Kinematics of Polymer Chains with Freely Rotating Bonds in a Restrictive Environment. 1. Theory

A mathematical formulation is developed for the study of the kinematics or geometry of motion of polymer chains with freely rotating bonds. The chains are assumed to be in a dense continuous medium constraining the spatial displacements of atoms. The basic postulate of the proposed model is that, following any perturbation of an equilibrium configuration, the atoms rearrange cooperatively in space so as to minimize their overall square displacements. This postulate is equivalent to minimizing the energy spent against the surroundings during the motion of the chain from one configuration to another. The present formulation allows for the calculation of the changes in all of the degrees of freedom of the chain, internal and external, in response to a change in the dihedral angle of an internal bond. By the use of Lagrange multipliers, the formulation may be generalized to study the results of perturbations on a chain subject to other types of external constraints, such as uniaxial deformation.