Quantum chemistry and molecular dynamics studies of the entropic elasticity of localized molecular kinks in...

We investigate the thermodynamic consequences of the distribution of rotational conformations of polyisoprene on the elastic response of a network chain. In contrast to the classical theory of rubber elasticity, which associates the elastic force with the distribution of end-to-end distances, we find that the distribution of chain contour lengths provides a simple mechanism for an elastic force. Entropic force constants were determined for small contour length extensions of chains constructed as a series of localized kinks, with each kink containing between one and five cis-1,4-isoprene units. The probability distributions for the kink end-to-end distances were computed by two methods: (1) by constructing a Boltzmann distribution from the lengths corresponding to the minimum energy dihedral rotational conformations, obtained by optimizing isoprene using first principles density functional theory, and (2) by sampling the trajectories of molecular dynamics simulations of an isolated molecule composed of five isoprene units. Analogous to the well-known tube model of elasticity, we make the assumption that, for small strains, the chain is constrained by its surrounding tube, and can only move, by a process of reptation, along the primitive path of the contour. Assuming that the chain entropy is Boltzmann's constant times the logarithm of the contour length distribution, we compute the tensile force constants for chain contour length extension as the change in entropy times the temperature. For a chain length typical of moderately crosslinked rubber networks (78 isoprene units), the force constants range between 0.004 and 0.033 N/m, depending on the kink size. For a cross-linked network, these force constants predict an initial tensile modulus of between 3 and 8 MPa, which is comparable to the experimental value of 1 MPa. This mechanism is also consistent with other thermodynamic phenomenology.