J. Phys. Chem. B 2005, 109, 11559-11562
11559
Determination of Fracture Energy of High Strength Double Network Hydrogels Yoshimi Tanaka,† Rikimaru Kuwabara,‡ Yang-Ho Na,‡ Takayuki Kurokawa,‡ Jian Ping Gong,*,‡,§ and Yoshihito Osada‡ CreatiVe Research InitiatiVe “Sousei’’ (CRIS), Hokkaido UniVersity, Sapporo 001-0021, Japan, Graduate School of Science, Hokkaido UniVersity, Sapporo 060-0810, Japan, and SORST, JST, Sapporo 060-0810, Japan ReceiVed: January 6, 2005; In Final Form: April 9, 2005
The fracture energy G of double network (DN) gels, consisting of poly(2-acrylamido-2-methylpropanesulfonic acid) (PAMPS) as the first network and poly(acrylamide) (PAAm) as the second network, was measured by the tearing test as a function of the crack velocity V. The following results were obtained: (i) The fracture energy G ranges from 102 to∼103 J/m2, which is 100-1000 times larger than that of normal PAAm gels (100 J/m2) or PAMPS gels (10-1 J/m2) with similar polymer concentrations to the DN gels. (ii) G shows weak dependence on the crack velocity V. (iii) G at a given value of V increases with decreasing of cross-linking density of the 2nd network. The measured values of G were compared with three theories that describe different mechanisms enhancing the fracture energy of soft polymeric systems. A mechanism relating to a heterogeneous structure of the DN gel is convincing for the remarkable large values of G.
Introduction A wide range of medical, pharmaceutical, and prosthetic applications has been proposed for hydorgels. However, most of them are not realized because of the lack in mechanical toughness. Recently, we have reported a novel method to overcome this problem by inducing a double network (DN) structure for various combinations of hydrophilic polymers.1 The DN gels are comprised of two independently cross-linked networks. An optimal combination has been found when the first network is a rigid polyelectrolyte and the second one is a flexible neutral polymer. The DN gels synthesized with the optimal combination, containing about 90% water, exhibit fracture strength similar to that of natural cartilage (3-18 MPa). Also extremely low friction2-5 comparable to the cartilage has been realized in the DN gels. Beside the optimal combination of polymers for the first and second networks, we have found that (i) the cross-linking density of the two networks and (ii) the molar ratio of the two polymers are two crucial parameters in improving the resistance against stress: the strongest gel in our previous study is obtained when the first network is highly cross-linked and the second network is slightly cross-linked; the DN gels become very strong in mechanical properties only when the molar ratio of the second network to the first network is in a range of several to a few decades. The fracture strength mentioned above was determined as the compressive stress at which the fracture initiates. Although this quantity is convenient to qualitatively characterize the mechanical strength of the DN gels, it is inadequate to make quantitative consideration on the mechanism under high mechanical strength, because the maximum stress depends on the shape and the stickiness of the surface of the samples. On the other hand, the fracture energy G, defined as the work needed * To whom correspondence should be addressed. † CRIS, Hokkaido University. ‡ Graduate School of Science, Hokkaido University. § SORST.
