Fracture Process of Double-Network Gels by Coarse-Grained

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

Fracture Process of Double-Network Gels by Coarse-Grained Molecular Dynamics Simulation Yuji Higuchi,*,†,‡,§ Keisuke Saito,† Takamasa Sakai,‡,∥ Jian Ping Gong,⊥ and Momoji Kubo† †

Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai, Miyagi 980-8577, Japan PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan § Institute for Solid State Physics, The University of Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan ∥ Department of Bioengineering, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan ⊥ Faculty of Advanced Life Science and Soft Matter GI-CoRE, Hokkaido University, N21W11, Kita-ku, Sapporo 001-0021, Japan ‡

ABSTRACT: Double-network (DN) gels consisting of highly and slightly crosslinked networks exhibit good mechanical properties, complicated deformation behavior, and fracture processes owing to the existence of a large number of influential parameters. To determine the effects of these factors at the molecular level to improve the mechanical properties further, we studied the fracture processes of DN gels using a coarse-grained molecular dynamics simulation. First, we propose a modeling method for DN gels consisting of highly (first) and slightly (second) cross-linked networks. Then, we stretch the DN gels and investigate the effects of the network ratio, chain length, and first- and second-network structures on the mechanical properties. During the stretching, the stress increases with the bond breaking in the first network. Then, the stress further increases with the simultaneous bond breaking in the first and second networks when they are entangled with one another. Finally, the bond breaking in the first network stops, and only the bond breaking in the second network occurs. The second network remains at a high strain, which prevents rupture of the gel. We find that (i) a low concentration of the first network is necessary for the gel to exhibit the properties of both the first and second networks, (ii) a tense first network increases the Young’s modulus, and (iii) the second network with a long chain length and separated crosslinking points increases the peak stress and ductility. We have therefore successfully elucidated the effects of the network structures on the mechanical properties of DN gels.

■

INTRODUCTION Hydrogels possess a high water content within their polymer chain networks and exhibit good biological compatibility, making them promising biomaterials for medical applications.1−5 However, conventional hydrogels are weak and brittle and therefore difficult to apply to biomaterials. Recently, tough hydrogels have been developed by the design of polymer networks such as topologically designed networks,6 ideal network structures,7 nanocomposite-containing hydrogels,8 and double-network (DN) gels.9 Since DN gels consist of two kinds of networks, the mechanism responsible for their toughness is difficult to understand but interesting owing to the complex network structure. DN gels consist of a mixture of highly and slightly crosslinked networks, which are referred to as the first and second networks, respectively. The essential criteria were suggested in the previous review10 and can be summarized as follows. (i) The first network should be rigid and brittle, which indicates a high level of cross-linking (when the cross-linking concentration is high, the chains between the cross-linked points are tight). (ii) The second network should be soft and ductile, which means a low level of cross-linking. (iii) The concentration of the second network should be 20−30 times higher than that of the first network. (iv) The molecular mass of polymer chains in the second network should be high. To © XXXX American Chemical Society

satisfy this requirement, the chain length of the second network should be long enough to possess many cross-linking points. For DN gels, the design of the network structure is most important for increasing the toughness. Indeed, the same idea can be applied to elastomers. Ducrot et al. showed that elastomers consisting of the same network structure as DN gels also exhibited high Young’s modulus and ductility.11 In the deformation and fracture processes of DN gels, the stress increases with bond breaking in the first network at a low strain. This process was confirmed by observing large hysteresis between the first and second loads.12 At low strain, the covalent bonds in the first network serve as sacrificial bonds, which prevents crucial fractures such as large crack propagation. Then, necking and ductile properties are observed at high strain owing to the second network.13 To improve the mechanical properties and develop gels with high performance and toughness, it is essential to elucidate the effects of the two kinds of polymer chains at the molecular level. However, the influential parameters such as the ratio of the networks, molecular mass, cross-linking concentration, and network structures are too numerous to understand the deformation and fracture Received: January 18, 2018 Revised: April 2, 2018

A

DOI: 10.1021/acs.macromol.8b00124 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

where l is the bond length between neighboring monomers, θ is the angle between adjacent bond vectors, r is the distance between two monomers, and a is the diameter of the beads. The length and energy are measured in units of a and kBT, respectively, where kB is the Boltzmann constant and T is the absolute temperature. The potential for bonding was also used to study the fracture processes of polymers.15,29 On the basis of the previous studies, we selected the average bond length as approximately 1.0a, the length of bond breaking as 1.5a, and the energy barrier to bond breaking as 75kBT. We set the values for Ubond to kbond = 3600, R0 = 1.5a, R1 = (2.5/3.0)a, and U0 = 75.0. For the bending elasticity, we chose kθ = 4, 16, 64, and 4000, where the persistence lengths are Np ≃ kθ ≃ 2.0a, 4.0a, 8.0a, and 63a, respectively. The excluded-volume and attractive effects were included in the Lennard-Jones potential, ULJ. The value of ε was set to 0.4. We excluded the interaction of ULJ for the bonded pairs. The force, f, was calculated from eqs 1−3. The stress against the stretching σ was calculated from the forces and kinetic energy as σ = −(1/V)∑[ri f i + (1/m)pipi] = σbond + σbend + σLJ + σkinetic, where V is the volume of the cell and m = 1.0 and pi are the mass and momentum, respectively. The concentration in this calculation means the number concentration of monomers in the first and second networks, which is proportional to molar concentration in the experiment. The monomers obey the stochastic dynamics described by the Langevin equation

processes. There are still problems to be solved in the molecular theory. The key questions that remain unanswered include how these important parameters influence the mechanical properties such as the Young’s modulus, yielding, and rupture and how they can be used to obtain the desired mechanical properties. Molecular simulations are useful for elucidating complicated processes in polymers. The deformation and fracture processes of polymers have been actively studied. Coarse-grained simulations are useful for revealing polymer mechanisms at the molecular level. The fracture processes that occur upon stretching and during adhesion and separation have been well studied in elastomers and polymer glasses.14−21 Although the behavior of homogeneous structures such as elastomers and polymer glasses has been revealed at the molecular level, the study of inhomogeneous structures is more challenging. However, the inhomogeneous structures of polymers have recently been studied via molecular simulations. One example of an inhomogeneous structure is a lamellar structure consisting of elastomer and glass layers.22,23 Crystalline polymers consisting of coexisting crystal and amorphous regions are more difficult to investigate and have been studied by only a few research groups.24−30 In hydrogels composed of polymers, the progress in studying the deformation and fracture processes at the molecular level has been similar. While single gels have been well studied,31−37 inhomogeneous structures, namely, DN gels, have not been investigated at the molecular level. Jang et al. studied the mechanical properties of DN gels using molecular dynamics simulations.38 However, they used allatomic models and considered only small networks. Edgecombe et al. investigated swelling and mechanical properties of two interpenetrating polymer networks by a Monte Carlo simulation.39 They adopted a coarse-grained model and focused on the charge of polymers and counterions in the solvent. Since the calculation of electrostatic interaction is too large, the small networks were constructed. Thus, the issue of how the important factors affect the mechanical properties of DN gels has not yet been revealed. In this study, we investigated the fracture processes of DN gels using a coarse-grained molecular dynamics simulation. First, the details of the coarse-grained molecular dynamics simulation are introduced. Next, the proposed modeling method for DN gels is discussed. The fracture processes of DN gels are then presented, followed by a discussion of the implications of our work and a comparison with other studies.

m

Ubend k = θ (1 − cos θ )2 kBT 2

(2)

⎡⎛ a ⎞12 ⎛ a ⎞6 ⎤ = ϵ⎢⎜ ⎟ − 2⎜ ⎟ ⎥ ⎝r ⎠ ⎦ kBT ⎣⎝ r ⎠

(3)

= −γ

dri + fiU + ξi dt

(4)

0

0

length in the z direction and L0 is the initial length of L.

