Distance, digital simulations

The main process variables in differential contacting devices vary continuously with respect to distance. Dynamic simulations therefore involve variations with respect to both time and position. Thus two independent variables, time and position, are now involved. Although the basic principles remain the same, the mathematical formulation, for the dynamic system, now results in the form of partial differential equations. As most digital simulation languages permit the use of only one independent variable, the second independent variable, either time or distance is normally eliminated by the use of a finite-differencing procedure. In this chapter, the approach is based very largely on that of Franks (1967), and the distance coordinate is treated by finite differencing. 

For the purposes of considering diffusion at microelectrodes, it is convenient to introduce two categories of electrodes those to which diffusion occurs in a linear fashion and those to which diffusion occurs in a nonlinear fashion. The former category consists of cylindrical and spherical electrodes. As shown schematically in Figure 12.2A, the lines of flux (i.e., the pathway followed by material diffusing to the electrode) are straight, and the current density is the same at all points on the electrode. Thus, the diffusion problem is one-dimensional (i.e., distance from the electrode surface) and involves solution of the appropriate form of Fick s second law, Equation 12.7 or 12.8, either by Laplace transform methods or by digital simulation (Chap. 20). 

Fig. 6 Normalized probability density of nearest neighbor distances for 2.2 X 10 drops/cm of mercury, electrodeposited onto vitreous carbon at 220 mV from 0.01 mol dm aqueous solution (A) [58]. Also shown are the probability densities corresponding to uniformly distributed drops (thin line) and excluded nucleation for distances less than rj = [(87rcM//o) D(t — from each drop (heavy line), obtained from digital simulations [59].

However, it is not easy to calculate the exact relation between the probe current and the probe-sample distance in the feedback mode due to the geometric complexity in the experimental system. For quantitative analyses, digital simulation has been used in the feedback mode SECM both for conductor and insulator substrates [ 18]. Digital simulation is a powerful technique for analysis of relatively complicated electrochemical systems such as SECM. In the simulation, the space between the probe and the substrate are divided into small volimie elements. Mass transfer based on diffusion for each volume element is calculated. If one chooses suitable initial and boundary conditions, the experimental situation including the mass transfer and various chemical reactions can be simulated. Erom the simulation, one can determine the heterogeneous rate... 

In the past, Matsue et al. have discussed this issue in a study related to the characterization of diaphorase-pattemed surfaces by SECM. Using a digital simulation based on the explicit finite difference method that considered the heterogenous enzyme reaction at the substrate, they generated steady-state current vs. distance profiles that depended on the surface concentration of the enzyme. Using these working curves, they quantified the surface concentration of the active immobilized diaphorase (134). Using, the electrochemical... 

We introduce here the way in which we solve analogue (i.e. continuous) problems digitally, for diffusion processes in the bulk of the solution. That is to say, we digitally simulate diffusion in solution, given a boundary concentration at the electrode and one at some large distance from it. How large this must be will be discussed in this chapter also. The way in which we get Cq, the concentration at the electrode, is not a diffusion problem and will be dealt with in Chapt. 4. 

Wang and Sun (2001) developed another numerical method to simulate textile processes and to determine the micro-geometry of textile fabrics. They called it a digital-element model. It models yams by pin-connected digital-rod-element chains. As the element length approaches zero, the chain becomes fully flexible, imitating the physical behavior of the yams. The interactions of adjacent yarns are modeled by contact elements. If the distance between two nodes on different yarns approaches the yam diameter, contact occurs between them. The yarn microstructure inside the fabric is determined by process mechanics, such as yarn tension and interyam friction and compression. The textile process is modeled as a nonlinear solid mechanics problem with boundary displacement (or motion) conditions. This numerical approach was identified as digital-element simulation rather than as finite element simulation because of a special yam discretization process. With the conventional finite element method, the element preserves... 

Figure 2.17. It has been shown that track lengths between fineparticles in a random array are log-normally distributed, a) A random number table consists of the digits 0 through 9 chosen in random order to form a table, b) A simulated random field of view can be generated by turning every 9 in the random number table of (a) into a black square. The distance between black squares then generates various track lengths when examined with a series of parallel lines, c) Track length distribution generated firom the array of (b).

---