Yielding Criteria of Double Network Hydrogels - Macromolecules

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Yielding Criteria of Double Network Hydrogels Takahiro Matsuda,† Tasuku Nakajima,*,‡ Yuki Fukuda,§ Wei Hong,∥,‡ Takamasa Sakai,⊥ Takayuki Kurokawa,‡ Ung-il Chung,⊥ and Jian Ping Gong*,‡ †

Graduate School of Life Science, ‡Faculty of Advanced Life Science, and §School of Science, Hokkaido University, N10W8, Kita-ku, Sapporo, Hokkaido 060-0810, Japan ∥ Department of Aerospace Engineering, Iowa State University, 2271 Howe Hall, Ames, Iowa 50011-2271, United States ⊥ Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan S Supporting Information *

ABSTRACT: Double network (DN) gels, consisting of a brittle first and flexible second network, have been known to be extremely tough and functional hydrogels. In a DN gel subjected to force, the brittle first network breaks prior to the fracture of the flexible network. This process, referred to as internal fracture, dissipates energy and increases the energy required to completely fracture DN gels. Such internal fracture macroscopically appears as a yielding-like phenomenon. The aim of this paper is to investigate the relationship between the yield point and the first network molecular structure of DN gels to more deeply understand the internal fracture mechanism of DN gels. To achieve this goal, we synthesized DN gels having a tetra-PEG first network, which is known to be a nearly ideal and well-controlled network gel. We have found that yielding of the DN gels occurs when the first network strands reach their extension limit (finite extensibility), regardless of their deformation mode. This conclusion not only helps by further understanding the toughening mechanism of DN gels but also allows for the design of DN gels with precisely controlled mechanical properties.

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INTRODUCTION In the 21st century, various mechanically robust hydrogels have been created, which has enabled the practical use of hydrogels in various fields.1−3 Double network (DN) gels, consisting of two independent and asymmetric networks, are a representative tough hydrogel system.4,5 Despite containing up to 90% water, optimized DN gels show excellent strength, extensibility, and toughness, similar to some kinds of industrial rubbers. Also, these mechanical properties can be widely controlled by altering the composition of the gels.4−8 DN gels are not only tough but also have excellent functions like biocompatibility,9,10 bioactivity,11 low immune response,12 and low sliding friction.13 Based on these features, DN gels possess great potential, especially as medical materials, such as substrates for cell cultures,11 artificial articular cartilage,14,15 and substrates for in vivo cartilage regeneration.16 The high toughness of DN gels mainly originates from their contrasting double network structure. The first network of tough DN gels must be brittle, dilute, and weak, while the second network must be flexible, concentrated, and (relatively) strong. This results in a system where extensibility and strength of the first network are much poorer than those of the second network. DN gels become tough regardless of their chemical constituents if the two networks satisfy the above-mentioned conditions.17 Such contrasting structures induce dramatically large energy dissipation during stretching and fracture tests.5,8,18,19 When a DN gel is deformed, the first network © XXXX American Chemical Society

chains break internally at an early stage of deformation while the second network maintains the integrity of the whole DN gel. As a result, before the second network fracture, an extremely large quantity of energy is dissipated due to the internal fracture of the first network, which substantially increases the toughness and other mechanical properties of DN gels. Such internal fracture of the first network visually appears as a yielding-like phenomenon. Figure 1 shows the typical tensile stress−strain curves of a tough DN gel. A clear yielding point,

Figure 1. Typical yielding-like and necking phenomena of a tough double network hydrogel during a uniaxial tensile test. Received: November 30, 2015 Revised: February 4, 2016

