method using two-step polymerization has been widely employed to make the DN gels due to the advantage that a rich variety of chemical components can be employed as the constituents of the first and second networks.12 The excellent mechanical properties of the DN gels are characterized by the high values of elastic modulus of 0.1−1.0 MPa, ultimate tensile stress of 1−10 MPa, ultimate tensile

Polymer hydrogels have received considerable attention as a class of soft and wet materials in recent years.1 The hydrogels exhibit the unique properties such as large deformability with markedly low modulus, swelling/deswelling, shock absorbing, and low sliding friction.1 Because of these features, they have been applied to drug delivery,2 superabsorbents,3,4 actuators for optics and fluidics,5 contact lenses,6,7 artificial polymeric materials,8 and tissue engineering.9,10 Most of the hydrogels, however, are poor in mechanical strength due to high water content, which often becomes a major drawback in the industrial applications. Recently, much efforts have been devoted to developing the mechanically robust hydrogels. These approaches include double network hydrogels (DN gels),11−13 nanocomposite hydrogels,14−16 sliding-ring hydrogels,17 tetra-PEG hydrogels,18 and physical interaction enhanced hydrogels.19−21 The DN gels, which were first reported by some of present authors,11 are composed of the two contrasting and well-entangled chemically cross-linked networks: The first network is formed by rigid and brittle polyelectrolyte, while the second network is made of soft and ductile neutral polymer. © XXXX American Chemical Society

Received: May 15, 2018 Revised: June 21, 2018

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DOI: 10.1021/acs.macromol.8b01033 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Schematics of several types of deformation: uniaxial (U), planar (PE), unequal biaxial (UB), and equibiaxial extensions (EB). μ denotes the ratio of the two nominal strains (εi = λi − 1; i = x, y).

strains covers a wide range of accessible homogeneous deformation.40,41 The biaxial cyclic stretching data enabled to discuss the effects of the type and anisotropy of imposed strain field on the stress-softening phenomena. To our knowledge, the Mullins effect of tough DN gels has not yet been characterized by various modes of biaxial stretching. The biaxial cyclic loading experiments of the DN gels are expected to reveal the new aspects of the stress-softening phenomena resulting from the chain fracture. The biaxial cyclic stretching data will also provide significant information about the applicability of tough DN gels to industrial use because they are subjected to various complicated deformations when used in practical applications.12 The present study employs several types of biaxial deformation, i.e., planar extension (pure shear), unequal biaxial extension (with a ratio of 1/2 for the nominal strains in the two directions), equibiaxial extension, and uniaxial tension for the characterization of the internal fracture in tough DN gels. The classical types of DN gel, which are synthesized by a twostep free radical polymerization, are used as the specimens. The loading−unloading cycles are conducted in various deformation modes with increasing the maximum elongation in the i-direction (λim; i = x, y) in each cycle, using a custombuilt biaxial tester.36 The analysis of the loading−unloading curves provides the initial modulus (A), energy dissipation (D), the dissipation factor (defined as the ratio of D to the input strain energy), and ultimate elongation of network strands in each cycle (λim * ) as a function of λim in various deformation modes. We reveal how these characteristics in each cycle depend on λim as well as the deformation type and discuss the underlying mechanism of the stress-softening phenomena in tough DN gels. The results in present study will also provide the important basis of novel molecular design of tough hydrogels.

strain of 1000−2000%, and tearing fracture energy of 100− 2000 J m−2.12,26 A key of the high mechanical performance of the DN gels is the internal fracture of the chains in the rigid and brittle first network which are served as the sacrificial chains.27 The large fracture energy is governed by the dissipated energy of the internal fracture of the sacrificial network chains in the damage zone.1,13,27,28 The fracture of sacrificial network strands is clearly recognized by the mechanical softening in the loading−unloading stretching29,30 as well as the yielding and necking in uniaxial extension.13,31,32 The stress softening in cyclic stretching has been known commonly as the Mullins effect, especially in filler-reinforced elastomers:33,34 The stresses during reloading and unloading become considerably smaller than those during the virgin loading. In the case of filled elastomers, the Mullins effect mainly originates from the destruction of the physical interaction in inherent structures such as filler networks and filler−polymer interfaces.35−38 This is why most of the filled elastomers subjected to cyclic stretching can recover the original tensile properties by appropriate annealing treatments.33,34 In contrast, the mechanical softening in DN gels is irreversible because it results from the fracture of the covalent bonds.27,29,30 The Mullins effect of tough DN gels was investigated by several researchers using uniaxial stretching and compression.29,30,39 They correlated the progress of the internal fracture of the first network with a decrease in initial Young’s modulus (E) accompanying an increase in maximum elongation in each stretching cycle (λm). Creton et al.29 showed that the λm dependence of E is not sensitive to whether the deformation type is stretching or compression when the compression is assumed as equibiaxial stretching. The authors also indicated that the energy dissipation under loading− unloading cycles in DN gels is totally attributed to the irreversible fracture of the chemical network strand of the first network without appreciable viscoelastic effect.29 Some of the present authors30 further investigated the uniaxial tensile behavior of the DN gels using loading−unloading cycles in the whole strain regime including prenecking, necking, and strainhardening aspects. Hong et al.39 proposed a damage function to characterize the partial damage caused by imposed stretching. They fitted the damage function, where the damage rate is expressed by a log-normal distribution function, to the data of the λm dependence of E and estimated the elongation at which the fracture of the network strands occurs most frequently. All of these earlier studies employed exclusively uniaxial deformation as the imposed deformation mode. Uniaxial deformation, however, gives a limited basis for the comprehensive understanding of the nonlinear mechanical behavior because it is only a particular one among physically admissible deformations of elastomeric materials. Biaxial stretching with various combinations of the two orthogonal

