Deformation Behavior and Failure of Bimodal Networks

Sep 19, 2017 - Using computer simulations, we have investigated the deformation and stress–strain behavior of a series of ideal gels without any def...
1 downloads 12 Views 2MB Size
Download PDF
Article pubs.acs.org/Macromolecules

Deformation Behavior and Failure of Bimodal Networks Natasha Kamerlin†,‡ and Christer Elvingson*,† †

Department of Chemistry - Ångström Laboratory, Physical Chemistry, Uppsala University, Box 523, S-751 20 Uppsala, Sweden Department of Mathematics, Uppsala University, Box 480, S-751 06 Uppsala, Sweden

‡

S Supporting Information *

ABSTRACT: Using computer simulations, we have investigated the deformation and stress−strain behavior of a series of ideal gels without any defects, with a bimodal molecular weight distribution, subject to tensile strains. These networks were prepared with a spatially homogeneous distribution of short and long chains, where all chains are elastically active, without needing to consider possible effects of chain aggregation or entanglements on the physical properties. For all fractions of short chains, the first chains to rupture were the short chains that were initially oriented along the strain axis. The average orientation of the short chains slightly increased with decreasing fraction of short chains. This could be explained by the detailed structure of the network at different compositions. Analysis of the stress−strain relation for the short and long chains showed that the stress was not uniformly shared. Instead, the short chains are more strongly deformed whereas the long chains only make a negligible contribution at smaller strains. The mechanical properties of the bimodal networks at lower fractions of short chains also deviated from the behavior of equivalent unimodal networks with the corresponding average chain length, showing that bimodality alone is sufficient to increase both the maximum extensibility and toughness.

■

INTRODUCTION Most standard polymer gels have relatively poor mechanical properties due to their large solvent content, which may be a limiting factor, for example, in tissue engineering or other biomedical applications where load bearing is required. Various methods have been employed to reinforce gels and improve their failure properties, including incorporating fillers.1,2 One approach to improving the mechanical properties is to use unconventional network structures. For example, hydrogels developed by Gong et al. consisting of two components with very different degrees of cross-linking, so-called doublenetworks, may exhibit enhanced mechanical properties and have received considerable attention as promising candidates in bioengineering.3−8 Another way of taking advantage of a bimodal molecular weight distribution is to end-link short and long chains into one network. The practical appeal of such bimodal networks is their potential for substantial improvements in strength and toughness when the chain lengths of the two components are considerably different. While bimodal poly(dimethylsiloxane) (PDMS) elastomers have been extensively studied by e.g. Mark et al. in uniaxial tension experiments,9−15 to the best of our knowledge, no such systematic study has been conducted on the deformation behavior of bimodal hydrogels. In the studies by Mark et al., bimodal networks were prepared by end-linking mixtures of low- and high-molecularweight precursor chains. Unlike conventional free radical polymerization reactions which may lead to a broad distribution of chain lengths,16 end-linking methods give a more welldefined structure with a predictable strand length distribution. In some cases, these bimodal networks were found to be reinforced, with both a high ultimate strength, common to a © XXXX American Chemical Society

short-chain network, and a high maximum extensibility, common to networks of long chains.9−15 The stress−strain curves for the reinforced bimodal networks showed an upturn in stress at higher elongation ratios, which was attributed to the limited extensibility of the short chains, with the long chains delaying the rupture process. Furthermore, the bimodal reinforcement was still present after swelling with linear dimethylsiloxane oligomers.17 Among the requirements observed by Mark et al. for such a bimodal enhancement, it was noted that the system should contain a high concentration of short chains and that the two molecular weights should differ by at least a factor of 10,13 although Rath et al. recently showed that bimodal networks with a molar mass ratio less than 5 could still display enhanced mechanical properties and an upturn in stress.18 The early works by Mark et al. did not consider the role of the spatial distribution of short and long chains within the bimodal network. Experiments using small-angle neutron scattering,19,20 dynamic light scattering,21 and NMR22 have, however, shown that bimodal networks appear heterogeneous, as the short chains tend to segregate into clusters, and it was speculated that these clusters could be responsible for the enhanced properties. Highly heterogeneous networks, with clusters of cross-linked short chains interconnected by long chains, have been synthesized and found to have lower elastic moduli than both the corresponding homogeneous bimodal networks and the unimodal networks.15,23,24 In more recent work, Mark et al. studied the effect of nanoscale clusters.15,25,26 Received: August 1, 2017 Revised: September 8, 2017

A

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

the chains are homogeneously distributed. Starting from these model networks, we impose a uniaxial strain and explore the deformation behavior and ultimate properties of the networks. Our goal is to study how the macroscopic deformation is partitioned between the different types of chains, by following the chain orientation and rupture as well as the stress−strain behavior under an applied strain. These bimodal networks are also compared with equivalent unimodal networks of a similar cross-linking density.

