One-dimensional mass transport

The UMEs used in bioarrays can be divided into three types disk, ring, and strip electrodes. The theory of the disk, ring, and strip UMEs has been extensively studied [97-100], Due to the edge effect, the profile of the mass diffusion to the ultramicroelectrode surface is three dimensional, and can significantly enhance the mass transportation in comparison to the conventional large electrode with one-dimensional mass transportation. The steady-state measurement at a planar UME can be expressed as... 

Convection refers to bulk directional (instead of random) motion of a fluid (see Chapter 3). In the presence of convection, a one-dimensional mass transport (including both diffusion and convection) equation can be obtained by adding a convective term to the diffusion equation ... 

The one-dimensional mass transport equation for plug flow with dispersion, and a... 

In the following, we will analyze the fluxes and transport at the moving boundary b. For inhomogeneous single phase solids, the one dimensional mass transport balance in the z-direction (without a reaction term) reads... 

The component fluxes N entering into Eqs. (A1)-(A3) are determined based on the mass transport in the film region. Because the key assumptions of the film model result in the one-dimensional mass transport normal to the interface, the differential component balance equations including simultaneous mass transfer and reaction in the film are as follows ... 

The flux, Jo(x, t), is defined as the transport of O per unit area (mol s cm ). It can be divided into three components, diffusion, migration, and convection, as originally expressed in the Nernst-Planck equation, written for one-dimensional mass transport along the x-axis in Eq. 18. 

In other words, the law describes the time development in the concentration profiles. Note that Pick s two laws have been written in forms that apply to one-dimensional mass transport processes, but the extension of the equations to three dimensions is straightforward. 

The one-dimensional mass transport equation of Off includes two additional reaction terms, where denotes the concentration of Off" participating in water autoionization reaction, and represents the part that is consumed in reaction with lead. Equation 2.29 presents lead hydroxide precipitation/dissolution reaction in aqueous phase, pff is the major factor that influences the reaction direction and production rate. The equilibrium requires 2 mol of Off- per 1 mol of Pb" to generate 1 mol of Pb(Off)2. This means that the change in hydroxyl molar concentration due to the precipitation reaction is twice as the change in Pb+ s molar concentration. 

The classical approach to the oxidation of metals has been as a one-dimensional mass transport problem, in which the transport is across a growing, planar film of the product and is driven by the free energy difference between the anodic and cathodic processes occurring at the metal-oxide and oxide-atmosphere interfaces, respectively. 

Mass transport to an electrode can be described by the Nernst-Planck equation. For one-dimensional mass transport along the x-axis, it can be written as... 

As we have seen earlier, the mass transport in the majority of simple electrochemical systems approximates to one-dimensional mass transport to a semiinfinite planar electrode, and it is the simulation of such systems that will be discussed here. The extension to cylindrical and spherical geometries is fairly straightforward, but systems such as the finite disc, or the ring disc electrode, where two-dimensional mass transport has to be considered, are more difficult to handle. 

Fig. A.6 - Schematic diagram of the space grid used in the simulation of systems involving one dimensional mass transport to a semi-infinite planar electrode.

A (hemi)spherical electrode has been applied to estimation of charge transfer kinetics, effect of electric migration, and kinetics of chemical steps. Spherical diffusion takes place through an increasing area as the radial coordinate r, measured from the center of the spherical electrode, increases. The main differential equation for three-dimensional mass transport is given in Table 3.1. On substitution of c=y r, this relationship can be reduced to the one-dimensional mass transport yielding... 

Electrochemical experiments in the laboratory are generally designed so that they may be described adequately by a one-dimensional mass transport model. The key dimension is always that perpendicular to the electrode surface and is here designated the x-dimension. 

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