Macromolecules 1992,25, 6315-6321 6316 Kinematics of Polymer Chains with Freely Rotating Bonds in a Restrictive Environment. 2. Conformational and Orientational Correlations I. Bahar' and B. Erman Polymer Research Center and School of Engineering, Bogazici University, and TUBZTAK Advanced Polymeric Materials Research Center, Bebek 80815, Istanbul, Turkey L. Monnerie Ecole Superieure de Physique et Chimie Indwtrielle, PCSM, 10 rue Vauquelin, Ceder 05, Paris 75231, France Received February 13, 1992; Revised Manuscript Received June 17, 1992 ABSTRACT: The mathematical formulation developed in the preceding paper is used to study the kinematics of polymer chains in a restrictive environment. Conformational and orientational correlations along the chain are analyzed. Calculations are performed by generating an initial configuration for a 25-bond freely rotating chain, changing the dihedral angle for the middle bond, and studying the resulting changes in all of the degrees of freedom, internal and external. Results from Monte Carlo chains generated in this manner are averaged over a sufficient number of initial configurations. The following conclusions are reached (i) A change in the dihedral angle of a given internal bond in small steps up to f120° is accommodated by the spatial readjustments of about three neighboring bonds on each of ita sides. Spatial configurations of more distant neighbors are negligibly affected. Thus, the conformational correlation length along the freely rotating chain extends over 6-8 bonds. (ii) In general, the rotational motion of a given bond is accompanied by counterrotations of ita first neighbors. (iii) The response of second neighbors is lightly stronger than that of the first neighbors on the average, though occurring randomly in the positive or negative sense. (iv) The kinematics of motion imply the potential occurrence of transitions of the form g*tt - ttg* and ttt - g*tgr, in agreement with previous predictions of Helfand. (v) Bonds in the close neighborhood of the rotating bond exhibit a broad distribution of angular displacemenb in space aa a result of the compensating effect of the cooperative motions. Introduction The cooperative nature of the conformational transitions of chain units in a moving segment predominantly stems from the requirement to localize or compensate the original motion. The propositions of crankshaft' and 3- or 4-bond motions2 as the mechanism of local relaxational processes in polymers conform with the idea of immobilizing the ends of a moving segment which, in turn, is of the smallest possible size compatible with the tetrahedral geometry. These motions involve simultaneous isomeric rotations of more than one skeletal bond. Later experimental obser- vation of activation energies of about one rotameric barrier height3 invalidates, however, this picture of highly localized motions. A classification of local motions based on the spatial rearrangement of the so-called "tails" surrounding the mobile segment has found widespread use in describing local chain dynami~s.~95 Accordingly, correlated motions giving rise to the translation of the tails are referred to as type I1 motions and appear from Brownian dynamics simulations6J to be relatively favored. Gauche migration and pair gauche production/annihilation are the two major transitions belonging to this category. Their respective kinetic schemes are g*tt - ttg* and ttt * g*tgT, where t, g+, and g refer to the rotational isomeric states trans, gauche+, and gauche-. On the other hand, crankshaft motions and 3- or 4-bond jumps, which are classified as type I, are highly improbable, as confirmed by simulations. The third group of transitions, referred to as type 111, constitutes the major fraction of operating transitions. Those are individual bond flips between rotational isomeric states, which are accommodated by the coordinated small- amplitude rearrangements of neighboring units. Thus, the concept of a kinetic segment of a few backbone bonds 0024-9297/92/2225-6315$03.00/0 over which the new orientation resulting from the rotating bond is spread and dissipated without recourse to a coupled rotameric jump of a nearby unit is introduced as a probable mechanism of conformational relaxation. In general, backbone bonds in real chains are expected to undergo several oscillatory motions about a given rotational energy minimum before eventually jumping to another rotational isomeric ate. The magnitude of these oscillations depends both on the shape of the intramo- lecular potential well and on the immediate environment of the chain. Recent molecular dynamics studies of the isolated poly(methy1ene) chain6 indicate that the oscil- lations are in the range of about f15-30° about the minima. Molecular dynamics simulations in the bulk state9 lead to similar results. In the case of chains with freely rotating bonds, identical to the model adopted in the present study, the simulations of Takeuchi and Roes show that the oscillations in the dihedral angles may be much larger. Such oscillatory motions were first observed in WNMR in polymer melts10 and more recently by two-dimensional NMR experiments of Spiess and ~ollaborators1~-13 for chains in the bulk state at temperatures down to the glass transition temperature. In NMR experiments, the re- orientation of the transition moment vector-along the C-H bond, for example-is expected to accompany the motion of the backbone bond on which it is rigidly attached. The experimentally observed broad distribution of angular reorientation in space for a given transition moment vector invites attention to the limits of the validity of rotational isomeric jumps of about 120° amplitude as the basic mechanism of local relaxation. Instead, cooperative small- amplitude motion of several adjoining units, smoothening to a large extent the distortions induced by individual jumps, is brought into consideration. 0 1992 American Chemical Society6316 Baharet al. Macromolecules, Vol. 25, No. 23, 1992 A chain of 25 bonds is considered. This chain is sufficiently long to permit the internal reorganization of the atoms without substantial displacement of the tails, as will be presented below. Bonds are originally assigned three types of equally probably torsional states, trans, gauche+, and gauche-, by a random number generator subroutine. The original configuration of