Polymer chain of freely‐jointed segments in nematic molecular mean‐field

Abstract

The distribution function of orientations of a segment, which interacts with the orienting field and is within the chain with given the end-to-end distance-vector, is calculated. The number of segments per chain is finite. The Lagrange method of conditional minimization of the chain free energy (the Helmholtz function) functional is used. Constraints for the segment orientations stem from fixed the chain end-to-end distance-vector. Hence, Lagrange multipliers, energy, free energy, and entropy, for the chain with given the end-to-end distancevector, are calculated. Then, the distribution function of values of that vector is obtained. Furthermore, an average free energy per chain inside the polymer network with given a topological structure, the system self-deformation, and modulus of elasticity are calculated and discussed in Gaussian limit, that is, for the number of segments per chain tending to infinity.

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

“…In general, the chain entropy is a function of the vector t and depends on the parameter u, and is proportional to n. In spherical coordinates we use variables t, Q and F for description of the vector t. Because of the three-chain model the calculation of an average for variables Q and F produces the following values: Q ¼ 0 and any F, and Q ¼ p=2; F ¼ 0 and (3) and (4). The plot tends to minus infinity for t ¼ 1.…”

“…Discussion is presented in. [4,5] As discussed, we use a three-chain model of the system. Therefore, further calculations will be performed for three chains for which we have: t z ¼ t, t x ¼ t y ¼ 0 (termed further parallel chain), and t x ¼ t, t y ¼ t z ¼ 0 and t y ¼ t, t x ¼ t z ¼ 0 (termed further perpendicular chains).…”

“…That problem has been studied by many works. [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.…”

“…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.…”

Internal orientations and elastic properties of non-Gaussian nematic polymer network

“…The first of considered methods was that proposed by Kuhn. In Gaussian limit, the function F obtained by Kuhn 4,5 and by Kuhn and Grün 9 is proportional to the average ͗r 2 ͘. Short-range covalent chemical interactions in macromolecule are involved into calculations via a length of the statistical segment.…”