■

RESULTS AND DISCUSSION Modeling of Double-Network Gels. Previous simulation studies have focused on a single-gel model31−37 or a small double-network model.38,39 There has been no study on DN gels using a coarse-grained model to date. Thus, in this subsection we discuss the development of the modeling method for DN gels. We first consider the mechanical properties of single gels to validate the results of the coarsegrained model. We then propose a method for modeling DN gels and subsequently stretch the DN gels. In the initial state, a radical-like polymerization method40 was used to generate the models. Then, the polymer chains were randomly cross-linked in any given value. The cell was relaxed without external pressure using the NPT ensemble. In experimental studies, poly(2-acrylamido-2-methyl-1-propanesulfonic acid) (PAMPS) polyelectrolyte and polyacrylamide (PAAm) neutral polymers are often used to generate the first and second networks of DN gels, respectively. In the case of elastomers, double networks possessing similar characteristics

METHOD To investigate the fracture process of DN gels, we performed coarse-grained molecular dynamics simulations using a bead− spring model with a three-dimensional boundary condition. The potential energy of this system can be represented by the following bonding, bending, and attractive/repulsive terms: (1)

dt

2

where γ = 1.0 is the drag coefficient. The constant τ = a(m/ ϵ)1/2 was selected as the unit for the time scale. We set the time step as dt = 7.5 × 10−3τ. The Brownian force ξi satisfies the fluctuation−dissipation theorem ⟨ξi(t)ξi(t′)⟩ = 6kBTγδijδ(t − t′). In the stretching simulation, the temperature was set to kBT = 1.00. The cell was drawn in the z direction with a velocity of v = 0.06a/τ. The cell lengths in the x and y directions were fixed. This drawing process was adopted from the previous studies15,17,29 to reveal the mechanical properties of polymers. L−L ΔL The strain was calculated using L = L 0 , where L is the cell

■

Ubond = k bond(l − R 0)3 (l − R1) + U0 kBT

d2ri

ULJ

B


Article

Macromolecules were used.11 Here, we model the double networks as the highly cross-linked first (first) and the slightly cross-linked second (second) networks. The number of monomers per chain, M, number of polymers, N, and concentration of cross-linking, C, are summarized in Table 1. The concentration was calculated Table 1. Modeling Parameters for the First and Second Networksa network

M

N

C (%)

lp (a)

first second

100 100

200 200

4 1

4 2

a

M, N, C, and lp indicate the number of monomers per chain, number of polymers, concentration of cross-linking points, and persistence length, respectively.

using C = Nc/(M × N), where Nc is the total number of crosslinking points in the system. The persistence length, lp, is also indicated. To mimic the properties of the rigid first network, we selected the high cross-linking concentration of 4%. Because electrolyte gels such as PAMPS are very often used as the first network and these gels are tense, we set the polymer chain as slightly rigid, with a persistence length of 4a. In contrast, neutral gels are commonly used as the second network, and these gels are ductile. Therefore, the cross-linking concentration and persistence lengths were set as 1% and 2a, which are less than those of the first network. To confirm the validity of the gel models for the first and second networks, the gels consisting of 100% first and 100% second networks were stretched in one direction (z direction). Figures 1a and 1b show typical snapshots of the fracture process of the 100% first and 100% second networks, respectively. The stress−strain curves for the model gels are presented in Figure 1c. In the 100% first network model, the stretching of the gel caused low-density regions to appear at a strain of 3.07. The gel ruptured at a strain of 5.01. In the stress−strain curve, a peak was observed at a strain of around 3.0. The maximum stress was almost 2.5kBT/a3. Then, the stress decreased toward zero at a strain of approximately 5.6. It became zero at a strain of 7.0, indicating the fracture of the gel. In the stretching of the 100% second network model, lowdensity regions were observed at a strain of 4.35; however, the gel did not rupture even at the high strain of 7.95. In the stress−strain curve, a peak was observed at a strain of 6.5. The stress was approximately 0.5kBT/a3, which is less than that of the 100% first network model. At a strain of 8.0, the stress did not reach zero, which indicated that this gel does not break even at a high strain. These results demonstrate that the 100% first network and 100% second network models are hard but brittle and ductile but soft, respectively. These trends are in agreement with the previous experimental study of PAMPS and PAAm gels.9 Therefore, the simple models described above can successfully reproduce the mechanical properties of the gels. In this study, the gels were stretched along one direction while the other two directions were fixed. This is a very severe condition because deformation perpendicular to the stretched direction is not greatly allowed, which results in the fracture of the gels. We consider that this model calculation corresponds to the fracture process around the crack by tearing in experimental studies. The model calculation indicates that the small area around the crack in the experiment is enlarged in the simulations. Similar simulations have been performed for polymer glasses15 and semicrystalline polymers,29 which

Figure 1. Fracture processes of (a) 100% first and (b) 100% second network gels. The size of the snapshot for the 100% second network gel at a strain of 7.95 was reduced by 33.3% because the cell length along the stretching direction was too large. (c) Stress−strain curves for the 100% first network and 100% second network models.

successfully revealed the dynamics at the molecular level. The correspondence between the experiments and simulations of the rupture was briefly discussed. In experiments, for a typical single-network gel, the sample ruptures suddenly when some network strands break. This indicates that the bond breaking in the networks leads to the crucial fracture. For a DN gel, the sample does not rupture abruptly when some network strands break. In contrast, in the simulations, sudden rupture was not observed even for single-network gels owing to the simulation size. When one part of the network strands breaks, the gel does not suddenly rupture but instead breaks gradually. This is because the broken part is large compared to the simulation size. Therefore, the rupture stress in the simulations cannot be directly compared with that in the experiments. We should also mention the compressibility. The osmotic pressure was not considered in the simulation, and therefore the stretching simulations were performed under the incompressible condition. The characteristics of DN gels have also been observed in double-network elastomers.11 Here, we also only consider the network structure. However, this is sufficient to clarify the important role that the network structures play in the mechanical properties. Next, we describe the proposed modeling method for DN gels. In experimental studies, the initial step is the synthesis of the first network by copolymerization of the appropriate monomers and cross-linkers. Prior to the synthesis of the second network, the first network is expanded by dilution with a solution of the monomers for the second network. After the C


