Kuhn Chain Model

As seen above, the key result of the freely jointed chain statistics and incorporation of local conformational details such as restricted bond angles and dihedral angles is that R ) is always proportional to the chain length, that is, the size exponent v = 1/2, as long as the excluded volume effect is absent. The only 

This conclusion therefore suggests that we could build an equivalent freely jointed chain, without compromising the Afo-dependence of the polymer size given by Equation 2.20, by defining an effective bond length, which is renormalized by incorporating the consequences of Coo. Such a model is the Kuhn chain model and is the basis of all coarse-grained models of polymer chains. 

The Kuhn parameters N and are related to the number of actual skeletal bonds and bond length by requiring that both Kuhn and real chains have the same chain length L at full extension. For the Kuhn chain, L = Nt since the model chain is freely jointed. For the real chain of the type discussed in Section 2.1, L = Noacos(6/2), where (180 - 9) is the bond angle. Therefore, 

Also the mean square end-to-end distance for the Kuhn chain (being a freely jointed chain) is 

Some Examples of Relationship between Kuhn Length and Bond Length 

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Figure 2.9 Kuhn chain model a real chain conformation of No bonds with bond length a is mapped into a freely jointed chain of N steps, each of length .

In the rest of the book, we shall use N to denote interchangeably the number of Kuhn segments, the degree of polymerization of the chain, and chain length. By using the same symbol R to denote the various measures of the polymer radius, one of the main results of the Kuhn chain model is the universal law. 

Cyclisation of long-chain molecules is a field where theory has far preceded experiment. In his pioneering treatment of flexible chains in terms of the freely-jointed chain model, Kuhn (1934) derived for the local concentration Ceff of one chain end in the neighbourhood of the other (see p. 7) expression (56) where Aa is Avogadro s number and Ceff is given in moles per... 

This assumption is equivalent to considering the polymer molecule as a Gaussian chain. For a Gaussian chain the probability of the two ends colliding in three-dimensional space is proportional to its length to the power -3/2. For the Kuhn (or freely-jointed chain) model the same assumption maybe taken for sufficiently long chains [60]. For linear polymers in good solvents, no similar simple assumption can be adopted. To study cyclization one has to resort to more sophisticated mathematical treatments (see, e.g. [61]). 

In the diaryl-substituted polysilane, the global dimensions are approximated by a wormlike chain model with a persistence length of 100 A. Thus, for the highest molecular weight studied, LIq — 100, and the chain dimensions are similar to those of a flexible-model Kuhn chain with bond lengths IkOf —200 A, that is,... 

The first model (Kuhn chain) is built up by planar segments of limited conjugation length which are separated by defects, e.g. cis double bonds. The second concept of a worm-like chain (Porod-Kratky chain) visualizes a continuous... 

The proportionality constant C is the stress-optic coefficient. From the freely jointed chain model, one can derive its value (Kuhn and Griin 1942 Treloar 1975) ... 

However, the freely jointed Kuhn chain is the most idealised model for a real macromolecule therefore, there exist several corrections taking into account specific features of the polymer structure. Thus, for chains in which LJA<10, condition (11.30) no longer holds. The modified condition is written as follows [94] ... 

The model used to describe these results was the freely jointed chain model of the polymer molecule, originally due to Kuhn and Gruen, which treats the molecule as a chain of independent segments of length Z. The force of retraction is then entropic and given by the Langevin function... 

The freely jointed chain model (known also as random flight model) was proposed for polymers by Kuhn in 1936. The chain is assumed to consist of n bonds of equal length I, jointed in linear succession, where the directions 6, cf>) of bond vectors may assume all values (0 tt ... 

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