in creating a unit area of the fracture surface, is usually used in the field of fracture mechanics to quantify the intrinsic resistance of various materials for fracture, including rubbers6 and gels.7,8 Also in the field of adhesive science, G is used to evaluate the strength of adhering interfaces.9,10 For the soft matter systems, i.e., rubbers, gels, and soft adhesives, to control the crack velocity V is relatively easy; thus, it is possible to measure G as a function of V. G(V) measured under different physical or chemical conditions, such as temperature and cross-linking density, brings us important information on the fracture mechanisms of the soft matter systems. In this work, to clarify the extremely high mechanical strength of the DN gels, we measured, using the tearing method, the fracture energy G(V) of the DN gels with different cross-linking densities of the second network. The following results are obtained: (i) The fracture energy G increases with decreasing the cross-linking density of the second network. (ii) G shows weak dependence on the crack velocity V. (iii) The typical value of G is on the order of 102-103 J/m2: this is much larger than the fracture energy of normal PAAm7 and PAMPS8 gels. We also compare the measured values of G with three theories that propose different mechanisms enhancing G of the soft polymeric systems. Experimental Section a. Materials. 2-Acrylamido-2-methylpropanesulfonic acid (AMPS; Tokyo Kasei Co., Ltd.) and acrylamide (AAm; Junsei Chemical Co. Ltd) were used as received. N,N′-Methylenebis(acrylamide) (MBAA; Tokyo Kasei Co., Ltd.), a cross-linking agent for both AMPS and AAm gels, was recrystallized from ethanol. 2-Oxoglutaric acid (Wako Pure Chemical Industries, Ltd.), a radical initiator for the gelation reactions, was used as received. b. Synthesis of Double Network Gel. The DN gels are synthesized through a two-step sequential free-radical polymerization. In the first step, 4 mol % of MBAA and 0.1 mol %
10.1021/jp0500790 CCC: $30.25 © 2005 American Chemical Society Published on Web 05/19/2005
11560 J. Phys. Chem. B, Vol. 109, No. 23, 2005
Figure 1. Rheological determination of the storage shear modulus (filled square), the shear loss modulus (filled triangle), and the loss tangent (open circle) as a function of frequency for the DN gels with different cross-linking densities x of the second network.
of 2-oxoglutaric acid were added to 1 M AMPS solution (the molar percentages, 4 and 0. 1mol %, are determined with respective to the AMPS monomer). After being bubbled with argon gas for 30 min, the solution was poured into molds consisting of two parallel glass plates and silicone spacers that maintain a gap of 20 mm between the glass plates. Photopolymerization was carried out under argon atmosphere with an ultraviolet (UV) lamp for 10 h. (The distance between the lamp and the sample chamber was about 200 mm.) In the second step, after the gelation was completed, the PAMPS gel was immersed into 2 M AAm solution containing 0.1 mol % of 2-oxoglutaric acid and 0-0.2 5mol % of MBAA for at least 2 days until the equilibrium was reached. By irradiation with the UV lamp for 10 h, the second network was subsequently synthesized in the presence of the first network. Samples were swollen in water for 2 weeks. The cross-linking density of the first network was fixed to 4 mol % with respect to the monomer (AMPS) and only that of the second network is changed systematically. Hereafter we use an index “x” to indicate the molar percentage of the cross-linker of the second network with respect to the monomer (AAm). As reported in a previous paper,1 the PAMPS/PAAm DN gels with different x show almost the same solvent content (90 wt %), molar ratio of the second network to the first network (20), and elastic modulus (0.1 MPa) at the swelling equilibrium. Figure 1 shows a rheological characterization of the DN gels in the shear deformation mode with a strain amplitude of 0.1%. Their elastic (storage) modulus is much larger than their viscous (loss) modulus in a frequency range from 10-2 to 10 Hz. Hence, the bulk response of the DN gels to an applied deformation is nearly elastic within the range. c. Measurements. Fracture Stress. The fracture stress σm was determined by compressive stress-strain measurements, using a tensile-compressive tester (Tensilon RTC-1310A, Orientec Co.). The samples were water-swollen cylindrical gels with a diameter of 9 mm and a height of 4 mm. The compressive strain rate was 0.1%/min. Fracture Energy. To measure the fracture energy G(V), defined as the work required to create a unit area of fracture surface, the tearing test was performed with a commercial test machine (Tensilon RTC-1150A, Orientec Co.). The gels were cut into the shape shown in Figure 2a, which has the standardized JIS-K6252 1/2 sizes (w ) 5-5.5 mm, d ) 7.5 mm, L ) 30 mm, the length of the initial notch is 20 mm), with a gel cutting machine (Dumb Bell Co., Ltd.). The two arms of a test piece (Figure 2b) were cramped and the upper arm was pulled upward at constant velocity Vp (the lower arm was fixed). The tearing force F was recorded. G is calculated by the following
Tanaka et al.
Figure 2. (a) Sample shape (w ) 5 mm, L ) 50 mm, d ) 7.5 mm, the length of the initial notch is 20 mm) and (b) the tearing method to measure fracture energy. F is the tearing force, Vp is the pulling velocit,y and V is the crack velocity.