Article

Macromolecules dilution, the second network is formed within the first network. In this process, each of the two networks is cross-linked separately. To mimic this step in the simulation, we modeled the DN gel networks as summarized below and in the schematic images shown in Figures 2a−d: (i) The polymer

Three models were prepared here, where the ratios of the first and second networks were 20% and 80%, 50% and 50%, and 80% and 20%, respectively. The modeling parameters are summarized in Table 2. The chain lengths, concentrations, and Table 2. Modeling Parameters Used To Investigate the Influence of the Ratio of the Two Networksa M of first M of second N of first N of second C of first (%) C of second (%) lp of first (a) lp of second (a)

first 80%

first 50%

first 20%

100 100 160 40 4 1 4 2

100 100 100 100 4 1 4 2

100 100 40 160 4 1 4 2

a

M, N, C, and lp indicate the number of monomers per chain, number of polymers, concentration of cross-linking points, and persistence length, respectively.

persistence lengths of the first and second networks were kept constant in these calculations. Figures 3, 4, and 5 show

Figure 2. (a−d) Schematic images of the modeling method for DN gel networks. (e) Sample snapshot of a DN gel showing the two kinds of networks.

chains were generated via a radical-like polymerization method40 (Figures 2a,b). (ii) The polymer chains were then divided into the first and second networks in the desired ratio (Figure 2c). (iii) The persistence length and cross-linking concentration were set separately. The first and second networks are not cross-linked with one another (Figure 2d). By using separate parameters, the rigid and soft networks were formed. This model permits alterations of the ratio of the two different networks, chain length, persistence length, distance between cross-linked points, and so on. Figure 2e shows a typical snapshot of a DN gel consisting of 50% of the first network and 50% of the second network. Ratio of the Two Networks. Two different networks are present in DN gels. Therefore, changes in their properties such as the cross-linking concentration, a tense of network, and chain length greatly influence the mechanical properties of the resulting DN gels. The existence of such a large number of influential parameters makes theoretical studies of these gels very complicated. To elucidate the underlying mechanisms responsible for the toughness of DN gels, it is essential to break down the problem and determine the influence of each parameter individually. Initially, to simplify the problem, we changed the ratio of the two networks to find the conditions that best represent the properties of DN gels.

Figure 3. Fracture processes of the DN gel consisting of 80% and 20% of the first and second networks, respectively. (a) Cross-sectional snapshots and (b) stress−strain curve and numbers of bond-breaking events in the first and second networks.

snapshots and stress−strain curves of the fracture process when the ratios of the first and second networks were 20% and 80%, 50% and 50%, and 80% and 20%, respectively. During the stretching, the numbers of bond-breaking events in the first and second networks are also shown in Figures 3, 4, and 5 because the bond breaking in the networks is a dominant factor that influences the mechanical properties, according to experimental studies.12 As shown in Figure 3a, when the ratios of the first and second networks were 80% and 20%, respectively, the networks extended and then almost broke at a strain of 4.66. A void was formed with a small number of chains connecting the two parts of the fractured gel. This process was similar to that observed for the 100% first network as shown in Figure 1a. The stress D


Article

Macromolecules

confirm that the bond breaking is an important parameter for unraveling the mechanical properties of the networks during stretching. After the initial increase in the stress, it decreased toward zero at a strain of 6.0. The number of bond-breaking events in the second network was only 1. This indicates that the second network has only a negligible influence on the mechanical properties. As shown in Figure 4a, when the ratios of the first and second networks were 50% and 50%, respectively, the networks extended and then almost broke at a strain of 4.74, whereupon only a few chains still connected the two parts of the fractured gel. The stress increased and a peak was observed at a strain of 3.0. The stress then decreased toward 0 at a strain of 6.5. The bond breaking in the bonds in the first network increased at a strain of 1.5. In contrast, the bond breaking in the second network did not occur. These trends are almost identical to those observed for the DN gel consisting of 80% and 20% of the first and second networks, respectively. As shown in Figure 5, for the gel comprising 20% and 80% of the first and second networks, respectively, the networks extended at a strain of 4.03; however, they did not separate. Remarkably, the first and second networks were entangled with one another, which was not observed in the previous two cases shown in Figures 3 and 4. At a high strain of 7.08, the second network still connected the gel. The bond breaking in the first network began at a strain of 1.0, and the stress increased. At strains from 4.0 to 5.0, the bond breaking in the second network simultaneously occurred, and the stress decreased. The bond breaking in both networks occurred and worked at the same time. Then, the bond breaking in the first network stopped, and only the bond breaking in the second network occurred at strains from 6.0 to 8.0, while the stress against the stretching remained. These results demonstrate that DN gels gradually exhibit the characteristics of the first and second networks when the concentrations of the first and second networks are low and high, respectively. However, the two networks are not entangled with each other, leading to less bond breaking and less works of the second network, when the concentrations of the first and second networks are high and low, respectively, or the same. Entanglements between the first and second networks are important, as indicated by experimental results.41 Therefore, for a DN gel to exhibit the properties of both the first and second networks, the concentrations of the highly cross-linked first network and the slightly cross-linked second network should be low and high, respectively. Previous experimental results have also demonstrated that tough DN gels are composed of low and high concentrations of the highly and slightly cross-linked networks, respectively.42 As our results are in agreement with the experimental studies, the proposed modeling method described in this article can be considered validated. Chain Length. We next investigated the influence of the chain length (molecular mass) on the mechanical properties. This is because the entanglements between the two networks are an important factor influencing the mechanical properties as discussed above, and these entanglements increase with increasing chain length. Here, four models were prepared, where the chain lengths of the first and second networks were set to 100 or 200 monomers, as summarized in Table 3. Figure 6 shows the stress−strain curves for models A, B, C, and D. The chain lengths of the first and second networks were shortest in model A. Upon increasing the chain length of the second network from 100 monomers to 200 monomers per

Figure 4. Fracture processes of the DN gel consisting of 50% of each of the first and second networks. (a) Cross-sectional snapshots and (b) stress−strain curve and numbers of bond-breaking events in the first and second networks.

