The role of pressure in rubber elasticity

We describe a series of molecular dynamics computations that reveal an intimate connection at the atomic scale between difference stress (which resists stretches) and pressure (which resists volume changes) in an idealized elastomer, in contrast to the classical theory of rubber elasticity. Our simulations idealize the elastomer as a "pearl necklace," in which the covalent bonds are stiff linear springs, while nonbonded atoms interact through a Lennard-Jones potential with energy epsilon(LJ) and radius sigma(LJ). We calculate the difference stress t(11)-(t(22)+t(33))/2 and mean stress (t(11)+t(22)+t(33))/3 induced by a constant volume extension in the x(1) direction, as a function of temperature T and reduced density rho(*)=Nsigma(IJ) (3)/nu. Here, N is the number of atoms in the simulation cell and nu is the cell volume. Results show that for rho(*)<1, the difference stress is purely entropic and is in good agreement with the classical affine network model of rubber elasticity, which neglects nonbonded interactions. However, data presented by van Krevelen Properties of Polymers, 3rd ed. (Elsevier, Amsterdam, 1990), p. 79] indicate that rubber at standard conditions corresponds to rho(*)=1.2. For rho(*)>1, the system is entropic for kT/epsilon(LJ)>2, but at lower temperatures the difference stress contains an additional energy component, which increases as rho(*) increases and temperature decreases. Finally, the model exhibits a glass transition for rho(*)=1.2 and kT/epsilon(LJ) approximately 2. The atomic-scale processes responsible for generating stress are explored in detail. Simulations demonstrate that the repulsive portion of the Lennard-Jones potential provides a contribution sigma(nbr)>0 to the difference stress, the attractive portion provides sigma(nba) approximately 0, while the covalent bonds provide sigma(b)<0. In contrast, their respective contributions to the mean stress satisfy Pi(nbr)<0, Pi(nba)>0, and Pi(b)<0. Analytical calculations, together with simulations, demonstrate that mean and difference stresses are related by sigma(nbr)=-APi(nbr)P(2)(theta(b)), sigma(b)=BPi(b)P(2)(theta(b)), where P(2)(theta(b)) is a measure of the anisotropy of the orientation of the covalent bonds, and A and B are coefficients that depend weakly on rho(*) and temperature. For high values of rho(*), we find that sigma(nbr)]>sigma(b)], and in this regime our model predicts behavior that is in good agreement with experimental data of D.L. Quested et al. J. Appl. Phys. 52, 5977 (1981)] for the influence of pressure on the difference stress induced by stretching solithane.