While the macroscopically phase-separated bimodal networks were weak, their stress−strain relation still showed a stress upturn, similar to the ones previously observed for reinforced bimodal networks. The nanostructurally clustered bimodal networks, formed from very short chains, were well reinforced but did not show a distinct stress upturn. On the basis of their findings, they suggested that there may also be a structural component to the bimodal enhancement effect, where the cluster formation may contribute to the reinforcement. In another approach, Kondo et al. recently used tetra-poly(ethylene glycol) (PEG) prepolymers to form bimodal networks with a more homogeneous structure than those obtained using conventional end-linking methods.27 The mechanical properties of these bimodal networks coincided with predictions by theoretical models for unimodal networks having the same number-average chain length, suggesting that bimodality alone does not account for the bimodal enhancement but that the latter may instead be due to the presence of entanglements and short-chain clusters. To study the correlation between the macroscopic properties and the molecular scale structure, computer simulations are particularly useful by offering the possibility to have precise control of the microstructural network characteristics. Although the mechanical response of polymer systems under various conditions has been extensively studied in simulations,28−40 only a few studies have focused on the deformation behavior of bimodal networks, mainly using lattice models.41−45 A twodimensional lattice was used by Termonia to examine the stress−strain behavior of bimodal networks and could reproduce the high toughness seen in experiments,41 while Sotta used a cubic lattice with chains fixed at both ends to measure the induced orientation of chain segments under uniaxial strain. 42 Both experiments and Monte Carlo simulations were used by Genesky et al. to examine the role played by the limited extensibility and of clustering of short chains in the ultimate mechanical properties of bimodal networks.43−45 Reasonable agreement was found between the experimental results and simulation data, with some bimodal networks at higher fractions of short chains displaying improved properties even compared to equivalent networks with the same number-average chain length. Some caution is, however, required in interpreting the simulation results beyond the elastic region, as the model did not take into account bond breaking. The study found that short chains tend to cluster when the concentrations of short chains is low, while networks with higher short-chain concentrations appear more homogeneous, although some heterogeneity might have been present also at high concentrations. Since the networks with clustering at lower concentrations of short chains did not show any increased toughness or upturn in stress, it was concluded that the limited extensibility of the short chains at high elongation ratios plays the most significant role in improving the ultimate properties. In the present study, we focus on the effect of a bimodal molecular weight distribution on the deformation behavior of polymer gels, without needing to consider possible aggregation effects. Various factors, such as short-chain clustering, structural defects, and trapped entanglements, may influence the mechanical properties. By decoupling these factors, it is possible to find which of these makes the greatest contribution to the reinforcement mechanism. In a first approach, we generate bimodal networks of various compositions in a controlled manner, which are free from defects and where

■

COMPUTATIONAL DETAILS Constructing a Network. The details related to the generation of an ideal, cross-linked network have been described previously.46,47 In summary, tetravalent cross-linking nodes are inserted at random positions within a cylinder, subject to a minimum distance constraint in order to achieve a more homogeneous node−node distribution. Each node is formally connected to its four nearest unsaturated cross-linking nodes. The polymer chains corresponding to the short and long chains are then generated by placing Ns or Nl number of beads, respectively, along the connecting paths to form a closed network. The number of permanent cross-links in this model is thus well-defined, and there are no trapped entanglements, loops, or dangling ends; i.e., all chains are elastically active. For the polymer chains, we have used a bead−spring model, where the beads have a radius of a = 0.5 nm. The stretching of the bonds is described by a harmonic potential, where the stretching force constant is chosen such that the second moment of the bond length distribution at equilibrium is b2/ 1000, where b = 2a is the distance at minimum stretching potential. To enable bond breaking and material fracture, bonds that extend beyond a center-to-center distance of 1.5b during a simulation are removed, i.e., the corresponding interaction is not included, and the chain becomes elastically inactive. The stiffness of the chains is modeled by a bending potential Ub =

kb ∑ (cos θi − cos θ0)2 2 i

(1)

where the angle between consecutive bonds corresponding to the minimum bending potential has been set to θ0 = 0 and the bending force constant, kb, is determined from the persistence length,48 which is set to P = 2b. The nonbonded forces representing an excluded volume between the units are modeled using a soft repulsive potential49 Firep =