Figure 3. Dependence of the fracture energy at V ) 0.5 × 10-2 m/s (O) and the fracture stress (0) on the cross-linking density x of the second component, PAAm.
equation,
G)
Fave 2w
where Fave is the average of F during the tear and w is the width of the gels. We ignored the elongation of the arms, i.e., we regard the crack velocity V as equal to (1/2) × Vp, and did not take account of the change of elastic energy stored in the pulled arms. (G and V receive only a few percent correction, even if we take into account this effect.) The tearing velocity V was changed from 0.5 × 10-5 to 0.5 × 10-2 m/s. Results and Discussions Figure 3 shows plots of G at V ) 0.5 × 10-2 m/s and σm versus the cross-linking density x of the second network (PAAm). Both G and σm show similar behavior, i.e., these quantities decrease with increasing x. It should be emphasized that the σm data have relatively large errors, especially in the small x region where remarkable enhancement of the strength is observed. On the other hand, the error in G data is small for all x. This fact represents an advantage of measuring G. (However, when x is larger than 0.2, the DN gel was too brittle for determining G by the tearing method.) Figure 4 shows G(V) for x ) 0, 0.02, 0.1, and 0.25. We can find the following results: (E1) G at a given value of V decreases with increasing x. (E2) The maximum value of G(V) ranges from 102 to 103 J/m2. (E3) G is insensitive to V: while V changes over three decades, G for each x changes to less than three times.
Fracture Energy of Double Network Gels
Figure 4. Dependence of the fracture energy of the DN gels on the crack velocity V.
The fracture energy G of the DN gels is much larger than that of normal PAAm gels7 (10° J/m2) or PAMPS gels8 (10-1 J/m2) with similar polymer compositions to the DN gels. The maximum value of G for the DN gels is larger than the reported values for rubbers in the slow crack velocity limit,6 101-102 J/m2. What is the origin of the remarkable enhancement of G(V)? In general, the following two factors are relevant for the fracture energy G of solids: (a) microscopic irreversible processes in the vicinity of crack tips, such as cutting of chemical bonds, and (b) dissipation processes on semimacroor macroscopic scales around the crack tips, such as plastic deformations. Furthermore, (c) heterogeneous structures on semimacro or macro scales can affect the fracture energy; socalled composites are reinforced by the mechanism. For soft polymeric systems, the mechanisms corresponding to a, b, and c have been proposed by Lake-Thomas,11 de Gennes,12 and Okumura,13 respectively. Below, we compare the values of G of the DN gels with the predictions of the theories. The classical Lake-Thomas theory11 predicts that for polymer network systems, the contribution from mechanism (a) is given by FnU, where F is the areal density of polymer chains on fracture surfaces, U is the energy of covalent bond of the chains, and n is the average number of monomer units between crosslinkers. The factor n, playing the role of an enhancement factor in G, arises from the following reason: the bond energy U is much larger than the thermal energy kBT (U is about 6.4 × 10-19 J for the C-C covalent bond, about 100 times kBT at room temperature); to cut a covalent bond of the polymer chain stretched at the crack tip, an energy U′ (≈U - kBT ≈ U) must be supplied to the bond; since the chain contains n monomers connected in series, the total energy needed in cutting the bond is about nU′ ≈ nU; after the cutting, the energy nU′ dissipates and participates in the fracture energy. If the PAAm chains are highly entangled with the first PAMPS network and behave like long partial chains in rubbers, G should be enhanced by the Lake-Thomas mechanism. However, the mechanism is not sufficient to explain the remarkably large G of the DN gels as estimated below. From GPC measurement14 on the PAAm chains synthesized in the linear PAMPS solution, the average number n of repeating units of a PAAm chain is about 2.8 × 104. We have nU ≈ 1.8 × 10-14 J. To estimate F, we must know whether the PAAm chains overlap or not. The size R of a random coil of the chain is given by R = an1/2 ≈ 40 nm, where we employ the length the monomer unit of AAm, 0.25 nm, as the size of the segment a. The density of the DN gels is closed to that of water, and AAm
