Ensembles of self avoiding five-way cubic lattice chains consisting of 50 segments were generated for athermal and &conditions, and the instantaneous shape and the orientation of each chain were characterized by evaluation of the squared three axes L;, L;, and L: of its equivalent ellipsoid and their orientation in space; the variation of these quantities within allowed pair configurations (relative to their values in isolated chains) was studied as a function of the separation R between the two centers of gravity of the two members of all isolated chain pairs, which could be formed from the ensemble. For small R (when R is smaller than the average radius of gyration of isolated chains (2);'*) L:, the squared longest axis of the equivalent ellipsoid, is strongly expanded for athermal as well as for &conditions, while the squared two shortes axes, L: and L:, are much less enlarged or even decreased in this range, thus emphasizing the prolate character of the ellipsoids on approaching R = 0. Furthermore the angle between the two L,-axes of the two chains forming a compatible pair is remarkably increased at vanishing separation (by ca. 10" in athermal systems and by ca. 7" in &systems). Thus, chains forming compatible pairs at R = 0 not only have to increase their dimensions (as already described earlier) but also have to change their instantaneous shape (towards a still higher anisotropy of the equivalent ellipsoids) and their relative orientation (the longest axes L, tend towards orthogonal orientation) in order to relieve the thermodynamic stress which they impose on each other. By integration of the effects over all separations the concentration dependence of all quantities was determined in the limit of zero concentration. Most quantities are subject to an appreciable, non-zero concentration dependence. Contrary to most theoretical predictions this is also true for &conditions.1,2 times the value at infinite separation when the centers of gravity of the chains coincide (see Fig. 1). &chains in addition exhibit a slight increase of 9 at very high separations (see Fig. 3). Splitting 3 into its components parallel and perpendicular to the axis connecting the centers of gravity of the two chains, 4, and J", , revealed that except for the increased value at vanishing separation, which is the same for 4 , a n d 9lit is the parallel component which causes this behavior, while the perpendicular 0025-1 16X/83/$03.00
Using a Metropolis‐Monte Carlo‐process equilibrated self‐avoiding unbranched model chains consisting of 50–400 segments have been generated on a 5 way cubic lattice, no attractive potential being operative between the segments (“athermal” chains). By checking all possible pairs of chains for absence of overlaps (double occupancies of lattice points) for several intermolecular separations R within an ensemble of such chains consisting of 100–200 individuals the pair distribution function G(R) has been evaluated as the fraction of pairs which are free of intermolecular overlaps. G(R) qualitatively has the form which is predicted by the Flory‐Krigbaum‐theory (FK‐theory), a quantitative inspection, however, shows that it is substantially higher than it should be according to the FK‐theory, which is also reflected in much lower excluded volumes compared to the FK‐prediction. ‐ This disagreement can be explained by inspection of the frequency distribution of the overlaps in the chain pairs: While it is implicit in the FK‐treatment that the overlaps form independently of each other and, as a consequence, their distribution should obey a Poisson‐statistics, the actual distributions show that the existence of one (or more) overlaps favors the formation of further ones. As on the other hand the average number of overlaps within a pair, Z. which has been evaluated numerically, is well consistent with the FK‐theory this leads to an increase of the fraction of pairs which are free of overlaps, i.e. G(R). relativ to a Poisson‐distribution. ‐ It is proposed, therefore, to subdivide the chains not ‐ as in the FK‐treatment ‐ into n segments of volume V, but into n' clusters of segments of volume V′(nV = n′V′) where n' should be chosen in a way that the resulting clusters can be considered to behave actually independently in the overlap formation process. On this basis a substantial improvement of the theory of G(R) for linear chains will be possible.
The knowledge of the average configuration of a linear polymer chain embedded in an amorphous matrix built up of identical chainsa so-called bulk polymerwithout doubt is of fundamental importance for the understanding of many properties of polymers. Admittedly, in the meantime the progress in neutron small angle scattering techniques has led to considerable information on dimensions andwith a lesser degree of certaintyon the shape of the chains. The access to further details, however, seems to be barred by the enormous experimental and instrumental expenses associated with this method.Another way of obtaining detailed information on such systems consists in their computer simulation. In order to keep the calculation time within fairly reasonable limits, this method is restricted to the investigation of model systems, which in addition are placed on lattices. Usually calculations were performed by first constructing a more or less biased multiple chain system by generating (or deliberately placing) a certain number of chains on a finite part of the chosen lattice adopting periodic boundary conditions. As a next step the initial bias was removed by subjecting the chains to rearrangement by "relaxation processes" (in the widest sense). After equilibration had been accomplished the same relaxation processes were then used to manage the proper movement of the system in the phase space during which the statistical averages of the relevant chain properties were sampled.For a long time the relaxation mechanisms used in these calculations could be considered to be simplified imitations of the physical processes leading to changes in chain conformation. One way consisted in moving one, two, or (at most) very few segments at one time3y4) The "modes" of microrelaxations belonging to this group can be described as end-group rotations (a), transformation of L-structures (b), and rotation of U-structures (c).In a modification of this method simultaneous movement of more segments was achieved by rotating one part of the chain about a dihedral angle'). a) Part of a presentation by 0. F. Olaj and W. Lantschbauer at the "Makromolekulares b, Part of the doctoral thesis of W. Lantschbauer, Universitat Wien, 1979. Kolloquium", Freiburg/Brsg., Germany, March 1 -3, 1979. 01 73-2803/82/12 0847-12/$01 .OO 848 0. F. Olaj, W. LantschbauerWith a second method6) (the so called "slithering snake" method), which has the features of a reptation process, a segment is removed from one end of the chain and attached to the other (d).Two serious difficulties arise when the simulation of more highly concentrated systems is attempted by these methods. First, it is impossible to exceed a certain limiting volume fraction 11 when making up the initial system of (even strongly biased) excluded volume pseudo-random flight chains (this volume fraction depends on chain length and is about 0,75 for chains covering 50 lattice points on a simple cubic lattice). Further, the relaxation mechanisms listed so far can operate successfully only when there are voids (l...
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.