Root-mean-square length

Eq. (16 ) reverts to (8) for sufficiently small values of the ratio, r/r, of the displacement length to the maximum length. The bracketed quantity in the exponential of Eq. (16 ) may be looked upon as a correction factor on the value of which should have been used in the simpler Eq. (8). For r/vm, less than about one-half, this correction is negligible. The ratio of the root-mean-square length to the maximum length varies inversely as the square root of the maximum length, according to Eq. (13), i.e., / / m = from which it... 

The ratio of the root mean square lengths is called the chain expansion factor ... 

In any case, it can be demonstrated with the aid of the dumb-bell model that eq. (5.10) is a much better approximation for statistical coil molecules than eq. (5.11 a) for rigid rods. Two cases are considered for the purpose A rigid dumb-bell of fixed length hr and an elastic dumb-bell, according to the usual definition, possessing a root mean square length (Kyi. ... 

The characteristic ratio, Coo, is the square of the ratio of the actual root-mean-square length to the nx 2l ... 

What is the root-mean-square length of a molecule of polypropylene of MW = 5000 Compare this to the contour length of the molecule. 

Exponent in Mark-Houwink equation [77] = K M a = 0.5 signifies a rigid sphere or compact coil and a = 1.4 a rodlike or highly extended molecule. d (< 2>)1/2 end-to-end, root-mean-square length derived from the Flory-Fox equa-()3/2... 

Using the Monte Carlo method, for walks that can intercept, confirms this result. The root mean square length of the 100 step walks in Fig. 3.7 is 14.135. The theoretical distribution in the figure is derived in Section 3.4. It is the product of a 4irr term (the surface area of a sphere of radius r on which the chain end lies) and the Gaussian distribution of Eq. (3.13). 

Calculate the root-mean-square length of a polyethylene chain of M = 250000 g moP assuming that the equivalent random link corresponds to 18.5 C—C bonds. 

As shown in Appendix 9.A, from considering the polymer chain as a Gaussian chain consisting of No segments with the root mean square length b, the functional form for 5L t) is derived as... 

The time-correlation function 5L 0)5L t)) of Eq. (9.3) will be derived by considering the polymer chain as a Gaussian chain consisting of No segments each with the root mean square length b. Let 5 (t) be the contour position of the nth bead relative to a certain reference point on the primitive path. Then the contour length of the primitive chain at time t is given by... 

A distinction must be drawn between gels which are diluted first and then cross-linked and those which are cross-linked first and then swollen (Section F below). In the former, the network strands have their average random configurations in a more or less unstrained state (except perhaps for gels linked at quite high dilution, where the polymer coils overlap each others domains only to a limited extent). In the latter, however, the strands are all extended beyond their normal root-mean-square lengths, in proportion to the cube root of the swelling factor p/c = uj. ... 

The theory of cyclization dynamics was first presented by Wileaski and Fixman [WF] (5). A number of curious features of the theory prompted detailed attention by Doi (11), by Perico and Cuniberti (12), and by others (13). The theory is developed in terms of the bead-and-spring Rouse-Zimm [RZ] model (14). Unrealistic in detail, this model is quite useful for describing low frequency, large flmq[>litude chain motions. The RZ model, figure 2, treats the chain as a series of n beads connected by (n-1)harmonic springs of root-mean-squared length b. 

The only parameter, b, is the inverse of the most probable chain end separation and is under the above conditions Z3/n. The extended length of the chain is, of course, equal to nC, whereas the average chain end separation (the root-mean-square length /W equals /n. 

Fig. 6. Freely jointed chain of Kuhn segments and head/spring chain model. For coarse-grain treatments for long time and length scales, the polymer may be modeled by a chain of beads and massless entropic springs. The beads are assumed to have a hydrodynamic radius a. The Kuhn segment length, b, is defined as the root mean squared length of the springs. The root mean squared chain end-to-end distance is called the Flory radius Rp...

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