Statistical thermodynamics of networks at large deformations

A general theory of non-Gaussian elasticity is presented for real polymeric chains having fixed bond angles and restricted internal rotations. The theory contains the displacement-vector distribution given by Nagai, and the Flory-Wall-Hermans procedure is used for the calculation of network properties. Whereas the treatment is valid for all types of polymer chains, it is not totally satisfactory from a practical standpoint because of a slow series convergence if the chains are stiff. It is best utilized for flexible polymers under conditions of light crosslinking. Detailed network behavior is investigated only for polyethylene type chains having uncorrelated internal rotations. In this instance the fractional contribution f_e/f of the internal energy of the total force f is found to be a function of elongation at high degress of stretching. It may decrease, or increase, depending upon the sign of f_e/f at low elongations. Furthermore, the variation of f_e/f with elongation is independent of the fixed bond angle of the chain backbone. Stress–strain behavior and energy–strain behavior are in opposition, i.e., when the non-Gaussian contribution to the stress is greatest, it is the least for the ratio f_e/f, and vice versa. The presence of correlated internal rotations would not be expected to greatly alter these general conclusions.