Proper Accounting of Conformations of a Polymer Near a Surface

It is well known that in free space the conformations of a freely jointed chain (n-mer) can be generated with proper equal a priori probabilities by means of a particle performing an n-step random walk. However, near a boundary the method of a random walk with reflecting barriers weights too heavily those paths that touch the boundary r times by a factor (w)r where w is greater than 1. The essence of a proper accounting is to place a completely absorbing site just beyond the boundary and to count only those chains that do not terminate on the absorbing site. For example, in the limit of large n where the diffusion equation becomes valid, the proper boundary condition is that of complete absorption at the boundary (concentration equals zero) rather than complete reflection (gradient of concentration equals zero) as has been assumed previously. In considering the problem of a polymer confined to a finite length strip of a one-dimensional lattice one ends up considering nonstochastic submatrices of the matrix of transition probabilities. Characteristic polymer dimensions near a surface are found to be larger than they would be away from the surface. The diffusion equation is then generalized so that it describes a polymer chain with self-excluded volume. Energy effects are also discussed.