ϵrepkBT b

2

⎛b

⎞ − 1⎟⎟rij n̂ ij ⎝ rij ⎠

∑ ⎜⎜ j

(2)

where ϵ is a dimensionless force constant set to 200 and n̂ij is the unit vector from particle j to i. Brownian Dynamics Simulation. To simulate the deformation of gels in a solvent, we have used Brownian dynamics. In these simulations, the time evolution of the position of unit i is given by50,51 rep

ri(t + Δt ) = ri(t ) +

D0Δt Fi + A i kBT

(3)

where Δt is the time step, D0 = kBT/(6πηa) is the diffusion coefficient, η is the solvent viscosity, and Fi is the total direct force on unit i. The term Ai is a Gaussian random vector with the properties ⟨Ai⟩ = 0 and ⟨Ai·Aj⟩ = 6D0Δtδij. All simulation results presented are carried out at a constant temperature of T B

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules = 293.15 K, viscosity of η = 1.002 × 10−3 Pa s, and a time step Δt = 1.0 ps. The model networks (generated as described above) consist of 5000 chains. The short and long chain lengths were set to Ns = 6 and Nl = 30, respectively. The number fraction of short chains, fs, was varied between 0.5 and 1.0. A pure long-chain network (i.e., fs = 0.0) was, however, too computationally intensive to simulate within a practical amount of time. The initial configurations were relaxed until no significant changes could be observed in the size and shape of the system. Unless otherwise stated, results represent averages over three independent starting structures. Uniaxial Elongation. The deformation of the equilibrated gels was studied by applying a continuous uniaxial strain until failure. Analogous to an experimental setting, a boundary driven deformation is accomplished by introducing two grips in the form of two infinite planes at the gel cylinder bases,52,53 at initial positions z = ±L0/2 (determined from the longitudinal density profile, Figure S1). At each time step, the planes, together with all units i such that |z(i)| > L0/2, are incrementally shifted to move the sample ends apart at a rate of 1.7 × 10−6b per time step, or 100 mm/min, corresponding to strain rates of (3.3−5.8) × 104 s−1. The deformation behavior and stress−strain response were investigated for strain rates of (3−7) × 104 s−1 and found to be identical, with only a marginally larger ultimate stress with increasing strain rate (Figure S2). During elongation, the alignment of the chains along the strain axis was determined by the segmental orientation, which is described by the second-order Legendre polynomial54 P2 = ⟨3 cos2 ϕ − 1⟩/2

Figure 1. Simulation snapshots using VMD56 taken during uniaxial elongation of a system with fs = 0.90. Short and long chains are colored blue and purple, respectively.

(4)

where ϕ is the angle between the direction of extension and the local chain axis of the polymer segment. The stress tensor was evaluated according to55 σ=−

1 ∑ ∑ fij(ri − rj) V i j>i

Figure 2. Chain tortuosity for the short (solid) and long (dashed) chains in bimodal networks with fs = 0.50−1.00.

(5)

where fij is the force exerted on unit i due to unit j and V is the volume over which the stress is averaged. The results below will be presented using the dimensionless stress σ′ = (4πa3/ (3kBT))σ. The strain is given by α = (Lz − L0)/L0, where Lz is the current distance between the grips.

chains in a unimodal short-chain network with a long chain. This long chain would necessarily have a smaller end-to-end distance (i.e., a larger tortuosity) than if it were part of a unimodal long-chain network. Similarly, it could be expected that the greater structural rigidity which results from increasing the fraction of short chains would also progressively perturb some of the short chains compared to the conformations assumed by a few short chains dispersed in a relatively flexible long-chain network. The long chains will also tend to swell the rigid short-chain network at larger fs, potentially pulling some short-chain ends together and others apart. Compared to the unimodal network, the slight increase in the end-to-end distance for fs = 0.97 would suggest that the resulting contribution of the long-chain expansion is to pull apart the short chains. The decrease in τ with decreasing fraction of short chains, however, is not nearly as significant as for the long chains. Applying a uniaxial tension reduces the tortuosity, as the chain strands are pulled apart and their end-to-end distances are increased, and eventually leads to stretching of the chain backbone, which can be observed as an increase in stress (cf. Figure 4). At sufficiently large deformations, a further increase in strain leads to bond breaking (Figure S3). Depending on the fraction of short chains, the systems can be stretched to

■

RESULTS AND DISCUSSION Simulation snapshots of one of the systems taken at various stages during the tensile deformation are shown in Figure 1. Visualizing the elongation process reveals the formation of cavities at larger strains that grow and ultimately lead to macroscopic fracture. Unless otherwise stated, the results are presented up to the point of network failure, which is taken as the strain at which the stress starts to significantly decline. Network Structure. The extent to which the short and long chains have been stretched during the elongation can be measured by the tortuosity, τ, which is the ratio of the chain contour length to the end-to-end distance. The tortuosity is shown in Figure 2 for a series of bimodal networks under uniaxial strain. The initial tortuosity of both the short and long chains decreases with decreasing cross-linking density as the restrictions on the motion of the cross-linking nodes are reduced. For the long chains, this decrease in tortuosity may be understood by looking at the effect of replacing one of the short C

