Small Gaussian chain

For small deviations from equilibrium, we average r — rm 1 by the statistical segment distribution. For a Gaussian chain which is realized under... 

Thus all seems perfect. We have constructed an RG mapping, wliich indeed shows a fixed point. However, the expression (8.32) for / is not satisfactory. It must be independent of A, otherwise dilatation by A2 does not lead to the same result as repeated dilatation by A. Now Eq. (8.32) is only approximate since in Eq. (8,31) we omitted terms O 0 2. This is justified only if 0 is small. We thus need a parameter which allows us to make. If arbitrarily small, irrespective of A. Only e — 4 — d can take this role. In all our results the dimension of the system occurs oidy in the form of explicit factors of d or It thus can be used formally as a continuous parameter. To make our expansion a consistent theory, we have to introduce the formal trick of expanding in powers of e — 4 — d. 3 vanishes for = 0, consistent with the observation (see Chap, fi) that the excluded volume is negligible above d = 4, not changing the Gaussian chain behavior qualitatively. For e > 0 Eq. (8.32) to first order in yields... 

For small overlap a virial expansion holds. For large overlap screening suggests that Ry(n, c j is proportional to n (Gaussian chain behavior). Thus... 

For a network of uniform length chains. Lake and Thomas substituted for L with an Equation predicted from rubber elasticity theory. They also derived an alternate expression for L for a network of random Gaussian chains. The two expressions differ only by a small numerical constant. Making either substitution, and rearranging terms, it can be shown that... 

Note that. In a sense, the polyelectrolyte behaves now as If it were an "ideal" (Gaussian) chain with a relatively small number of Kuhn lengths. This coil size Is determined by the local stiffness, but not by long-range excluded volume. Should q Increase even furher (approaching L), then the wormlike chain would be better described by a slightly curved rod (as expressed by (5.2.21)) than by a random-flight chain. 

In order to simplify the theory further it is usual to assume that, for all chains that need to be considered, the actual end-to-end distance is very much less than the fully extended length, i.e. r nl, which becomes true for all chains when n is sufficiently large. A chain for which the assumption is valid is called a Gaussian chain. Consider such a chain with one end fixed at the origin and let the other end, P, be free to move (see fig. 3.7). With OP nl it can then be shown that the probability p x, y, z) that P lies in the small element of volume dx dy dz at (x, y, z) is... 

Calculate the ratio of the probability of the end-to-end separation of a Gaussian chain being within a small range dr near (a) 2r s and (b) 3rnns to that for it being within a small range of the same size near rms-... 

Kloczkowski, A. Mark, J. E. Erman, B., Fluctuations, Correlations and Small Angle Neutron Scattering from Endlinked Gaussian Chains in Regular Bimodal Networks. Macromolecules 1991, 24, 3266-3275. 

First, we know that polymer chains show the properties of a Gaussian chain in the narrow region near the theta temperature where the excluded volume parameter z is sufficiently small. We have... 

Region Ifo appears for semifiexibie chains (p > 1) only at p > 1, the overlap concentration c for Gaussian chains is small as a result, the individual chain conformation remains unperturbed by interactions with other drains in spite of significant overlap in the region IIq. 

The static stmcture factor of a Gaussian chain simplifies in the limit of small 4 < Rj as... 

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