Figure 5. Fracture processes of the DN gel consisting of 20% and 80% of the first and second networks, respectively. (a) Cross-sectional snapshots and (b) stress−strain curve and numbers of bond-breaking events in the first and second networks. An enlarged snapshot at a strain of 4.03 is also shown. The size of the snapshot at a strain of 7.08 was reduced by 8% because the cell length along the stretching direction was too large.

increased, and a peak was observed at a strain of around 2.0. The stress then decreased. When the number of bond-breaking events in the first network increased at a strain of 1.0, the stress also sharply increased. Upon bond breaking, the bonds were extended and the stress increased. This indicates that the first network works against the stretching and is responsible for the mechanical properties of the system. Therefore, these results E


Article

Macromolecules

leads to the higher fracture strength of DN gels containing longer chains in the second network. When the number of monomers per chain for the second network was 100, the number of cross-linking points was approximately 1 because the concentration of cross-linking points was 1%. The second network was difficult to percolate, and the gel was hardly formed. However, topological entanglements of polymer chains act as physical cross-links against stretching. Thus, the stress in response to stretching is low when the chain length for the second network is short. Upon increasing the polymer chain length for the second network to 200 monomers, the second network percolated and the gel was formed. Indeed, the threshold of the cross-linking concentration for the percolation is estimated as pc ≃ M−1 in linear polymer chains.43 Therefore, the second network percolates for M = 200. The percolation of the second network enhances the bond breaking in the second network and the entanglements between the first and second networks. Therefore, the stress increases. According to the above results, when the concentrations of the first and second networks are low and high, respectively, and the chain length of the second network is long, the DN gels contain many entanglement points. Upon stretching, the bond breaking in the first network interpenetrated in the second network occures first. However, the DN gels do not break because the first and second networks are entangled with one another. The entanglement of two networks leads to deconcentration of the stress. In one review of DN gels,10 it was discussed that entanglement between the first and second networks is important for the force transmission from the second network to the first network to form the softened zone at the mesoscale. We hypothesize that the first network structure is broken and optimized at the same time owing to the entanglements during the stretching. These entanglements are important for the fracture process of DN gels and exhibit pulley effects. At a high strain, the chains in the second network are extended flexibly and then the bond breaking occurs. Network Structure of the First Gel. Although the effects of network ratio and chain length on the fracture process of DN gels have been revealed, the mechanical properties of these gels are still not consistent with experimental results.44 In particular, the Young’s modulus is higher in the experiments than in the simulations. In an attempt to improve the mechanical properties, the design of the network structures was investigated experimentally using tetra-PEG gels.45 Indeed, changes in the structure of the first network were found to lead to changes in the mechanical properties.46−48 According to our results, the bond breaking in the first network occurred at a low strain. Thus, we changed the structure of the first network, which could conceivably affect the Young’s modulus. In experimental studies, the first network is rigid.10 To realize this rigidity in the simulations, we established the tense first network by increasing the persistence length. Prior to preparing the cross-linked networks, more rigid chains were chosen as the first network, with a persistence length of 63a. The polymer chains in this first network were very rigid and almost straight. This afforded a tense network between the cross-linking points. In experimental terms, this model corresponds to a highly swollen state due to osmotic pressure. After the cross-linking, the persistence length was decreased to 8a to avoid an unrealistically rigid state. The relaxation calculation was then performed. This system is distinct from the previously discussed models, where the chains between the cross-linking

Table 3. Modeling Parameters Used To Investigate the Influence of the Chain Lengtha model A

model B

model C

model D

100 100 80 320 4 1 4 2

100 200 80 160 4 1 4 2

200 100 40 320 4 1 4 2

200 200 40 160 4 1 4 2

M of first M of second N of first N of second C of first (%) C of second (%) lp of first (a) lp of second (a) a

M, N, C, and lp indicate the number of monomers per chain, number of polymers, concentration of cross-linking points, and persistence length, respectively. The total number of monomers was 4 × 104 in all models: M(1st) × N(1st) + M(2nd) × N(2nd) = 4 × 104. The ratio of monomers in the first network was set at 20% in all models.

Figure 6. Stress−strain curves of fracture processes for different chain lengths. The polymer chains of the first network in models A, B, C, and D consisted of 100, 100, 200, and 200 monomers, respectively. The polymer chains of the second network in models A, B, C, and D consisted of 100, 200, 100, and 200 monomers, respectively.

chain in model B, the maximum stress increased. In contrast, increasing the chain length of the first network from 100 monomers to 200 monomers per chain in model C did not substantially affect the stress. In the case of model D with the longest chain lengths for both networks, the stress was almost identical to that in model B. These results clearly demonstrate that increasing the chain length of the second network increases the stress in response to the strain, whereas increasing the chain length of the first network does not. To determine the roles of the first and second networks, we also investigated the number of bond-breaking events in each network, as summarized in Table 4. The number of bond-breaking events in the first Table 4. Numbers of Bond-Breaking Events in the First and Second Networks for Models A, B, C, and D network

model A

model B

model C

model D

first second

33 14

36 49

27 9

33 39

network was almost the same in all of the models. The chain lengths of the first and second networks did not affect the work of the first network. In contrast, the number of bond-breaking events in the second network was higher in models B and D than in models A and C. The chain length of the second network influences the work of the second network. Upon increasing the chain length of the second network, the number of entanglements between the two networks increases. This F


Article

Macromolecules

to a strain between 0.05 and 0.25. The stress was treated as a linear function and fitted by

points were loose. The parameters used in the stretching simulation are summarized in Table 5.

σzz = Y (ΔL /L0) + A

Table 5. Modeling Parameters Used To Investigate the Influence of the Network Structure of the First Gela M of first M of second N of first N of second C of first (%) C of second (%) lp of first (a) lp of second (a)

loose network

tense network

100 200 80 160 4 1 4 2

100 200 80 160 4 1 8 2

(5)

where Y is the Young’s modulus and A is a constant representing the stress at a strain of 0.05. The estimated Young’s moduli for the loose and tense networks were 0.16 and 0.41, respectively. When the first network is tense, the Young’s modulus increases because a tense network is hardly deformable at a low strain. We have therefore demonstrated that a tense highly cross-linked network is important for the Young’s modulus. Network Structure of the Second Gel. Finally, we focused on the structure of the second network and prepared two models. In addition, on the basis of the above results, we reconstructed and enlarged the model. Here, the chain length of the second network was increased to enhance the entanglement between the first and second networks. The concentration of cross-linking points in the first network was also increased to improve the Young’s modulus. The parameters are summarized in Table 6.

a

M, N, C, and lp indicate the number of monomers per chain, number of polymers, concentration of cross-linking points, and persistence length, respectively. The total number of monomers was 4 × 104 in both models. The ratio of monomers in the first network was set at 20% in both models.

Figures 7a and 7b show the stress−strain curves for the loose and tense networks, respectively. The numbers of bond-

Table 6. Modeling Parameters Used To Investigate the Influence of the Network Structure of the Second Gela network

M

N

C (%)

lp (a)

first second

100 400

160 160

8 0.7

8 2

a

M, N, C, and lp indicate the number of monomers per chain, number of polymers, concentration of cross-linking points, and persistence length, respectively. The total number of monomers was 8 × 104. The ratio of monomers in the first network was set at 20%.