J. Phys. Chem. B, Vol. 109, No. 23, 2005 11561 occupies about 10 wt % of the DN gel. The number Ω of the PAAm chains contained in the unit volume (1 m3) of the DN gel is given by Ω ) (1000 × 0.1)/(Mwn/1000) × (6 × 1023) ≈ 3 × 1022, where Mw ) 71 is the molecular weight of the monomer unit of PAAm. The total volume of the coils is given by ΩR3 ≈ 2 m3; this is on the same order of the unit volume. Thus, we may roughly estimate F by assuming the coils are closely packed, F ≈ 1/R2, and we obtain G ≈ nU/R2 ≈ 10 J/m2. This is much smaller than the observed values of G. The effect of viscoelastic dissipation on the fracture energy G has been discussed by de Gennes.12 The de Gennes theory was originally proposed for the fracture in loosely cross-linked rubbers in which high-frequency modulus µh is much larger than low-frequency modulus µs (λ ≡ µh/µs ∼ 100). The outline of the theory is as follows. When a crack with a velocity V propagates in such a viscoelastic material with a single relaxation time τ, a strongly deformed region near the crack tip satisfying r < Vτ (r is the distance from the crack tip to the region) behaves as a hard solid with the high elastic modulus µh, and a low strained region far from the tip satisfying λVτ < r behaves as a soft solid with low µs. The viscoelastic dissipation arises only in the intermediate region, satisfying τV < r < λτV. The theory predicts that due to the dissipation in the intermediate region, the observed fracture energy G is enhanced by the factor λ ) µh/µs, G = λG0, where G0 is the intrinsic fracture energy determined by mechanism (a) mentioned above. Although the viscoelastic enhancement seems to be applicable to the DN gels, it cannot explain the extremely large values of G in the DN gels: we may approximately regard µs as being the elastic modulus of the PAMPS single network gel before the second polymerization; the PAMPS is a very rigid polyelectrolyte polymer, and the PAMPS gel has an elastic modulus (0.1 MPa) on the same order of the DN gels; thus, the factor λ is not sufficient to increase G to 103 times, even if the mechanism works. Okumura13 has discussed the Griffith criterion for heterogeneous materials consisting of a soft phase with an elastic modulus µs and a hard phase with µh to propose an essential mechanism for the extremely large G of the DN gels. (Note that the notations µs and µh here represent a different meaning from those in the above: in the present context, they represent the spatial variation of the elastic modulus.) Important assumptions of the Okumura theory are as follows: (i) Breaking of the hard phase determines the fracture of the whole system. (ii) Divergence of stress at crack tips predicted by the continuum (linear elstic) mechanics,15 σ(r)∼r-1/2 (r is the distance from the crack tip, see Figure 5), should be cut off at the characteristic size ξ of the heterogeneity, rather than at the lower limit of the continuum description ah (ξ . ah; ah is, for example, the mesh size of polymer network systems). The predicted form of the fracture energy is G ) λG0, where G0 is the intrinsic fracture energy of the hard phase, and the enhancement factor λ depends on the method of distributing the stress to the two phases. For the equal stress condition, where both soft and hard phases sustain a common stress, the enhancement factor is given by λ = (µh/µs)(ξ/ah). For the equal strain condition, where the strain is common to both phases, λ = ξ/ah. The factor ξ/ah contained in λ for both conditions comes from the following physical reason: to compensate the reduction of the cutoff stress by the heterogeneity, the excess elastic energy must be supplied to the material. This results in the enhancement of the effective fracture energy. The factor µh/µs characteristic of the equal stress condition comes from the following reason: When the soft and hard phases sustain the common stress, the soft phase stores an
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Tanaka et al. the DN gels is measured in steady state fractures. In order for the enhancement mechanism can work in the steady state fracture, a mechanism to steadily dissipate the excess elastic energy is required. The PAAm chains filling the voids may play the role of the absorber of the excess elastic energy. The experimental result of (E3) is consistent with this possibility, i.e., for large x, the dissipation of the excess elastic energy is suppressed, and the enhancement mechanism works insufficiently. Conclusion
Figure 5. Schematic of an infinite plate containing a crack stretched by a uniform remote stress σrem. The plate consists of soft and hard phases with a characteristic size ξ of the heterogeneity. The stress near the crack tip behaves like σ(r) ∼ 1/r1/2 for r . ξ.