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

become increasingly shielded from the applied strain, such that their end-to-end distances remain virtually unchanged during the elongation. In the networks considered here, where the chains are distributed homogeneously, no such shielding effect is present even at low molar concentrations of short chains. Instead, the average orientation of the short chains increases slightly when fs is reduced (for fs > 0.50). Although the average orientation of both types of chains is zero in the undeformed state, since there is no preferred direction of the bond vectors, one can analyze the initial orientation of only the chains which have ruptured at failure. The short chains that have ruptured have on average an initial segmental orientation between P2 = 0.083−0.16 (and an initial end-to-end orientation between P2 = 0.21−0.43) for fs = 0.50− 0.97, and the initial orientation is even higher for the first chains that rupture. In other words, it is shown that the first chains to break are the ones that are already somewhat aligned along the strain axis before imposing a strain. Mechanical Properties. A summary of the mechanical properties is shown in Table 1. The normalized stress−strain

different extents before the network fails. The stiffer short chains, however, have a lower chain tortuosity compared to the longer chains and more quickly reach their maximum extensibility. All the systems reach a plateau value greater than one, indicating that a significant portion of the chains are not stretched at failure, due to topological constraints and local chain rupturing. The decrease in tortuosity during elongation is associated with a greater average orientation of the chains in the direction of the strain axis. Figure 3 shows the chain segmental

Table 1. Properties of the Different Bimodal Networks (with Ns = 6 and Nl = 30) as Well as a Set of Unimodal Networksa fs

nc

ϕ

αf

σ′f

γ

0.50 0.65 0.80 0.90 0.95 0.97 1.00 1.00 (N = 8) 1.00 (N = 10) 1.00 (N = 16)

0.027 0.034 0.044 0.056 0.065 0.069 0.077 0.058 0.048 0.030

0.10 0.12 0.13 0.14 0.15 0.15 0.15 0.13 0.11 0.08

4.2 4.0 3.3 1.7 1.3 1.4 1.4 1.7 2.1 3.0

0.12 0.15 0.21 0.38 0.52 0.52 0.59 0.46 0.35 0.18

0.39 0.46 0.40 0.20 0.12 0.16 0.19 0.18 0.25 0.27

a Fraction of short chains (fs), cross-linking density (nc), volume fraction (ϕ), ultimate strain (αf), ultimate stress (σ′f), and area under the corresponding stress−strain curves (γ).

curves for different fs are shown in Figure 4. When the bimodal networks contain only a small amount of long chains (fs > 0.90), the stress−strain relation is essentially controlled by the

Figure 3. (a) Segmental chain orientation for networks with fs = 0.50− 1.00 and (b) the corresponding average orientation of the chain segments for the short (solid) and long (dashed) chains, shown separately.

orientation as the bimodal networks are deformed. The qualitative behavior is the same also for the average orientation of the chain end-to-end vectors, although the orientation of the end-to-end vector is larger compared to averaging over the individual links in each chain (Figure S4). The average orientation of the network chains at any given strain generally decreases as more long chains are incorporated into the network (Figure 3a). The separate orientations of the short and long chains are also shown in Figure 3b. Deformation causes the short chains to become highly stretched while the long chains are still being unravelled, which is consistent with observations in experiments and simulations at higher mol % of short chains.42,44,57 Simulations using dynamically end-linked networks have shown that decreasing the short-chain concentration ( fs) reduces the average orientation of the short chains.44 For those end-linked networks, the formation of clusters during the generation of networks with lower concentrations of short chains means that the short chains

Figure 4. Scaled stress−strain curves for networks with fs = 0.50−1.00. The dashed lines correspond to unimodal networks with similar crosslinking densities (N = 8 (yellow), 10 (green), and 16 (purple)). Note that the unimodal network with N = 16 has a cross-linking density lying between fs = 0.50 and fs = 0.65. D