A

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Macromolecules

To investigate the yielding criteria of DN gels, we will systematically tune the conformation of the first TPEG network. In particular, the original structure of the TPEG network (such as connectivity, average molecular weight between cross-linking points, etc.) is kept constant, while the prestretching ratio of the TPEG network was widely varied by the molecular stent method. As shown in Figure 2,

accompanied by macroscopic necking, can be found. The mechanical character of DN gels dramatically changes at the yield point, being dominated by the rigid network prior to and the flexible network after yielding. Thus, the yield point can be considered to represent the point of the first network internal fracture. Based on these phenomena, the hypothesis that the first network breaks into discontinuous fragments at the yield point has been proposed.5,7,20,21 While it has been understood that yielding is an extremely important phenomenon for the toughening and fracture mechanisms of DN gels, one remaining problem is making a connection between the fracture behavior and the molecular structure of DN gels. To access this problem, comparing the yield point of DN gels and the first network structure is required, since the yield point is the representative point of the internal fracture of the first network. To date, however, such comparison has been especially difficult since the first network of DN gels is typically synthesized by a free-radical copolymerization of vinyl monomer and divinyl cross-linker. This reaction is uncontrolled and results in an essentially random structure of the first network, which prevents correlation between the yield point and the first network molecular structure in DN gels. The aim of this study is to investigate the relationship between the yield point and the first network molecular structure of double network gels by using a well-defined first network to further our understanding of the toughening mechanism of DN gels. In recent years, tetra-PEG (TPEG) gels have attracted great attention as model gels since they have a well-defined and homogeneous network structure.22−25 TPEG gels are prepared by the cross-end coupling of two mutually reactive tetra-arm polymers. As polymerization degree of each arm is well-controlled, the length distribution of network chains is kept in a narrow range. Small-angle neutron scattering studies showed that TPEG gels had practically no noticeable excess scattering in the region below 200 nm.23 Infrared spectroscopy revealed a reaction conversion of up to 0.9, which suggested near absence of dangling chains.24 Mechanical testing of TPEG gels suggested that formations of trapped entanglements or elastically ineffective loops are negligible.25 These results strongly suggest that TPEG gels constitute a near-ideal polymer network. Thus, TPEG gels are the most suitable candidate for creating well-defined first networks in DN gels. We have previously synthesized tough DN gels using a tetraPEG first network.26 As TPEG gels have a relatively flexible and condensed network in water, they cannot be directly used as the brittle first network. Therefore, we introduced strong, linear polyelectrolytes, called molecular stents, into the TPEG gels to increase the overall osmotic pressure. As high osmotic pressure increases their swelling ratio, TPEG network chains are stretched to an extended conformation, and the properties of the TPEG gels changed to become brittle and weak, which are suitable properties for the first network of tough gels. These swollen and brittle TPEG network-based DN gels, called StTPEG DN gels, also showed high toughness and yielding-like behaviors like conventional DN gels with a random first network, which indicates that homogeneity of the first network structure does not affect the essence of the fracture mechanism of DN gels. On the other hand, the total amount of internal fracture of the St-TPEG gels before the yield point is much less than that of conventional DN gels due to the first network inhomogeneity of the latter.

Figure 2. Synthesis process of St-TPEG/PAAm DN gels having various prestretching ratios of the first network, which is controlled by the molecular stent (polyelectrolyte) concentration.

polyelectrolytes trapped in a gel (molecular stent) generate extra osmotic pressure due to their counterions to swell the gel. By tuning the polyelectrolyte concentration in the TPEG gel, the swelling ratio of the gel can be varied, which permits us to finely tune the prestretching ratio and the network density of the Tetra-PEG network and investigate the effect of these two structural factors on the yielding of St-TPEG/PAAm DN gels.