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EXPERIMENTS

Materials. 2-Acrylamido-2-methylpropanesulfonic acid sodium salt (NaAMPS, Toa Gosei, Co., Ltd.) was used as received. Acrylamide (AAm, Junsei Chemicals, Co., Ltd.) was recrystallized from acetone. N,N′-Methylenebis(acrylamide) (MBAA, Wako Chemical Industries, Co., Ltd.) was used as received. 2-Oxogultaric acid (Wako Chemical Industries, Co., Ltd.) was used as received. Soda-lime glass plates (thickness: 3 mm) were used for the gel synthesis. Synthesis of DN Gels. The DN gels were synthesized by a twostep free radical polymerization. First, the first network precursor aqueous solution containing 1 M NaAMPS monomer, 3 mol % MBAA as cross-linking agent, and 1 mol % 2-oxoglutaric acid as photoradical initiator was prepared. All the molar percentages are referred to the monomer concentration. The solution was poured into the mold prepared with the two glass plates and the silicone rubber (thickness: 0.5 mm) as a spacer. The PNaAMPS gels were then synthesized by irradiation of 365 nm UV light (4 mW/cm2) to the glass mold from two sides for 8 h under an argon atmosphere. The resulting PNaAMPS gels were then immersed in the second network B


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were completely dried at 80 °C in a vacuum, and the weights in the dry states (wd) were measured. The degree of swelling (Q) was evaluated by the relation of Q = ws/wd. The effect of the degree of elongation on Q was also investigated using the subsamples prestretched by uniaxial elongation.

precursor aqueous solution containing 2 M AAm monomer, 0.01 mol % MBAA, and 0.01 mol % 2-oxogultaric acid for at least 12 h until swelling equilibrium was reached and then sandwiched with the two glass plates. The PAAm networks were then synthesized in the presence of the PNaAMPS networks by irradiation of 365 nm UV light (4 mW/cm2) for 8 h under an argon atmosphere. The synthesized PNaAMPS/PAAm DN gels were soaked in pure water before measurements for at least 7 days. Tensile Measurements. The stretching measurements were performed using three types of biaxial deformation, i.e., planar (PE), unequal biaxial (UB), and equibiaxial (EB) extensions, and uniaxial extension (U) (Figure 1). Biaxial and uniaxial extensions were conducted at 25 °C by a custom-built biaxial tester and a TC-500NA3 (T.SE. Co.), respectively. In the biaxial extensions, the sheet samples with dimensions of 65 × 65 × 1.5 mm3 (gauge length; L0 = 50 mm) were deformed with the constant crosshead speeds of Vx and Vy in two orthogonal directions (x- and y-directions), and the resulting tensile force was measured in each direction. The details of the biaxial tester were described elsewhere.36,42 In EB, the specimens were stretched equally in the x- and y-directions with Vx = Vy = 1.0 mm s−1. In PE, the specimens were stretched uniaxially in the xdirection with Vx = 1.0 mm s−1, while the dimension in the y-direction was kept unchanged. In UB, the specimens were stretched unequally with Vx = 1.0 mm s−1 and Vy = 0.5 mm s−1, so that the ratio of nominal strains εy/εx could be kept 0.5. In U, the rectangular specimens with dimensions of 65 × 6.0 × 1.5 mm3 (L0 = 35 mm) were stretched with Vx = 0.70 mm s−1 so that the initial strain rate (Vx/L0) could be the same as that in the biaxial extensions. It should be noted that no strain-rate effect on the stress−strain behavior in loading and unloading processes is observed, which reflects the purely elastic response of the DN gels.29,30,43 The real values of elongation during biaxial and uniaxial stretching were evaluated from the dimensional changes of several local grids in the sample sheets. The detailed methods for the evaluations of the real elongation and the effective cross section for the detected load are given in the Supporting Information of a previous paper.36 The value of the gauge length (L0 = 50 mm) is so large that the area of the inhomogeneous strain field around the clamps can be negligibly small relative to that of the uniform strain field. For the investigation of the internal fracture during loading, the specimens were stretched with the loading−unloading cycles using various degrees of the maximum elongation (λm) in each type of extension. The same crosshead speeds were employed in the loading and unloading processes. The values of λm were increased stepwise from 1.04 up to 1.62, 1.88, 1.79, or 1.97 for EB, UB, PE, or U, respectively. The examined range of λm in each type of extension corresponds to the prenecking region, the details of which will be described later. To avoid drying the gel specimens during the measurements, the loading−unloading cycles were conducted successively with no interval time between every cycle. The time required to complete the total cycles was less than 10 min, which was short enough to exclude the effect of drying on the stress−elongation behavior. The onset time for the pronounced effect of drying on the mechanical properties was examined in the preliminary experiments (Figure S2 of the Supporting Information). The cross section and gauge length in the undeformed state of the virgin specimens were employed to calculate the nominal stress and elongation throughout the loading−unloading cycles. The residual strains immediately after the complete unloading were also evaluated in each cycle for all types of deformation. The satisfactory reproducibility of the stress−strain behavior in the virgin loading as well as the cyclic stretching was confirmed by repeating the measurements at least three times for each type of deformation. Swelling Measurement. Swelling experiments were performed using the specimens prestretched by a single loading−unloading cycle at λx,m = 1.5 with various types of deformation, i.e., U, PE, UB, and EB. The subsamples were cut from the area subjected to uniform strain field in the prestretched samples, and they were allowed to swell in water for 1 week to attain the equilibrium. After the measurements of the weights of the fully swollen subsamples (ws), the subsamples

