Diffusion of charged species

There are many applications in chemical engineering where diffusion of charged species is involved. Examples include ion exchange, metals extraction, electrochemical reactors, and membrane separations. There is an excellent textbook in this area (Newman, 1991). Here we will be content to show that the treatment of electrolyte diffusion follows naturally from the generalized treatment of diffusion given in Section 2.3. 

Diffusion of charged species is by independent paths. In other words, it is assumed that the flux of species / is proportional to its electrochemical potential gradient solely and is independent of the gradient in the electrochemical potential of the other components. 

Electrochemical Impedance Spectroscopy (ElS) is a method used to characterize electron-transfer reactions by perturbing the system in a sinusoidal manner over a wide range of frequencies. This method, which is very sensitive to the properties of the electrode interface, provides information regarding electron-transfer kinetics, diffusion of charged species, charging/discharging, and system conductance. 

It should be stressed that Figure 7 represents a static picture. The lipid and protein molecules in the membrane are capable of rapid lateral movement, with the lipids having diffusion constants around 10 cm 7s. This indicates that in a time period of 1 /is a lipid molecule could diffuse over a distance of 2 nm, which would certainly have the effect of smearing out the detail presented in Figure 7, but only if the time period of interest is of 1 /IS or greater. The time for diffusion of charged species across the membrane is less than 1 /is for electrons and protons, and approaches this value for ions. Considerations of the discreteness of the surface charges are therefore of relevance for such processes. 

Relative rates of diffusion of charged species where diffusion in pure water = 1... 

The fundamentals of this deposition process rely on the diffusion of charged species in the solution toward the working electrode. At this electrode, the ECD process is governed by the Nernst equation ... 

For an ion to move through the lattice, there must be an empty equivalent vacancy or interstitial site available, and it must possess sufficient energy to overcome the potential barrier between the two sites. Ionic conductivity, or the transport of charge by mobile ions, is a diffusion and activated process. From Fick s Law, J = —D dn/dx), for diffusion of a species in a concentration gradient, the diffusion coefficient D is given by... 

The aqueous diffusivities of charged permeants are equivalent to those of uncharged species in a medium of sufficiently high ionic strength. The product DF(r/R) is the effective diffusion coefficient for the pore. It is implicit in k that adsorption of the cations does not occur, so that the fixed surface charges on the wall of the pore are not neutralized. Adsorption is more likely to occur with multivalent cations than with univalent ones. 

Note that Deff depends on rii and that its temperature dependence involves that of r. One can of course imagine many more complicated situations, in which the diffusion of the different species is more inextricably coupled or in which motion of charged species is important. 

Ambipolar diffusion involves the transport of charged species, and in such cases overall electric charge neutrality must be maintained during diffusion. Moreover, during ambipolar diffusion the difference in the mobilities of the diffusing species sets up a field, the Nernst field, that influences the rates of motion of the particles. 

As suggested before, the role of the interphasial double layer is insignificant in many transport processes that are involved with the supply of components from the bulk of the medium towards the biosurface. The thickness of the electric double layer is so small compared with that of the diffusion layer 8 that the very local deformation of the concentration profiles does not really alter the flux. Hence, in most analyses of diffusive mass transport one does not find any electric double layer terms. For the kinetics of the interphasial processes, this is completely different. Rate constants for chemical reactions or permeation steps are usually heavily dependent on the local conditions. Like in electrochemical processes, two elements are of great importance the local electric field which affects rates of transfer of charged species (the actual potential comes into play in the case of redox reactions), and the local activities... 

Principal differences between bulk media water and membrane water partition coefficients are listed in Table 2. These differences are essentially based on the several orders of magnitude difference in surface-to-volume ratio. In the liposomal system, charges built up due to sorption of charged species can be electrically neutralised by counter-ions from the electrolyte (diffuse double... 

Diffusion of the electroactive species within the electrode toward or away from the interface with the electrolyte is an irreversible process. The sum of the products of the forces and fluxes corresponds to the entropy production. In order to avoid space charge accumulation, the motion of at least two types of charged species has to be considered for charge compensation. Onsager s equations read in the isothermal case (neglecting energy fluxes)... 

Diffusion, of molecular species as well as colloidal particles, plays perhaps a more dominant role in many topics of interest to us. For example, without diffusion of ions we will not have the diffuse electrical double layers next to charged surfaces (discussed in Chapter 11). At the colloidal level, diffusion plays a central role in the transport and collision of particles in colloidal stability (discussed in Chapter 13). There are many more such examples. 

Integration of the stationary electro-diffusion equations in one dimension. The integration of the stationary Nernst-Planck equations (4.1.1) with the LEN condition (4.1.3), in one dimension, for a medium with N constant for an arbitrary number of charged species of arbitrary valencies was first carried out by Schlogl [5]. A detailed account of Schlogl s procedure may be found in [6]. In this section we adopt a somewhat different, simpler integration procedure. 

The flow of electrons from the semiconductor into the chemisorbed layer, and vice versa, without any diffusion of ionic species at the same time, induces a space charge between the interior of the semiconductor and its surface. This space charge will exist only for a small depth into the sohd near the surface. The physical and chemical behavior of this... 

A gradient in electrostatic potential can produce a driving force for the mass diffusion of a species, as discussed in Section 2.2.2. Two examples of this are the potential-gradient-induced diffusional transport of charged ions in ionic conductors such as those used in solid-electrolyte batteries and the electron-current-induced diffusion of interstitial atoms in metals. 

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