In the initial model, the structure of the second network was constructed using the same procedure as applied previously. To characterize the structure of the second network, the distances between the cross-linked points were calculated and these are shown in Figure 8 (indicated by “short”). The distances were

Figure 7. Stress−strain curves and numbers of bond-breaking events in the first and second networks for the (a) loose and (b) tense first networks.

breaking events in the first and second networks are also indicated. In the loose network (Figure 7a), the stress increased gradually. The first peak of the stress was observed at a strain of 1.5. At this strain, the bond breaking in the first network also started. Then, the stress increased and a second peak occurred at a strain of 3.0. The bond breaking in the second network started before those in the first network had finished. Both networks worked well in this model. In the tense network (Figure 7b), the stress increased sharply. The first peak of the stress was observed at a strain of 0.75, while the bond breaking in the first network started at a strain of 0.6. Then, the stress increased with the bond breaking in the second network. A second peak was observed at a strain of 4.0. Both networks also worked well in this model. Therefore, the tense structure of the first network only influenced the Young’s modulus. We next evaluated the Young’s modulus by fitting the stress in response

Figure 8. Histogram showing the distances between the cross-linked points obtained from the “short” and “long” models.

calculated by counting the number of monomers between the cross-linked points. The average value was found to be 9.4a. The distances were distributed around zero. This indicated that the distances between the cross-linked points in the second network were short and unnatural. To improve the second network, we therefore applied a second model where we increased the distances between the cross-linked points and reduced the distribution around zero. The distances from this improved model are also shown in Figure 8 (indicated by G


Article

Macromolecules “long”). The average value was 73.8a, and the distribution had become Gaussian. We consider this distance distribution to be more realistic compared with that obtained from the “short” model. Figure 9 shows the stress−strain curve and numbers of bondbreaking events in the first and second networks with the short

Figure 10. Stress−strain curve and numbers of bond-breaking events in the first and second networks for the second network with the long cross-linked distance (“long” model).

shown in Figure 9. This demonstrates that an improvement in the structure of the second network enhanced the properties of the first network. To elucidate the mechanism, we evaluated the entanglements between the first and second networks by counting both the number of monomers in the second network around a bond-breaking chain and the number of chains crosslinked with the bond-breaking chain in the first network, when the length between two monomers was less than a and the repulsive force acted upon them. We define this number as Nrep. The degree of entanglements was calculated using Nent = Nrep/Nfirst, where Nfirst is the number of first-network chains cross-linked with bond-breaking chains from the same network. Figure 11 shows the degree of entanglement and the stress−

Figure 9. Stress−strain curve and numbers of bond-breaking events in the first and second networks for the second network with the short cross-linked distance (“short” model).

cross-linked distance (“short” model). The stress increased, and the first peak was observed at a strain of 0.6. Then, the stress decreased, and no second peak was observed. The bond breaking in the first network occurred at a strain of 0.25, and the bond breaking ceased at a strain of 2.1. The bond breaking in the second network started at the low strain of 1.6. The range in which bond breaking in both networks occurred at the same time was narrow. These features indicated that the two networks do not work well. The estimated Young’s modulus of 1.76 is high compared with the previous results discussed in this study. Meanwhile, the chain length of the second network is sufficiently long compared with those of the previous results. Therefore, the problem seemed to lie with the structure of the second network. The short distance between the cross-linked points in the second network causes the bond breaking at a lower strain for a larger simulation size. It should be noted that the stretched cell length in the z direction, ΔL = L − L0, increases with an increasing simulation size because the initial cell length is large and the large stretched length is essential even though the same strain is compared. For the previous systems discussed in this study, the short distance between the cross-linked points was not problematic owing to the small simulation size and the small stretched cell length. However, the second network does not work well with the short distance between the cross-linked points in the larger model. This originates from the simulation size. However, we hypothesize that short cross-linking distances may be problematic in experimental studies. The bond breaking in networks with short distances between the cross-linking points can easily occur, which leads to the rupture of the gel in experimental studies. Figure 10 shows the stress−strain curve and numbers of bond-breaking events in the first and second networks for the improved model (“long” model). The stress increased, and the first peak was observed at a strain of 0.6. Then, the stress increased again, and a second peak was observed at a strain of 4.0. The bond breaking in the first network occurred initially, followed by those in the second network. At strains from 3.2 to 5.2, the bond breaking in both networks occurred. This indicated that the two networks work well. Interestingly, the number of bond-breaking events in the first network increased considerably compared with the results for the short model

Figure 11. Stress−strain curve and degree of entanglement between the first and second networks for the second network with the long cross-linked distance.

strain curve. At strains from 0.6 to 2.0, the number of entanglements for a bond-breaking chain in the first network was around 9. Then, the number of entanglements increased at strains from 2.0 to 5.0. In Figure 9, the bond breaking in the chains in the first network did not occur at a high strain. In contrast, in Figure 10 the bond breaking in the first network occurred at a high strain. Therefore, at strains from 2.0 to 5.0, the increased entanglement between the first and second networks enhanced the elongation of the first network, leading to the bond breaking. When the cross-linked distance was short, the bond breaking in the second network occurred at low strain, and the network broke apart. This indicated that the entanglement between the first and second networks was not enhanced. Furthermore, the bond breaking in the second network did not occur at a high strain because the bond breaking had already occurred at a low strain. Therefore, the polymer chains in the second network do not work. When the cross-linked distance was long, the bond breaking in the second network did not occur at a low strain, and the second network still remained at a high strain. The entanglement between the first and second networks was enhanced, and the bond breaking H


Article

Macromolecules

second network. To evaluate this process quantitatively, the probability densities of the monomers in the first and second networks were calculated along the stretching direction, as shown in Figures 12b and 12c, respectively. The cell lengths were normalized. At a strain of 0.00, the monomers in the first and second networks were distributed homogeneously. At a strain of 0.77, the probability density of the first network at a normalized distance from 0.8 to 0.9 decreased slightly, while the monomers of the second network remained homogeneously distributed. At a strain of 2.01, the probability density of the first network greatly decreased at a normalized distance from 0.7 to 1.0. Therefore, the decrease at a strain of 0.77 indicated the start of void formation. At a strain of 2.01, the probability density of the second network in the same area decreased. These results indicated that the first network broke first and then the second network broke in the same region. At a strain of 4.02, two peaks were observed in the probability density of the first network, indicating a block structure. The density of the first network became almost zero in two areas; however, the density of the second network still had a value of 0.6. This indicated that the second network did not break. We have therefore successfully elucidated the fracture process at the molecular level. A similar fracture process was suggested by the experimental results,10 where block structures consisting of the first network were connected by the second network in the fracture process of DN gels. Our simulation results confirm the experimental results and reveal the details of the underlying processes at the molecular level. To reveal the reason for the generation of void, we counted the number of breaking points in the first network along the stretching direction until the low-density region appeared. Figure 13a shows the number of breaking-events in the first network along the stretching direction for a strain from 0.00 to 0.77. The cell length was normalized. There are two peaks at around 0.2 and 0.8. The values are consistent with the voids, which are lowest parts at a strain of 4.02 in Figure 12b. This

in the second network could occur and increase the stress at the high strain. Therefore, the polymer chains in the second network work well. Finally, we will now present the detailed fracture process for the best model obtained in this study. We will compare our results with the experimental results and discuss the processes. Figure 12a shows snapshots of the fracture process. Initially,

Figure 12. Fracture processes of well-designed DN gels consisting of 20% and 80% of the first and second networks, respectively. The first network was tense, and each chain consisted of 100 monomers. The cross-linked distance of the second network was sufficiently long and each chain consisted of 400 monomers. (a) Cross-sectional snapshots and probability densities of the (b) first and (c) second networks.

