Segment Distribution Function for Loops

For a loop in which the first segment starts at the interface (z = 0) and the i-th segment first returns to the surface, Hoeve43) derived the probability P(z, k i) of finding its k-th segment at the distance z by use of a method analogous to that employed in deriving the end-to-end distribution of free chains. His formula is 

This result is valid for z larger than 6, the mean thickness of train segments that are in contact with the surface. Eq. (B-25) indicates that gt(z) is a simple exponential function of z. 

For a system with Na adsorbed polymers, each having n segments, the volume fraction p2(z) of segments at z 6 is given by 

The root-mean-square distance (z2)122 of segments from the interface is calculated from 

Since -An changes little with n for small A, the root-mean-square thickness of segments (z2)112 is approximately proportional to n1 2. In Fig. 3, the dimensionless quantity ft z2)1/2 is plotted against n1/2 for c = 0.1, o = 1.0 and v2(0) = 0.04 and 0.00442). The calculated curves are almost linear. 

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