Deformation Behavior of Homogeneous and Heterogeneous Bimodal

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Deformation Behavior of Homogeneous and Heterogeneous 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: In this study, the effect of spatial heterogeneities on the deformation behavior during uniaxial elongation as well as the ultimate properties of bimodal gels consisting of both short and long chains was investigated by molecular simulations. Defect-free networks were created containing dense short-chain clusters and compared with gels having a homogeneous distribution of chains. In both cases, the first chains to rupture were the ones already aligned along the strain axis prior to imposing a strain. The presence of clusters was generally not found to improve the ultimate stress or toughness; the short chains within the clusters were effectively shielded from deformation, even at large fractions of short chains. The heterogeneous network tended to be weaker than the corresponding homogeneous network at a given fraction of short chains, fracturing before any significant deformation of clusters had taken place. The deformation behavior was, however, found to be sensitive to the degree of heterogeneity and the number of intercluster connections. At large fractions of short chains, clustering offered an improvement in the ultimate strain compared to a homogeneous bimodal network and also an equivalent unimodal network with the corresponding number-average chain length, thus providing a small improvement in toughness.

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during network formation,17−22 and it was suggested that these clusters might play a role in enhancing the mechanical properties. While most studies have used a one-step process of end-cross-linking a mixture of short and long chains, highly heterogeneous networks have also been formed in a dual-step process, by first prereacting the short chains below the gelation threshold and subsequently cross-linking with the long chains.13,23−26 The resulting network morphology may be envisioned as heavily cross-linked short-chain clusters embedded within a loosely cross-linked mesh of long chains. These macroscopically phase separated networks were generally found to be weaker, with significantly lower elastic moduli both in comparison to the corresponding unimodal networks and also with respect to more homogeneous bimodal networks with similar short-chain concentrations.24,26 However, some later studies by Mark et al. would suggest that there may still be a structural component to the reinforcement mechanism, depending on the size of clusters and degree of cluster formation. Heterogeneous bimodal networks that were templated to contain clusters on a nanoscale were found to be well reinforced, despite the lack of a pronounced upturn in

INTRODUCTION Hydrogels are generally mechanically weak, a factor which limits their use in many applications where high mechanical strength is required, such as in load-bearing tissue-engineering applications. Several efforts have been undertaken to develop gels with enhanced mechanical performance. For example, one approach that has received a lot of interest in recent years has been the formation of double network hydrogels. These gels combine networks with different cross-linking densities to achieve high mechanical strength and toughness, exceeding those of the individual components.1−6 Another way of using a bimodal distribution of chain lengths is by end-linking short and long chains into one network. While bimodal poly(dimethylsiloxane) (PDMS) elastomers have been extensively studied and found to display enhanced mechanical properties,7−13 no systematic study of the deformation behavior of bimodal hydrogels has been conducted to the best of our knowledge. In a series of studies by Mark et al., bimodal PDMS networks were found to be unusually tough compared to the constituent unimodal networks, and the observed upturn in stress at large strains was attributed to the limited extensibility of the short chains.7−9,11−13 The toughening mechanism, however, has been speculated.7−9,11−16 Numerous experiments have shown that end-linked bimodal networks appear heterogeneous, possessing large-scale structures due to the aggregation of short chains © XXXX American Chemical Society

Received: September 29, 2017 Revised: November 22, 2017

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DOI: 10.1021/acs.macromol.7b02112 Macromolecules XXXX, XXX, XXX−XXX

