Gas‐Like Theory of Polymer Networks

Abstract

The theory presented starts from the model of gas of statistical segments with potential interactions between them. The fact that the system considered is an elastomer is incorporated into the model by constraints imposed on orientations of segments. Constraints are caused by the elastomer structure. The structure is determined by a connection of segments in linear chains, while chains are connected in the network structure. The statistical and thermodynamic description of the system is obtained by use of the conditional optimization of its free energy (Helmholtz function). The Lagrange multipliers for that optimization are calculated by use of cumulants and the multivariate method. In general, results of the theory presented are formulated for any type of potential interactions.

Numerical Verification of Analytical Results for Statistical Description of Polymer Chains in Nematic Systems

“…Results in this respect are obtained in. [6,7] As shown in, [6] for g x , g y and g z equal to zero the thermodynamic functions in Equations (4) and (5) reduce to those obtained by Maier and Saupe for the nematic system of unconnected hard rods.…”

“…Three Lagrange multipliers g x , g y and g z are in a relation to constraints of the l orientations, which are other than those determined by the orienting field only. For segments within the linear chain that constraint is given by the fact that the sum of all segment-vectors l is equal to the end-to-end distancevector r. In this case we have the following equations for the Lagrange multipliers: [6] …”

“…It was recognized in [6] that for the molecular mean-field approach to the description of orienting interactions between hard-rod particles contained in a large system the density probability function v u; f; g x ; g y ; g z of orientations of the hard-rod particle represented by the vector l is given by the following formula:…”

“…[4][5][6][7] Calculations presented here are based on results of the general theory presented in. [6] As discussed, the Porod and Kratky worm-like model of the chain is not used here. However, their three-chain representation of the three-dimensional polymer system is applied.…”

“…The reason why we prefer that model rather than the Porod and Kratky model is discussed in. [4][5][6][7][8] Kuhn and Gr€ un considered pure entropic isolated chains without energetic interactions between statistical segments. In the work presented here a chain is within a threedimensional system of polymer chains.…”