Connectivity and Entanglement Stress Contributions in Strained Polymer Networks

Bridge-rich and loop-less hydrogel networks through suppressed micellization of multiblock polyelectrolytes

Most triblock copolymer-based physical hydrogels form three-dimensional networks through micellar packing, and formation of polymer loops represents a topological defect that diminishes hydrogel elasticity. This effect can be mitigated by maximizing the fraction of elastically effective bridges in the hydrogel network. Herein, we report hydrogels constructed by complexing oppositely charged multiblock copolymers designed with a sequence pattern that maximizes the entropic and enthalpic penalty of micellization. These copolymers self-assemble into branched and bridge-rich network units (netmers), instead of forming sparsely interlinked micelles. We find that the storage modulus of the netmer-based hydrogel is 11.5 times higher than that of the micelle-based hydrogel. Complementary coarse grained molecular dynamics simulations reveal that in the netmer-based hydrogels, the numbers of charge-complexed nodes and mechanically reinforcing bridges increase substantially relative to micelle-based hydrogels.

A Comparison between Predictions of the Miller-Macosko Theory, Estimates from Molecular Dynamics Simulations, and Long-Standing Experimental Data of the Shear Modulus of End-Linked Polymer Networks

Long-standing experimental data on the elastic modulus of end-linked poly(dimethylsiloxane) (PDMS) networks are employed to corroborate the validity of the Miller-Macosko theory (MMT). The validity of MMT is also confirmed by molecular dynamics (MD) simulations that mimic the experimentally realized networks. It becomes apparent that for a network formed from bulk, where the fractions of the loops are small, it is sufficient to account for the topological details of a reference tree-like network, i.e., for its degree of completion, junction functionalities, and trapped entanglements, in order to practically predict the modulus. However, a mismatch is identified between the MMT and MD simulations in relating the fraction of the soluble material to the extent of reaction. A large contribution of entanglements to the modulus of PDMS networks prepared with short precursor chains is presented, suggesting that the elastic modulus of commonly used end-linked PDMS networks is in fact entanglement-dominated.

Microphase behaviors and shear moduli of double-network gels: The effect of crosslinking constraints and chain uncrossability

By performing coarse-grained molecular dynamics simulations, we study the effect of crosslinking and chain uncrossability on the microphase behaviors and mechanical properties of the double-network gels. The double-network systems can be viewed as two separate networks interpenetrating each other uniformly, and the crosslinks in each network are generated, forming a regular cubic lattice. The chain uncrossability is confirmed by appropriately choosing the bonded and nonbonded interaction potentials. Our simulations reveal a close relation between the phase and mechanical properties of the double-network systems and their network topological structures. Depending on the lattice size and the solvent affinity, we have observed two different microphases: one is the aggregation of solvophobic beads around the crosslinking points, which leads to locally polymer-rich domains, and the other is the bunching of polymer strands, which thickens the network edges and thus changes the network periodicity. The former is a representation of the interfacial effect, while the latter is determined by the chain uncrossability. The coalescence of network edges is demonstrated to be responsible for the large relative increase in the shear modulus. Compressing and stretching induced phase transitions are observed in the current double-network systems, and the sharp discontinuous change in the stress that appears at the transition point is found to be related to the bunching or debunching of the network edges. The results suggest that the regulation of network edges has a strong influence on the network mechanical properties.

Multiscale equilibration of highly entangled isotropic model polymer melts

We present a computationally efficient multiscale method for preparing equilibrated, isotropic long-chain model polymer melts. As an application, we generate Kremer-Grest melts of 1000 chains with 200 entanglements and 25 000-2000 beads/chain, which cover the experimentally relevant bending rigidities up to and beyond the limit of the isotropic-nematic transition. In the first step, we employ Monte Carlo simulations of a lattice model to equilibrate the large-scale chain structure above the tube scale while ensuring a spatially homogeneous density distribution. We then use theoretical insight from a constrained mode tube model to introduce the bead degrees of freedom together with random walk conformational statistics all the way down to the Kuhn scale of the chains. This is followed by a sequence of simulations with carefully parameterized force-capped bead-spring models, which slowly introduce the local bead packing while reproducing the larger-scale chain statistics of the target Kremer-Grest system at all levels of force-capping. Finally, we can switch to the full Kremer-Grest model without perturbing the structure. The resulting chain statistics is in excellent agreement with literature results on all length scales accessible in brute-force simulations of shorter chains.

