Diffusion migration equation

The concentrations of the reactants and reaction prodncts are determined in general by the solution of the transport diffusion-migration equations. If the ionic distribution is not disturbed by the electrochemical reaction, the problem simplifies and the concentrations can be found through equilibrium statistical mechanics. The main task of the microscopic theory of electrochemical reactions is the description of the mechanism of the elementary reaction act and calculation of the corresponding transition probabilities. 

The simulation model includes the coupled reaction-diffusion-migration equations for interstitial atoms, interstitial clusters and related defects. These equations have the general form for species i ... 

The conductivity of liquid and glass membranes is determined by ion-migration (absence of an excess of supporting electrolyte is assumed) in the diffusion layer. Equation (25) should then be written as ... 

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

The EOT flow has a flat profile, almost as a piston therefore, its contribution to dispersion of the migrating zones is small. The EOT positively contributes to the axial diffusion variance (Equation (31)), when it moves in the same direction as the analytes. However, when the capillary wall is not uniformly charged, local turbulence may occur and cause irreproducible dispersion. ... 

Equation (44) is the expression of the resolution for CE in electrophoretic terms. However, the application of this expression for the calculation of Rs in practice is limited due to D,. The availability of the diffusion coefficient of different compounds in different media is not always straightforward. The practical calculation of the resolution is frequently done with an expression that uses the width of the peaks obtained in an electropherogram. This way of working results in Rs values that are more realistic as all possible variances are considered (not only longitudinal diffusion in Equation (44)). Assuming Gaussian distribution for the migrating zones, the resolution can be expressed as follows ... 

A specific feature of the black sphere model is trivial functional discrimination of terms entering the kinetic equations depending if they are related on the defect production or on the spatial correlations. The r.h.s. of equations (7.1.50) to (7.1.52) describe decay of newly-created defect if they find themselves in the recombination volumes of dissimilar defects. If this is not the case, newly created defects can further disappear during diffusive migration. The latter problem was already considered in [14] (see equations (2.1) to (2.3) therein). 

A common experimental situation is the electrolysis of a neutral species to form an ion. The one-electron oxidation of ferrocene to ferricenium is an example of such a process. Since the current arises from the flux of a neutral to the electrode, the diffusion-limited current is unaffected by lowering the electrolyte concentration (Fig. 12.5, left panel). However, evaluation of the diffu-sion/migration equations shows that the charged product of the electrochemical reaction is removed from the diffusion layer by ion migration while inert ions of opposite charge are drawn in by the process of migration, again to maintain electroneutrality [68]. 

In several studies by Kaminskii and his co-workers [130 133], the time dependences of heterogeneous processes involving reactants migrating over the surface was described by the diffusion kinetics equations but the par-... 

In order to use the migration equations, especially the generally accepted equation (7-51), values for the partition coefficient K of the migrant between P and L and the diffusion coefficient DP of the migrant in P are needed. For migrants with a high solubility in the foodstuff or simulant, the value K = 1 can be used and a worst case estimation is obtained in this way. 

GPPS = general purpose polystyrene HIPS = high impact polystyrene a) 1 1 HIPS GPPS diffusion coefficient equation In D = 15.61 - 14 500- b) apparent diffusion coefficient calculated from experimental migration data. (1 /T(K))... 

In this chapter, we present most of the equations that apply to the systems and processes to be dealt with later. Most of these are expressed as equations of concentration dynamics, that is, concentration of one or more solution species as a function of time, as well as other variables, in the form of differential equations. Fundamentally, these are transport (diffusion-, convection-and migration-) equations but may be complicated by chemical processes occurring heterogeneously (i.e. at the electrode surface - electrochemical reaction) or homogeneously (in the solution bulk chemical reaction). The transport components are all included in the general Nernst-Planck equation (see also Bard and Faulkner 2001) for the flux Jj of species j... 

Mass transport processes - diffusion, migration, and - convection are the three possible mass transport processes accompanying an - electrode reaction. Diffusion should always be considered because, as the reagent is consumed or the product is formed at the electrode, concentration gradients between the vicinity of the electrode and the bulk solution arise, which will induce diffusion processes. Reactant species move in the direction of the electrode surface and product molecules leave the interfacial region (- interface, -> interphase) [i-v]. The - Nernst-Planck equation provides a general description of the mass transport processes. Mass transport is frequently called mass transfer however, it is better to reserve that term for the case that mass is transferred from one phase to another phase. 