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and entanglements. Such homogeneous bimodal networks were the focus in a recent paper.29 In the current study, heterogeneous bimodal networks are formed by introducing a bias in the distribution of crosslinking nodes, such that a given number of cross-linking nodes are initially distributed within a number of fictitious spheres, resulting in short chains that are segregated in small clusters connected by long chains. Heterogeneous domains in bimodal networks have recently been visualized using confocal fluorescence microscopy, and the clusters were found to be rather evenly distributed.26 Furthermore, the average cluster size was shown to vary only moderately at low concentrations of short chains, whereas networks with more than 90 mol % short chains had formed larger agglomerates of short-chain domains. In an experimental two-step process, it is possible to influence the degree of heterogeneity by adjusting the reaction time during which the short-chains are prereacted, with larger times resulting in an increased size of short-chain clusters and eventually leading to large sol fractions and networks with defects.23 For the networks simulated here, the degree of heterogeneity may be influenced by adjusting the volume fraction of the fictitious spheres, φspheres, as well as by the number of cross-linking nodes contained in each cluster. In order to capture the same qualitative trend as observed in experiments,26 a larger φspheres has been used at larger fractions of short chains to promote the formation of agglomerates. Heterogeneous bimodal networks were formed containing short chains of length Ns = 6 and long chains of length Nl = 30, with a total of 5000 chains. Three different fractions of short chains were investigated, namely fs = 0.65, 0.80, and 0.90. For each set of parameters, three independent starting configurations were used. The heterogeneous bimodal networks simulated here are compared with the corresponding homogeneous bimodal networks presented in ref 29, as well as with a short-chain unimodal network containing N = 6 in each chain and unimodal networks with N = 8, 10, and 16 that have a similar cross-linking density as the bimodal networks considered in this work. Polymer Chain Model. The polymer chains that connect the cross-linking nodes are modeled as bead−springs, with beads of radius a = 0.5 nm. The stretching force constant for the harmonic potential is chosen such that the second moment of the bond length distribution at equilibrium is b2/1000, where b = 2a. Bonds that exceed a center-tocenter distance of 1.5b during an elongation simulation are considered broken. A bending potential is also included

stress normally attributed to the limited extensibility of the short chains.13,25 The presence of clusters in bimodal networks has also been observed in computer simulations.15,27,28 Genesky et al. used a cubic lattice to form dynamically end-linked bimodal networks and then, transforming to off-lattice coordinates, studied the structure and stress−strain relation under a uniaxial strain.15,28 At low molar concentrations of short chains, they found that the short chains aggregate into domains during the end-linking process, while networks at higher concentrations appear more homogeneous. Since the networks containing clusters did not show any significant enhancement in mechanical properties, it was suggested that the finite extensibility of the short chains, rather than cluster formation, is responsible for the improvement in ultimate properties of bimodal networks at high concentrations of short chains. It was, however, noted that some degree of heterogeneity was likely to have been present also at high short-chain concentrations. In a recent paper,29 we used molecular simulation methods to study the mechanical properties of ideal bimodal gels under uniaxial tension, where the short and long chains were distributed homogeneously. The purpose was to isolate the effect of a bimodal molecular weight distribution on the deformation behavior, without needing to consider possible aggregation effects, network defects, or entanglement effects. In the present contribution, we extend the study to explore the relationship between structure and toughening mechanism by also looking at the effect of inhomogeneities in the form of cluster formation. Our main goal is to investigate how the mechanical properties of a bimodal network are affected as a result of local fluctuations in the cross-linking density, by performing a direct comparison between the properties of a highly homogeneous network and a highly heterogeneous network containing short-chain clusters. We show that clustering does have an impact on the deformation behavior and ultimate properties also at large fractions of short chains.

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SIMULATION DETAILS Ub =

Generating a Heterogeneous Bimodal Network. A detailed description of the generation of bimodal networks with a homogeneous and heterogeneous distribution of chain lengths is provided in the Supporting Information, and a schematic of such networks is shown in Figure 1. In summary, a homogeneous bimodal network is formed by randomly distributing the tetravalent crosslinking nodes within a cylinder and subsequently connecting these with short and long chains of fixed lengths, based on a distance criterion, to form a cross-linked network that is free from any defects

kb 2

∑ (cos θi − cos θ0)2 i

(1)

where θi corresponds to the angle between two consecutive bonds, θ0 = 0 corresponds to the angle at minimum bending potential, and the bending force constant, kb, is determined from the persistence length,30 which is set to P = 2b. Nonbonded units interact via a soft repulsive potential31

Firep =

ϵrepkBT b

2

⎛b

⎞ − 1⎟⎟rij n̂ ij ⎝ rij ⎠

∑ ⎜⎜ j

(2)

where ϵrep is a dimensionless force constant set to 200, and n̂ij is the unit vector from particle j to i. Network Deformation. Using Brownian dynamics simulations,32,33 a continuous uniaxial strain is applied on the equilibrated networks. Following the procedure in ref 29, a boundary-driven deformation is initiated by introducing two grips in the form of infinite planes at the gel cylinder bases, which transmit the strain through the sample when they are pulled apart, mimicking an experimental extension setup. At each time step, the planes are shifted apart at a rate of 1.7 × 10−6b per time step or 100 mm/min. All simulations are carried out at a constant temperature of T = 293.15 K, viscosity of η = 1.002 × 10−3 Pa s, and time step of Δt = 1.0 ps. The resulting deformation is analyzed with respect to the chain alignment and stress−strain relation. The segmental orientation is described by the second order Legendre polynomial34