elastic energy much larger than that of the hard phase. Thus, further excess elastic energy is required to initiate the crack growth, and the effective fracture energy becomes even larger. The heterogeneity mechanism is feasible for the remarkably large G of the DN gels: the structural analysis of the DN gels by dynamic light scattering (DLS) showed that in the relaxation time spectrum of DLS, slow modes probably corresponding to the Rouse mode of the PAAm solution exist,16 which suggests the existence of large voids in the first PAMPS network filled with the second PAAm polymers. However, problems relating to the role of the second PAAm chains remain to be clarified. First, according to the Okumura theory, the enhancement factor ξ/ah arises even when the voids are empty or filled with solvent. On the other hand, the normal PAMPS gels are very weak and brittle, i.e., they have very small fracture energies. Second, deformation of the DN gels is close to the equal strain condition in the theory (deformation of the soft voids passively obeys that of the surrounding first network), and under this condition, the enhancement factor is independent of the mechanical property of the soft phase (voids). On the other hand, G of the DN gels is sensitive to the cross-linking density x of the second network (E3). To consider the above discrepancies, the difference in the method of determining the fracture energy may be important: the enhancement of G in the Okumura theory is concerned with the criterion for onset of crack growth; on the other hand, G of
We measured the fracture energy G of the DN gels by a tearing test in order to qualify the mechanical strength of the DN gels. The measured values of G for small x are on the order of 102-103 J/m2, much larger than G for normal PAMPS or PAAm gels with similar polymer concentrations. A possible mechanism for the large values of G is the reduction of stress concentration due to the heterogeneity of the DN gels. Acknowledgment. This research was supported in part by Grant-in-Aid for the Basic Research A and the Creative Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan. References and Notes (1) Gong, J. P.; Katsuyama, Y.; Kurokawa, T.; Osada, Y. AdV. Mater. 2003, 15, 1155. (2) Gong, J. P.; Higa, M.; Iwasaki, Y.; Katsuyama, Y.; Osada, Y. J. Phys. Chem. B 1997, 101, 5487. (3) Gong, J. P.; Osada, Y. J. Chem. Phys. 1998, 109, 8062. (4) Gong, J. P.; Iwasaki, Y.; Osada, Y. J. Phys. Chem. B 2000, 104, 3423. (5) Kaneko, D.; Tada, T.; Kurokawa, T.; Gong, J. P.; Osada, Y. AdV. Mater. 2005, 17, 5335. (6) Greensmith, H. W. J. Polym. Sci. 1959, 21, 175. (7) Tanaka, Y.; Fukao, K.; Miyamoto, Y. Eur. J. Phys. E 2000, 3, 395. (8) Zarzycki, J. J. Non-Cryst. Solids 1988, 100, 359. (9) Brown, H. R. Macromolecules 1993, 26 (3), 1666. (10) Ondarcuhu, T. J. Phys. II 1997, 7, 1893. (11) Lake, G. J.; Thomas, A. G. Proc. R. Soc. London 1967, 300, 108. (12) de Gennes, P. G. Langmuir 1996, 12, 4497. (13) Okumura, K. Europhys. Lett, 2004, 67, 470. (14) Tsukeshiba, H.; Huang, M.; Na, Y.-H.; Kurokawa, T.; Kuwabara, R.; Tanaka, Y.; Furukawa, H.; Osada, Y.; Gong, J. P. J. Phys. Chem. B, submitted for publication. (15) Slepyan, L. I. Models and Phenomena in Fracture Mechanics; Springer-Verlag: Berlin, Germany, 2002. (16) Na, Y.-H.; Kurokawa, T.; Katsuyama, Y.; Tsukeshiba, H.; Gong, J. P.; Osada, Y.; Okabe, S.; Karino, T.; Shibayama, M. Macromolecules 2004, 37 (14), 5370.