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. 

The calculation of the transfer processes in the near-electrode diffusion layer is based on the set of equations of anisothermic ionic mass transfer, which is caused by diffusion, migration, convection, and homogeneous chemical reactions ... 

Fick s Second Law shows that the rate of extraction increases with decreasing particle size. If one remembers that the rate-controlling step for extraction is the migration of the solute through the pores of the particles to the particle surface, a smaller particle will have a shorter path for the solute to reach the surface. A shorter diffusion path equates to a faster extraction rate. 

The Importance of Concentration Polarization As noted earlier, concentration polarization occurs when the effects of diffusion, migration, and convection are insufficient to transport a reactant to or from an electrode surface at a rate that produces a current of the magnitude given by Equation 22-2. Concentration polarization requires applied potentials that are larger than calculated from Equation 22-2 to maintain a given current in an electrolytic cell (see Figure 22-2). Similarly, the phenomenon causes a galvanic cell potential to be smaller than the value predicted on the basis of the theoretical potential and the IR drop. 

Equation (8) indicates that the current measured at the SECM tip, it(z = 0), is directly proportional to ft. At steady state, mass continuity requires that the rate of transport from the pore into the receptor compartment be equal to the rate of transport within the pore. Thus, it(z = 0) is also proportional to the rate of transport at any point within the pore. The molecular flux in the pore, N, is obtained by simply dividing ft by the cross-sectional area of the pore. Note that in deriving Eq. (8), no restrictions have been placed on the mechanism of transport within the pore. Thus, it(z = 0) is proportional to the flux in the pore, independent of whether the flux is due to diffusion, migration, or convection. It can be shown that the tip current at any arbitrary separation distance, z, is also proportional to the flux in the pore. 

The molar flux (N,) equation for charged ions in solution, with diffusion, migration, and convection terms,... 

Frumkin was the first to point out that while examining wetting one must account for the formation of the adsorption layer or thin film present in equilibrium with the macroscopic liquid phase. Such a layer may be formed either by the transfer of substance from a liquid phase via vapor, or due to the diffusion (migration) of molecules of a liquid along the solid surface. In the latter case Young s equation should be written as... 

In this section, we discuss the general partial differential equations governing mass transfer these will be used frequently in subsequent chapters for the derivation of equations appropriate to different electrochemical techniques. As discussed in Section 1.4, mass transfer in solution occurs by diffusion, migration, and convection. Diffusion and migration result from a gradient in electrochemical potential, Ji, Convection results from an imbalance of forces on the solution. 

As can be concluded from Eqs. (7.14) and (7.21), the diffusion-migration problem is non-linear. The Newton-Raphson method has been applied successfully to the resolution of the Nernst-Planck-Poisson equation system although the convergence is slower than for the kinetic-diffusion problems studied in Chapter 6. Thus, the unknown vector x corresponds to... 

C. Maximum Current Density. When a direct current is apphed to a two-compartment cell divided by an ion-exchange membrane, as illustrated in Fig. 4.8.23, ionic species may flow across the membrane. Equation (13) gives the flux of species i. The three terms on the right-hand side of this equation are the components associated with diffusion, migration, and convection, respectively. 

The mass transport of anions across a membrane [95] is governed by diffusion, migration, and electroosmotic water convection and can be described using the Nemst-Planck equation. 

Equations (20 and 21) describe ionic transport as a combination of electrolyte diffusion and ionic conduction, and hence the name diffusion-conduction flux equation is suggested. Its comparison with the diffusion-migration flux equation, Eq. (7), evidences the difference between migration and conduction. Migration is due to electric fields, either external or internal, and does not require a nonzero current density. Conduction is the ionic motion associated to the part of the electric field that is controlled externally. Note that the electric current density is a measurable magnitude but not the local electric field. Conduction requires a nonzero electric current density. 

Here, is a constant with unit of current density and D wx) is an effective dimensionless diffusion coefficient. The migration equation reads... 

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