MIGRATION MOLAR FLUX

In addition, the terms dCjdx, dP/dx, and dT/dx are absent in eq. (4.10) as contributing gradients to the total migration flux. Therefore, eq. (4.10) represents an approximation model However, dfi/dx strongly contributes to Jm and it can be defined as 

A more convenient approach to derive B is by combining eqs. (4.4) and (4.12) along with eq. (2.1c). Thus, 

Diffusion can be defined as the process in which the transfer of matter is due to random motion. The mathematics of diffusion can be found in Crank s book [23], in which different diffusion processes are analyzed. Let s assume an isotropic and homogeneous medium in which the diffusivity D is constant and the rate of transfer of matter (ions, atom or naolecules) is described by Pick s first law of diffusion. Despite that diffusion may be treated as a threeHlimensional process, it may be assumed that it occurs in isotropic media. 

In general, the conditions for steady-state dC/dt = 0) and for transient-state dC/dt 0) are governed by Pick s laws of diffusion 

Pick s laws are significant in measuring diffusivity in isotropic and anisotropic media. Confining a diffusion problem to a one-dimensional treatment suffices most approximations in diffusion. Hence, eqs. (4.16) and (4.17) can be sin Iified for diffusion flow along the x-direction in isotropic media 

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As for the migration terms, a link Is also made between the migration molar flux densities of the two Ions at all points, since the electric field Is the same for both ions ... 

The transient period also denotes a change in the distribution between the migration and diffusion processes. At steady state, an equal distribution has been achieved, as already demonstrated above based on the assumption that the Nernst-Einstein equation applies. Figure A.20 illustrates this phenomenon by showing the changing curves for the diffusion and migration molar flux densities ratio for Ag", throughout the electrolyte. 

The ferrous Fe+ ions are not included in the analysis since the crevice cathodic protection is for preventing the formation of this t5 e ion. For a coupled diffusion and migration molar flux under steady-state conditions, dCjIddt = 0, the molar flux becomes the continuity equation for mass transfer under steady state conditions... 

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

For the case of a binary electrolyte (i.e., a single salt that dissociates in solution into one cation and one anion species), we can rewrite the molar flux equations for positive and negative ions in terms of a salt concentration gradient diffusion term, a migration term explicit in the current density (as opposed to the VO driving force term in Equation (26.54)), and a bulk convection term ... 

A statement of the constitutive relation analogous to those for mass, heat, and momentum is that the flux due to migration in an electric field is proportional to the force acting on the particle multiplied by the particle concentration. The molar flux in stationary coordinates is then... 

In the approximation of an extremely diluted solution, the molar flux of i-th component is a sum of fluxes caused by migration, diffusion, and convection, and in view of the expressions (4.27) and (4.32), is equal to... 

The equation which presents molar flux densities as a function of the electrochemical potential implies that the proportionality coefficient for the driving forces is identical in the case of both migration and diffusion. 

As will be discussed later, this equation underlines the interdependence of the coefficients for the migration and diffusion terms. Bear in mind that just because there is a connection made between the proportionality factors in the migration and diffusion components it does not mean that the corresponding molar flux densities are collinear and in the same direction. There is a priori no connection between the respective forces (related to the gradients of electric potential or of activity). For example, one may come across cases where the migration and diffusion currents for a species i share the same direction, and other cases where the directions are opposite . ... 

Relative contributions of the migration and diffusion molar flux densities at steady state... 

In other words, since the Interfaclal molar fluxes are zero, then the migration and diffusion components at the Interface have opposite values different from zero. 

Each different molar flux density can be written as the sum of two terms (diffusion and migration, see section 4.2.1.4) ... 

Writing the 3 components of molar flux densities or of current densities EQI ii=-DiZi 9 grad E + 9 cOmedium, diffusion migration convection... 

Theories of mass transport in electrolytes or elec-trolyttic solutions take into account that motion of dissolved species / can be driven by gradients in electric potential O (migration), as well as by gradients in molar concentration c, (diffusion) and by motion of material at the bulk velocity v (convection). The most commonly deployed model for electrolyte transport is the Nemst-Planck theory [1], developed in detail by Levich [2]. Within this theory, one constituent of the solution - typically a neutral species in relative excess - is identified as a solvent . The total molar flux of any remaining solute species i, Ni, is then expressed relative to a stationary coordinate frame as... 

The model indicates that a plane of area dA containing specie j moves in the x-direction from position 1 to position 2, and then to position 3. This motion is influenced by modes of mass transfer, such as diffusion due to a molar concentration gradient, migration due to an electrical field, natural or forced convection due to the kinematic velocity or a combination of these modes, mass transfer of species j can be quantified by the absolute value of the molar flux J or the mass flux J. Notice that J is perpendicular to the moving plane of species j and represents the absolute value of the vector molar flux J. The total flux can be defined as... 

If mass transfer is aided by convection, which is the case for most practical purposes, then can be predicted from eq. (7.96). However, both diffusion, migration and convection mass transfer influence the electrolytic deposition in electrowinning, electrorefining and electroplating. In this case, the total molar flux is predicted by eq. (4.2) and the current density by eq. (4.9). 

The mathematics of oxidation kinetics involves diffusion and migration mass transfer. According to Pick s first law of diffusion, the molar flux is related to the rate of oxide thickness growth (dx/dt) is given by... 

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