5-13. Partitioning of Polymers between Pores and Bulk Solution One of the simplest models for the configuration of a linear, flexible polymer is a random: flight chain. In this model the molecule is assumed to be a freely jointed chain consisting of N mass points connected by rectilinear segments of length . The position of each successive mass point is selected at random, with reference only to the position of the preceding mass point. Accordingly, te shape of the chain is described by a random walk. The analogy between the resulting set of polymer configurations and the random walks of diffusing molecules allows the probability of certain chan configurations to be calculated using an equation like that used to describe transient diffusion. Certain configurations of long chains are unable to fit in small pores. This is reflected in a partition coefficient K that is less than unity, where K is the equilibrium ratio of the polymer concentration in the pores to that in bulk solution. For a random-flight chain and cylindrical pores the partition coeff- cient is given by K=201P(r,1)rdr once all chain configurations that intersect the pore wall have been excluded. The probability Figure P5 - 14. Steady conduction in a cylin cal fiber of material B that is surrounded by al fiber A terial A. Far from the fiber, there is an imposed temperature gradient in the x direction. A distribution P(r,t) is governed by tP=r2r(rrP),P(r,0)=1,rP(0,t)(r)0=P(1,t)=0A.x=p where is the radius of gyration of the polymer divided by the pore radius. In this model, which assumes that N is large, the radius of gyration is given by (N2/6)1/2. The variable t is position along the chain, expressed as a fraction of the total contour length (i.e., 0t1 ). Show that K()=4i=1i2exp(i22) where i are the roots of the Bessel function J0. This result was derived originally by Casassa (1967)