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A matrix method is used to determine fluctuations of junctions and points along the polymer chains making up a phantom Gaussian network that has the topology of an infinite, symmetrically grown tree. The functionalities of the junctions alternates between [lowercase_phi_synonym]1 and [lowercase_phi_synonym]2, such that one end of each network chain has functionality [lowercase_phi_synonym]1, while the opposite end has functionality [lowercase_phi_synonym]2. Quantities calculated include fluctuations of [lowercase_phi_synonym]1-functional and [lowercase_phi_synonym]2-functional junctions, and fluctuations of points along network chains, as well as correlations of these fluctuations. This was done for points and junctions along any path in the network, where these points and junctions were separated by no junctions or several junctions, Fluctuations have also been calculated for the distances between points and junctions. The present results represent significant generalizations of earlier work in this area [Kloczkowski et al., Macromolecules 22, 1423 (1989)]. These generalizations and extensions should be very useful in a number of contexts, such as interpreting small-angle neutron scattering results on labeled paths in polymer networks, or fluctuations of loops in the Gaussian network model of proteins. [ABSTRACT FROM AUTHOR]

An analytical solution to the master equation governing the conformational dynamics of linear polymer chains is formulated. Symmetric chains with N bonds subject to independent rotational potentials are considered. The eigenvalues of the transition rate matrix, which characterize the frequencies of the various relaxation modes, and the corresponding eigenvectors and eigenrows are obtained in closed form. A simple recurrence equation permits one to express the eigenvalues of the N-bond motion in terms of the nonzero eigenvalues associated with the isomeric transitions of single bonds. This leads to a clear understanding of the increase in conformational mobility with N. [ABSTRACT FROM AUTHOR]

Abstract: We derived the analytical solution of the eigenvalue problem for stereo-regular vinyl chains, such as stereo-regular polypropylene chains. The solution is applicable to all stereo-regular polymers of the type (AB) x which do not have symmetry between gauche + and gauche − states, and to polymers, such as polyoxymethylene or polydimethylsiloxane, for which symmetry between the gauche + and gauche − states does exist. For symmetric chains, the general solution of the eigenvalue problem is reduced to the known solution for polyoxymethylene chains. To illustrate the method the calculations have been performed for the three rotational states (trans, gauche + and gauche −) model, but the general algebraic solution is applicable for any ν rotational states models of polymer chains. We used the analytical solution of the eigenvalue problem to calculate numerically elastic properties of stereo-regular polypropylene chains within the framework of Mark–Curro theory (J Chem Phys, 79, 5705, 1983). [Copyright &y& Elsevier]

Abstract: This review illustrates how elastomer reinforcement can be modeled using Monte Carlo simulations on rotational isomeric state chains to characterize their spatial configurations in the vicinity of filler particles. The results are distributions of the chain end-to-end distances as perturbed by this excluded-volume effect, and the results obtained are in agreement with experimental results gotten by neutron scattering. The use of these distributions in standard molecular theories of rubberlike elasticity then produces stress–strain isotherms suitable for comparison with those in elongation experiments. Such simulations have now been carried out for elastomeric matrices reinforced by spherical filler particles (either on a cubic lattice or randomly dispersed), or by prolate or oblate particles on cubic lattices (either with their axes oriented or randomized). The simulated mechanical properties are consistent with experimental results available at the present time, and should provide encouragement and guidance for additional simulations and experiments. [Copyright &y& Elsevier]

Abstract: The elastomeric properties of networks of stereoregular polypropylenes (PP) filled with spherical nanoparticles have been modeled in an attempt to obtain better insights into elastomer reinforcement. The polymers were either isotactic or syndiotactic PP in the amorphous state, and the simulations were based on rotational isomeric state (RIS) theory combined with the largest eigenvalue method for deriving conditional bond probabilities. Monte Carlo simulations gave distributions of the end-to-end distance of these chains in the presence of the particles, and these were used in the Mark–Curro theoretical approach to calculate values of the normalized stress, and the reduced stress (shear modulus) under uniaxial stretching. The simulations were calculated for PP chains having 100–200 skeletal bonds, for several temperatures from 481 to 650K, and for varying filler particle sizes (up to 100Å). The presence of the filler nanoparticles was found to influence chain conformations, frequently leading to significant chain extensions, which significantly affect the elastomeric properties of the nanocomposites. [Copyright &y& Elsevier]