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EXPERIMENTS

Materials. Tetra-amine-terminated PEG (TAPEG) and tetra-NHSglutarate-terminated PEG (TNPEG) (NOF Corporation) were used as received. Average molecular weights of the two tetra-arm-PEGs were 10 kg/mol. 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 chloroform. N,N′-Methylenebisacrylamide (MBAA, Wako Chemical Industries, Co., Ltd.) was recrystallized from ethanol. 2-Oxogultaric acid (αketo, Wako Chemical Industries, Co., Ltd.) was used as received. Sodalime glass plates (thickness: 3 mm) were used for the gel synthesis. Glass molds were prepared from two glass plates spaced by silicone rubber (thickness: 0.5−2 mm). Synthesis of the Tetra-PEG Network. 74.6 mg/mL of TAPEG was dissolved in phosphate buffer (pH 7.4), and 74.6 mg/mL of TNPEG was dissolved in phosphate−citric acid buffer (pH 5.8) to obtain c* solutions.27 Equal volumes of the TAPEG and the TNPEG c* solutions were mixed together, poured into the glass mold, and kept for 12 h at room temperature to obtain platelet TPEG gels. The resulting average molecular weight of a network strand in the TPEG gels is 5 kg/mol, which corresponds to an average polymerization degree of 114. The same TPEG gels were used for synthesis of all the St-TPEG DN gels. B

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Macromolecules Synthesis of the Molecular Stent. The TPEG gels were then immersed in 0.2−2.5 M of 2-acrylamido-2-methylpropanesulfonic acid sodium salt (NaAMPS) and 0.1 mol % of α-keto aqueous solutions for at least 1 day. The gels were then sandwiched by the two glass plates, wrapped by a plastic film, and moved into an argon blanket. 365 nm UV (4 mW/cm2) polymerization was then carried out for 3 h to synthesize linear PNaAMPS within the first TPEG network. TPEG gels containing PNaAMPS are called St-TPEG gels. Synthesis of the Second Network. The St-TPEG gels were immersed in the second network precursor aqueous solutions containing 2−6 M of acrylamide (AAm), 0.01 mol % of MBAA, and 0.01 mol % of α-keto for about 12 h and then sandwiched with the two glass plates. After that 365 nm UV (4 mW/cm2) polymerization was carried out for 9 h to synthesize the second PAAm network within the St-TPEG gels in an argon blanket. A part of the obtained St-TPEG/ PAAm DN gels was immersed in pure water before measurements (called swollen samples) while the rest was directly used for testing (called as-prepared samples). One-Dimensional Swelling Ratio Measurement. The asprepared TPEG gel was chosen as the reference state. The onedimensional swelling ratio of the DN gels, α, was defined as α = t/t0, where t is the thickness of the DN gel and t0 is at reference state. Thickness of the gel was measured with calipers. Uniaxial Tensile Test. Uniaxial tensile tests were performed on dumbbell-shaped samples standardized as JIS K 6261-7 (12 mm in length, 2 mm in width, 0.5−2.7 mm in thickness) with an Instron 5965 tensile tester accompanied by a noncontact extensometer AVE (Instron Co.). Tensile velocity was fixed at 100 mm/min. Nominal tensile stress, σtens, was defined as the force divided by the original cross-sectional area. The deformation ratio, λtens, was defined as the length of the deformed sample divided by that of the sample in the relaxed state. The tensile yield point was defined as where the slope of the stress−strain curve is zero. Tensile yield stress, σy‑tens, and yield deformation ratio, λy‑tens, were defined as σtens and λtens at the yield point, respectively. Uniaxial Compression Test. Uniaxial compression tests were performed on cylinder-shaped gels (17 mm in diameter, 2.5−6.2 mm in thickness) with a Tensilon RTC-1310A (Orientec Co) tensilecompressive tester. The samples were compressed by two parallel metal plates. Silicone oil was added between the sample and the metal plates to prevent adsorption and friction between them. The compression rate was fixed at 0.1 min−1. The definitions of σcomp and λcomp are the same as those of the tensile test, so λcomp = 1 represents the undeformed state and the compression proceeds with decreasing λcomp. The compressive yield point was defined as a flection point of the stress−strain curves. Compressive yield stress, σy‑comp, and yield deformation ratio, λy‑comp, were defined as σ and λ at the yield point, respectively.

Figure 3. One-dimensional swelling ratio, α, of the St-TPEG/PAAm DN gels with various concentrations of molecular stent in the asprepared and swollen states. Unless visible, error bars are smaller than the symbol size (number of experiments, n = 3).

