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

DE Hanson and RL Martin, JOURNAL OF CHEMICAL PHYSICS, 133, 084903 (2010).

DOI: 10.1063/1.3475522

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. (C) 2010
American Institute of Physics. **doi: 10.1063/1.3475522**

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