both networks were distributed homogeneously at a strain of 0.00. The DN gel was then stretched, and the distribution appeared to be homogeneous at a low strain of 0.77. A void appeared at a strain of 2.01 (see the right part of the snapshot). Around the void, the first network almost disappeared; however, the second network remained intact in this region and prevented the rupture of the gel. At a strain of 4.02, the second network still existed, and the DN gel had not broken. Remarkably, block structures of the first network were observed, and these block structures were connected by the

Figure 13. (a) Summation of number of bond breaking events in the first network for a strain from 0.00 to 0.77. (b) Probability density of cross-linking points in the first and second networks along the stretching direction at a strain of 0.00. I


Article

Macromolecules means that bond breaking in the first network is the reason for the generation of voids. To find the defect, we analyzed the density of cross-linking points in the first network. Figure 13b shows the density of cross-linking points along the stretching direction at a strain of 0.00. The cell length was normalized. Since the value is fluctuated much, we could not find the relation with the generation of voids, indicating no defect originated in the density of cross-linking points. Furthermore, the density of monomers in the first-network is almost flat as shown in Figure 12b. Therefore, the defect does not originate in the density of monomers and cross-linking points. We think that the small network consisting of four cross-linking points is defect and preferentially breaks. This interesting topic should be solved in the future. The simulated stress−strain curve exhibited the same trend as typical experimental stress−strain curves.44 Although the simulation results cannot be directly compared with the experimental results, we believe that the mechanism revealed in this study is the same as that which occurs experimentally. In experimental studies, strain hardening was observed prior to sample failure.13,49 We believe that the hardening in the simulation corresponds to the increase in the stress at strains from 2.0 to 4.0 shown in Figure 10. The stress against the stretching still remained at strains from 4.0 to 7.0; however, we believe that the fracture state of the gel at strains over approximately 4.0 corresponds to the rupture that occurs in experiments. As discussed above, in the simulations the gel gradually breaks after rupture owing to the limited simulation size. The rupture observed experimentally is difficult to observe in the simulations. There are many parameters to construct DN gels. This is one of the important problems to be solved, which is our purpose in this article. Thus, we revealed its effects one by one. Finally, we found the best condition and that the stress−strain curve and fracture processes are consistent with experiments. Therefore, we believe that our modeling method can reproduce the experimental condition and can reveal the detailed fracture process at the molecular level. To validate our modeling method and show the repeatability, three different models in the same modeling conditions were constructed from the different initial conditions. Figure 14a shows the stress−strain curves. The stress−strain curve shown in Figure 10 is also indicated as trajectory 1. The Young’s moduli are 1.61, 1.83, 1.63, and 1.99 for trajectory 1, 2, 3, and 4, respectively. It is clearly shown that all stress−strain curves are almost same, where the stress increases sharply at a low strain, the stress increases again, and the DN gels fractured at a high strain. Figures 14b and 14c show the numbers of breaking events in the first and second networks, respectively. The events also show same tendency in all trajectories. Therefore, we successfully showed the repeatability and validated our modeling method. The experiment50 showed that the chemical bonding between the first and second networks, called internetwork cross-linking (INC), assists the transfer of the applied force, resulting in the high ductility and high strength of the DN gel, when the number of INC points and cross-linking points in the second network is over the threshold of network percolation. Therefore, the effect of INC is similar to that of the crosslinking in the second networks. It was also revealed that the fracture energy is largest when INC is suppressed and the second network is successfully constructed. At first, we reproduced the above experimental condition, where the

Figure 14. Four different trajectories in the fracture process of DN gels. Trajectory 1 is same as the trajectory in Figure 10. (a) Stress− strain curve. Number of bond breaking events in (b) the first and (c) second networks.

second network was not cross-linked and the number of INC points was increased. Here, we changed the number of INC points, NINC c , and the number of cross-linking points in the second network, N2nd c . The concentration of INC points was calculated using CINC = NINC c /(M2nd × N2nd), where M2nd is a number of monomers per chain in the second network and N2nd is the number of polymers in the second network. At first, CINC was set at 0.0%, 0.35%, 0.70%, and 3.5%, while the second network was not cross-linked. Figure 15a shows the stress− strain curve with the different concentrations of INC points. For the reference, the stress in Figure 10 was indicated. When the concentration of INC points is 0.0% or 0.35%, the stress decreases after the first peak, which indicates that the DN gels are brittle. This is because the second network was not constructed and the applied force was not transferred. The stress increases with the increase in the concentration of INC points to 0.60% and 3.5%. However, it was brittle and was not ductile. This shows that INC indeed transfers the applied force but does not realize the best condition of hard and ductile DN gels. In the experiment,50 it was shown that the best condition is that INC is suppressed and the second network is constructed, INC assists the transfer of applied force, and DN gels become brittle when INC points are too many. Our calculation is consistent with the experimental results and is validated. Next, we also tried another idea, where the first and second networks were cross-linked after the construction of their networks with an expectation of an increase in the transfer of applied force. The concentration of INC points was set at J


Article

Macromolecules

the molecular level, the method reported here allowed the observation of (i) the bond breaking in the first and second networks during stretching, (ii) the formation of block structures of the first network during the fracture process of the DN gel, and (iii) the topological entanglement between the first and second networks, all of which are difficult to observe directly through experiments.

■

CONCLUSIONS Elucidating the fracture processes of double-network gels on the molecular scale is essential for improving the properties of these materials. We have studied the fracture process by focusing on polymer network structures and chain dynamics via coarse-grained molecular dynamics simulations. We found that (i) a low concentration of the highly cross-linked first network is necessary for the gels to exhibit the properties of both the first and second networks, (ii) the tense first network increases the Young’s modulus, and (iii) the second network with a long chain length and separated cross-linking points increases the peak stress and ductility. We also revealed the fracture process at the molecular level. At a low strain, the tense first network lengthens and then the bond breaking occurs. Then, entanglement between the first and second networks enhances further bond breaking in the first network and prevents the breakage of the gel. The bond breaking in the first network generates a void and separates, leading to the formation of block structures connected through the second network. At a high strain, the bond breaking in the second network leads to an increase in the stress and high ductility. The voids generated by the separation of the first network continue to grow. We have successfully revealed the effects of the network structures on the mechanical properties and the fracture process at the molecular level.

Figure 15. Stress−strain curve with the difference concentration of INC points. The concentration of INC points was calculated using INC is the number of INC points, CINC = NINC c /(M2nd × N2nd), where Nc M2nd is a number of monomers per chain in the second network, and N2nd is the number of polymers in the second network. For the reference, the stress in Figure 10 is also indicated. (a) Without crosslinking points in the second network and with cross-linking between the first and second networks. (b) With cross-linking between the first and second networks after the construction of the first and second networks.

