Fracture of model end-linked networks

Advances in polymer chemistry over the last decade have enabled the synthesis of molecularly precise polymer networks that exhibit homogeneous structure. These precise polymer gels create the opportunity to establish true multiscale, molecular to macroscopic, relationships that define their elastic and failure properties. In this work, a theory of network fracture that accounts for loop defects is developed by drawing on recent advances in network elasticity. This loop-modified Lake–Thomas theory is tested against both molecular dynamics (MD) simulations and experimental fracture measurements on model gels, and good agreement between theory, which does not use an enhancement factor, and measurement is observed. Insight into the local and global contributions to energy dissipated during network failure and their relation to the bond dissociation energy is also provided. These findings enable a priori estimates of fracture energy in swollen gels where chain scission becomes an important failure mechanism.

Models that link materials structure to macroscopic behavior can account for multiple levels of molecular structure. For example, the statistical, affine deformation model connects the elastic modulus E to the molecular structure of a polymer chain,Eaff=3νkbT(ϕo13Roϕ13R)2,1]where ν is density of chains, ϕ is polymer volume fraction, R is end-to-end distance, ϕo and R_o represent the parameters taken in the reference state that is assumed to be the reaction concentration in this work, and k_bT is the available thermal energy where k_b is Boltzmann’s constant and T is temperature (1–6). Refinements to this model that account for network-level structure, such as the presence of trapped entanglements or number of connections per junction, have been developed (7–11). Further refinements to the theory of network elasticity have been developed to account for dynamic processes such as chain relaxation and solvent transport (12–17). Together these refinements link network elasticity to chain-level molecular structure, network-level structure, and the dynamic processes that occur at both size scales.While elasticity has been connected to multiple levels of molecular structure, models for network fracture have not developed to a similar extent. The fracture energy G_c typically relies upon the large strain deformation behavior of polymer networks, making it experimentally difficult to separate the elastic energy released upon fracture from that dissipated through dynamic processes (18–26). In fact, most fracture theories have been developed at the continuum scale and have focused on modeling dynamic dissipation processes (27). An exception to this is the theory of Lake and Thomas that connects the elastic energy released during chain scission to chain-level structure,Gc,LT=ChainsArea×Energy DissipatedChain=νRoNU,2]where NU is the total energy released when a chain ruptures in which N represents the number of monomer segments in the chain and U the energy released per monomer (26).While this model was first introduced in 1967, experimental attempts to verify Lake–Thomas theory as an explicit model, as summarized in SI Appendix, have been unsuccessful. Ahagon and Gent (28) and Gent and Tobias (29) attempted to do this on highly swollen networks at elevated temperature but found that, while the scalings from Eq. 2 work well, an enhancement factor was necessary to observe agreement between theory and experiment. This led many researchers to conclude that Lake–Thomas theory worked only as a scaling argument. In 2008, Sakai et al. (30) introduced a series of end-linked tetrafunctional, star-like poly(ethylene glycol) (PEG) gels. Scattering measurements indicated a lack of nanoscale heterogeneities that are characteristic of most polymer networks (30–32). Fracture measurements on these well-defined networks were performed and it was again observed that an enhancement factor was necessary to realize explicit agreement between experiment and theory (33). Arora et al. (34) recently attempted to address this discrepancy by accounting for loop defects; however, different assumptions were used when inputting U to calculate Lake–Thomas theory values that again required the use of an enhancement factor to achieve quantitative agreement. In this work we demonstrate that refining the Lake–Thomas theory to account for loop defects while using the full bond dissociation energy to represent U yields excellent agreement between the theory and both simulation and experimental data without the use of any adjustable parameters.PEG gels synthesized via telechelic end-linking reactions create the opportunity to build upon previous theory to establish true multiscale, molecular to macroscopic relationships that define the fracture response of polymer networks. This paper combines pure shear notch tests, molecular dynamics (MD) simulations, and theory to quantitatively extend the concept of network fracture without the use of an enhancement factor. First, the control of molecular-level structure in end-linked gel systems is discussed. Then, the choice of molecular parameters used to estimate chain- and network-level properties is discussed. Experimental and MD simulation methods used when fracturing model end-linked networks are then presented. A theory of network fracture that accounts for loop defects is developed, in the context of other such models that have emerged recently, and tested against data from experiments and MD simulations. Finally, a discussion of the local and global energy dissipated during failure of the network is presented.