Structural Relaxation and Vitrification in Dense Cross-Linked Polymer Networks: Simulation, Theory, and Experiment

Abstract

We present a coordinated experimental, simulation, and theoretical study of how polymer network permanent cross-links impact the segmental relaxation time over a wide range of temperatures and different criteria for defining the glass transition temperature, T g. The simulations adopt a coarse-grained model calibrated to represent the specific polymer chemistry of interest. The elastically collective nonlinear Langevin equation (ECNLE) theory of activated segmental relaxation is extended to explicitly treat chain semiflexibility and network cross-linkers, with the latter modeled as locally pinned or vibrating sites. Our key findings include the following: (i) tight cross-linking leads to very large increases of the segmental relaxation time and elevation of T g, which grows roughly linearly with cross-link fraction beyond a low threshold, (ii) a remarkably good (but not perfect) collapse of Angell plots of the alpha relaxation time for all cross-link densities studied based on using the cross-link fraction dependent dynamic T g, which applies for very different dynamic vitrification time scale criteria, and (iii) construction of a microscopic understanding of the experimental and simulation observations based on the central idea of ECNLE theory that activated structural relaxation involves cross-link fraction dependent coupled local cage and nonlocal collective elastic barriers. Overall, excellent agreement between experiment, theory, and simulation is found. We suggest that our study of how quenched chemical cross-links strongly modify the alpha relaxation is more generally valuable as a distinct probe of the basic physics of glassy polymer dynamics and as a flexible tool to manipulate small-molecule diffusion in membrane applications.

Theory of the Effects of Specific Attractions and Chain Connectivity on the Activated Dynamics and Selective Transport of Penetrants in Polymer Melts

“…For polymer melts, the theoretical alpha relaxation process involves Kuhn segment hopping over a local cage barrier ( F B ) coupled with small collective long-range displacements of all segments outside the cage characterized by a nonlocal elastic barrier ( F el ), as sketched in Figure . Recent studies support the predictions of this approach for a large number of real-world polymeric liquids, including olefins, dienes, siloxanes, and vinyl polymer chemistries, deduced based on a simplified mapping of chains to a disconnected on the Kuhn length scale effective HS fluid model. ,,, …”

“…The polymer–polymer and polymer–penetrant equilibrium intermolecular site–site pair correlation functions are computed with PRISM theory for the chosen polymer and penetrant models, − intermolecular interactions, and packing fractions. The single scalar PRISM integral equation in Fourier space for the homopolymer liquid is h mm ( q ) = ω m ( q ) C mm ( q ) S mm ( q ), where h mm ( r ) = g mm ( r )-1, g mm ( r ) is the interchain site–site radial distribution function, S mm ( q ) is the static collective structure factor, and C mm ( q ) is the direct correlation function.…”

“…The starting point for constructing a microscopic predictive theory is to understand the polymer matrix structural (segmental alpha) relaxation. Major progress − has been achieved in this direction based on the elastically collective nonlinear Langevin equation (ECNLE) theory originally constructed for spherical particle [mainly hard sphere (HS)] fluids , where kinetic constraints and barriers are quantified by a dynamic free energy constructed directly from knowledge of the liquid pair packing structure. For polymer melts, the theoretical alpha relaxation process involves Kuhn segment hopping over a local cage barrier ( F B ) coupled with small collective long-range displacements of all segments outside the cage characterized by a nonlocal elastic barrier ( F el ), as sketched in Figure .…”

“…(i) First is to model the matrix as a liquid of connected semiflexible chains, which modifies the polymer–polymer and polymer–penetrant packing correlations and hence the dynamical constraints experienced by penetrants and the Kuhn segment scale alpha relaxation process. The technical details of how to do this for the pure polymer melt (and also cross-linked polymer networks) are fully worked out in refs and , and have been successfully confronted with simulations and experiments. This advance will also allow an assessment of the importance of the polymer local structure on penetrant mobility by comparing the new theoretical results with the prior ones obtained using the HS mixture model. , (ii) Next is to introduce specific penetrant–matrix attractions which result in the formation of transient physical bonds.…”

“…We emphasize that many (not all) of the technical details, equations, derivations, numerical implementations, and discussion of limitations of the equilibrium and dynamic statistical mechanical theory tools employed in this article have been thoroughly documented previously, especially in refs , , , and will not be repeated here. This includes the polymer reference interaction site model (PRISM) integral equation theory formulated at the elementary site or bead level (Figure ) for the structural pair correlation functions of a homopolymer liquid and the dynamic theories formulated at the Kuhn length scale.…”