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

Abstract

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.

The molecular kink paradigm for rubber elasticity: Numerical simulations of explicit polyisoprene networks at low to moderate tensile strains

“…The value chosen for K ε to achieve agreement with both experiments (0.09 nN strain −1 ) is within the range of values found from MD simulations of isolated kinks (0.06 to 0.6 nN strain −1 ). This value actually corresponds to that from the MD simulations 19 for a kink comprised of 3 isoprene units. However, the width of the kink end-to-end distance distributions found in the MD studies ranged between 0.4 and 1.4%, FWHM.…”

“…Our ansatz is that the force extension curve for polyisoprene molecules mimics this behavior; i.e., there are also three distinct extension regions for chain stretching. For two of these regions, MD and ab initio simulations 18,19 have provided quantitative predictions for the relevant chain forces, albeit for isolated chains. The simulations showed that isoprene units have 18 unique rotameric states and those with shorter end-to-end distances tend to be ∼1 kcal/mole higher in energy than the more extended conformations.…”

“…This has also been confirmed by MD simulations for kinks comprised of 1-5 isoprene units. 19 Because the kink end-toend distances are directly related to the rotameric state populations and the distributions are Gaussian, we expect that the entropy may be obtained from the natural logarithm of p(x). The entropic force can then be calculated from the change in free energy with respect to the end-to-end distance of the kink.…”

“…18 The force required to rupture an isoprene bond was found to be about 6.8 nN and the corresponding strain surprisingly large, greater than 40%. Ab initio and MD studies 19 have also established that entropic forces are inherent in local molecular kinks, comprised of a few isoprene units. Entropy changes were associated with the non-energetically degenerate rotational conformations of the isoprene units.…”

“…19 This paradigm differs from that of classical rubber elasticity theory (which is based on the Gaussian chain end-to-end distance distribution) in that we assume that an elastic chain force arises only from an increase in the end-to-end distance, beyond its initial, zero-strain value. We will present two new chain force extension models, applicable to low and moderate strains, and combine them with a previously published high strain model.…”