Figure 4. (a) Tensile stress−deformation ratio curves of the asprepared St-TPEG/PAAm DN gels with various α. The curves with an asterisk mean that the measurements were stopped before sample failure due to the measurement limitation of the video extensometer. (b) Tensile stress−deformation ratio plot highlighting the small deformation region. (c, d) Yield deformation ratio, λy‑tens, and yield stress, σy‑tens, dependence on α of the St-TPEG/PAAm DN gels. Error bars are less than the size of the symbols (number of experiments n = 3), except for a sample of α = 2.1 (n = 1 due to the experimental difficulty). Dashed lines represent the observed power-law relationships.

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RESULTS Swelling Properties. Figure 3 shows the one-dimensional swelling ratio, α, of the St-TPEG/PAAm DN gels with various concentrations of molecular stent, PNaAMPS. α increased with the NaAMPS feed concentration as the large number of counterions of PNaAMPS induces high osmotic pressure. α of the DN gel was successfully controlled over a very wide range from 1.0 to 3.9 (in the as-prepared state) and 1.6 to 5.4 (in the swollen state). At small α, the first network TPEG chains are in a coiled conformation. With increasing α, the chains are gradually prestretched to their extended state. Tensile Yield Point of the St-TPEG DN Gels. Figure 4a shows the tensile stress−deformation ratio curves of the asprepared St-TPEG/PAAm DN gels with various α. Mechanical properties of the gels dynamically changed with varying α. In general, the DN gels with larger α showed lower stress than those with smaller α. This occurs because decreasing α results

in dilution of the first network, which mainly bears stress in DN gels. Figure 4b shows the tensile stress−deformation ratio curves of the as-prepared St-TPEG/PAAm DN gels at small deformation. Most of the DN gels showed a yield point except those with very small α. The yield deformation ratio, λy‑tens, and yield stress, σy‑tens, monotonically decrease with an increase in α. In order to obtain a quantitative relationship, λy‑tens and σy‑tens of both the as-prepared and swollen St-TPEG/PAAm DN gels were plotted against α with logarithmic axes as shown in Figure 4c,d. Two definite scaling relationships, which are λ y‐tens ∝ α −1

(1)

and C

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Macromolecules σy‐tens ∝ α −2

similarity implies that the yield point of DN gels corresponds to the ultimate stretching point of the first network chains in the tensile direction. However, the value of L (= 9.0) is slightly larger than the theoretical value of 7.6. Three possible reasons can be raised. One is the change of bond angle or length in the highly stretched PEG chains. It is well-known that when covalent bonds are stretched, their bond angle and length significantly change.29,30 This idea is also applicable to PEG chain fracture. Oesterhelt et al. have proposed that when PEG chains reach their maximum extension, their bond angle and length should change, based on AFM measurements of a PEG single chain.31 Also, Heymann et al. have implied similar assumptions based on their theoretical study on PEG.32 Another possible reason is fluctuation of the relative position of the network chain-ends, which is not assumed in the affine network model. The phantom network model is the simplest network model which takes into account positional chain-end fluctuations in a network.33,34 It has been found that the phantom network model is more applicable than the affine network model for the tetra-PEG gels synthesized at relatively low concentration.27 In the phantom network model, a single network strand having N monomers with chain-end fluctuation can be considered as a longer chain having Nf/(f − 2) monomers without fixed chain-ends, where f is the functionality of the network.32 In this work, f is 4 and therefore Nf/(f − 2) = 2N. Simply assuming a Gaussian chain, extensibility of such a single chain is bN1/bN0.5 = N0.5, where b is the Kuhn length and should be constant. Thus, a theoretically phantom network chain can be stretched (2N)0.5/N0.5 = 1.44 times larger than the affine network chain. While this idea (applying the phantom network model) can explain the larger experimental L, a problem of this idea is whether chain-end fluctuation still exists at such highly stretched states or not. The other possible reason is imperfect network connectivity. Defects in the network lead to a decrease of substantive cross-linking density, which may be a reason for the difference between experimental and theoretical L. Indeed, infrared spectroscopy studies have shown that the conversion of the cross-end coupling reaction, p, of the TPEG gels used in this work was 0.91, which is less than 1.35 However, our previous experiments and simulations imply that p does not affect to the maximum extensibility dramatically when p is significantly high (>0.7).28,35 Thus, at this time, the authors believe the first effect (bond angle/length change) is most probable. Subsequently, we will discuss the origins of the relationship σy‑tens ∝ α−2. Figure 5b shows auxiliary illustrations for the following discussion. Based on the proposed assumption that yielding of DN gels is due to finite extensibility of the TPEG network in the tensile direction, yield stress, σy‑tens, is ideally expressed as

