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Although the linear randomflight chain is the basic model in the theory of dilute polymer solutions, it is unrealistic and incomplete from the point of view of the structural restrictions or the conformational statistics of polymer chains. Thus, in this section, we shall describe the statistical properties of those chains which are not placed in a category termed the linear random-flight chain; that is, chains with short-range interferences, branches, or high stiffness. 9a. Short-Range Interferences and Unperturbed Molecular Dimensions In order to make a random-flight chain model more realistic, account should be taken of the fact that the valence angle between successive bonds in the chain is actually a fixed quantity and the angle of rotation about each bond is not uniformly distributed owing to steric hindrances caused by interactions between atomic groups attached to the spine of the chain.

28) SEC. 5. Distribution of the End-to-End Distance and Related Quantities 15 It should be noted that Eq. 28) is valid over the whole range of R, 0 ≤ R ≤ na (full extension). The problem was also solved by Kuhn and Gr¨ un12 and by James and Guth13 in a different manner. Their procedure is based on the fact that the problem is essentially equivalent to that of finding the polarization of a gas due to the orientation of permanent magnetic or electric dipoles in an external field of sufficient strength to produce effects approaching complete orientation.

If λk are the eigenvalues of the matrix A with elements Akl and ψ j is the column vector of the ψjk with transpose ψ jT = (ψj1 · · · ψjn ), an orthogonal transformation Q which diagonalizes A gives n +∞ Γj (σx , ρ) = k=1 −∞ exp −λk ξk2 + iσx (ψ jT Q)k ξk dξk . 16) As shown below, the real parts of the λk are all positive, and hence the integral in Eq. 16) is convergent. Thus we find n Γj (σx , ρ) = π n/2 −1/2 λk k=1 exp − σx2 4 λk−1 (ψ jT Q)k2 . 17) k From Eqs. 20) l where the Qkl are the normalized components of Q.