Molecular Characterization of Polymer Networks

Polymer networks are complex systems consisting of molecular components. Whereas the properties of the individual components are typically well understood by most chemists, translating that chemical insight into polymer networks themselves is limited by the statistical and poorly defined nature of network structures. As a result, it is challenging, if not currently impossible, to extrapolate from the molecular behavior of components to the full range of performance and properties of the entire polymer network. Polymer networks therefore present an unrealized, important, and interdisciplinary opportunity to exert molecular-level, chemical control on material macroscopic properties. A barrier to sophisticated molecular approaches to polymer networks is that the techniques for characterizing the molecular structure of networks are often unfamiliar to many scientists. Here, we present a critical overview of the current characterization techniques available to understand the relation between the molecular properties and the resulting performance and behavior of polymer networks, in the absence of added fillers. We highlight the methods available to characterize the chemistry and molecular-level properties of individual polymer strands and junctions, the gelation process by which strands form networks, the structure of the resulting network, and the dynamics and mechanics of the final material. The purpose is not to serve as a detailed manual for conducting these measurements but rather to unify the underlying principles, point out remaining challenges, and provide a concise overview by which chemists can plan characterization strategies that suit their research objectives. Because polymer networks cannot often be sufficiently characterized with a single method, strategic combinations of multiple techniques are typically required for their molecular characterization.

Effect of Chain Polydispersity on the Elasticity of Disordered Polymer Networks

Due to their unique structural and mechanical properties, randomly cross-linked polymer networks play an important role in many different fields, ranging from cellular biology to industrial processes. In order to elucidate how these properties are controlled by the physical details of the network (e.g., chain-length and end-to-end distributions), we generate disordered phantom networks with different cross-linker concentrations C and initial densities ρinit and evaluate their elastic properties. We find that the shear modulus computed at the same strand concentration for networks with the same C, which determines the number of chains and the chain-length distribution, depends strongly on the preparation protocol of the network, here controlled by ρinit. We rationalize this dependence by employing a generic stress-strain relation for polymer networks that does not rely on the specific form of the polymer end-to-end distance distribution. We find that the shear modulus of the networks is a nonmonotonic function of the density of elastically active strands, and that this behavior has a purely entropic origin. Our results show that if short chains are abundant, as it is always the case for randomly cross-linked polymer networks, the knowledge of the exact chain conformation distribution is essential for correctly predicting the elastic properties. Finally, we apply our theoretical approach to literature experimental data, qualitatively confirming our interpretations.

Towards realistic simulations of polymer networks: tuning vulcanisation and mechanical properties

Simulations of coarse-grained network models have long been used to test theoretical predictions about rubber elasticity, while atomistic models are still largely unexplored. Here we devise a novel algorithm for the vulcanisation of united-atom poly(cis-1,4-butadiene), characterize the topology of the resulting networks and test their mechanical properties. We observe clear changes in the network structure when using slower vulcanisation, contrary to the traditional view that cross-linking simply freezes the melt configuration. Non-ideality of our networks reverberates on the distribution of strand length and on the strands deformation, which is highly non-affine, especially for short strands. Nevertheless, we do recover some of the trends observed on ideal bead-and-spring networks and controlled laboratory experiments, such as the linear relationships linking the degree of cross-linking and the density. We also compare different deformation methods and find step-equilibrium protocols to be more reliable. Regardless of the adopted method, it is advisable to precede the deformation by a pre-stretching cycle in order to release internal stresses accumulated during the vulcanisation.

Complex Network Representation of the Structure-Mechanical Property Relationships in Elastomers with Heterogeneous Connectivity

The complicated structure-property relationships of materials have recently been described using a methodology of data science that is recognized as the fourth paradigm in materials science. In network polymers or elastomers, the manner of connection of the polymer chains among the crosslinking points has a significant effect on the material properties. In this study, we quantitatively evaluate the structural heterogeneity of elastomers at the mesoscopic scale based on complex network, one of the methods used in data science, to describe the elastic properties. It was determined that a unified parameter with topological and spatial information universally describes some parameters related to the stresses. This approach enables us to uncover the role of individual crosslinking points for the stresses, even in complicated structures. Based on the data science, we anticipate that the structure-property relationships of heterogeneous materials can be interpretatively represented using this type of "white box" approach.