(2)

were found. Interestingly, the results of the as-prepared DN gels and the swollen DN gels overlapped, although their second network structure should be different due to swelling of the latter. This fact implies that the first network structure is the dominant factor of the yielding criterion of the St-TPEG/ PAAm DN gels. Next the origins of these yield-related relationships and the coefficients were studied. First, we will discuss the origins of the relationship of λy‑tens ∝ α−1. This equation can be rewritten as αλy‑tens = L, where L is a proportional constant. L can be determined from the slope of the plot of λy‑tens vs α−1 as 9.0. Here, let us revisit the physical meaning of α and λtens. Figure 5a

Figure 5. Schematic illustrations for determining factors of (a) the yield deformation ratio and (b) the yield stress of DN gels.

shows auxiliary illustrations for the following discussion. α is the one-dimensional swelling ratio of the St-TPEG/PAAm gels compared to the as-prepared TPEG gels. The TPEG network in the St-TPEG/PAAm DN gels is already prestretched α times by swelling. On the other hand, λtens is the uniaxial deformation ratio of the St-TPEG/PAAm DN gels possessing a prestretched TPEG network. Thus, the total deformation ratio of the TPEG network in the stretched St-TPEG DN gels compared to the asprepared TPEG gels is α × λtens. Considering these relationships, the equation αλy‑tens = L means that yielding of the StTPEG/PAAm DN gels occurs when the total deformation ratio of the TPEG network in the tensile direction reaches a certain constant value, L. This means that the yield point of the StTPEG/PAAm DN gels is determined only by the deformation ratio of the TPEG network in the tensile direction, regardless of the deformation in the other two directions. Then, what happens to the TPEG network at the point αλy‑tens = L? We simply consider the finite extensibility of a single polymer chain and the affine deformation model. It is assumed that average structure of an ideal TPEG network with perfect connectivity is a diamond lattice structure. By using the preparation condition and lattice parameter, the original endto-end distance of one TPEG network strand in the reference state is calculated as 5.4 nm.28 At a theoretical fully stretched state, each TPEG network strand reaches its contour length, which also can be calculated as 41 nm without consideration of bond length or bond angle change (for details, see Supporting Information). By dividing the latter by the former, the theoretical limit extension ratio of a single chain is calculated to be 7.6. This value is almost consistent with L = 9.0. This

σy‐tens = Fd

(3)

where F (N) is the force required to break one TPEG network strand by stretching and d (m−2) is the area density of the TPEG chains in the relaxed DN gels in the tensile direction. As F is theoretically constant, σy‑comp should be proportional to the area density, d. d decreases with increasing one-dimensional swelling ratio, α, of the gel with a relationship of d ∝ α−2. By substitution of this relationship into eq 3, we obtain σy−tens ∝ α−2. In our system, d can be calculated since the TPEG network structure has been well-determined. Area density, d, of the TPEG network is calculated as D

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Macromolecules d=

⎛ v0 ⎞2/3 ⎜ ⎟ ⎝ α3 ⎠

(4)

where v0 is the number density of elastically effective network strands in the TPEG network in the reference state (1/m3). v0 can be calculated by adopting the theory of a treelike structure for tetrafunctional networks, such that 3 ⎛ ⎛1 ⎞1/2 ⎞⎛ 3 ⎛1 ⎞1/2 ⎞ 1 3 3 v0 = n0⎜⎜ + ⎜ − ⎟ ⎟⎟⎜⎜ − ⎜ − ⎟ ⎟⎟ 4 ⎠ ⎠⎝ 2 4⎠ ⎠ ⎝p ⎝p ⎝2