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RESULTS Virgin Loading in Various Types of Extension. Figure 2 shows the nominal stress (σi (i = x, y))−elongation (λx)

Figure 2. Nominal stress−elongation curves of DN gels for uniaxial (U), planar (PE), unequal biaxial (UB) with εy/εx = 0.5, and equibiaxial (EB) extensions. The cross symbols denote the rupture of the specimens. The uniaxial data in the entire strain range are given in Figure S1.

relationships for the DN gels in various types of deformation, i.e., uniaxial (U), planar (PE), unequal biaxial (UB), and equibiaxial (EB) extensions. The x- and y-directions correspond to the directions of the larger and smaller strains, respectively. As the stresses in the two directions in EB agree with each other within experimental error, only the data in the x-direction are shown in the figure. The value of σ in EB is the largest and followed by UB-x, PE-x, U, UB-y, and PE-y when compared at the same value of λx. The cross symbols in the figure represent the point of macroscopic rupture. In the case of uniaxial stretching, the specimens undergo inhomogeneous necking (yielding) deformation in the large λx region of λx > 2, which is featured by the plateau region in the σ−λ curve. The shoulder in the σ−λ curves is a sign of the onset of yielding, and it can also be recognized near the rupture point in biaxial stretching. The necking propagation during biaxial stretching, however, was not clearly observed because the specimens underwent macroscopic rupture immediately after the onset of the necking around the chucks. Thus, the present study focuses on the “prenecking region” in each deformation, i.e., the strain region up to the rupture point for biaxial stretching, and the region of λx < 2 for uniaxial stretching. The detailed investigation of the uniaxial yielding behavior of the DN gels was made in earlier studies.30,31 The uniaxial σ−λ curve in the entire strain region is given in the Supporting Information (Figure S1). Loading−Unloading Cycles in Various Types of Extension. Figures 3−6 illustrate the σ−λ curves in the cyclic stretching with various values of the maximum elongation (λm; numerals in the figures) in uniaxial, planar, unequal, and equal biaxial extensions, respectively. The σ−λ data of U, PE, and UB for the small and large values of λm are separately displayed in the two panels. The features of the stress softening seen previously in uniaxial cycle stretching29,30 are also observed in every type of biaxial stretching: The stresses during reloading C


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Figure 3. Nominal stress−elongation relation of loading−unloading cycles for DN gels under uniaxial extension in the regions of (a) 1.04 ≤ λm ≤ 1.44 and (b) λm > 1.44. The numerals denote the values of λm in each cycle. The solid and dashed curves indicate the loading and unloading processes, respectively.

Figure 4. Nominal stress−elongation relation of loading−unloading cycles for DN gels under planar extension in the stretching direction in the regions of (a) 1.04 ≤ λm ≤ 1.44 and (b) λm > 1.44 and in the constrained direction in the regions of (c) 1.04 ≤ λm ≤ 1.44 and (d) λm > 1.44. The numerals denote the values of λm in each cycle. The solid and dashed curves indicate the loading and unloading processes, respectively.

onset of residual strain is observed in each deformation: λm ≈ 1.15 for EB and λm ≈ 1.2 for other types of stretching. Swelling Behavior after Various Types of Preextension. Figure 8a shows the swelling degrees (Q) as a function of λx,m for the specimens prestretched by uniaxial extension. The swelling degree reflects the number of elastic network chains per unit volume (ν) in cross-linked polymer networks.40 The comparison of Q for the specimens with and without prestretching provides the information about the degree of first network bond fracture by prestretching. When λx,m is larger than a threshold value (ca. 1.2), Q becomes larger than that for the virgin specimen (Qvir). The values of Q almost linearly increases with an increase in λx,m, which qualitatively agrees with the results in earlier studies.30 Figure 8b displays Q as a function of λy,m for the specimens prestretched by various

and unloading processes are lower than those during the virgin loading; After λ exceeds λm in the previous loading, the σ−λ relation follows the path of the virgin loading. Importantly, the reloading curve coincides with the corresponding unloading curve at each λm in all types of extension. This coincidence indicates that the stress softening of the present DN gels is purely elastic in origin, and it is mainly governed by the internal brittle fracture of the chains without an appreciable viscoelastic effect.30 Figure 7 shows the residual strains (ε r ) evaluated immediately after complete unloading in each cycle. Importantly, the magnitudes of εr remain very small (less than 5%), although they increase with an increase in λx,m in each type of stretching. A finite threshold value of λm for the D


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Figure 5. Nominal stress−elongation relation of loading−unloading cycles for DN gels under unequal biaxial extension (εy/εx = 0.5) in (a) the direction of larger strain (x-direction) and (b) the y-direction. The numerals denote the values of λm in each cycle. The solid and dashed curves indicate the loading and unloading processes, respectively.

