Polymer random walk

Eyring,H., Ree.T., Hirai,N. The viscosity of high polymers — Random walk of a group of connected segments. Proc. Natl. Acad. Sci. 44,1213-1217 (1959). 

Let P(i, s) be the unnormalized probability that the sth step of a polymer random walk (s + 1 segments) ends at a lattice site in layer i. The proceeding step s — 1 must reside in one of the layers i — 1, i, or i + 1, suggesting the recurrence relation... 

In the limit that the number of effective particles along the polymer diverges but the contour length and chain dimensions are held constant, one obtains the Edwards model of a polymer solution [9, 30]. Polymers are represented by random walks that interact via zero-ranged binary interactions of strength v. The partition frmction of an isolated chain is given by... 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

As a consequence of these various possible conformations, the polymer chains exist as coils with spherical symmetry. Our eventual goal is to describe these three-dimensional structures, although some preliminary considerations must be taken up first. Accordingly, we begin by discussing a statistical exercise called a one-dimensional random walk. 

Next let us apply random walk statistics to three-dimensional chains. We begin by assuming isolated polymer molecules which consist of perfectly flexible chains. 

When we discussed random walk statistics in Chap. 1, we used n to represent the number of steps in the process and then identified this quantity as the number of repeat units in the polymer chain. We continue to reserve n as the symbol for the degree of polymerization, so the number of diffusion steps is represented by V in this section. 

Fig. 22.4. The random walk of o chain in a polymer melt, or in a solid, glassy polymer means that, on average, one end of the molecule is -yJn)A away from the other end. Very large strains (=4) are needed to straighten the molecule out.

Vincent analyzed the tensile fracture stress o, of a broad range of polymers as a function of the number of backbone bonds per cross sectional area ( 2) and found a nearly linear relation, o 2, as shown in Fig. 12. 2 is related to via the theory of entanglements for random walk chains as [74]... 

Polymer molecules in fabricated items rarely adopt a true random walk. Manufacturing processes stretch out molecules and then freeze them in an extended configuration before they have time to relax to the random state. Manufacturers exploit orientation in order to control physical properties. 

Name three factors that affect a polymer s random walk configuration. How does each factor affect molecular volume ... 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

The large-scale structure of polymer chains in a good solvent is that of a self-avoiding random walk (SAW), but in melts it is that of a random walk (RW).11 The large-scale structure of these mathematical models, however, is... 

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