Migration Poisson equation

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... 

Transport, often restricted to diffusion (Pick s laws), but in principle also including convection (Navier-Stokes equation) and migration (Poisson-Boltzmann equation)... 

Under conditions relevant for fuel cell operation, the reaction current density of the ORR is small compared to separate flux contributions caused by proton diffusion and migration in Equation 3.62. Therefore, the electrochemical flux termNjj+ on the left-hand side of the Nernst-Planck equation in Equation 3.62 can be set to zero. In this limit, the PNP equations reduce to the Poisson-Boltzmann equation (PB equation). This approach allows solving for the potential distribution independently and isolating the electrostatic effects from the effects of oxygen transport. 

For the deduction of this expression, it has been assumed that Poisson s equation holds (i.e., V2 = 0). If the migration contribution can be suppressed (for example, due to the addition of supporting electrolyte), Eq. (1.172) simplifies to Fick s second law ... 

The bulk transport of ions in electrochemical systems without the contribution of advection is described by Poisson-Nernst-Planck (PNP) equations (Rubinstein, 1990).The well-known Nernst-Planck equation describes the processes of the process that drives the ions from regions of higher concentration to regions of lower concentration, and electromigration (also referred to as migration), the process that launches the ions in the direction of the electric field (Bard and Faulkner, 1980). Since the ions themselves contribute to the local electric potential, Poisson s equation that relates the electrostatic potential to local ion concentrations is solved simultaneously to describe this effect. The electroneutrality assumption simplifies the mathematical treatise of bulk transport in most electrochemical systems. Nevertheless, this no charge density accumulation assumption does not hold true at the interphase regions of the electric double layer between the solid and the Uquid, hence the cause of most electrokinetic phenomena in clay-electrolyte systems. 

The analysis of mass transport by diffusion under chemical (dCldx) and by migration under electrical (dd>ldx) gradients in dilute solutions—for which the interactions between individual species can be neglected—is described by the Nemst-Planck (Eq. 2.1) and the Poisson s (Eq. 2.2), together referred to as the PNP equations ... 

VOINOV You said the Debye length you calculate in the case of solid electrolyte is smaller than atomic dimension and consequently diffuse layer effects are irrelevant- Presumably, you use for this calculation the Debye length formula developed for the case of aqueous electrolytes. As you have stressed, in many solid electrolytes, only one species can migrate and in that case the solution of Poisson s equation is not the same as in the case of liquid electrolytes. As long as you do not have this solution and have shown it can be approximated by an exponential, and that this exponential is the same as in the case of liquid electrolytes, it seems to me difficult to calculate a Debye length in solid electrolytes. 

Finite-element simulations are useful to understand the mechanism of NDR and its dependence on the composition in the internal and external solutions, pore geometry, and nanopore surface charge density. Similar to modeling flow effects on nanopore ICR described earlier, the Nernst-Planck equation governing the diffusional, migrational, and convective fluxes of ions (Equation 2.18), the Navier-Stokes equation for low-Reynolds number flow engendered by the external pressure and electroosmosis (Equation 2.20), and Poisson s equation relating the ion distributions to the local electric field (Equation 2.19) were simultaneously solved to obtain local values of the fluid... 

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