curve at small strain for all types of extension. Figure 9a illustrates Ai (i = U, PE-x, PE-y, UB-x, UB-y, and EB) as a function of λx,m. The values of A for the virgin specimens (A0) in each type of extension are also indicated in the figure. The PE‑y UB‑x UB‑y EB :A0 :A0 in the experiments is relation of AU0 :APE‑x 0 :A0 :A0 close to the expectation of linear elasticity theory for incompressible materials, i.e., 3:4:2:5:4:6.40,41,44 The values of A in all types of extension decrease with an increase in λx,m, and they become almost similar in the sufficiently large λx,m regime. The reduction in A by loading is characterized by the dependence of the ratio Ai/A0i on λx,m (Figure 9b). The ratio Ai/Ai0 in each type of extension decreases dramatically with an increase of λx,m, when λx,m exceeds the threshold value (λcx,m). The presence of λcx,m was observed in the previous studies investigated by uniaxial stretching.30,45 It is important to note here that λcx,m is sensitive to the type of extension; i.e., λcx,m for U (λcx,m = 1.12) is the largest and followed by PE (λcx,m = 1.08), UB (λcx,m = 1.06), and EB (λcx,m = 1.03), which is shown in the inset of Figure 9b. The details of the effect of stretching mode on λcx,m will be discussed later. The reduction in A with an increase in λx,m is attributed to the fracture of PNaAMPS chains (first network component). In fact, the Young’s modulus of the single PAAm gels (second network component) is negligibly small (about 18 kPa)46 relative to that of the DN gels (260 kPa), and the single PAAm gels show no appreciable mechanical hysteresis under the uniaxial loading−unloading process.47 Some earlier studies29,30,48 directly correlated the reduction in AU with the fraction of the fractured PNaAMPS chains (first network component) out of the elastically effective PNaAMPS chains (ϕm), i.e., ϕm ≈ 1 − (AU/AU0). This assumption, however, is crude because the first polyelectrolyte network undergoes a markedly high swelling from the preparation state, with the result that the modulus in the swollen state does not obey the conventional picture of rubber elasticity; i.e., A is not only simply proportional to the number of elastically effective network strands. The modulus of the highly swollen polyelectrolyte gels is pronouncedly influenced by the finite extensibility effect of highly stretched chains49,50 and the stiffening effect of the network strands caused by the internal Donnan equilibrium pressure.51 In fact, the Young’s modulus of the present specimen in the fully swollen state (AU0 = 260 kPa) is more than 10 times higher than that of the as-prepared first network with the corresponding composition (19 kPa), despite the dilution by swelling. Moreover, a broad size

Figure 6. Nominal stress−elongation relation of loading−unloading cycles for DN gels under equibiaxial extension in the x-direction. The numerals denote the values of λm in each cycle. The solid and dashed curves indicate the loading and unloading processes, respectively.

Figure 7. Residual strains in the x-direction evaluated immediately after complete unloading (εr) as a function of λx,m in uniaxial (U), planar (PE), unequal biaxial (UB), and equibiaxial (EB) extensions for the DN gels.

types of extension with λx,m = 1.5. At the same λx,m, Q increases with an increase in λy,m. This demonstrates that the degree of bond fracture is influenced by the degrees of strain in the two directions.

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DISCUSSION Reduction of Initial Modulus in Various Types of Extension. Initial elastic modulus (Ai; i = U, PE, UB, and EB) in each cycle is evaluated from the initial slope of the reloading E


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Figure 8. (a) Swelling degree (Q) as a function of λx,m for the DN gels prestretched by uniaxial stretching. Qvir denotes Q for the virgin specimen. (b) Swelling degree as a function of λy,m for the DN gels prestretched by various types of extension with λx,m = 1.5.

Figure 9. (a) λx,m dependence of initial elastic modulus for the reloading process in uniaxial (AU, Young’s modulus), planar (APE‑x and APE‑y in xand y-directions, respectively), unequal biaxial (AUB‑x and AUB‑y in x- and y-directions, respectively), and equibiaxial (AEB) extensions. The value of A for the virgin specimen (A0) in each type of extension is A0EB = 0.57 MPa, A0UB‑x = 0.45 MPa, A0UB‑y = 0.35, A0PE‑x = 0.33 MPa, A0U = 0.26 MPa, and A0PE‑y = 0.22 MPa. (b) Ratio Ai/Ai0 as a function of λx,m for each type of extension (i = U, PE-x, PE-y, UB-x, UB-y, or EB).

first networks as a random process and expresses the rate of the damage using the log-normal distribution function:39 ÄÅ ÉÑ 2Ñ ÅÅ η0 ÅÅ 1 ijj λx ,m − 1 yzz ÑÑÑ dη expÅÅÅ− 2 jjln =− z ÑÑ ÅÅ 2d jk λ 0 − 1 zz{ ÑÑÑ dλx ,m 2π λx , md ÅÇ ÑÖ (1)

