Molecular modeling of methane diffusion in glassy atactic polypropylene via multidimensional transition state theory

A multidimensional transition state theory (TST) approach is formulated for the study of elementary jumps involved in the diffusion of a gaseous penetrant in a glassy polymer, taking explicitly into account the coupling between polymer and penetrant degrees of freedom along each jump. In this approach, an initial picture of states (local minima of the potential energy of the polymer + penetrant system) and "macrostates" (i.e., collections of states communicating over barriers small relative to k(B)T) is obtained through geometric analysis of accessible volume in representative glassy polymer configurations. Saddle points of the potential energy are computed using the "necks" between accessible volume clusters as initial guesses and progressively augmenting the set of degrees of freedom with respect to which the saddle point is calculated. Starting from the saddle points, states and reaction paths are then mapped out in the multidimensional space of penetrant and polymer degrees of freedom, using Fukui's intrinsic reaction coordinate (IRC) approach, cast in a subset of flexible generalized coordinates. Finally, rate constants for the interstate transitions are computed by multidimensional TST, after invoking a harmonic approximation. Application of this approach to methane at low concentration in glassy atactic polypropylene at 233 K gives well-converged reaction paths involving ca. 350 degrees of freedom of the polymer and confirms the macrostate hypothesis. The rate constant distribution for intermacrostate jumps is found to be very broad, ranging between 10(-12) and 10(6) mu s(-1). The jump length distribution for jumps between states belonging to different macrostates is fairly narrow and centered about 5 Angstrom, which is comparable to the range of motion of the penetrant within a macrostate. Energy barriers for intermacrostate jumps exhibit a broad, asymmetric distribution with mean around 5 kcal/mol, while entropy barriers are much more narrowly and symmetrically distributed with mean around -4k(B). The detailed information on elementary jumps gathered by these atomistic TST calculations can form the basis for a coarse-grained model of diffusion and for estimating the diffusivity.