Molecular Dynamics Validation and Applications of the Maximum Entropy Homogenization Procedure for Predicting the Elastic Properties of Gaussian Polymer Networks

Interplay of Spatial and Topological Defects in Polymer Networks

Polymer networks are widely used in applications, and the formation of a network and its gel point can be predicted. However, the effects of spatial and topological heterogeneity on the resulting network structure and ultimately the mechanical properties, are less understood. To address this challenge, we generate in silico random networks of cross-linked polymer chains with controlled spatial and topological defects. While all fully reacted networks investigated in this study have the same number of end-functionalized polymer strands and cross-linkers, we vary the degree of spatial and topological heterogeneities systematically. We find that spatially heterogeneous cross-linker distributions result in a reduction in the network's primary loops with increased spatial heterogeneity, the opposite trend as observed in homogeneous networks. By performing molecular dynamics simulations, we investigated the mechanical properties of the networks. Even though spatially heterogeneous networks have more elastically active strands and cross-linkers, they break at lower extensions than the homogeneous networks and sustain slightly lower maximum stresses. Their shear moduli are higher, i.e., stiffer, than theoretically predicted, and higher than their homogeneous gel counterparts. Our results highlight that topological loop defects and spatial heterogeneities result in significantly different network structures and, ultimately, different mechanical properties.

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.

Molecular dynamics studies of entropic elasticity of condensed lattice networks connected with uniform functionality f = 4

To study the linear region of entropic elasticity, we considered the simplest physical model possible and extracted the linear entropic regime by using the least squares fit and the minimum of the mean absolute error. With regard to the effect of the fluctuation of the strand length N_s, the strand length with fluctuation was set to a form proportional to (1.0 + C (R - 0.5)), where R is a uniform random number between 0 and 1 and C is the amplitude of fluctuation. This form enabled us to analytically calculate the fluctuation dependence of the elastic modulus G. To reveal the linear regions of entropic elasticity as a function of the strand length between neighboring nodes in lattices, molecular dynamics (MD) simulations of condensed lattice networks with harmonic bonds without the excluded volume interactions were performed. Stress-strain curves were estimated by performing uniaxial stretching MD simulations under periodic boundary conditions with a bead number density of 0.85. First, we used a diamond lattice with functionality f = 4. The linear region of the entropic elasticity was found to become larger with the increasing number of beads in a strand N_s. For N_s = 100, the linear region had a strain of up to 8 for a regular diamond lattice. We investigated the effect of strand length fluctuation on the diamond lattice, and we confirmed that the equilibrium shear modulus G increases as the obtained analytical prediction and the linear entropic region in the stress-strain curves becomes narrower with increasing fluctuation of N_s. To investigate the difference in network topology with the same functionality f and uniform strand length N_s, we performed MD simulations on regular networks of the BC-8 structure with f = 4 prepared from the ab initio DFT calculations of carbon at high pressure. We found that the elastic behavior depends on the network connectivity (i.e., topology). This indicates that the network topology plays an important role in the emergence of nonlinearity owing to the crossover from entropic to energetic elasticity.

Adding the Effect of Topological Defects to the Flory-Rehner and Bray-Merrill Swelling Theories

The Flory-Rehner and Bray-Merrill swelling theories are venerable theories for calculating the swelling of polymer networks and are widely applied across polymer materials. Here, these theories are revised to include cyclic topological defects present in polymer networks by using a modified phantom network model. These closed-form equations assume defect contributions to the swelling elasticity to be linear and additive and allow different assumptions regarding prestrain of larger loops to be incorporated. To compare to the theories, swelling experiments are performed on end-linked poly(ethylene glycol) gels in which the topological defects (primary and secondary loops) have been previously measured. Gels with higher loop densities exhibit higher swelling ratios. An equation is derived to compare swelling models independent of knowledge of the Flory-Huggins χ parameter, showing that the revised swelling models for loop defects are more accurate than both the phantom network model that neglects loops and the Bray-Merrill equation.

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.

Molecular Dynamics Simulation for Evaluating Fracture Entropy of a Polymer Material under Various Combined Stress States

Herein, the stress-state dependence of fracture entropy for a polyamide 6 material is investigated through molecular dynamics simulations. Although previous research suggests that a constant entropy increase can be universally applied for the definition of material fracture, the dependence of stress triaxiality has not yet been discussed. In this study, entropy values are evaluated by molecular dynamics simulations with varied combined stress states. The calculation is implemented using the 570,000 all-atom model. Similar entropy values are obtained independently of stress triaxiality. This study also reveals the relationship between material damage, which is correlated with void size, and the entropy value.