distribution of the network strands in the considerably heterogeneous structure of the first network further complicates the relation between Ai/Ai0 and ϕm. Nevertheless, as the reduction in A is certainly caused by the fracture of PNaAMPS chains, we employ here the reduction of A as a simple (qualitative) measure of the degree of chain fracture caused by loading, without correlating directly it with ϕm. The ratio Ai/Ai0 depends on the type of extension when compared at the same λx,m (>λcx,m): EB shows the lowest values, followed by UB and PE, and U, which is consistent with the results in the swelling experiments for the prestretched samples (Figure 8b). At sufficiently high λx,m, the values of Ai/Ai0 become almost constant (ca. 0.3) independently of the type of extension. It should be remembered that the corresponding high λx,m region is close to the onset strain for inhomogeneous necking deformation. This result implies that almost the same fraction of elastically effective PNaAMPS chains in the virgin specimens undergo fracture in every type of extension just before the onset of necking deformation. A model of partial damage for the uniaxially stretched DN gels was proposed by Hong et al.39 The function η, which characterizes the partial damage, measures the initial stiffness of the damaged state relative to that of the virgin state and is intended to correlate with the fraction of the intact polymer chains in the first network: η (η ≡ AU(λx,m)/AU0 ) is a monotonically decreasing function of maximum stretch λx,m in uniaxial stretching and η = 1 in the virgin specimens. This model treats the damage process of the individual chains in the

where η0 is a dimensionless numerical factor for normalization, d is a parameter governing the width of the probability distribution, and λ0 approximately corresponds to the stretch at which the rate of the damage, dη/dλx,m, becomes maximum. The function η(λx,m) is obtained by integrating eq 1 as η(λx ,m) ≡

AU (λx ,m) A 0U

=1−

ÅÄ ÑÉÑ λx ,m − 1 yz η0 ÅÅÅ ij 1 Ñ zz + 1ÑÑÑ ÅÅerfjj ln j z ÑÑ 2 ÅÅÅÅÇ jk d 2 λ 0 − 1 z{ ÑÑÖ

(2)

A similar form but using the maximum free energy density as a variable was proposed by Ogden et al.52 to describe the Mullins effect in filled elastomers. Equation 2 using λx,m as a variable was fitted to the data in uniaxial stretching30,39 and compression.39 Figure 9b, however, shows that the values of A/ A0 considerably depend on the type of extension when compared at the same λx,m. The λx,m dependence of A/A0, which is featured by the slope in the figure, becomes stronger in the order of U, PE-x, UB-x, and EB. Obviously, the ratios A/ A0 (i.e., η) in various stretching modes are not a single function of λx,m, and they require the other measures of the degree of deformation. This is not surprising because the input mechanical work (W), which is given by W = ∫ σx dλx + ∫ σy F


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Figure 10. Initial modulus ratios (Ai/Ai0) as a function of (a) (I1,m − 3) and (b) (mm − √3) in each type of extension (i = U, PE, UB, or EB). The solid line in (b) represents the fitting curve by eq 5.

dλy, increases in order U < PE < UB < EB when compared at the same λx,m. The principal invariants Ii (i = 1, 2, 3) of the deformation gradient tensor (F) and the magnitude (m) of the left Cauchy−Green deformation tensor (B; B = FFT where FT is the transpose tensor of F) have been used as a measure of the degree of deformation to describe the hyperelastic behavior.33,53,54 The principal invariants Ii (i = 1, 2) and m for incompressible materials (I3 = λx2λy2λz2 = 1) are expressed using the principal ratios as I1 = tr B = λx 2 + λy 2 + λz 2

(3a)

I2 = [I12 − tr(B2 )] /2 = λx 2λy 2 + λy 2λz 2 + λz 2λx 2

(3b) Figure 11. Rate of the damage, dη/dmm, as a function of (mm − √3) obtained from the fitted curve in Figure 10b.

and m= B =

(B · B ) ≡

(tr B2 ) ≡

(I12 − 2I2)

(4)

λx,m as the stretch at which the elastically effective first network strands in the DN gels break most frequently. As mentioned before, however, it is crude to relate directly the reduction in A with the fraction of the fractured chains due to the significant effects of finite chain extensibility and stiffening of the network strands on modulus in the highly swollen polyelectrolyte gels. Thus, the physical meaning of mm,max becomes less evident, and mm,max should be regarded as the degree of deformation at which the damage rate is maximized by the chain fracture. The success of eq 5 in every type of extension and independent of the strain direction implies that the induced partial damage can be treated to be isotropic even if the imposed deformation is anisotropic. The earlier study, however, points out the presence of finite mechanical anisotropy in the uniaxially prestretched DN gels on the basis of their anisotropic swelling behavior.30 The effect of the induced mechanical anisotropy is not definitely observed in the values of A/A0 obtained from the two directions in anisotropic biaxial extensions (PE and UB). This may be because the ratio of the moduli evaluated by the same stretching mode before and after prestretching is not sensitive to the induced mechanical anisotropy. The induced anisotropy by the Mullins effect in filled elastomers has clearly been revealed by uniaxial stretching of the ribbon specimens cut out in various directions from the samples predeformed in various stretching modes38,42,56,57 or on the strain-induced light emission using cross-linker molecules with mechanoluminescent ability.58 The characterization of the induced mechanical anisotropy in the prestretched DN gels by these methods will be made in separate study.

where tr denotes the trace operations of a tensor. The quantity m corresponds to the length of the vector Λ = (λx2, λy2, λz2).55 Figures 10a and 10b show the ratios Ai/Ai0 (i = U, PE-x, PEy, UB-x, UB-y, and EB) as a function of I1,m and mm, respectively, each of which is the maximum of I1 and m in each cycle. In the figures, the quantities (I1,m − 3) and (mm − √3) are employed in the horizontal axis in order that they shall vanish automatically in the undeformed state (I1 = 3 and m = √3). Compared to Figure 9b using λx,m as a variable, the data in various stretching modes tend to overlap each other in both figures. In particular, all data are well collapsed into a single curve when mm is used as a variable (Figure 10b). This shows that the ratios A/A0 in all types of extension are expressed as a single function of mm. The following equation, which is obtained by replacing λx,m with mm in eq 2, successfully describes the data in Figure 10b. ÄÅ ÉÑ ÑÑ η0 ÅÅÅ ij 1 Ai (mm) mm − 3 yzz Ñ j Å zz + 1ÑÑÑ η(mm) ≡ = 1 − ÅÅerfjj ln i j z Å ÑÑ 2 ÅÅÇ k d 2 m0 − 3 { A0 ÑÖ (i = U , PE‐x , PE‐y , UB‐x , UB‐y , and EB)