(5)

where n0 is the number concentration of the tetra-arm polymer in the TPEG pregel solution and p is the reaction conversion (fraction of connected bond).36,37 For this study, n0 was determined from the preparation condition as 8.4 × 1024 m−3 and p is 0.91. Thus, v0 can be calculated as 6.9 × 1024 m−3. On the other hand, F is determined by the weakest bond in a TPEG network strand, which consists of ether bonds and an amide bond.22 Our previous study shows that activation enthalpies of the ether bond (∼86 kJ/mol) and the amide bond (∼89 kJ/mol) are very similar, suggesting forces required for cleavage of these bonds are also similar.38 For an ether bond in PEG, Aktah et al. have calculated with the DFT method that the force required for cleavage of a PEG chain in water is around 3 nN.39 Thus, this time we roughly adopt this value as F. By using these values, the experimental yield stress, σy‑tens, and the ideal yield stress, Fd, can be compared. As a result, in all cases, the ideal value, Fd, was much larger than the experimental value σy‑tens. For example, in the case of α = 2.3, calculated Fd is 21 MPa, whereas measured σy‑tens was only 0.83 MPa, which is 25 times smaller than the ideal value. This result means that not all the TPEG chains are fully stretched and bear stress at the yield point of the DN gels. The large difference between σy‑tens and Fd can be explained by the assumption that stress is concentrated on a part of the TPEG chains in the StTPEG/PAAm DN gels. The MALDI-TOF mass spectrum of 10k TAPEG (see Figure S1 of Supporting Information) implies that even though the dispersity (Mw/Mn) of tetra-PEG is very small (1.05), a distribution in molecular weight exists. In addition, abrupt strain hardening of a PEG chain only occurs when the chain approaches its finite extensibility.31 Thus, it is easy to imagine that stress is concentrated on the relatively short TPEG chains, and such chains mainly bear the stress at the yield point of the St-TPEG/PAAm DN gels. Compressive Yield Point of the St-TPEG DN Gels. Figure 6a shows the compressive stress−deformation ratio curves of the St-TPEG DN gels having various α. Clear flection points can be found in the curves. As the gels show softening at this point, such flection points may correspond to yielding. Thus, in this paper, such points are called the compressive yield point. Figure 6b,c shows a scaling plot of the compressive yield deformation ratio, λy‑comp, and yield stress, −σy‑comp, against α. Two definite scaling relationships, which are λ y‐comp ∝ α 2

Figure 6. (a) Compressive stress−deformation ratio curves of the asprepared St-TPEG/PAAm DN gels with various α. The measurements were stopped at σcomp ∼ −43 MPa due to limitations of the load cell. (b) Compressive stress−deformation ratio plot highlighting the area near the yield points. (c, d) Yield deformation ratio, λy‑comp, and yield stress, σy‑comp, dependence on α of the St-TPEG/PAAm DN gels. Error bars are smaller than the size of the symbols (n = 3). Dashed lines are observed power-law relationships. The solid line in (d) is the theoretical power-law relationship.

yield point may also be associated with TPEG chain rupture in the first network when finite extensibility is reached. A sample under axial compression also extends in the radial direction. For an incompressible gel, the radial stretch λr and the axial stretch λz are related as λzλr2 = 1. Substituting λz for the yield stretch, λy‑comp, in eq 6, we arrive at the scaling relation in terms of the radial stretch at the yield point, λy‑r λ y‐r ∝ α −1

