Geometrical properties of a Kuhnian polymer chain

We calculate the form factor H[q], i.e. the generating function of the gyration radii, and the generating function G [q] of the end-to-end radii, to first order in the dimensionless second viral coefficient g, or in 03B5 = 4 2014 d, d being the space dimension. These functions are calculated in the asymptotic limit of Kuhnian chains, i.e. very long chains in a good solvent. They are used to determine universal geometrical properties of the Kuhnian chain, like the average gyration radii RG[2n]of order 2 n and average 2 nth powers R [2n] of the end-to-end distance. The universal ratios RG[2n] /R [2n] are calculated and shown to have a regular g or 03B5expansion, even for large n. In this approach, one obtains the coefficients of the g expansion as general functions of the dimension d. This allows us to calculate the numerical values of the geometrical quantities either by a pure 03B5-expansion to first order, setting d = 4 in the coefficients, or, better, in the g-expansion by using the values of the coefficients at d = 3 and the best known value of the Kuhnian fixed point g*. The probability distribution for the internal distances between any two points of the Kuhnian chain is reconsidered, and its normalized universal form is given. In particular we get very simple forms for three limiting cases : 1) the two points form a finite segment inside a very long chain, or 2) a finite segment at the extremity of the very long chain, or 3) are the extremities of the whole chain. We finally compare the form factor H [q] to the experimental data obtained recently by Noda et al. It is shown that in the large q region, it is possible to take into account polydispersity effects and obtain apparently a good agreement between theory and experiments. J. Physique 47 (1986) 1633-1656 OCTOBRE 1986, Classification Physics Abstracts 05.20 05.40 64.70