Figure 1. Schematic of a homogeneous and heterogeneous bimodal network, respectively, where the filled circles represent the crosslinking nodes and the bold and thin lines correspond to short and long chains, respectively. B

DOI: 10.1021/acs.macromol.7b02112 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules P2 = 3cos2 ϕ − 1 /2

(3)

where ϕ is the angle between the direction of extension and the local chain axis of the polymer segment. The stress tensor is evaluated according to35

σ=−

1 V

∑ ∑ fij(ri − rj) i

j>i

(4)

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

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RESULTS AND DISCUSSION Unless otherwise stated, the results are presented up to the point of macroscopic failure, which we define as the strain at which the stress starts to significantly decrease. By looking at the distribution of bond angles for all investigated systems (not shown), one can note that the strain rate is small enough to ensure that the bond angle distribution remains unperturbed up to (at least) a strain of α = 0.5. Only at strains close to failure does the distribution become slightly skewed toward smaller angles, especially for the homogeneous bimodal networks. Network Structure. The probability distribution of the chain end-to-end distance (Ree) before and after elongation as well as the short-chain segmental orientation (P2) in the direction of the strain axis are shown in Figure 2. In the unperturbed state, phase separating the short chains into domains removes some of the constraints at the junction points of the long chains, allowing these to assume slightly larger endto-end distances in the heterogeneous bimodal networks compared to when they are randomly distributed within the network (Figure 2a). After elongation up to failure, both the short and long chains in the homogeneous network are to some extent pulled apart due to the intimate coupling between all chains, and the distribution maxima shift to larger values. In contrast, a short chain that resides deep within a cluster, surrounded by other short chains, is essentially shielded from stretching. The end-to-end distribution for the short chains in the heterogeneous network is almost unchanged after elongation, aside from the appearance of a minor shoulder, which likely reflects the deformation of short chains near the periphery of the clusters. This shielding effect can also be seen following the short-chain segmental orientation during elongation in Figure 2b; the orientation of the short chains within the densely cross-linked regions is smaller than that of their counterparts in the homogeneous network, even at large fractions of short chains. The heterogeneous networks thus begin to break down before the clusters are significantly deformed. As previously noted,29 the orientation of the short chains in a homogeneous network during elongation is not substantially changed when the fraction of short chains is increased, due to the presence of percolating short-chain structures between the grips inducing the tensile force. It can be seen in Figure 2b that, although the short-chain orientation is severely reduced as a consequence of clustering, P2 increases with fs in the heterogeneous networks. This latter trend has previously been observed in simulations of end-linked bimodal networks that were not specifically templated to contain clusters.15 In our case, increasing the fraction of short chains is associated with a higher degree of intercluster connections and a transition to a percolating short-chain structure formed by larger agglomerates

Figure 2. (a) Probability distribution for the chain end-to-end distances (scaled with the particle diameter) for the short and long chains in a homogeneous and heterogeneous bimodal network with fs = 0.65. The solid lines show the undeformed equilibrium distribution and the dashed lines show the distribution for the deformed networks at failure. (b) Segmental chain orientation with increasing strain for the short chains in a homogeneous (dashed lines) and heterogeneous (solid lines) bimodal network with fs = 0.65, 0.80, and 0.90.

of clusters. This is also reflected in the end-to-end distribution after elongation, which now approaches the corresponding distribution of a homogeneous network as more short chains are deformed (see Figure S1). The orientation of the short and long chains is shown in Figure 3. In homogeneous networks, the short chains are consistently more strongly oriented along the strain axis than the long chains. The chain orientation in the heterogeneous networks, however, depends on the network composition. At small fractions of short chains, primarily the long chains are deformed while the short chains within the clusters are only marginally perturbed from their equilibrium conformations due to the shielding effect. At large fractions, the short chains start to experience the imposed strain more strongly at low to intermediate strains. We further investigated the sensitivity of these results to the degree of cluster branching at large fractions of short chains by increasing the number of cross-linking nodes per cluster and reducing the volume fraction of the fictitious spheres used during network formation, thus effectively suppressing intercluster connections. Reducing the number of intercluster connections results in an even stronger shielding effect experienced by the short chains within the clusters as these become less firmly embedded within the long-chain C