(5)

The solid line in Figure 10b represents the fitted result using the parameter values η0 = 0.8, d = 0.9, and m0 = 2.34. Figure 11 displays the rate of the damage, dη/dmm, which is obtained from eq 5 with the fitted values, as a function of mm. The damage rate increases steeply with an increase in mm in the small deformation region and reaches the maximum at mm,max ≈ 2.0. The condition mm,max ≈ 2.0 corresponds to λx,m ≈ 1.17, 1.22, 1.28, and 1.30 for EB, UB, PE, and U, respectively. The earlier studies30,39 considered the corresponding value of G


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Figure 12. (a) Schematics for the evaluations of energy dissipation (D) on the basis of virgin loading and reloading curves with a single stretching cycle for general biaxial deformation. The λx,m dependence of (b) D and (c) the input work in the virgin loading (W0) in various types of deformation, i.e., uniaxial (U), planar (PE), unequal biaxial (UB), and equibiaxial (EB) extensions, for the DN gels.

Energy Dissipation in Various Types of Extension. The energy dissipation (D) in each type of extension is evaluated graphically from the area enclosed by the virgin loading and reloading curves, which is schematically shown in Figure 12a.29,30,42,53 The dissipation D is also expressed by the difference in the input work to the virgin sample and reloading processes with the maximum elongation λm (designated as W0 and Wr, respectively): D = W0 − Wr

(6)

These input work W0 and Wr are obtained by the relation of Wi = Wi,x + Wi,y = ∫ σx dλx + ∫ σy dλy (i = 0, r) where Wi,x and Wi,y are measured from the σj−λj (j = x, y) curves, respectively. It should be recalled that the unloading and reloading curves in the prenecking region of interest are identical (Figures 3−6) due to the purely elastic character of the present DN gels, and thus we do not need to distinguish the unloading and reloading processes. Figures 12b and 12c illustrate D and W0 as a function of λx,m for each type of extension, respectively. The values of D and W0 depend on λx,m and the type of extension in a qualitatively similar way, excepting the presence of a threshold value of λcx,m for the emergence of D. Both quantities increase with an increase in λx,m, and EB shows the highest values, followed by UB, PE, and U for the same λx,m. Dissipation factor, which is defined by the ratio of D to W0, corresponds to the fraction of the dissipated energy to the input work in the virgin loading (W0):36 Δ=

D W0

Figure 13. Dissipation factor (Δ) as a function of λx,m in various types of deformation. The inset shows the data in the small regime of λx,m.

an increase in λx,m when λx,m > λcx,m, and Δ is sensitive to the type of deformation when compared at the same λx,m (>λcx,m). The value of λcx,m for Δ in each type of deformation is almost similar to those for Ai/A0i (Figure 9b) and εr (Figure 7). At sufficiently large values of λx,m, Δ in each type of extension becomes almost constant, Δ ≈ 0.65. This means that ca. 65% of the input work dissipates at the strains near the onset strain of necking deformation. Figures 14a and 14b illustrate Δ as a function of (I1,m − 3) and (mm − √3), respectively, in order to compare the values of Δ in different types of extension at the same magnitude of deformation. As in the case of Ai/Ai0 (Figure 10b), all data of Δ in various types of extension fall on a single curve when mm is used as a variable (Figure 14b), although the corresponding D−mm relations are not represented by a single curve (Figure S3b). This clearly demonstrates that Δ in the present DN gels

(7)

Figure 13 shows Δ as a function of λx,m in each type of extension. As in the case of D (Figure 12b), Δ increases with H


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Figure 14. Dissipation factor (Δ) as a function of (a) (I1,m − 3) or (b) (mm − √3) in various types of deformation. The inset of (b) shows the data in the small deformation regime. The solid line represents the fitted result of eq 8.

Figure 15. Dissipation factor (Δ) versus (1 − Ai/Ai0) at the same values of λx,m in each type of extension for (a) the DN gel and (b) the silica-filled styrene−butadiene rubber (reloading). The data in ref 36 are used in (b).

