Peter H. Verdier

Monte Carlo Studies of Lattice‐Model Polymer Chains. I. Correlation Functions in the Statistical‐Bead Model

A method is presented for obtaining correlation functions in the free‐draining statistical‐bead model (Rouse model) of a polymer chain. Autocorrelation functions for the squares of end‐to‐end length and radius of gyration are given as functions of the number of statistical segments in the chain.

Monte Carlo Studies of Lattice‐Model Polymer Chains. II. End‐to‐End Length

The relaxation and the equilibrium behavior of lattice‐model polymer chains are studied by simulation on a digital computer. Results are presented for the behavior of the square of end‐to‐end length l2 for chains of 8, 16, 32, and 64 beads, with and without excluded volume restrictions. It is found that the relaxation of the lattice‐model chains without excluded volume is remarkably similar to that of statistical‐bead models. The introduction of excluded volume restrictions causes drastic qualitative changes in the relaxation behavior of the longer chains, and lengthens the time required for their relaxation by factors of up to 15. While the distribution in l2 for the longer chains with excluded volume departs noticeably from Gaussian form at quite small and quite large values of l2, it appears close to Gaussian in a range from one‐third to three times the mean‐square value of l2.

Monte Carlo studies of lattice‐model polymer chains. III. Relaxation of Rouse coordinates

The relaxation of the seven lowest Rouse coordinates for simple lattice models of polymer chains of up to 64 beads, with and without excluded volume, is studied by simulation on a digital computer. The similarity between the relaxation of the lattice‐model chains without excluded volume and that of a statistical‐bead model, noted in previous studies of end‐to‐end length, is confirmed and examined in greater detail. The effect of excluded volume in slowing down the relaxation of the Rouse coordinates is examined, and a simple picture is suggested which accounts qualitatively for the results obtained. The nonnormal coordinate nature of the Rouse coordinates for chains with excluded volume is demonstrated by their nonexponential autocorrelation functions. However, the results suggest that for each chain length, there is a unique longest internal relaxation time, corresponding to an internal coordinate closely resembling the lowest Rouse coordinate.

Relaxation of the aspherical shapes of random‐coil polymer chains

The conformations of random‐coil polymer chains are known to be appreciably more aspherical than might have been expected intuitively. However, this instantaneous asphericity will only affect the measured physical properties of flexible chains in solution if the relaxation of the asphericity requires a time longer than the inherent sampling time of the experiment. It is therefore important, in the analysis of phenomena which depend upon chain shape, to know the time scale over which deviations from spherical symmetry persist. In this paper we extend our previous work on the relaxation of asphericity in chains without excluded volume interactions and present corresponding results for chains with excluded volume. Autocorrelation functions for the radius of gyration squared, its principal components, and the moments of inertia are determined using a dynamical model for linear polymer chains which simulates excluded volume effects. The results suggest that the asphericity of a random‐flight chain is increased...

Relaxation Behavior of the Freely Jointed Chain

A method is presented for treating the relaxation behavior of the freely jointed chain model of a random coil polymer. Exact results are exhibited for the relaxation of quantities linear in chain coordinates. For the treatment of quantities quadratic in chain coordinates, a numerical approach is employed and exemplified by obtaining the autocorrelation in the square of end‐to‐end length for chains of up to 16 beads. In both cases, the rapid approach of the behavior of the freely jointed chain of N beads to that of the Rouse model of N statistical segments is demonstrated.

Monte Carlo Studies of the Relaxation of Vector End‐to‐End Length in Random‐Coil Polymer Chains

The effects of excluded volume interactions upon the dynamical behavior of random‐coil polymer chains are studied by obtaining autocorrelation functions for vector end‐to‐end length of lattice‐model chains of 9, 15, 33, and 63 beads by a Monte Carlo simulation technique. It is found that relaxation of the vector end‐to‐end length requires from 4 to 7 times as long as relaxation of its square, in contrast to the predictions of simple models without excluded volume effects.

A simulation model for the study of the motion of random-coil polymer chains

Abstract A lattice model of the dynamical behavior of a random-coil polymer chain in solution is described. Simulation of the model by a high-speed digital computer is discussed. The model appears especially suitable for the study of the effects of excluded volume interactions upon the motions of random-coil polymer chains.

Separating connectivity and expansion effects in polymer single chain dynamics

The effects of chain volume and connectivity upon the motions of flexible polymers in dilute solution have been studied by computer simulation of simple off-lattice bead-flip models of from 9 to 99 beads. Long internal relaxation times are given for free-draining chains with bead diameters from zero to 0.93 times the stick lengths. Moves are forbidden which would result either in bead overlap (excluded volume) or in one stick passing through another (chain connectivity). In the extreme case of zero bead diameter, where there is no expansion of the chains by excluded volume, the long relaxation time varies as about the 2.1 power of chain length, as opposed to the 2.0 power for similar chains without connectivity constraints. As bead diameter is increased until it equals stick length, the exponent increases to the value of 2.48 established by previous work. Over the range of bead diameters employed, the chain-length dependence of long relaxation times and translational diffusion constants can be described b...

Interpretation of quasi-elastic light scattering data for flexible chains: model dependence

Abstract The autocorrelation functions and corresponding relaxation times obtained from the forward depolarized quasi-elastic light scattering experiment are exhibited for two quite similar models of flexible polymer chains in solution. A very small change in the chain dynamics is found to be sufficient to change the relaxation time from a relatively short time independent of chain length, with an autocorrelation function suggestive of an unweighted sum of contributions from all the relaxation times in the spectrum of chain motion, to a long time with an autocorrelation function identical with that for the end-to-end vector, strongly dependent upon chain length and dominated by the longest relaxation time in the spectrum. These results raise the question whether widely-used models in which information about short-range chain structure and motion is deliberately omitted can be expected to be appropriate for the interpretation of depolarized scattering experiments.

Monte Carlo studies of lattice‐model polymer chains. IV. Equilibrium dimensions and distributions in Rouse coordinates

Distributions in the squares of the seven lowest Rouse coordinates for simple lattice models of polymer chains of up to 64 beads, with and without excluded volume, have been obtained by dynamical simulation on a digital computer. The expansion of the mean‐square values of the coordinates is found to depend primarily, though not entirely, upon the ratio of the number of beads in the chain to the index of the Rouse coordinate. The distribution functions for the squares of the Rouse coordinates for chains both with and without excluded volume constraints are close to Gaussian with the sole exception of the lowest Rouse coordinate for chains with excluded volume, which exhibits a pronounced depletion at small values. This depletion is very similar to that previously observed in distributions in the square of end‐to‐end length.