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Macromolecules Table 1. Properties of the Different Networksa fs (nc) 0.65 (0.034) 0.80 (0.044) 0.90 (0.056) 1.00 N= N= N=

(0.077) 8 (0.058) 10 (0.048) 16 (0.030)

αf

σf′

Υ

4.0 4.1 3.3 3.5 1.7 2.7 1.4 1.7 2.1 3.0

0.15 0.063 0.21 0.081 0.38 0.16 0.59 0.46 0.35 0.18

0.46 0.13 0.40 0.19 0.20 0.24 0.19 0.18 0.25 0.27

homogeneousb heterogeneous homogeneousb heterogeneous homogeneousb heterogeneous unimodalb unimodalb unimodalb unimodalb

a

Fraction of short chains ( fs), cross-linking density (nc), ultimate strain (αf), ultimate stress (σ′f ), and area under the stress−strain curves until failure (Υ). bValues taken from ref 29. Figure 3. Segmental chain orientation for homogeneous and heterogeneous networks during elongation for fs = 0.65 and 0.90, where the solid lines correspond to the short chains and the dashed lines to the long chains.

network, which can be seen as a drop in the short-chain orientation (see Figure S2). So far, we have considered the orientation as a function of strain. Looking instead at the initial average segmental orientation of the chains, P2 for both the short and long chains is essentially zero at equilibrium (i.e., at α = 0). If the orientation is determined for only the chains that have ruptured at failure, then it can be seen that the initial chain segmental (α=0) orientation of the ruptured short chains (P2,seg ) in the homogeneous bimodal networks is between P(α=0) 2,seg = 0.10− 0.12 (and the initial end-to-end orientation between P(α=0) 2,ee = 0.25−0.34) for fs = 0.65−0.90. For the long chains, the (α=0) corresponding values are P(α=0) 2,seg = 0.021−0.043 (and P2,ee = 0.26−0.58). In other words, the first chains to rupture are the ones that are already aligned along the strain axis in the undeformed state. This is also true for the heterogeneous bimodal networks, where the corresponding initial orientation (α=0) of the short chains is P(α=0) 2,seg = 0.10−0.13 (and P2,ee = 0.20− (α=0) 0.28), and for the long chains P2,seg = 0.06−0.092 (and P(α=0) 2,ee = 0.41−0.45). The difference in network morphology is also reflected in the fraction of ruptured chains as a function of strain (see Figure S3). In both types of networks, the short chains start to rupture under a relatively small applied strain (of α ≈ 0.7); however, the short chains rupture at a faster rate in the homogeneous networks. On the other hand, a larger fraction of long chains are ultimately ruptured in the heterogeneous networks as these chains sustain most of the load. Mechanical Properties. The mechanical properties of the networks are summarized in Table 1. Figure 4 shows the stress as a function of strain for the homogeneous and heterogeneous networks. Both types of network morphologies display the same qualitative behavior: networks with a smaller fraction of short chains are weaker than those with larger fractions but can reach larger strains before failing. The highest ultimate stress is obtained when the short chains are homogeneously distributed and thus intimately coupled. The heterogeneous networks are much weaker in comparison, even at large fractions of short chains. This behavior is in agreement with the results above, where it was noted that short chains within the highly crosslinked domains do not deform to any large extent when the system is subjected to elongation (cf. Figure 2). Analogous to

Figure 4. Stress−strain curves for bimodal networks with fs = 0.65− 1.0. The solid lines correspond to the heterogeneous networks, the dashed lines correspond to the homogeneous networks, and the dotted lines correspond to the unimodal networks of the corresponding cross-linking densities (see Table 1).