This relationship is consistent with the relationship derived in the tensile case (eq 1). Furthermore, a calculated proportionality coefficient of eq 8, which corresponds to L in the tensile case, is 8.9, which is very close to the obtained L of 9.0. Such consistency suggests that the same yielding criterion may be applied to the DN gels regardless of deformation mode: the finite extensibility of the first network chains. Now let us turn to the compressive yield stress. Because of volume incompressibility, the yielding phenomenon (or the rupture of the first network) should not be affected by the hydrostatic stress state of the sample. By adding to a sample a hydrostatic tension, sr = sθ = sz = s, which equals the axial compression in magnitude, the uniaxial compression state is converted to an equivalent state of equal biaxial tension: sz = 0, sr = sθ = s (9) It should be noted that the hydrostatic stress, s, is the true stress measured with respect to the deformed geometry. The true stresses (force per unit deformed area) are related to the nominal stress (force per unit original area) by the simple geometric relations

(6)

and

−σy‐comp ∝ α −4

(8)

(7)

can be found. The power indexes are different from those of the tensile tests. Let us look further into the origin of the compressive yield point. In analogy to the tensile yield point, the compressive

sr = λrσr ,

sθ = λθ σθ ,

sz = λzσz

(10)

In terms of the nominal stresses, the equal biaxial tension takes the form E

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Macromolecules σr = σθ = −

s = − λz s λr

in our case, the two curves are almost overlapping. It may be attributed to a significant strain hardening effect of the first network. We think such large strain hardening effectively “hides” the effect of the other two axes. For further discussion on this topic, an in-depth study of biaxial stretching of DN gels is required. Normalization of the Stress−Strain Curves. Based on the above-mentioned discussion about the extraordinary strain hardening effect of the first network, it can be expected that stress−deformation behavior of DN gels is almost only dominated by the deformation ratio of the first network chains in the stretching axis to its reference state and their density. To confirm this idea, we normalized all the stress−deformation ratio curves of the St-TPEG/PAAm DN gels shown in Figures 4a and 7a based on the first network structure. In particular, σ was normalized by initial area density of the first network chains (normalized σ: σα2), and λ is normalized by the deformation ratio of the first network polymer strand length to its reference state (normalized λ: λα). Figure 8 shows the

(11)

where s is related to the nominal compressive stress, σz, by s = λσz. Consequently, at the yield point, the equivalent biaxial stress, σy‑biax, is related to the compressive yield stress, σy‑comp, as σy‐biax = −λ y‐comp3/2σy‐comp

(12)

Under consideration of the above-mentioned yielding criterion of DN gels, if we assume that yielding under biaxial tension follows the same scaling relation as that of uniaxial tension, σy‑biax ∝ α−2, the corresponding compressive yield stress is −σy‐comp ∝ α −5

(13)

This scaling relation is very close to that of eq 7. The difference may be attributed to technical difficulties in experiments such as friction between plates and gels, which may induce nonideal compression (sometimes called barreling problem), although silicone oil was used to prevent it. On the basis of the above results, we can conclude that the proposed yielding criterion of DN gels, which is that a DN gel yields when the first network strands reach their extension limit, is general and can be applied to both tensile and compressive states. Comparison of Uniaxial and Equal Biaxial Stretching. Ideally, the deformation modes of a uniaxial compression tests are equivalent to that of the equal biaxial tensile tests. Based on the above-mentioned idea shown in eq 11, compressive stress, σcomp, can be converted to equal biaxial (radial) stress, σr. Also, the compressive deformation ratio, λcomp, can be converted to the equal biaxial (radial) deformation ratio, λr. We calculated the equal biaxial tensile stress−deformation ratio curves of the St-TPEG/PAAm DN gels from the compression test results as shown in Figure 7a. Interestingly, the calculated curves have all

Figure 8. Normalized stress−deformation ratio curves, both uniaxial and equal biaxial stretching, of the St-TPEG/PAAm gels with various α.

normalized stress−deformation ratio curves (both uniaxial and equal biaxial) of the St-TPEG/PAAm DN gels. As we expected, all the curves collapse almost perfectly. Amazingly, they not only show similar yield points but also similar strain hardening behavior started from the normalized deformation ratio of around 7. This overlapping is also proof of the dominancy of the first network structure on the mechanical behavior of DN gels. Effect of the Second Network on the Yield Point. We also studied the effect of the second network on the yield point of the St-TPEG/PAAm DN gels. The feed concentration of acrylamide for the second network was controlled from 2 to 6 M while α was kept constant (2.4−2.5). Figure 9 shows the