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules matrix of short chains. These bimodal networks behave similarly to the pure short chain unimodal network (i.e., fs = 1.00). The stress increases rapidly with the applied strain, but the networks fail after a relatively small additional strain, at a high ultimate stress. Introducing extensibility by reducing the fraction of short chains leads to a plateau at a lower stress value and a leveling off. During this time, primarily the short chains continue to rupture throughout the sample (Figure S3), weakening the network, but the eventual fracture is delayed by the presence of long chains. Computing the area under the stress−strain curves up to point of failure, corresponding to the toughness, shows a peak for fs = 0.65 (see Table 1). Although this could not be compared with a pure long-chain network due to CPU time, some previous experimental studies of bimodal elastomers have found an improvement over a range of compositions, with the best improvement in toughness for rather high short-chain concentrations, typically over 85 mol %, depending on, for example, how the networks are synthesized and which precursor chain lengths are used.9,11,17,43,44 In order to compare the stress−strain relation of the bimodal networks with unimodal networks of similar cross-linking densities, we also generated a number of equivalent unimodal networks (see Table 1 and Figure 4). Considering previous works, Genesky et al. compared bimodal networks at fs = 0.95 with equivalent unimodal networks and found an almost identical stress−strain behavior at small strains.43 Their bimodal networks, however, could be stretched much further than the unimodal networks before reaching fracture. Kondo et al. found, contrary to the results by Mark et al. and Genesky et al., that for tetra-PEG bimodal networks the ultimate strain corresponded to predictions by models for unimodal tetraPEG networks,27 suggesting that the stress is shared uniformly within the network and that the short chains do not rupture prior to the long chains. It should be noted that the former work used chains with a molecular weight ratio of 20 whereas Kondo et al. used a ratio up to 4. Although the ratio between our two chain lengths is only slightly larger than the one used by Kondo et al., our results are quite different. For the networks considered here, the bimodal networks with lower fs can be stretched much further than the corresponding unimodal network with the same number-average molecular weight of strands. This greater extensibility also leads to a larger area under the stress−strain curves and thus a larger toughness for the bimodal networks compared to the unimodal network with the same cross-linking density. The difference in the maximum extensibility between the bimodal and equivalent unimodal networks is, however, reduced at higher fractions of short chains. For fs = 0.90, a bimodal molecular weight distribution alone is not sufficient to improve the maximum extensibility compared to an equivalent unimodal network. It should be noted though that previous studies looked at networks with a possible presence of trapped entanglements and chain slipping, not included in our present model. Although the mechanical stress is a macroscopic quantity, it has its origin in microscopic terms,58 and it is useful to consider how the stress varies on a local scale by looking at the contribution to the stress from the short and long chains, separately. These separate contributions are shown in Figure 5, where it can be seen that the stress is not uniformly shared among the two types of chains for most parts of the deformation. In the absence of clustering and entanglements, the long chains do not contribute significantly to the mechanical properties at low elongation ratios. From the

Figure 5. Stress−strain curves for the same set of systems as in Figure 3, separately showing the stress from the short and long chains. The solid lines correspond to the short chains and the dashed lines to the long chains.

orientation in Figure 3, it was seen that the average orientation of the short chains increases slightly when some of these chains are substituted by long chains (i.e., for lower fs). The same behavior is also observed in Figure 5: at smaller strains (approximately 0.5 < α < 1), before any significant chain rupturing has taken place, the short-chain stress is generally larger for smaller fs, at least for fs down to 0.65. This could partly be due not only to the lower initial short-chain tortuosity but also to the connectivity of the network. We note that a cross-linking node in a bimodal network with fs = 0.97 almost always has four short chains attached, whereas a cross-linker in the fs = 0.65 network typically has two or three short chains. All four (nonentangled) chains at a cross-linking site are, however, not equally deformed in a uniaxial extension. By visualizing the chains that are more strongly oriented along the strain axis, it is seen that the stress is mainly sustained by the two short chains of each cross-linking site, in succession along the strain axis, forming percolated structures between the two moving surfaces (see Figure 6). The other two (short or long) chains at the junction will be only marginally deformed. Since a network with fs = 0.65 has fewer short chains per crosslinking site than fs = 0.97, a larger fraction of the short chains will be deformed in the small strain regime for fs = 0.65, before any significant chain rupturing has taken place, which could be seen above as a slightly larger average short-chain segmental orientation and stress. For fs = 0.50, there will, however, also be a number of cross-linking sites that have fewer than two short chains. Once a short chain ruptures, the next chain to rupture in fs = 0.97 will likely be another short chain that will also rupture with a small additional strain, which can be seen as a slightly more rapid increase in the fraction of ruptured chains (Figure S3). When a short chain in e.g. fs = 0.65 ruptures, the strain will be reapportioned to one of the flexible long chains at the junction, which will require a larger strain before rupturing. This reapportioning can be seen in the results above by correlating the short-chain rupturing with the deformation behavior; the short chains start to rupture at a strain of α ≈ 0.8 (Figure S3), and this is also when the long chains start to become more significantly affected by the applied strain (see Figures 3 and 5). E

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules

■

Article

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b01653. (i) Density profile of a bimodal network at equilibrium; (ii) stress−strain relation for a bimodal network being elongated at different strain rates; (iii) fraction of ruptured chains for the short and long chains, separately; (iv) orientation of the end-to-end vectors for networks with fs = 0.50−1.00 (PDF)

■

AUTHOR INFORMATION

Corresponding Author

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

Christer Elvingson: 0000-0001-5115-5481 Notes

The authors declare no competing financial interest.

