Power Of Chain Length Research Articles

Note: A simple picture of subdiffusive polymer motion from stochastic simulations

Entangled polymer solutions and melts exhibit unusual frictional properties. In the entanglement limit self-diffusion coefficient of long flexible polymers decays with the second power of chain length and viscosity increases with 3–3.5 power of chain length.1 It is very difficult to provide detailed molecular-level explanation of the entanglement effect.2 Perhaps, the problem of many entangled polymer chains is the most complex multibody issue of classical physics. There are different approaches to polymer melt dynamics. Some of these recognize hydrodynamic interactions as a dominant term, while topological constraints for polymer chains are assumed as a secondary factor.3,4 Other theories consider the topological constraints as the most important factors controlling polymer dynamics. Herman and co-workers describe polymer dynamics in melts, as a lateral sliding of a chain along other5,6 chains until complete mutual disentanglement. Despite the success in explaining the power-laws for viscosity, the model has some limitations. First of all, memory effects are ignored, that is, polymer segments are treated independently. Also, each entanglement/obstacle is treated as a separate entity, which is certainly a simplification of the memory effect problem. In addition to that, correlated motions of segments are addressed within the framework of renormalized Rouse-chain theory,7 without calling any topological entanglements in advance. This approach leads to the generalized Langevin equation characterized by distinct memory kernels describing local and nonlocal segment correlations8–10 or to the Smoluchowski equation in which the segments’ mobility is treated as a stochastic variable.11 Both models describe the polymer segments motion at a microscopic level. An interesting alternative is to solve the integrodifferential equation for the chain relaxation with a sophisticated kernel function.12 The design of the kernel function is based on a mesoscopic description of the polymer melt. These theories explain some experimental data, although the description of the crossover between the Rouse and non-Rouse behavior is not satisfactory. Obviously, within the scope of a short note we cannot review all theoretical concepts of the polymer melt dynamics. Here we focus just on the interpretation of the observed single segment autocorrelation function.

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 by the sum of two terms, the first due to chain swelling by excluded volume and consistent with the predictions of scaling theory, the second due only to chain connectivity.

Statics and dynamics of dense polymer systems studied by monte carlo simulation

AbstractMonte Carlo simulations of coarse–grained models of macromolecules offer a unique tool to study the interplay between coil conformations, thermodynamic properties, and chain configurational relaxation and diffusion.Two examples are discussed where the chain conformation strongly differs from a gaussian coil: (i) collapsed chains in a bad solvent, where anomalous diffusion occurs in the Rouse limit and the relaxation time increases at least with the third power of chain length. (ii) Expulsion of a chain from a semidilute polymer brush. The initially stretched chain contracts to a gaussian coil and the center of mass moves outward with constant velocity until it reaches the region of the “last blob” where crossover to diffusive behavior occurs.

Monte Carlo simulation of dense polymer systems on a lattice

At present the collective motion algorithm is the only Monte Carlo technique which allows the simulation of monodisperse polymer melts on a lattice at a volume fraction of unity. Here we describe the algorithm in detail and show that it fulfils the condition of detailed balance. It yields chain dimensions which are slightly smaller than chain dimensions obtained for polydisperse systems or for systems at volume fractions smaller than unity, as expected. The computational expense to obtain an independent sample containing a constant number of chains increases as about the 3.6th power of chain length.

Simulation of the relaxation of random-coil polymer chains by lattice models with excluded volume: The effect of bead movement rules

Earlier work showed that excluded volume constraints lengthen the long relaxation times of lattice-model polymer chains using single-bead elementary motions by a factor proportional to about the first power of chain length. This paper reports the extension of this work to chains employing double-bead elementary motions and to chains employing random mixtures of both kinds of motion. For pure two-bead motion, the chain-length dependence of the lengthening of relaxation time is essentially identical with that previously obtained for pure single-bead motion. For models using both kinds of move, this chain-length dependence is suppressed at shorter chain lengths. However, for chains longer than about 30 beads, contrary to the prediction of Deutch and others, the lengthening of the relaxation time by excluded volume again becomes proportional to about the first power of chain length. Other studies that have not exhibited this first power dependence have either examined relaxation from highly nonequilibrium configurations or have not employed sufficiently long chains for the effect to be observed.