■

0.035%, 0.07%, 0.35%, and 1.4%. Figure 15b shows the stress− strain curves. For the reference, the stress in Figure 10 was indicated. When the concentration of INC points is 0.035%, the stress is almost same as that without INC. This indicates that INC does not affect the mechanical properties of DN gels when the concentration of INC points is low. With an increase in the concentration of INC points, the stress increases but the strain at the maximum stress decreases and DN gels become brittle. Therefore, in both cases, the chemical bonding between the first and second networks exhibits disadvantages. We successfully reveal that chemical bonding between the first and second networks is defect in DN gels. The effect of the stretching velocity is not trivial in these simulations. Therefore, we also examined the stretching simulation with velocities of v = 0.10a/τ and 0.01a/τ for the model shown in Figure 10. We confirmed that the stress against the stretching and the mechanism of the fracture processes did not change in this simulation. Experiments have shown that the fracture energy does weakly depend on the crack velocity, but the tensile stress−strain behavior does not.51 Therefore, our simulation results are in agreement with the experimental findings. In the previous molecular simulations, single gels31−37 or double-network gels consisting of small networks38,39 were studied. In this article, we have proposed a method for modeling DN gels that shows good agreement with the experimental results. We constructed simple models and successfully separated the numerous complicated factors to allow their effects to be determined individually. This is one of the key advantages of molecular simulations. Furthermore, at

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (Y.H.). ORCID

Yuji Higuchi: 0000-0001-8759-3168 Takamasa Sakai: 0000-0001-5052-0512 Jian Ping Gong: 0000-0003-2228-2750 Momoji Kubo: 0000-0002-3310-1858 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This research was supported by JST-PRESTO “Molecular Technology and Creation of New Functions” (Project Nos. JPMJPR13KF and JPMJPR14K7) and MEXT as “Exploratory Challenge on Post-K Computer” (Challenge of Basic Science− Exploring Extremes through Multi-Physics and Multi-Scale Simulations).

■

REFERENCES

(1) Nicodemus, G. D.; Bryant, S. J. Cell Encapsulation in Biodegradable Hydrogels for Tissue Engineering Applications. Tissue Eng., Part B 2008, 14, 149−165. (2) Hoffman, A. S. Hydrogels for Biomedical Applications. Adv. Drug Delivery Rev. 2002, 54, 3−12. (3) Drury, J. L.; Mooney, D. J. Hydrogels for Tissue Engineering: Scaffold Design Variables and Applications. Biomaterials 2003, 24, 4337−4351.

K


Article

Macromolecules

Shear Using Molecular Dynamics Simulation. Macromolecules 2014, 47, 2515−2528. (26) Yeh, I.-C.; Andzelm, J. W.; Rutledge, G. C. Mechanical and Structural Characterization of Semicrystalline Polyethylene under Tensile Deformation by Molecular Dynamics Simulations. Macromolecules 2015, 48, 4228−4239. (27) Yamamoto, T. Molecular Dynamics in Fiber Formation of Polyethylene and Large Deformation of the Fiber. Polymer 2013, 54, 3086−3097. (28) Jabbari-Farouji, S.; Rottler, J.; Lame, O.; Makke, A.; Perez, M.; Barrat, J.-L. Plastic Deformation Mechanisms of Semicrystalline and Amorphous Polymers. ACS Macro Lett. 2015, 4, 147−150. (29) Higuchi, Y.; Kubo, M. Coarse-grained Molecular Dynamics Simulation of the Void Growth Process in the Block Structure of Semicrystalline Polymers. Modell. Simul. Mater. Sci. Eng. 2016, 24, 055006. (30) Higuchi, Y.; Kubo, M. Deformation and Fracture Processes of a Lamellar Structure in Polyethylene at the Molecular Level by a CoarseGrained Molecular Dynamics Simulation. Macromolecules 2017, 50, 3690−3702. (31) Hosono, N.; Masubuchi, Y.; Furukawa, H.; Watanabe, T. A Molecular Dynamics Simulation Study on Polymer Networks of EndLinked Flexible or Rigid Chains. J. Chem. Phys. 2007, 127, 164905. (32) Zidek, J.; Milchev, A.; Jancar, J.; Vilgis, T. A. DeformationInduced Damage and Recovery in Model Hydrogels - A Molecular Dynamics Simulation. J. Mech. Phys. Solids 2016, 94, 372−387. (33) Sliozberg, Y. R.; Mrozek, R. A.; Schieber, J. D.; Kröger, M.; Lenhart, J. L.; Andzelm, J. W. Effect of Polymer Solvent on the Mechanical Properties of Entangled Polymer Gels: Coarse-Grained Molecular Simulation. Polymer 2013, 54, 2555−2564. (34) Schneider, S.; Linse, P. Swelling of Cross-Linked Polyelectrolyte Gels. Eur. Phys. J. E: Soft Matter Biol. Phys. 2002, 8, 457−460. (35) Schneider, S.; Linse, P. Monte Carlo Simulation of Defect-Free Cross-Linked Polyelectrolyte Gels. J. Phys. Chem. B 2003, 107, 8030− 8040. (36) Schneider, S.; Linse, P. Discontinuous Volume Transitions in Cross-Linked Polyelectrolyte Gels Induced by Short-Range Attractions and Strong Electrostatic Coupling. Macromolecules 2004, 37, 3850− 3856. (37) Edgecombe, S.; Schneider, S.; Linse, P. Monte Carlo Simulations of Defect-Free Cross-Linked Gels in the Presence of Salt. Macromolecules 2004, 37, 10089−10100. (38) Jang, S. S.; Goddard, W. A.; Kalani, M. Y. S. Mechanical and Transport Properties of the Poly(ethylene oxide)−Poly(acrylic acid) Double Network Hydrogel from Molecular Dynamic Simulations. J. Phys. Chem. B 2007, 111, 1729−1737. (39) Edgecombe, S.; Linse, P. Monte Carlo Simulation of Two Interpenetrating Polymer Networks: Structure, Swelling, and Mechanical Properties. Polymer 2008, 49, 1981−1992. (40) Perez, M.; Lame, O.; Leonforte, F.; Barrat, J.-L. Polymer Chain Gneration for Coarse-Grained Models Using Radical-Like Polymerization. J. Chem. Phys. 2008, 128, 234904. (41) Huang, M.; Furukawa, H.; Tanaka, Y.; Nakajima, T.; Osada, Y.; Gong, J. P. Importance of Entanglement between First and Second Components in High-Strength Double Network Gels. Macromolecules 2007, 40, 6658−6664. (42) Ahmed, S.; Nakajima, T.; Kurokawa, T.; Haque, M. A.; Gong, J. P. Brittle−Ductile Transition of Double Network Hydrogels: Mechanical Balance of Two Networks as the Key Factor. Polymer 2014, 55, 914−923. (43) de Gennes, P. Scaling Concepts in Polymer Physics; Cornell University Press: 1979. (44) Matsuda, T.; Nakajima, T.; Fukuda, Y.; Hong, W.; Sakai, T.; Kurokawa, T.; Chung, U.-i.; Gong, J. P. Yielding Criteria of Double Network Hydrogels. Macromolecules 2016, 49, 1865−1872. (45) Kondo, S.; Sakurai, H.; Chung, U.-i.; Sakai, T. Mechanical Properties of Polymer Gels with Bimodal Distribution in Strand Length. Macromolecules 2013, 46, 7027−7033.