■

Figure 6. Simulation snapshots using VMD at a fixed configuration at α = 1.45, showing only the short chains for clarity, from left to right for chains with the lowest to highest orientation along the strain axis, in bimodal networks with fs = 0.65 (upper panel) and fs = 0.97 (lower panel). The chains with the higher orientations collectively form nonbranched percolating structures between the two walls. 56

ACKNOWLEDGMENTS The computations were performed on resources provided by SNIC through Uppsala Multidisciplinary Centre for Advanced Computational Science (UPPMAX) and the National Supercomputer Centre (NSC) at Linköping University.

■

■

REFERENCES

(1) Gaharwar, K.; Peppas, N. A.; Khademhosseini, A. Nanocomposite hydrogels for biomedical applications. Biotechnol. Bioeng. 2014, 111, 441−53. (2) David, A.; Teodorescu, M.; Stanscu, P. O.; Stoleriu, S. Novel Poly(ethylene glycol) Composite Hydrogels with Hydrophilic Bentonite Nanoclay as the Filler. Materiale Plastice 2014, 51, 113−118. (3) Gong, J. P.; Katsuyama, Y.; Kurokawa, T.; Osada, Y. Doublenetwork hydrogels with extremely high mechanical strength. Adv. Mater. 2003, 15, 1155−1158. (4) Tsukeshiba, H.; Huang, M.; Na, Y. H.; Kurokawa, T.; Kuwabara, R.; Tanaka, Y.; Furukawa, H.; Osada, Y.; Gong, J. P. Effect of polymer entanglement on the toughening of double network hydrogels. J. Phys. Chem. B 2005, 109, 16304−9. (5) Gong, J. P. Why are double network hydrogels so tough? Soft Matter 2010, 6, 2583−2590. (6) Nonoyama, T.; Gong, J. P. Double-network hydrogel and its potential biomedical application: A review. Proc. Inst. Mech. Eng., Part H 2015, 229, 853−863. (7) Chen, J.; Ao, Y.; Lin, T.; Yang, X.; Peng, J.; Huang, W.; Li, J.; Zhai, M. High-toughness polyacrylamide gel containing hydrophobic crosslinking and its double network gel. Polymer 2016, 87, 73−80. (8) Fathi, A.; Lee, S.; Zhong, X.; Hon, N.; Valtchev, P.; Dehghani, F. Fabrication of interpenetrating polymer network to enhance the biological activity of synthetic hydrogels. Polymer 2013, 54, 5534− 5542. (9) Andrady, A. L.; Llorente, M. A.; Mark, J. E. Model networks of end-linked polydimethylsiloxane chains. VII.Networks designed to demonstrate non-Gaussian effects related to limited chain extensibility. J. Chem. Phys. 1980, 72, 2282. (10) Mark, J. E.; Tang, M.-Y. Dependence of the elastomeric properties of bimodal networks on the lengths and amounts of the short chains. J. Polym. Sci., Polym. Phys. Ed. 1984, 22, 1849−1855. (11) Tang, M.-Y.; Mark, J. E. Effect of composition and cross-link functionality on the elastomeric properties of bimodal networks. Macromolecules 1984, 17, 2616−2619. (12) Clarson, S. J.; Galiatsatos, V.; Mark, J. E. An investigation of the properties of non-Gaussian poly(dimethylsiloxane) model networks in the swollen state. Macromolecules 1990, 23, 1504−1507.

CONCLUSION Computer simulations have been used to investigate the deformation behavior and failure of ideal bimodal gels with a homogeneous distribution of chains, subject to a uniaxial strain. It could be shown that the first chains to rupture are short chains that are initially aligned along the strain axis. While simulations of dynamically end-linked networks have found that short-chain aggregation is more prominent at lower mole fractions, leading to a lower orientation with decreasing fraction of short chains, for our spatially homogeneous networks, such a shielding effect is not present even at lower fractions of short chains. Instead, the average orientation of the short chains increased slightly when the fraction of short chains was reduced. It was also shown that the stress is not uniformly shared among the short and long chains. The short chains are more strongly deformed while the long chains make a negligible contribution at smaller strains. A bimodal molecular weight distribution alone, however, is sufficient to strongly alter the mechanical properties of our ideal networks at lower fractions of short chains compared to unimodal networks of the corresponding number-average chain length, giving rise to a greater maximum extensibility and toughness. This difference in behavior is reduced with increasing fraction of short chains. In this work, we have considered ideal networks with a homogeneous distribution of chains in order to isolate the effect of bimodality. In real gels, there may also be collateral effects from a polydisperse short-chain length, network defects, and a structural component. Trapped entanglements, in particular, may also play a significant role and alter the deformation behavior. The effect of chain aggregation and entanglements will be systematically investigated in future work. F