Molecular dynamics simulation of thermo-mechanical behaviour of elastomer cross-linked via multifunctional zwitterions

Coarse-grained (CG) molecular dynamics simulations have been employed to study the thermo-mechanical response of a physically cross-linked network composed of zwitterionic moieties and fully flexible elastomeric polymer chains.

Highly Transparent and Toughened Poly(methyl methacrylate) Nanocomposite Films Containing Networks of Cellulose Nanofibrils

Cellulose nanofibrils (CNFs) are a class of cellulosic nanomaterials with high aspect ratios that can be extracted from various natural sources. Their highly crystalline structures provide the nanofibrils with excellent mechanical and thermal properties. The main challenges of CNFs in nanocomposite applications are associated with their high hydrophilicity, which makes CNFs incompatible with hydrophobic polymers. In this study, highly transparent and toughened poly(methyl methacrylate) (PMMA) nanocomposite films were prepared using various percentages of CNFs covered with surface carboxylic acid groups (CNF-COOH). The surface groups make the CNFs interfacial interaction with PMMA favorable, which facilitate the homogeneous dispersion of the hydrophilic nanofibrils in the hydrophobic polymer and the formation of a percolated network of nanofibrils. The controlled dispersion results in high transparency of the nanocomposites. Mechanical analysis of the resulting films demonstrated that a low percentage loading of CNF-COOH worked as effective reinforcing agents, yielding more ductile and therefore tougher films than the neat PMMA film. Toughening mechanisms were investigated through coarse-grained simulations, where the results demonstrated that a favorable polymer-nanofibril interface together with percolation of the nanofibrils, both facilitated through hydrogen bonding interactions, contributed to the toughness improvement in these nanocomposites.

Nonlinear mechanics of thermoreversibly associating dendrimer glasses

We model the mechanics of associating trivalent dendrimer network glasses with a focus on their energy dissipation properties. Various combinations of sticky bond (SB) strength and kinetics are employed. The toughness (work to fracture) of these systems displays a surprising deformation-protocol dependence; different association parameters optimize different properties. In particular, "strong, slow" SBs optimize strength, while "weak, fast" SBs optimize ductility via self-healing during deformation. We relate these observations to breaking, reformation, and partner switching of SBs during deformation. These studies point the way to creating associating-polymer network glasses with tailorable mechanical properties.

Topological versus rheological entanglement length in primitive-path analysis protocols, tube models, and slip-link models

We show that the front factor appearing in the shear modulus of a phantom network, G(ph) = (1-2/f)(ρk(B)T)/N(s), also controls the ratio of the strand length, N(s), and the number of monomers per Kuhn length of the primitive paths, N(ph)(PPKuhn), characterizing the average network conformation. In particular, N(ph)(PPKuhn) = N(s)/(1-2/f) and G(ph) = (ρk(B)T)/N(ph)(PPKuhn). Neglecting the difference between cross-links and slip-links, these results can be transferred to entangled systems and the interpretation of primitive path analysis data. In agreement with the tube model, the analogy to phantom networks suggest that the rheological entanglement length, N(e)(rheo) = (ρk(B)T)/G(e), should equal N(e)(PPKuhn). Assuming binary entanglements with f = 4 functional junctions, we expect that N(e)(rheo) should be twice as large as the topological entanglement length, N(e)(topo). These results are in good agreement with reported primitive path analysis results for model systems and a wide range of polymeric materials. Implications for tube and slip-link models are discussed.

Combined molecular algorithms for the generation, equilibration and topological analysis of entangled polymers: methodology and performance

We review the methodology, algorithmic implementation and performance characteristics of a hierarchical modeling scheme for the generation, equilibration and topological analysis of polymer systems at various levels of molecular description: from atomistic polyethylene samples to random packings of freely-jointed chains of tangent hard spheres of uniform size. Our analysis focuses on hitherto less discussed algorithmic details of the implementation of both, the Monte Carlo (MC) procedure for the system generation and equilibration, and a postprocessing step, where we identify the underlying topological structure of the simulated systems in the form of primitive paths. In order to demonstrate our arguments, we study how molecular length and packing density (volume fraction) affect the performance of the MC scheme built around chain-connectivity altering moves. In parallel, we quantify the effect of finite system size, of polydispersity, and of the definition of the number of entanglements (and related entanglement molecular weight) on the results about the primitive path network. Along these lines we approve main concepts which had been previously proposed in the literature.