(4) Pogue, B. W.; Patterson, M. S. Review of Tissue Simulating Phantoms for Optical Spectroscopy, Imaging and Dosimetry. J. Biomed. Opt. 2006, 11, 041102. (5) Kojima, C.; Suehiro, T.; Watanabe, K.; Ogawa, M.; Fukuhara, A.; Nishisaka, E.; Harada, A.; Kono, K.; Inui, T.; Magata, Y. DoxorubicinConjugated Dendrimer/Collagen Hybrid Gels for Metastasis-Associated Drug Delivery Systems. Acta Biomater. 2013, 9, 5673−5680. (6) Okumura, Y.; Ito, K. The Polyrotaxane Gel: A Topological Gel by Figure-of-Eight Cross-links. Adv. Mater. 2001, 13, 485−487. (7) Sakai, T.; Matsunaga, T.; Yamamoto, Y.; Ito, C.; Yoshida, R.; Suzuki, S.; Sasaki, N.; Shibayama, M.; Chung, U. Chung, U.-i. Design and Fabrication of a High-Strength Hydrogel with Ideally Homogeneous Network Structure from Tetrahedron-like Macromonomers. Macromolecules 2008, 41, 5379−5384. (8) Haraguchi, K.; Takehisa, T. Nanocomposite Hydrogels: A Unique Organic−Inorganic Network Structure with Extraordinary Mechanical, Optical, and Swelling/De-swelling Properties. Adv. Mater. 2002, 14, 1120−1124. (9) Gong, J.; Katsuyama, Y.; Kurokawa, T.; Osada, Y. DoubleNetwork Hydrogels with Extremely High Mechanical Strength. Adv. Mater. 2003, 15, 1155−1158. (10) Gong, J. P. Why are double network hydrogels so tough? Soft Matter 2010, 6, 2583−2590. (11) Ducrot, E.; Chen, Y.; Bulters, M.; Sijbesma, R. P.; Creton, C. Toughening Elastomers with Sacrificial Bonds and Watching Them Break. Science 2014, 344, 186−189. (12) Webber, R. E.; Creton, C.; Brown, H. R.; Gong, J. P. Large Strain Hysteresis and Mullins Effect of Tough Double-Network Hydrogels. Macromolecules 2007, 40, 2919−2927. (13) Na, Y.-H.; Tanaka, Y.; Kawauchi, Y.; Furukawa, H.; Sumiyoshi, T.; Gong, J. P.; Osada, Y. Necking Phenomenon of Double-Network Gels. Macromolecules 2006, 39, 4641−4645. (14) Rottler, J.; Barsky, S.; Robbins, M. O. Cracks and Crazes: On Calculating the Macroscopic Fracture Energy of Glassy Polymers from Molecular Simulations. Phys. Rev. Lett. 2002, 89, 148304. (15) Rottler, J.; Robbins, M. O. Growth, Microstructure, and Failure of Crazes in Glassy Polymers. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 68, 011801. (16) Rottler, J. Fracture in Glassy Polymers: A Molecular Modeling Perspective. J. Phys.: Condens. Matter 2009, 21, 463101. (17) Makke, A.; Perez, M.; Lame, O.; Barrat, J.-L. Mechanical testing of glassy and rubbery polymers in numerical simulations: Role of Boundary Conditions in Tensile Stress Experiments. J. Chem. Phys. 2009, 131, 014904. (18) Papakonstantopoulos, G. J.; Riggleman, R. A.; Barrat, J.-L.; de Pablo, J. J. Molecular Plasticity of Polymeric Glasses in the Elastic Regime. Phys. Rev. E 2008, 77, 041502. (19) Morita, H.; Yamada, M.; Yamaguchi, T.; Doi, M. Molecular Dynamics Study of the Adhesion between End-Grafted Polymer Films. Polym. J. 2005, 37, 782−788. (20) Morita, H.; Miura, H.; Yamada, M.; Yamaguchi, T.; Doi, M. Molecular Dynamics Study of the Adhesion between End-Grafted Polymer Films II - Effect of Grafting Density. Polym. J. 2007, 39, 73− 80. (21) Sides, S. W.; Grest, G. S.; Stevens, M. J.; Plimpton, S. J. Effect of End-Tethered Polymers on Surface Adhesion of Glassy Polymers. J. Polym. Sci., Part B: Polym. Phys. 2004, 42, 199−208. (22) Makke, A.; Lame, O.; Perez, M.; Barrat, J.-L. Influence of Tie and Loop Molecules on the Mechanical Properties of Lamellar Block Copolymers. Macromolecules 2012, 45, 8445−8452. (23) Makke, A.; Lame, O.; Perez, M.; Barrat, J.-L. Nanoscale Buckling in Lamellar Block Copolymers: A Molecular Dynamics Simulation Approach. Macromolecules 2013, 46, 7853−7864. (24) Lee, S.; Rutledge, G. C. Plastic Deformation of Semicrystalline Polyethylene by Molecular Simulation. Macromolecules 2011, 44, 3096−3108. (25) Kim, J. M.; Locker, R.; Rutledge, G. C. Plastic Deformation of Semicrystalline Polyethylene under Extension, Compression, and L


Article

Macromolecules (46) Nakajima, T.; Sato, H.; Zhao, Y.; Kawahara, S.; Kurokawa, T.; Sugahara, K.; Gong, J. P. A Universal Molecular Stent Method to Toughen any Hydrogels Based on Double Network Concept. Adv. Funct. Mater. 2012, 22, 4426−4432. (47) Nakajima, T.; Fukuda, Y.; Kurokawa, T.; Sakai, T.; Chung, U.-i.; Gong, J. P. Synthesis and Fracture Process Analysis of Double Network Hydrogels with a Well-Defined First Network. ACS Macro Lett. 2013, 2, 518−521. (48) Saito, J.; Furukawa, H.; Kurokawa, T.; Kuwabara, R.; Kuroda, S.; Hu, J.; Tanaka, Y.; Gong, J. P.; Kitamura, N.; Yasuda, K. Robust Bonding and One-Step Facile Synthesis of Tough Hydrogels with Desirable Shape by Virtue of the Double Network Structure. Polym. Chem. 2011, 2, 575−580. (49) Liang, S.; Wu, Z. L.; Hu, J.; Kurokawa, T.; Yu, Q. M.; Gong, J. P. Direct Observation on the Surface Fracture of Ultrathin Film DoubleNetwork Hydrogels. Macromolecules 2011, 44, 3016−3020. (50) Nakajima, T.; Furukawa, H.; Tanaka, Y.; Kurokawa, T.; Osada, Y.; Gong, J. P. True Chemical Structure of Double Network Hydrogels. Macromolecules 2009, 42, 2184−2189. (51) Tanaka, Y.; Kuwabara, R.; Na, Y.-H.; Kurokawa, T.; Gong, J. P.; Osada, Y. Determination of Fracture Energy of High Strength Double Network Hydrogels. J. Phys. Chem. B 2005, 109, 11559−11562.

M


Fracture Process of Double-Network Gels by Coarse-Grained

Recommend Documents