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

(35) Carrillo, J.-M. Y.; MacKintosh, F.; Dobrynin, A. Nonlinear Elasticity From Single Chain to Network and Gels. Macromolecules 2013, 46, 3679−3692. (36) Lyulin, A. V.; Balabaev, N. K.; Mazo, M. A.; Michels, M. A. J. Molecular dynamics simulation of uniaxial deformation of glassy amorphous atactic polystyrene. Macromolecules 2004, 37, 8785−8793. (37) Grest, G.; Putz, M.; Everaers, R.; Kremer, K. Stress-strain relation of entangled polymer networks. J. Non-Cryst. Solids 2000, 274, 139−146. (38) Mukherji, D.; Abrams, C. F. Microvoid formation and strain hardening in highly cross-linked polymer networks. Phys. Rev. E 2008, 78, 050801. (39) Hagita, K.; Morita, H.; Doi, M.; Takano, H. Coarse-grained molecular dynamics simulation of filled polymer nanocomposites under uniaxial elongation. Macromolecules 2016, 49, 1972−1983. (40) Nazarychev, V. M.; Lyulin, A. V.; Larin, S. V.; Gurtovenko, A. A.; Kenny, J. M.; Lyulin, S. V. Molecular dynamics simulations of uniaxial deformation of thermoplastic polyimides. Soft Matter 2016, 12, 3972−3981. (41) Termonia, Y. Molecular model for the mechanical properties of elastomers. 2. Synergic effects in bimodal cross-linked networks. Macromolecules 1990, 23, 1481−1483. (42) Sotta, P. Local order in monomodal and bimodal polymer networks: Monte Carlo simulations. Macromolecules 1998, 31, 8417− 8422. (43) Genesky, G. D.; Aguilera-Mercado, B. M.; Bhawe, D. M.; Escobedo, F. A.; Cohen, C. Experiments and simulations: Enhanced mechanical properties of end-linked bimodal elastomers. Macromolecules 2008, 41, 8231−8241. (44) Genesky, G. D.; Cohen, C. Toughness and fracture energy of PDMS bimodal and trimodal networks with widely separated precursor molar masses. Polymer 2010, 51, 4152−4159. (45) Aguilera-Mercado, B. M.; Genesky, G. D.; Duncan, T. M.; Cohen, C.; Escobedo, F. A. H NMR and simulation studies of chain segment orientation in PDMS bimodal networks. Macromolecules 2010, 43, 7173−7184. (46) Kamerlin, N.; Elvingson, C. Tracer diffusion in a polymer gel: simulations of static and dynamic 3D networks using spherical boundary conditions. J. Phys.: Condens. Matter 2016, 28, 475101. (47) Kamerlin, N.; Elvingson, C. Collapse dynamics of core-shell nanogels. Macromolecules 2016, 49, 5740−5749. (48) Allison, S.; Sorlie, S.; Pecora, R. Brownian dynamics simulations of wormlike chains: dynamic light scattering from a 2311 base pair DNA fragment. Macromolecules 1990, 23, 1110−1118. (49) Elvingson, C. A general Brownian dynamics simulation program for biopolymer dynamics and its implementation on a vector computer. J. Comput. Chem. 1991, 12, 71−77. (50) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: 1987. (51) Ermak, D. L. A computer simulation of charged particles in solution. I. Technique and equilibrium properties. J. Chem. Phys. 1975, 62, 4189−4196. (52) Hölzl, T.; Trautenberg, H. L.; Göritz, D. Monte Carlo simulations on polymer network deformation. Phys. Rev. Lett. 1997, 79, 2293−2296. (53) Hölzl, T.; Mesner, C.; Wittkop, M.; Kreitmeier, S.; Kain, S.; Göritz, D. Monte Carlo studies on the deformation behaviour of glassy model systems. Comput. Theor. Polym. Sci. 1999, 9, 99−109. (54) Bishop, M.; Michels, J. P. J. Polymer shapes in three dimensions. J. Chem. Phys. 1986, 85, 5961−5962. (55) Doi, M. Soft Matter Physics; Oxford University Press: 2013. (56) Humphrey, W.; Dalke, A.; Schulten, K. VMD-Visual Molecular Dynamics. J. Mol. Graphics 1996, 14, 33−38. (57) Besbes, S.; Bokobza, L.; Monnerie, L.; et al. Molecular orientation in deformed bimodal networks. 2.Fourier transform infrared measurements on poly(dimethylsiloxane) networks and comparison with theory. Macromolecules 1995, 28, 231−235.

