Mobile ion potential

Fig. 3.6 (a) Definition of bottleneck shortest distance and variation of mobile-ion potential with position with (solid line) and without (dotted line) relaxation of host structure, (h) Variation of mobile-ion potential for occupied sites separated by an array of inequivalent, empty sites, (c) Smoothing of the potential of (b) by introduction of mobile ions into second array of sites. 

In the more complex situations, (b) and (c) of Fig. 3.6, where the partially occupied sites are separated by inequivalent sites that are either empty or filled with mobile ions, fast ionic motion requires that the mobile-ion potential at the intervening sites be nearly the same as that at the partially occupied sites. (Of course, if the intervening sites are occupied by stationary ions, the ions in the partially occupied sites are immobilised.) From the constructions in Fig. 3.6, it is clear that the electrostatic forces between the mobile ions tend to smooth the potential where the inequivalent sites are filled with mobile ions whereas the local relaxation energy AH enhances the potential difference of the inequivalent sites where they are empty. 

The mobile ion potential depending on the concentration differences in both phases may be given as... 

The mobile ion potential results from the differences in the charge density between (solvent in the) gel and ambient solution. An increase of the mobile concentrations in the solution bath leads to a reduction of the eoneentration differences between gel and solution and thus to a decrease of the swelling. 

Another formulation of the chemical (mobile ion) potential is given in... 

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... 

In most cases of practically useful ionic conductors one may assume a very large concentration of mobile ionic defects. As a result, the chemical potential of the mobile ions may be regarded as being essentially constant within the material. Thus, any ionic transport in such a material must be predominantly due to the influence of an internal electrostatic potential gradient,... 

Equation (3.7) describes the equality of the chemical potentials of the mobile ions on both sides of the gel boundary expressed through the Donnan ratio KD and the ion charges z, Eq. (3.8) concerns the dissociation equilibrium of ionizable (carboxyl) groups of the network a is the degree of dissociation, eg is the concentration of the hydrogen ions in the gel Eq. (3.9) represents the gel electroneutrality condition. 

On the other hand, Doblhofer218 has pointed out that since conducting polymer films are solvated and contain mobile ions, the potential drop occurs primarily at the metal/polymer interface. As with a redox polymer, electrons move across the film because of concentration gradients of oxidized and reduced sites, and redox processes involving solution species occur as bimolecular reactions with polymer redox sites at the polymer/solution interface. This model was found to be consistent with data for the reduction and oxidation of a variety of species at poly(7V-methylpyrrole). This polymer has a relatively low maximum conductivity (10-6 - 10 5 S cm"1) and was only partially oxidized in the mediation experiments, which may explain why it behaved more like a redox polymer than a typical conducting polymer. 

A diffusion potential (p can develop in the membrane since in the case being considered, it contains two types of mobile ion. However, this potential is small. 

Thermochemistry. Chen et al.168 combined the Kohn-Sham formalism with finite difference calculations of the reaction field potential. The effect of mobile ions into on the reaction field potential Poisson-Boltzman equation. The authors used the DFT(B88/P86)/SCRF method to study solvation energies, dipole moments of solvated molecules, and absolute pKa values for a variety of small organic molecules. The list of molecules studied with this approach was subsequently extended182. A simplified version, where the reaction field was calculated only at the end of the SCF cycle, was applied to study redox potentials of several iron-sulphur clusters181. 

In solid-state electrodes the membrane is a solid disc of a relatively insoluble, crystalline material which shows a high specificity for a particular ion. The membrane permits movement of ions within the lattice structure of the crystal and those ions which disrupt the lattice structure the least are the most mobile. These usually have the smallest charge and diameter. Hence, only those ions that are very similar to the internal mobile ions can gain access to the membrane from the outside, a feature that gives crystal membranes their high specificity. When the electrode is immersed in the sample solution, an equilibrium is established between the mobile ions in the crystal and similar ions in the solution and the resulting potential created across the membrane can be measured in the usual manner. 

Freed of other restrictions, a mobile ion may be expected to diffuse down any concentration gradient that exists between porous solid and liquid. In the particular case of ion exchange, there is an additional requirement that the resin and liquid phases should remain electrically neutral. Any tendency for molecules to move in such a way as to disturb this neutrality will generate a large electrostatic potential opposing further movement, known as the Donnan potential. 

The occurrence of such ion trapping is clearly undesirable since it inevitably leads to a decrease in conductivity. In practice, in materials that contain potential traps such as charged aliovalent impurities/dopants, the conductivity values of a particular sample may actually decrease with time as the mobile ions gradually become trapped. Such ageing effects greatly limit the usefulness of a solid electrolyte in any device that needs to have a long working-life. 

It may be useful to think of the conduction pathway for a mobile ion as a series of potential wells and barriers. An example of a schematic energy profile is shown for a-AgI in Fig. 2.5(a), for sites connected in the sequence A-B-C-B -A, Fig. 2.4(b). Sites B are of somewhat lower energy than the A and C sites and form the preferred residences of Ag ions. However, the barriers for hopping between the different sites are all low. 

Where ions are disordered over a partially occupied array of energetically equivalent sites, their motion is diffusive. Fig. 3.6 illustrates the variation in ionic potential with intersite position for three possible situations in which a set of energetically equivalent sites are partially occupied by mobile ions. The partially occupied sites may (a) share common faces in a continuously connected network through the structure, (b) be separated from one another by an array of empty sites, or (c) be separated from one another by an array of sites that are occupied by the mobile ions. 

If we place an ionic conductor between parallel-plate blocking electrodes that produce an electric field E parallel to the x-axis, the electrostatic potential varies as — xE on passing from one electrode at x = 0 to the other. At equilibrium, the mobile-ion concentration Cj(x) is proportional to exp(qEx/kT), and the ionic drift-current density (7(E in the field is balanced by a diffusion current due to the concentration gradient (Fick s law) ... 

Since the polyelectrolytes contain only one type of mobile ion, the interpretation of conductivity data is greatly simplified. Polyelectrolytes have significant advantages for applications in electrochemical devices such as batteries. Unlike polymer-salt complexes, polyelectrolytes are not susceptible to the build up of a potentially resistive layer of high or low salt concentration at electrolyte-electrolyte interfaces during charging and discharging. Unfortunately flexible polyelectrolyte films suitable for use in devices have not yet been prepared. 

Figure 2.10 Scheme showingthe Donnan partition of mobile ions between the solution and a polymeric phase bearing an excess of negative charges. While positive ions are incorporated in the film to mantain electroneutrality, negative ions are excluded from it. This situation give rise to an interfacial potential (Donnan potential) at the interface. 

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