the orientation curves (Figure S2), the effect of reducing the number of intercluster connections is to further reduce the ultimate stress (see Figure S4). Although clustering does not have any significant effect on the maximum extensibility at small fractions of short chains, there is a clear difference at larger fractions. A homogeneous bimodal network with a large fs essentially consists of a matrix of short chains and the network behaves closely to a unimodal short-chain network. It can be seen that the maximum extensibility can be improved by incorporating more long chains into the bimodal network and also by the presence of clusters. These heterogeneous networks with fs = 0.90 have a significantly improved ultimate strain compared to the homogeneous bimodal or equivalent unimodal network. This difference in extensibility can also be visualized through simulation snapshots taken at approximately the same strain, shown in Figure 5. While the homogeneous network has fractured at a strain of α = 1.7, it can be seen that the long chains in the clustered network delay the fracture. The performance of the bimodal networks can be compared with that of unimodal networks with a similar cross-linking density in Table 1 and Figure 4. At small fractions of short chains ( fs = 0.65), the ultimate properties of the bimodal networks cannot be predicted from those of an equivalent unimodal network with the same number-average chain length D

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Figure 6. Stress−strain curves for the short (solid lines) and long (dashed lines) chains, for homogeneous and heterogeneous networks with fs = 0.65 and 0.90.

toughness for homogeneous bimodal networks increases with decreasing fs, reaching a peak for fs = 0.65, here the toughness decreases with decreasing fs for the heterogeneous networks. Since the ultimate strain is not improved by clustering at lower fs, the reduced ultimate stress inevitably leads to a lower toughness.

Figure 5. Simulation snapshots using VMD36 taken during uniaxial elongation of a system with fs = 0.90 for a homogeneous (upper panel) and heterogeneous (lower panel) network at approximately the same strain of α = 0.5, 2.5, and 3.0 (from left to right). Short and long chains are colored blue and purple, respectively.

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CONCLUSION In this work, we have studied the deformation of bimodal networks subject to a steady uniaxial elongation, with focus on the effect of the presence of dense short-chain clusters on the mechanical properties. It could be shown that, in the same way as in homogeneous bimodal networks, the first chains to rupture in the heterogeneous bimodal networks are the ones that are already aligned along the strain axis at equilibrium, prior to imposing a strain. It was further shown that the short chains in the heterogeneous networks are substantially shielded from deformation within the clusters, even at large fractions of short chains, with a reduced orientation along the strain axis. The formation of larger agglomerates of clusters, however, results in an increase in short-chain orientation with increasing fraction of short chains. The presence of clusters was generally not found to improve the ultimate stress or toughness in the bimodal networks considered here. Instead, the heterogeneous networks tend to be weaker than the corresponding homogeneous networks and appear to fracture before the strain becomes sufficiently large to initiate any significant deformation of the clusters. The results were shown to be sensitive to the degree of heterogeneity and the number of intercluster connections, such that decreasing the number of intercluster connections also decreased the ultimate stress. At larger fractions of short chains, however, small scale clustering does offer an improvement in the maximum extensibility, also compared to equivalent unimodal networks with the same number-average chain length, and subsequently also a small improvement in the toughness. This would suggest that an enhancement in the ultimate strain at larger fractions of short chains, compared to predictions from equivalent unimodal networks, could be a result of short-chain aggregation. In most gels, the presence of trapped entanglements, which is not included in the present model, could also be anticipated to influence the mechanical properties of bimodal networks, in

(N = 16). The equivalent unimodal network fails earlier at a strain of α = 3.0, compared to an ultimate strain of α ≈ 4 for the homogeneous and heterogeneous bimodal networks. At large fractions of short chains (fs = 0.90), the ultimate properties of the equivalent unimodal network (N = 8) more closely correspond to the ones found for the homogeneous bimodal networks. Since clustering at larger fractions of short chains improves the ultimate strain relative to a homogeneous network, this would suggest that an improved maximum extensibility of bimodal networks at larger fractions of short chains, compared to the corresponding equivalent unimodal network, such as the one observed by Genesky et al.15 for bimodal networks with fs = 0.95, might be a result of some degree of short-chain aggregation. Considering how the stress varies on a local scale, Figure 6 shows the separate stress−strain profiles of the short and long chains. As noted in ref 29, the stress is not shared uniformly between the short and long chains in a homogeneous network. Instead, the stress rises faster for the short chains, while the long chains only start to become taut at larger strains. The stress on the short chains is, however, significantly lower in the heterogeneous networks. Furthermore, while the short chains in the homogeneous networks follow a similar stress−strain profile for both values of fs, the short chains in the heterogeneous networks become increasingly deformed with increasing number of intercluster connections, i.e., at a higher fs, leading to a larger stress. The area under the stress−strain curves, corresponding to the toughness, is generally lower for the heterogeneous systems (see Table 1). At fs = 0.90, however, the increased ultimate strain is enough to slightly improve the toughness compared to a homogeneous bimodal network and a pure short-chain network. Furthermore, while we previously found29 that the E

