Conformation of an Isolated Polymer Molecule at an Interface. II. Dependence on Molecular Weight

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Abstract
The average conformation of polymer molecules adsorbed from very dilute solution is derived from the partition function Zn pertaining to an isolated molecule with at least one of its segments adsorbed. It is an extension of the treatment in the previous paper in which the generating function Γ (ζ) = ΣZnζn was derived, and from it the partition function Zn, in the limit of very large n, was obtained by the method due to Lifson. In the present paper Zn for finite n is obtained from Γ (ζ) by the use of Cauchy's theorem, the contour integration being evaluated by the steepest‐descent method. The numerical result thus obtained agrees well, even for n=5, with the exact results evaluated by direct power series expansion of Γ (ζ) by a computer. The results show that the conformation of adsorbed polymers can be classified into three distinct types according to the values of the segmental adsorption constant s. When s is large, the molecules lie flat on the surface and various intensive quantities are independent of the chain length n. When s is near the critical point sc, the fraction of adsorbed segments is proportional to n−½, in agreement with the earlier result by Simha, Frisch, and Eirich, but most of the desorbed segments are in the long‐chain ends rather than in the internal loops. For very small s, the number of adsorbed segments becomes independent of n, and all the remaining segments are in the free‐chain ends, with a negligible probability of loop formation. The present results apply to the helix—coil transition of DNA with some modifications.