Microscopic derivation of zero-flux boundary condition in one-dimensional random walk in presence of a totally reflecting barrier

**Marius Orlowski ^{}^{, }^{} **

DDL Laboratories, APRDL, Motorola, 3501 Ed Bluestein Boulevard, Austin, TX 78739, USA

Available online 24 June 2003.

In this paper a rigorous probabilistic two-probability-parameter model of a diffusion barrier is investigated that describes comprehensively reflection, absorption, and segregation phenomena at a diffusion barrier. As a special case, a rigorous analysis of counting paths for 1D random walk in the presence of a reflecting barrier is presented. This paper defines and makes distinction between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. A combinatorial formula is derived showing that in the presence of a totally reflecting barrier (at *m*_{b}=0) the probability of a particle departing from position *m*=2*j* and arriving at position *m*=2*k* on the positive axis after *N*=2*M* steps is given by *W*_{2j}(2*k*,*N*) = [*C*(2*M*, *M*-*j*+*k*)-*C*(2*M*, *M*-*j*-*k*-*l*)]/[*C*(2*M*,*M*)+2 _{i=0}^{j} *C*(2*M*,*M*+*i*)], where *C*(*n*,*m*) denotes the binomial coefficient. This formula enables easy computation of any random walk redistribution of a diffusing species near or at the totally reflecting barrier. The analysis shows that for a particle starting its random walk at the barrier, the probability of finding it at the interface is diminishing with the number of diffusion steps *N*=2*M* as 1/(*M*+1) and that the peak of the probability distribution is moving away from the barrier with the increasing number of steps as . Thus, the subsurface region is progressively depleted. The present analysis has bearing on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.

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