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demonstrate non-Gaussian effects related to limited chain extensibility. J. Chem. Phys. 1980, 72, 2282. (8) 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. (9) 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. (10) 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. (11) Mark, J. E. Elastomers with multimodal distributions of network chain lengths. Macromol. Symp. 2003, 191, 121−130. (12) Mark, J. E. Bimodal elastomeric networks. ACS Symp. Ser. 1982, 193, 349−366. (13) 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. (14) Mark, J. E.; Erman, B. Rubberlike elasticity: A molecular primer; Wiley: New York, 1988. (15) 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. (16) 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. (17) Wu, W. L.; Jong, L.; Hanyu, A.; Coyne, L. D.; Stein, R. S. Molecular structure of bimodal polymer networks. Macromolecules 1990, 23, 351−353. (18) Hecht, A.-M.; Horkay, F.; Geissler, E. Neutron scattering investigations on a bimodal polymer gel. J. Phys. Chem. B 2001, 105, 5637−5642. (19) Viers, B. D.; Mark, J. E. Large-scale structures in bimodal poly(dimethylsiloxane) elastomers. J. Inorg. Organomet. Polym. Mater. 2005, 15, 477−483. (20) Oikawa, H. Dynamic light scattering study on a bimodal endlinked polydimethylsiloxane network structure. Polymer 1992, 33, 1116−1119. (21) Saalwächter, K.; Ziegler, P.; Spyckerelle, O.; Haidar, B.; Sommer, J.-U.; Vidal, A. 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. (22) 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. (23) Galiatsatos, V.; Subramanian, P. R.; Klein-Castner, L. Designing heterogeneity into bimodal elastomeric PDMS networks. Macromol. Symp. 2001, 171, 97−104. (24) Bejenariu, A. G.; Yu, L.; Skov, A. L. Low moduli elastomers with low viscous dissipation. Soft Matter 2012, 8, 3917. (25) Viers, B. D.; Mark, J. E. Elastomeric Properties of Polysiloxane Networks: Birefringence Measurements on Bimodal Elastomers that are Presumed to be Spatially Inhomogeneous. J. Inorg. Organomet. Polym. Mater. 2007, 17, 283−288. (26) 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. (27) Michalke, W.; Kreitmeier, S.; Lang, M.; Buchner, A.; Göritz, D. Monte Carlo simulations of the spatial structure of end-linked bimodal polymer networks: part II. Comput. Theor. Polym. Sci. 2001, 11, 459− 466. (28) Aguilera-Mercado, B. M.; Genesky, G. D.; Duncan, T. M.; Cohen, C.; Escobedo, F. A. H NMR and simulation studies of chain

particular for long precursor chain lengths that could be wellentangled. Such entanglements might aid in reinforcing the heterogeneous networks by increasing the effective number of intercluster connections. The effect of trapped entanglements in both homogeneous and heterogeneous networks is currently being investigated.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b02112. Description of the construction of a homogeneous bimodeal network; (ii) description of the construction of a heterogeneous bimodal network; (iii) parameters used when generating the heterogeneous networks; (iv) probability distribution for the chain end-to-end distances for the short and long chains in a homogeneous and heterogeneous bimodal network with fs = 0.90; (v) segmental chain orientation as a function of strain for homogeneous (n = 0) and heterogeneous networks with fs = 0.90 and different degrees of clustering; (vi) fraction of ruptured short and long chains for homogeneous and heterogeneous networks with fs = 0.65 and 0.90; (vii) stress−strain curves for networks with fs = 0.90, with various degrees of clustering. (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Christer Elvingson: 0000-0001-5115-5481 Notes

The authors declare no competing financial interest.

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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.

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REFERENCES

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DOI: 10.1021/acs.macromol.7b02112 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.7b02112 Macromolecules XXXX, XXX, XXX−XXX