Figure 7. (a) Calculated equal biaxial stress−deformation ratio curves of the St-TPEG/PAAm DN gels with various α. (b) Normalized stress−deformation ratio curves of both uniaxial and equal biaxial tensile tests.

the features of the stress−deformation ratio curves of uniaxial tensile tests, which are a clear yield point, strain hardening (of the first network) before yielding, and a stress plateau region after the yield point. Further, the uniaxial and equal biaxial tensile curves of the St-TPEG/PAAm DN gels with the same α roughly overlap each other as shown in Figure 7b. This phenomenon is quite unusual for rubbery materials. The stress−deformation behavior of ideal rubbery materials obeying entropic elasticity depends not only on the deformation ratio of the stretching axis but also that of the other two axes. At the same strain, the equal biaxial stress of rubbery materials is theoretically always larger than the uniaxial stress.40 However,

Figure 9. Effect of the second network concentration on the stress− deformation ratio curves of the St-TPEG/PAAm gels. F

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Macromolecules

*Tel & Fax +81-11-706-4815; e-mail [email protected]. jp (J.P.G.).

stress−strain curves of the as-prepared St-TPEG/PAAm gels with various AAm concentrations. Although these DN gels had different concentrations of the second network, their yield points did not change significantly. This fact also clearly shows that the yielding behavior of DN gels, which corresponds to the fracture of the first network, is dominated strongly by the first network and hardly depends on the second network. Nevertheless, yield stress appears to slightly increase with increasing concentration of the second network. Although the fully stretched first network mainly bears the stress at the yield point, the soft second network also bears some stress. As a result, the apparent yield stress is the sum of the stress borne by the first and the second networks at the yield point. The former does not change, but the latter increases with increasing second network concentration. This might be the reason for the slight increase in the yield stress with increasing second network concentration. Also, the yield deformation ratio slightly decreases with increasing second network concentration. Entanglement between the two networks might slightly reduce the extensibility of the first network. Note that this is not meant to suggest that the second network is not important for toughening of DN gels. Existence of the second network is necessary for the occurrence of yielding of DN gels. If the relative amount of the second network to the first network is not enough, the DN gels become so brittle that they break before reaching the yield point.8

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was funded by a Grant-in-Aid for Scientific Research (S) (No. 124225006) from the Japan Society for the Promotion of Science (JSPS) and by the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). The authors thank Toa Gosei Co., Ltd., for the supply of NaAMPS (ATBS-Na) and Dr. Yuki Akagi (the University of Tokyo) for provision of the MALDITOF results. J. P. Gong thanks Prof. Michael Rubinstein (University of North Carolina) for his precious suggestions. T. Nakajima thanks Dr. Daniel R. King (Hokkaido University) for his careful proofreading.

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CONCLUSIONS We first clarified the yielding conditions of the double network gels by using St-TPEG/PAAm DN gels with a well-defined first network structure. We have found the following yielding criteria: (1) the yield point of DN gels is determined by the finite extensibility of the first network chains; (2) the yield stress (in the stretching direction) is determined by the area density of the first network; (3) the effect of the second network on the yield point is negligible. These important findings could not be realized without using TPEG gels with well-controlled network strand length and density. This study gives us clear insight into the yielding and toughening mechanism of DN gels and related tough soft materials having multiple network structure.41,42 Additionally, the yield point is sometimes considered to be the maximum allowable deformation of materials for practical use since materials begin deforming irreversibly at this point. Thus, clarification of the yield point of DN gels, done in this work, helps in the design of DN gels suitable for various applications such as medical and industrial ones.9−11,14,43

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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.5b02592. Calculation of contour length of the TPEG network strands and MALDI-TOF measurements of the TPEG prepolymer (PDF)

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DOI: 10.1021/acs.macromol.5b02592 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.5b02592 Macromolecules XXXX, XXX, XXX−XXX