the interfaces between filler and rubber and the slippage of polymer chains on the surfaces of filler.35,63−67 In the case of filled elastomers, the contributions of the fracture of polymer networks (i.e., matrix rubber) to Δ and the reduction in A are modest especially at moderate deformation (λ < 2),36,37,57,58 in contrast to the case of the DN gels, which explains the different results in Figure 15 between the stress-softening phenomena in the DN gels and the filled elastomers. As is naturally expected from the similarity of Δ and (1 − Ai/Ai0), the mm dependence of Δ for the DN gels is well described by the analogous equation to eq 5: ÅÄ ÑÉÑ ÑÑ Δ0 ÅÅÅÅ jij 1 mm − 3 zyz Δ= ln z + 1ÑÑÑ ÅÅerfjjj z z Å ÑÑ 2 ÅÅÇ k d 2 m0 − 3 { (8) ÑÖ

is governed by mm independently of the type of extension. In the plots using I1,m as a variable, none of the data of Δ (Figure 14a) and D (Figure S3a) form a single curve. These features are in contrast to the results of the apparently similar stress softening in silica-filled elastomers.36 For the silica-filled elastomers, the Δ−I1,m relations for various types of extension are well represented by a single curve, while the Δ−mm relations are not (Figure S6). The difference in governing factor for Δ between the DN gels and silica-filled elastomers results from the difference in the main origin of Δ. The difference in the main origin of Δ between these two types of material becomes obvious in the relation of Δ and Ai/Ai0. Figure 15 displays the relations of Δ and (1 − Ai/Ai0) at the same values of λx,m in each type of extension for the DN gel and the silica-filled styrene−butadiene rubber (silica/SBR). In the case of the DN gel, all data points are approximated by a straight line with a slope of unity which represents the equality of Δ and (1 − Ai/Ai0) (Figure 15a). This relation indicates that Δ has qualitatively the same meaning as (1 − Ai/Ai0) in the DN gels, although Δ is obtained in the entire strain regime while A reflects the linear elastic response at small deformation. The dissipation in the DN gels totally originates from the elastic fracture of the chains in the first network even at large deformation, independently of extension type. In contrast, the corresponding data in silica/SBR markedly deviate from the relation of the equality (Figure 15b). Evidently, Δ and the reduction in A are different in main origin. The reduction in A (at small deformation) mainly originates from the destruction of the filler network,59−62 while Δ (at large deformation) primarily reflects the destruction of

The solid line in Figure 14b depicts the fitted result by eq 8 with the parameter values Δ0 = 0.71, d = 0.9, and m0 = 2.34. The fitted values of d and m0 are the same as those employed in Figure 10b, and the values of Δ0 (= 0.71) and η0 (= 0.80) employed are almost similar. The rate of Δ, dΔ/dmm, with these parameters exhibits the maximum at mm ≈ 2.0 (Figure 16), which is similar to the corresponding result for Ai/Ai0 (Figure 11). The similarities in the fitting results for Ai/Ai0 and Δ confirm again that these quantities have the same origin, i.e., elastic fracture of the elastically effective chains in the first network, although they are evaluated at largely different degrees of strain. As is evident from the definitions eq 4, m explicitly includes the contribution of the cross-effect of strains between different axes λiλj (i, j = x, y, z; i ≠ j). The present results indicate that I


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ultimate elongation as a function of λx,m by fitting the Gent hyperelastic model40,68 to the uniaxial loading curves. The elastic free energy of the Gent model is given as W (I1) = −

I − 3 zyz G * ji zz (Im − 3) lnjjj1 − 1 j 2 Im* − 3 z{ k

(9)

where G and I*m are shear modulus and the maximum value of I1 where the stress becomes infinite. The σi−λi (i = x, y) relations in each type of deformation (EB, UB, PE, U) derived from eq 9, which are given in the Supporting Information (eq S4), are fitted to the reloading curves using I*m and G as two adjustable parameters. The fitted results for PE-x, PE-y, UB-x, and UB-y are shown in Figure 17, parts a, b, c, and d, respectively. The corresponding results for U and EB are also given in Figure S4. The present fitting attaches importance to the high λ region in order to evaluate the value of I*m. The model satisfactorily fits the data in each type of deformation excepting those in PE. In the fitted results for PE, a finite deviation is observed in the y-direction, when the fitted curve agrees with the data in the x-direction. This will reflect the presence of finite mechanical anisotropy in the specimen subjected to the prestretching of PE because PE has the highest anisotropy in strain field among the three types of biaxial deformation employed here. The earlier reports29,30,39 showed that the fitted values of Im * linearly varied with I1,m, i.e., the maximum I1 that the specimen experienced for the uniaxial stretching and compression cycles with various λx,m. Figure 18a shows the corresponding plots of the present results for various types of deformation. All data fall on a single straight line independently of the deformation type:

Figure 16. Rate of Δ, dΔ/dmm, as a function of (mm − √3) obtained from the fitted curve in Figure 14b.

the dissipation driven by the purely elastic fracture of network chains is significantly influenced by the cross-effect of strains. This is in contrast to that the dissipation in filled elastomers, which is caused by the destruction of the interfaces between polymer and filler, is governed by I1.36 The invariant I1 does not explicitly involve the cross-effect of strains, although the minimal cross-effect is implicitly included by the volume conservation (λxλyλz = 1). Strain Hardening at High Elongation in Reloading Process. As the earlier reports pointed out,29,30,39 the strain hardening behavior at high λ in each cycle also includes the information about the chain fracture during the loading of interest. The considerable upturn of stress in each reloading process provides a basis for the evaluation of the ultimate elongation at each λx,m. The earlier studies29,30,39 evaluated the

Figure 17. Fitted results for the reloading curves with various values of λx,m for DN gels by the Gent model under planar extension in the (a) xdirection and (b) y-direction and unequal biaxial extension in the (c) x-direction and (d) y-direction. The orange and black lines depict the fitted curves and the experimental data reproduced from Figures 4 and 5, respectively. The corresponding fitted results for equibiaxial and uniaxial extension are given in the Supporting Information (S4). J


