Planar diffusion, finite

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

Correct experimental conditions and preliminary evaluations of the data are of vital importance. The effects of non-kinetic factors—charging of double layer, non-planarity and finiteness of diffusion—have to be eliminated. 

Transient Finite (Symmetric) Planar Diffusion In this section, we progress from transient infinite diffusion problems to transient finite diffusion problems. In many cases, the approaches and solutions to finite problems are quite similar to those just discussed in the context of infinite diffusion problems. Transient finite diffusion problems can often be solved using the separation-of-variables technique, which... 

Solution This is a transient ID finite (symmetric) planar diffusion problem. The initial and boundary conditions for this problem are ... 

Transient Finite (Symmetric) Spherical Diffusion So far, we have only examined ID (Cartesian) examples of Fick s second law. Solving Fick s second law in alternative coordinate systems (e.g., for radial, spherical, 2D, or 3D problems) is not really any different. As an example, we examine here the case of transient finite spherical diffusion, which is essentially analogous to the transient finite planar diffusion problem that we just finished discussing. 

Pick s second law is a second-order partial differential equation. Solving it in order to predict transient diffusion processes can be fairly straightforward or quite complex, depending on the specific situation. In this chapter, analytical solutions were discussed for a number of cases, including ID transient infinite and semi-infinite diffusion, ID transient finite planar diffusion, and transient spherical finite diffusion as summarized in Table 4.4. In all cases, solution of Pick s second law requires the specification of a number of boundary conditions and initial conditions. 

Problem 4.3. Equation 4.43 provides the solution for transient finite (symmetric) planar diffusion in a plate of thickness L starting from a uniform initial concentration of c° when the concentrations at the edges of the plate are set to c at time t = 0 ... 

The conditions for planar diffusion are theoretically fulfilled only if the electrode surface is very large. In case of finite disk electrodes, edge effects arise and linear diffusion is no longer linear overall the electrode surface (Fig. 15.3). Diffusion also develops parallel to the electrode surface in the radial direction. However, if the radius of the disk electrode is large enough with respect to the diffusion layer thickness (as is the case of common employed disk electrode of millimeter size), edge effects can be neglected and Cottrell equation accurately accounts for the current profile at the electrode surface. These electrodes are nowadays called either conventional or macroelectrodes. ... 

The DigiSim program enables the user to simulate cyclic voltanunetric responses for most of the common electrode geometries (planar, full and hemispherical, and full and hemicylindrical) and modes of diffusion (semiinfinite, finite and hydrodynamic diffusion), with or without inclusion of IR drop and double-layer charging. 

Transport by combined migration—diffusion in a finite planar geometry can achieve a true steady state when only two ions are present, as we saw in Sect. 4.2. The same holds true when there are three or more ions present. Under simplifying conditions [see eqn. (89) below], it is possible to predict the steady-state behaviour with arbitrary concentrations of many ions. However, the corresponding transient problem is much more difficult and we shall not attempt to derive the general transient relationship, as we were able to do in deriving eqn. (82) in the two-ion case. 

When dealing with currents in ionic solutes, one must take into account the finite diffusion of ions within the electrolyte. As mentioned in Section 6.21, Fick s83 second law of diffusion states that the time-dependence of the concentration profile in a one-dimensional planar system Co(x,t) depends linearly on the derivative of the concentration gradient ... 

To illustrate the advantages gained in considering lattice symmetries, consider a target molecule B (or trap) positioned at an arbitrary site on a finite, 5x5 square-planar lattice. Calculation of the mean walklength ( ) before reaction (trapping) of a coreactant A diffusing on this lattice, and subject to specific boundary conditions, requires the specification of the matrix P and subsequent inversion of the matrix [I — P], If the trap is anchored at the centrosymmetric site on the lattice and periodic boundary... 

The first of the conclusions that can be drawn from these calculations is that for finite planar lattices of integral or fractal dimension, in the absence of any external bias, the diffusion-reaction process is more efficient when the target molecule is localized at a site of higher valency. This conclusion is consistent with the trends noted in Section IIIA, and with the results reported in Section IIIB. Further, the conclusion is consistent with results obtained in a lattice-based study of reactivity at terraces, ledges, and kinks on a (structured) surface [28], There the reaction... 

Fig. 3. Representations of the diffusive fields at (a) a semi-infinite planar electrode, (b) a hemispherical electrode, and (c) a finite disc electrode.

In this chapter we have established a mathematical model that fully describes a cychc voltammetry experiment of a one-electron reversible process at a planar macrodisc electrode where the diffusion coefficients of both chemical species are equal. The model consists of a one-dimensional partial differential equation that describes the evolution of the concentration of some chemical species in both time and space starting from some initial conditions at time t = 0, and boimded by some finite spatial region 0 < X < Xmax- At X = 0 is the electrode boundary which alters the concentration in a manner that depends on the potential applied to it. At X = Xmax the concentration is imaffected by the processes occurring at the electrode and so is equal to that of the bulk solution. The potential at the electrode is varied and the resultant current is recorded and plotted as a voltammogram. 

In the previous chapter finite difference methods were introduced for one of the simplest situations from a theoretical point of view cyclic voltammetry of a reversible E mechanism (i.e., charge transfer without chemical complications) at planar electrodes and with equal diffusion coefficients for the electroactive species. However, electrochemical systems are typically more complex and some refinements must be introduced in the numerical methods for adequate modelling. 

If the planar thickness of each phase were of monomolecular order, equilibrium would be prompt, resulting in a single uniform phase. Since the cake particles are finite, diffusion of solvent into the oil in the cake particle and diffusion of oil from within the particle out into the solvent/oil solution requires time. After a limited time period the somewhat strengthened (oil rich) miscella at the particle surface is replaced by miscella richer in solvent which reelevates the diffusion rate. This is accomplished by staged counter-current movement of the cells of press cake and the solvent. Figure 3 is a... 

In practical applications, very often diffusion is not semi-infinite. Such finite-length linear diffusion is observed, for example, for internal diffusion into mercury film deposited on a planar electrode, in deposited conducting polymers, for hydrogen diffusion into thin films or membranes of Pd or other hydrogen absorbing materials, or for a rotating disk electrode where the diffusion layer corresponds to the layer thickness. There are two cases of finite-length diffusion displayed Fig. 4.11 ... 

Ions diffuse toward the center of particles, therefore their diffusion path is limited, e.g. the boundary condition is reflective. Impedance analysis of the finite length diffusion for different electrode geometries and boundary conditions is summarized by Jacobsen and West [1995]. Particle geometries occurring in battery materials are thin plate (planar), spherical, and cylindrical. Below are equations for corresponding geometries, modified so that parameters are expressed in electrical terms. 

Bieniasz LK (2012) Automatic simulation of electrochemical transients assuming finite diffusion space at planar interfaces, by the adaptive Huber method for Volterra integral equations. J Electroanal Chem 684 20-31... 

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