The Kuhn Segment Length

The factor Cw the Flory characteristic ratio of the actual end-to end distance to that predicted on the basis of a random flight model, obviously depends on chain stiffness or bond rotational freedom (Equation 8-11). 

FIGURE 8-37 Schematic disgram depicting a 2-dimensional random walk. 

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Table 1. Molecular-weight averages, monomer lengths (/) and estimated upper value of the Kuhn segment lengths (/k) of the poly[2]catenanes 48, 49, and 59, and the poly[2]catenands 51, 56.

Beyond that, we succeeded in extracting unperturbed dimensions of NaPA chains under both solvent conditions. Expressed in terms of the Kuhn segment length, a value of l = 4.2 nm could be established for both solvent conditions [54]. Having thus available a constant value for 4 for inert salt levels... 

The position of the secondary minimum of the total free energy plus the thickness of a platelet (1 nm) was considered as the equilibrium separation between layers. The surface charge density was assumed to be constant. The Kuhn segment length was evaluated using the expression... 

It does not affect the exponents in the equation (i.e.. the dependence of on N), however, but simply introduces a prefactor. This suggests a different approach, where we consider the number of adjacent bonds whose combinations of allowed rotations essentially behave like a freely jointed unit when taken collectively. We would then have Nx effective segments each of length lp known as the Kuhn segment length (Figure 8-36), defined in Equation 8-12 ... 

A modification of the lattice model to account for semi-rigid polymers replaces the contour length, L, in the axial ratio with the Kuhn segmental length (34,35) i.e., x is replaced with xk = 2q/d... 

Figure 11.18. Effects of crosslinking on the brittle fracture stress of styrene-divinylbenzene copolymers. Note the catastrophic embrittlement at very high crosslink densities (i.e., at an average number of repeat units between crosslinks less than the Kuhn segment length of 8 repeat units). The data point (not shown) for the uncrosslinked limit (polystyrene) is at (°°,41).

Sophisticated experimental methods allow the development of models for polymers in dilute and semi-dilute solutions. Chain stiffness may be represented by the Kuhn segment lengths and determined in dilute solution. Models for cellulose and cellulose derivatives have recently been published whose main features are the irreversible aggregation of chains, if hydrogen bonding is possible even in dilute solutions. Trisubstituted cellulose derivatives or cellulose in hydrogen bond breaking solvents exist as molecular dispersed chains. How-... 

EMM Ac with /k=24.8 nm [11]. In contrast, if commercially available methyl cellulose (DS = 1.6-1.7) is substituted to DS=3 with the appropriate phenylcarbamate groups (3C1, 4C1), these derivatives exhibit no mesophase in dioxane, in fact only gels are formed, although the Kuhn segment length is comparable with those of 3C1-CTC (in dioxane this is 49.5 nm) with 32.8 nm (3C1) and 26.8 nm (4C1) [11]. 

Flory, P. J., Statistical Mechanics of Chain Molecules, Hanser Publisher, New York (1989). To express the stiffness of a chain, the worm-like chain is a useful model, which is characterized by two parameters the persistence length Ip and the contour length L. In the limit of L/lp —> oo, the Kuhn segment length as defined by Eq. (1.32) is twice the persistence length. 

This equation states that the statistical segment length L increases with increasing values of rj. Thus a stiffer chain has a larger 17, a larger L and a smaller Z. The Kuhn segment length is thus adjusted to account for chain stiffness. On the other hand, L depends on temperature since freedom of rotation, and hence, chain stiffiiess, is temperature dependent. 

From now on, for the simplicity of presentation, length is measured in units of the Kuhn segment length />. Thus R, L, A, etc. appearing below in this chapter are all dimensionless, i.e., real length divided by b. As Q(L,u,X) we first consider G(R L,u,A), the normalized distribution function of the end-to-end distance R of a d-dimensional Edwards chain. It can be shown that Gb(R Lo,vo) is given to first order in uo by... 

For quantitative description of the segmental dynamics of the components in the miscible blends, both concentration fluctuation and self-concentration effects should be consistently incorporated in the model. This approach has already been examined by Kumar et al. (1996), Kamath et al. (1999), and Colby and Lipson (2005). In particular, the model by Colby and Lipson adopted the value ( 1 nm) comparable to the Kuhn segment length to reasonably describe the segmental relaxation time and mode distribution for both PI and PVE components in PI/PVE blends. 

The polymer chain formed by ng and bK is also called the equivalently freely jointed chain. Therefore, the Kuhn segment length bK is... 

The resulting expression [8] appears to be astonishingly simple the dependences on the contour length L and the Kuhn segment length I are dropped out from the final formulae. This indicates to the universality of the theory, that is, its independence of the particular stmcture of the subchains and their possible polydispersity. The reason for this is connected with the fact that entropic elasticity is caused by large-scale properties of polymer coils, rather than by short-scale details of the chain stmctrue. 

What is the Kuhn segment length of 1 x 10 g/mol polystyrene What does the result suggest about the chain conformation ... 

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