^{1,a)}and Richard L. Martin

^{1}

### 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 rubberelasticity, 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.

We wish to thank Professor Tony Rappe and Dr. Enrique Batiste for their many helpful tutorials and suggestions and Dr. Neil Hensen for providing valuable assistance with the molecular dynamics simulation codes. This work was performed under the auspices of Los Alamos National Laboratory, which is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. DE-AC52-06NA25396.

I. INTRODUCTION

II. SIMULATION METHODOLOGY

III. DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Elasticity
- 21.0
- Rubber
- 14.0
- Probability theory
- 11.0
- Chemical bonds
- 9.0
- Molecular dynamics
- 9.0

## Figures

DFT optimized electronic energy as a function of the dihedral angle centered on the symmetric, center bond of isoprene.

DFT optimized electronic energy as a function of the dihedral angle centered on the symmetric, center bond of isoprene.

DFT optimized electronic energies, including zero point corrections, vs isoprene end-to-end distance as measured between the midpoints of the double bonds.

DFT optimized electronic energies, including zero point corrections, vs isoprene end-to-end distance as measured between the midpoints of the double bonds.

Distribution of contour lengths for chains constructed of 78 isoprene end-to-end distances randomly chosen from Boltzmann-weighted minimum energy rotational conformations (shown in Fig. 2) at temperatures 200, 250, 300, 350, and 400 K.

Distribution of contour lengths for chains constructed of 78 isoprene end-to-end distances randomly chosen from Boltzmann-weighted minimum energy rotational conformations (shown in Fig. 2) at temperatures 200, 250, 300, 350, and 400 K.

Most probable contour length for chains constructed of 78 isoprene end-to-end distances as a function of temperature, normalized to a single isoprene unit. Note that the contour length contracts with increasing temperature.

Most probable contour length for chains constructed of 78 isoprene end-to-end distances as a function of temperature, normalized to a single isoprene unit. Note that the contour length contracts with increasing temperature.

(a) Boltzmann-weighted end-to-end distance distribution for kinks of one and two isoprene units obtained from MD simulations. (b) Boltzmann-weighted end-to-end distance distribution for kinks composed of three, four, and five isoprene units obtained from MD simulations.

(a) Boltzmann-weighted end-to-end distance distribution for kinks of one and two isoprene units obtained from MD simulations. (b) Boltzmann-weighted end-to-end distance distribution for kinks composed of three, four, and five isoprene units obtained from MD simulations.

Distributions of contour lengths for chains containing isoprene units constructed from kinks of order 1–5 with kink end-to-end distances chosen randomly from the Boltzmann distributions shown in Figs. 5(a) and 5(b). Each chain is composed of only a single kink size comprised of between one and five isoprene units.

Distributions of contour lengths for chains containing isoprene units constructed from kinks of order 1–5 with kink end-to-end distances chosen randomly from the Boltzmann distributions shown in Figs. 5(a) and 5(b). Each chain is composed of only a single kink size comprised of between one and five isoprene units.

Distribution of contour lengths for chains randomly constructed from 78 single isoprene kink end-to-end distances, using DFT-B3lYP (blue) and MD-CHARRM force field (red). Note the close agreement of the two methods.

Distribution of contour lengths for chains randomly constructed from 78 single isoprene kink end-to-end distances, using DFT-B3lYP (blue) and MD-CHARRM force field (red). Note the close agreement of the two methods.

Least-squares fit to the derivative (with respect to chain contour length) of the natural logarithm of the distribution function shown in Fig. 8.

Least-squares fit to the derivative (with respect to chain contour length) of the natural logarithm of the distribution function shown in Fig. 8.

## Tables

DFT/B3LYP 6-31g(d) optimized parameter values for isoprene minimum energy rotational conformations.

DFT/B3LYP 6-31g(d) optimized parameter values for isoprene minimum energy rotational conformations.

Parameters for chains constructed from single kink type, 78 total isoprene units. : entropic force constant . : most probable chain length. Modulus: predicted initial tensile modulus for a network (see text).

Parameters for chains constructed from single kink type, 78 total isoprene units. : entropic force constant . : most probable chain length. Modulus: predicted initial tensile modulus for a network (see text).

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