Numerical Solution of SECM Diffusion Problems Using PDEase2 Program Package

NUMERICAL SOLUTION OF SECM DIFFUSION PROBLEMS USING PDEase2 PROGRAM PACKAGE 

The input hies for PDEase2 are regular text files, which can be created using any word processor. There is no case sensitivity, and spaces are not important. An input hie consists of a number of sections (e.g., Title, Select, Equations, and so on). Each section contains one or more statements. The following descriptor hie, disk.pde, contains the formulation required for PDEase2 to compute steady-state diffusion flux to an embedded microdisk and generate the graphical and numerical output. The statements in this hie are either self-explanatory or clarified by short comments written in curly brackets. 

The diffusion equation in disk.pde looks somewhat different from the corresponding Eq. (26). Nevertheless, these equations are completely equivalent, and eliminating the 1/R singularity makes the program work better. Symbol dR means the first derivative with respect to R, and dZZ represents the second derivative with respect to Z. Similarly, the boundary condition written in the form natural(C) = — M C R (where natural = inward diffusion flux X R and M is a sufficiently large number) is equivalent to Eq. (27a) but results in a significantly better computational efficiency. All variables in disk.pde are normalized according to Eqs. (4) to (6) and (14). The computed diffusion limited current is normalized by iTeo [Eq. (20)]. 

To simulate positive feedback situation one has to replace the boundary condition natural(C)=0 for the last segment of region 1 representing the substrate surface with natural(C)=M (l-C) R. Simulating positive feedback takes significantly less time, and larger errlim values can be used 

16 Simulation of the diffusion-controlled SECM process at an insulating substrate by PDEase2. (A) The final grid generated by the program consists of 8659 cells and 17588 nodes. (B) Distribution of concentration of redox species over R-axis. (C) Distribution of the flux over R-axis. For parameters see secm.pde file. 

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