DISTRIBUTION OF POLYMER-CHAIN LENGTHS INSIDE A SPHERICAL VOLUME - SOLUTION IN TERMS OF A DIFFUSION PROBLEM

The probability W(t) that a given number t of segments of an infinite chain lie within a given sphere can be expressed in terms of the single-pass length probability and the probability of reentrance into the sphere. The problem of calculating these two probabilities is equivalent to that of a diffusing particle exiting or entering the sphere after a given time, when the surface of the sphere is an absorbing wall. It is shown that the boundary condition, c equals 0, usually applied to an absorbing surface cannot be used for the present purpose. The boundary condition used instead is the so-called radiation condition alpha c/ alpha z equals kc; it is shown that when k approaches infinity the final answer for W(t), which is given in the form of an infinite series, approaches the correct limit.