Many Chains

These bead-spring models of Rouse and Kirkwood-Riseman-Zimm suffer from the artificiality of the beads and springs. The bead friction coefficient is an ad hoc phenomenological coefficient. This should arise naturally from the frictional forces coupling the polymer and solvent directly with the continuous version of the chain without beads and springs. 

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J. Low molecular weight symmetrical polyester polyols may be liquids as a result of many chain ends that disrupt crystalline packing entirely. 

Since these microscopic simulations typically can only treat short times and small samples, it is important to avoid surface effects. It is common to employ periodic boundary conditions. A special trick often used for these kinds of simulation is, instead of simulating a melt of many chains, to simulate one very long chain which falls back again and again into the box. In this way, the effect of the chain ends, which introduces artificially high free volume can be reduced. However, one should keep in mind that this chain interacts with its own periodic images. It is known that this may... 

Typical levels of MAH in grafted PP of 0.5-2 wt % correspond to only one or two units per chain. If the MAII units arc grouped it follows that many chains may contain no MAH. It has also been suggested that for PP all MAH may appear at the chain ends. This is rationalized in terms of the reaction of mid chain radicals with MAH always being followed by intramolecular chain transfer and chain... 

Frequently function R can be written as a single term having the simple form of equation 1. For Instance, with the aid of the long chain approximation (LCA) and the quasi-steady state approximation ((JSSA), the rate of monomer conversion, I.e., the rate of polymerization, for many chain-addition polymerizations can be written as... 

Transition from Single to Many Chain Behavior... 

Thus, with decreasing Q a crossover from the single-chain to the collective many-chain relaxation occurs at Q = 1/1 (c) (see Fig. 58a). 

The transition from single- to many-chain behavior already becomes obvious qualitatively from a line shape analysis of the NSE spectra (see Fig. 60) [116]. For dilute solutions (c = 0.05) the line shape parameter (3 is equal to about 0.7 for all Q-values, which is characteristic of the Zimm relaxation. In contrast, in semi-dilute solutions (e.g. c = 0.18), ft-values of 0.7 are only found at larger Q-values, whereas P-values of about 1.0, as predicted for collective diffusion [see Eq. (128)] are obtained at small Q-values. A similar observation was reported by [163]. 

Fig. 60. Crossover from single-chain to many-chain relaxation at T = 343 K. Lineshape analysis for PDMS/d-benzene at c = 5 and 18% double logarithmic plot of — ln/S(Q,t)/S(Q,0) vs. t/s. (Reprinted with permission from [116]. Copyright 1982 J. Wiley and Sons, Inc., New York)...

The experimental data also show that the crossover to the many-chain regime does not appear in the Q-window accessible to the method. Owing to this lack, the direct experimental evidence that the upturn has to be assigned to a single chain mode and does not result from a collective mode, is still missing. Nevertheless, assuming Dc c, and expressing Dg by DC/(Q/), from a simultaneous fit (c)/A = (6.0 0.5)c-1 and //A = 7.4 + 0.6 are derived. 

Fig. 65. Concentration dependence of the hydrodynamic screening length (c). The solid line represents the result of the simultaneous fit, the dashed line in correlation length (c) related to the transition from single to many chain behavior. (Reprinted with permission from [40]. Copyright 1984 American Chemical Society, Washington)...

Under good solvent conditions the crossover from single-chain relaxation to collective diffusion (many-chain behavior) can be observed by variation of the Q-value... 

The crossover from the single to the many-chain relaxation (Q(Q) Q2), which is expected to follow at further decreasing Q-values, has not yet been detected by the NSE technique... 

For example, if many chains are perfectly aligned, e.g. in the z-direction, the probability of the test chain also placing its segments in the z-direction becomes strongly enhanced. The oriented molecules leave an open corridor in the direction of the director. In terms of the vacancy probability, it is an event that it is enhanced with respect to the classical mean-field prediction. As a result, the test... 

Off-lattice models consider chains composed of interacting units in the free space. Single chains or simulation boxes containing many-chain systems can be investigated. Usually the solvent is only considered according to its quality effects in thermal systems. Therefore it is assumed to fill the remaining space act-... 

Fig. 2. The tube model replaces the many-chain system left) with an effective constraint on each single chain right). The tube permits diffusion of chains along their own contours only...

Since branched chains like LDPE have many chain ends, it is customary to use the radius of gyration, S, as a measure of the distribution of polymer segments. The radius of gyration is the distance of a chain end from the polymer s center of gravity rather than a measure of the RMS. Both measures are statistically related. 

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