(13) Mark, J. E. Elastomers with multimodal distributions of network chain lengths. Macromol. Symp. 2003, 191, 121−130. (14) Mark, J. E. Bimodal elastomeric networks. ACS Symp. Ser. 1982, 193, 349−366. (15) Viers, B. D.; Mark, J. E. Elastomeric properties of polysiloxane networks. Bimodal elastomers that are spatially inhomogeneous and others that are very broadly multimodal. J. Macromol. Sci., Part A: Pure Appl. Chem. 2007, 44, 131−138. (16) Di Lorenzo, F.; Seiffert, S. Nanostructural heterogeneity in polymer networks and gels. Polym. Chem. 2015, 6, 5515−5528. (17) Mark, J. E. Effect of swelling on the short-chain reinforcement observed in bimodal elastomeric networks. Macromolecules 1984, 17, 2924−2925. (18) Rath, S. K.; Sharma, S. K.; Sudarshan, K.; Chavan, J. G.; Patro, T. U.; Pujari, P. K. Subnanoscopic inhomogeneities in model endlinked PDMS networks probed by positron annihilation lifetime spectroscopy and their effects on thermomechanical properties. Polymer 2016, 101, 358−369. (19) Wu, W. L.; Jong, L.; Hanyu, A.; Coyne, L. D.; Stein, R. S. Molecular structure of bimodal polymer networks. Macromolecules 1990, 23, 351−353. (20) Hecht, A.-M.; Horkay, F.; Geissler, E. Neutron scattering investigations on a bimodal polymer gel. J. Phys. Chem. B 2001, 105, 5637−5642. (21) Oikawa, H. Dynamic light scattering study on a bimodal endlinked polydimethylsiloxane network structure. Polymer 1992, 33, 1116−1119. (22) Saalwächter, K.; Ziegler, P.; Spyckerelle, O.; Haidar, B.; Sommer, A. V. J.-U. 1H Multiple-Quantum Nuclear Magnetic Resonance Investigations of Molecular Order Distributions in Poly(dimethylsiloxane) Networks: Evidence for a Linear Mixing Law in Bimodal Systems. J. Chem. Phys. 2003, 119, 3468−3482. (23) Bejenariu, A. G.; Yu, L.; Skov, A. L. Low moduli elastomers with low viscous dissipation. Soft Matter 2012, 8, 3917. (24) Madsen, F. B.; Daugaard, a. E.; Fleury, C.; Hvilsted, S.; Skov, A. L. Visualisation and characterisation of heterogeneous bimodal PDMS networks. RSC Adv. 2014, 4, 6939. (25) Viers, B. D.; Mark, J. E. Genesky2. J. Inorg. Organomet. Polym. Mater. 2007, 17, 283−288. (26) Viers, B. D.; Mark, J. E. Large-scale structures in bimodal poly(dimethylsiloxane) elastomers. J. Inorg. Organomet. Polym. Mater. 2005, 15, 477−483. (27) Kondo, S.; Sakurai, H.; Chung, U.; Sakai, T. Mechanical properties of polymer gels with bimodal distribution in strand length. Macromolecules 2013, 46, 7027−7033. (28) Hanson, D. E. Numerical simulations of rubber networks at moderate to high tensile strains using a purely enthalpic force extension curve for individual chains. J. Chem. Phys. 2009, 131, 224904. (29) Hanson, D. E.; Barber, J. L. The theoretical strength of rubber: numerical simulations of polyisoprene networks at high tensile strains evidence the role of average chain tortuosity. Modell. Simul. Mater. Sci. Eng. 2013, 21, 075013. (30) Zhao, J.; Yu, P.; Dong, S. The influence of crosslink density on the failure behavior in amorphous polymers by molecular dynamics simulations. Materials 2016, 9, 234. (31) Zhao, J.; Lu, L.; Rabczuk, T. The tensile and shear failure behavior dependence on chain length and temperature in amorphous polymers. Comput. Mater. Sci. 2015, 96, 567−572. (32) Svaneborg, C.; Grest, G. S.; Everaers, R. Strain-dependent localization, microscopic deformations, and macroscopic normal tensions in model polymer networks. Phys. Rev. Lett. 2004, 93, 257801. (33) Termonia, Y. Molecular model for the mechanical properties of elastomers. 1.Network formation and role of entanglements. Macromolecules 1989, 22, 3633−3638. (34) Yagyu, H. Coarse-grained molecular dynamics simulation of the effects of strain rate on tensile stress of cross-linked rubber. Soft Mater. 2015, 13, 263−270. G

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules (58) Fenley, A. T.; Muddana, H. S.; Gilson, M. K. Calculation and visualization of atomistic mechanical stresses in nanomaterials and biomolecules. PLoS One 2014, 9, e113119.

H

DOI: 10.1021/acs.macromol.7b01653 Macromolecules XXXX, XXX, XXX−XXX