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Figure 18. (a) Fitted values of Im * − 3 for the reloading data in various types of deformation as a function of I1,m − 3. The dashed line depicts eq 10 with J0 = 0.2 and α = 1.2. (b) Values of λ*i,m (i = x, y) obtained from the fitted values of I*m using eq 3a as a function of λi,m (i = x, y) for various * = λi,m. deformation modes. The line represents the relation of λi,m

Im* − 3 = J0 + α(I1,m − 3)

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SUMMARY For the tough DN hydrogels, the initial modulus ratio (A/A0; the ratio of the initial modulus in the reloading and virgin loadings), energy dissipation (D), dissipation factor (Δ; the ratio of D to the input strain energy), and the ultimate * ) in each loading− elongation of network strands (λi,m unloading cycle have been evaluated as a function of the imposed maximum elongation in the i-direction (λi,m; i = x, y) in each cycle by various modes of stretching. In agreement with earlier results using uniaxial stretching, no difference in the unloading and reloading curves is observed in each stretching mode, confirming that the stress softening in the DN hydrogels is entirely attributed to the chain fracture of the PNaAMPS network (first network). The values of the modulus reduction (1 − A/A0), D and Δ depend on the type of extension when compared at the same λx,m: They increase in the order of U < PE < UB < EB. Each of A/A0 and Δ shows a universal relation independently of the stretching modes, when the magnitude of the left Cauchy− Green deformation tensor (mm) corresponding to the imposed maximum deformation in each cycle is used as a variable. In the case of filled elastomers showing apparently the similar stress softening, the corresponding universal relation for Δ is obtained using the first invariant of deformation tensor (I1) as a variable. This difference in governing variable indicates that the influence of the cross-effect of strains (λiλj; i, j = x, y, z and i ≠ j) on A/A0 and Δ is pronounced in the DN gels whereas it is minimal in the filled elastomers. Interestingly, the degrees of modulus reduction (1 − A/A0) and Δ in each cycle almost agree with each other, although A and Δ are obtained at very different strains, i.e., small- and entire-strain regimes, respectively. In the case of filled elastomers, they are considerably different [(1 − A/A0) > Δ]. This discrepancy reflects that the modulus reduction and Δ are different in main origin for the filled elastomers, while both of them totally stem from the elastic fracture of the chains in the first network in the entire strain range without appreciable thermal dissipation for the DN gels. * ) in each The ultimate elongation of network strands (λi,m cycle, which is evaluated from the reloading curve, is close to the maximum of the imposed elongation in the direction of interest (λi,m), independently of the stretching modes. This characteristic indicates that when the loading reaches λi,m, most of the chains with lower extensibility than λi,m are broken whereas most of the long chains with higher extensibility than

(10)

where J0 corresponds to the stretching limit in the virgin state, and α is a proportional dimensionless coefficient. The values of J0 and α in the present results (J0 = 0.19 and α = 1.1) are close to those in the earlier report (J0 = 0.2 and α = 1.2).39 The quantity I*m represents the maximum deformation of elastically effective polymer chains in each cycle. The increase in I*m with an increase in I1,m indicates that as λx,m increases, the loadbearing chains available for deformation during the loading of interest become longer whereas the short chains among the elastically effective chains in inhomogeneous first network undergo fracture.29 The present results demonstrate that the internal fracture process is well described by a linear relation of Im * and I1,m in every type of deformation. While Im * is evaluated from the strain hardening at large deformation, Im* is qualitatively similar to the small-strain modulus ratio (A/A0) as a measure of chain fracture in the prenecking regime examined here. An analogous linear relation is found in the plots of λi,m versus λ*i,m (i = x, y) for various types of deformation (Figure 18b). The quantity λ*i,m for each deformation mode, which represents the ultimate stretch along the i-axis, is obtained from the value of Im * using eq 3a. The linear relation in the figure indicates that the ultimate stretch in each principal direction is simply governed by the maximum of the imposed stretch in the corresponding direction in each cycle, regardless of the imposed stretch in other direction. A similar result was observed for the yield strain (onset strain of necking deformation) of the DN gels on the basis of the uniaxial stretching and compression data.31 In Figure 18b, the relation of λ*i,m ≈ λi,m is also notable. This relation indicates that (i) the imposed maximum stretch in each cycle (λi,m) causes the full stretch of the elastically effective chains in the i-direction whose ultimate extensibility corresponds to λi,m and (ii) in the end of the loading of interest most of the full stretched chains are broken whereas most of the long chains with higher extensibility than λi,m remain intact. It should also be noticed that λi,m * has no appreciable cross-effect of strains in contrast to the reduction in initial modulus (A/A0) as well as the dissipation factor (Δ). The modulus or the dissipation is controlled by the total number of intact elastically effective chains or of fractured chains in each cycle, respectively, while λi,m * is a directional quantity mainly governed by the chains in the i-direction whose ultimate extensibility corresponds to λi,m. K


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Macromolecules λi,m remain intact. The elongation λ*i,m has a directional character without appreciable cross-effect of strains, in contrast to the modulus reduction and Δ.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01033. Figures S1−S6 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (J.P.G.). *E-mail: [email protected] (K.U.). ORCID

Tasuku Nakajima: 0000-0002-2235-3478 Jian Ping Gong: 0000-0003-2228-2750 Kenji Urayama: 0000-0002-2823-6344 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was funded by the ImPACT Program of the Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). The authors thank Ms. Yukiko Hane for preparation of DN samples.

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REFERENCES

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