Transport diffusion-migration

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. 

As discussed in Sect. 2, there are three modes of transport diffusion, migration, and convection. We first rule out convection, not because it is unimportant, but because convective transport is the subject of Chap. 5. 

IONIC TRANSPORT BY MIGRATION AND DIFFUSION 4.3.1 Equations for the Total Flux... 

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... 

We will assume for all of the techniques discussed in this chapter that the analyte solution is quiet (that is, still and unstirred) in order to ensure that mass transport by convection is absent. Furthermore, we will also assume that an excess of ionic electrolyte has been added to the solution to ensure that mass transport by migration is also absent. We see that the only form of mass transport remaining is diffusion, and hence the subtitle to this chapter. 

In the previous chapter, we discussed dynamic electroanalytical techniques such as polarography and voltammetry. Each technique in that chapter was similar insofar as the principal mode of mass transport was diffusion. Mass transport by migration was minimized by adding an inert ionic salt to the electroanalysis sample and convection was wholly eliminated by keeping the solution still ( quiescent ). ... 

Electrons liberated at the anode (negative pole of the cell) by the electro-oxidation of the fuel pass through the external circuit (producing electric energy equal to —AG) and arrive at the cathode (positive pole), tvhere they reduce oxygen (from air). Inside the fuel cell, the electric current is transported by migration and diffusion of the electrolyte ions (H, OH, CO ), for example, in a PEMFC. 

In transport by migration or diffusion, the solute particle moves through a stationary solvent. Convection is a totally different process in which the solution as a whole is transported. Solute species reach or leave the vicinity of the electrode by being entrained in a moving solution. 

Bath, D.B., et al. 2000. Scanning electrochemical microscopy of iontophoretic transport in hairless mouse skin. Analysis of the relative contribution of diffusion, migration, and electroosmosis to transport in hair follicles. J Pharm Sci 89 1537. 

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... 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

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. 

In contrast to all the other techniques considered in this paper, in sorption experiments molecular migration is observed under nonequilibrium sorption conditions. Therefore, instead of self-diffusivities, D, in this case transport diffusivities. A, are derived. It is generally assumed (see, e.g.. Refs. 366) that the corrected diffusivities. Do,... 

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 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. 

Basically, three mechanisms are responsible for mass transport inside an electrochemical cell diffusion, migration, and convection. Diffusion is mass transport because of concentration gradients, i.e., variations in the concentration of a species with position. Diffusion occurs mainly near the electrode surface because of gradients created by the consumption of species that undergo redox reactions and are incorporated into the deposit. This incorporation process depletes the deposition species near the electrode, generating the concentration gradient. 

A particle migration model was proposed by Gadala-Maria and Acrivos to describe experimental shear-induced migration observations. This model allows for a better understanding of the shear effects on particle diffusion for concentrated suspensions. Based on these studies, a conservation equation for the solid phase was established by Phillips, Amstrong, and Brown, which takes into account convective transport, diffusion due to particle-particle interactions, and the variation of viscosity within the suspension, namely ... 

Back, R.P. 1987. Diffusion-migration impedances for finite, one-dimensional transport in thin-layer and membrane cells Part II. Mixed conduction cases Os(in)/Os(II)ClO4 polymer membranes including steady-state IV responses. Journal of Electroanalytical Chemistry 219, 23 8. 

The developed theory was extended to the case of salt dissolution in solution containing no supporting electrolyte (43e). Although the theory describing mass transport via diffusion/migration in such systems is more complicated, it was possible to ht the experimental current-distance curves for AgCl dissolution and to demonstrate the applicability of the second-order rate law to this process. A more complete discussion of the theory of the SECM induced dissolution can be found in Chapter 12. 

Transport within a pore can result from diffusion, migration, convection, or some combination of these mechanisms. For a relatively large (>1 /xm ra-... 

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. 

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