Freely-jointed chain, equivalent

Kuhn has shown how a real polymer chain may be approximated by an equivalent freely jointed chain. Instead of taking the individual bonds as statistical elements, one may for this purpose choose sequences of m bonds each. In Fig. 79, arbitrarily chosen statistical elements consisting of five bonds are indicated, the displacement vectors for these elements being shown by the dashed lines. The direction assumed by a statistical element will be nearly independent of the direction of the preceding element, provided the number m of bonds per... 

The introduction of vectors of constant displacement length to represent the individual elements, which actually vary in length, is rendered more plausible by inquiry into the effect of incorporating this artifice in the treatment of the freely jointed chain. In this case V = m H. Upon substitution of this expression together with n nlm in Eq. (17), the previous expression for / , Eq. (6), is recovered. Hence the calculated distribution is unaff ected by an arbitrary subdivision of the chain in this manner. We conclude that the value chosen for m in the reduction of the real chain to an equivalent freely jointed chain likewise is inconsequential (within the limits on m stated above). 

The corresponding relation for a real linear chain may be expressed in terms of the equivalent freely jointed chain comprising n bonds each of length V as follows ... 

Inasmuch as any real chain may be replaced by an equivalent freely jointed chain, provided the chain is sufficiently long, it is obvious that Eq. (14) is general. 

The real polymer chain may be usefully approximated for some purposes by an equivalent freely jointed chain. It is obviously possible to find a randomly jointed model which will have the same end-to-end distance as a real macromolecule with given molecular weight. In fact, there will be an infinite number of such equivalent chains. There is, however, only one equivalent random chain which will lii this requirement and the additional stipulation that the real and phantom chains also have the same contour length. 

The values of the Kuhn length b and corresponding molar mass of a Kuhn monomer Mq for various polymers are listed in Table 2.1. Throughout this book, we will use the equivalent freely jointed chain to describe all flexible polymers and will call N the degree of polymerization or number of monomers (short for Kuhn monomers) and call b the monomer length (instead of the Kuhn monomer length) and... 

The previous problem showed that the equivalent freely jointed chain follows random walk statistics even if the effective monomer is renormalized to be larger than b. What is the smallest effective monomer size for which this renormalization works ... 

Problem 2.8 A real polymer chain consisting of n bonds each of length I may be usefully represented by an equivalent freely jointed chain of N links each of length b such that it will have the same end-to-end distance and the same contour length. Obtain N and b in terms of the characteristic ratio Coo of ths polymer chain. 

From Eq. (P2.8.3) we obtain the length of each link of an equivalent freely jointed chain as... 

The equivalent freely jointed chain It will be seen below that the conformation of a particular bond in a polymer chain is determined inter alia by the conformation of the preceding and subsequent bonds in the chain. These bond correlations expand the dimensions of the chain to greater than those predicted by the freely jointed chain. Nonetheless, it is permissible in the limit of high molecular weight to represent the polymer conformation by an equivalent freely jointed chain. 

Fig. 4.2. Schematic representation of a section of a polymer chain and its representation by ai equivalent freely jointed chain.

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

The Kuhn length is the effective monomer size for the equivalent freely jointed chain (A Kuhn monomers of length b instead of n backbone bonds of length /)... 

As follows from eqn [4], the characteristic dimensions of the chain scale proportionally to the square root of the number of monomer units N irrespective of the correlations in orientation of consecutive segments. This enables an equivalent freely jointed chain of the same contour length L to be introduced but with renormalized number of the effeaive statistical segments of length I tl. 

Comparison of eqns [10] and [ 11 ] to eqns [2] and [3] points out that long wormlike chain acquires on the large scale the random coil conformation and its large-scale properties coincide with those of an equivalent freely jointed chain comprising Ni = L/2lp statistical segments each of length t = 2lp. Hence, both the freely jointed and the wormlike persistence chain models can be applied for desaiption of large-scale conformational properties of flexible and semiflexible chain polymers. 

These arguments suggest a strategy for treating real chains. The idea is to represent a real chain by an equivalent freely jointed chain. The equivalent chain has virtual bonds, or Kuhn segments, each of which represents more than one real chemical bond. The number of virtual bonds Nk and the length of each bond bK are determined by two requirements. First, the Kuhn model chain must have the same value of (r ) as the real chain, but it is freely jointed so its characteristic ratio equals one,... 

Up to this point we have discussed only the effects of the chemical structure—that is, bond angles, bond lengths, and steric factors—on the size of a polymer molecule in a solution or melt. These are the factors that go into r. It is also possible to describe the same factors in terms of an equivalent freely jointed chain. By this we mean the following. Suppose we write... 

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. 

Yet another length parameter that will be useful is the Kuhn length, b. Kuhn [6] imagined an equivalent freely-jointed chain that has the same extended length I as the actual molecule. Thus, if the equivalent chain has segments of length... 

Consider the polyethylene chain of Example 10.1. Determine the values n and V (in terms of n and /) of an equivalent freely jointed chain so that the... 

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