Diffusion, ambipolar

In the discussion so far, the diffusional and electrical fluxes of the ionic and electronic carriers were treated separately. However, as will become amply clear in this section and was briefly touched upon in Sec. 5.6, in the absence of an external circuit such as the one shown in Fig. 7.7, the diffusion of a charged species by itself is very rapidly halted by the electric field it creates and thus cannot lead to steady-state conditions. For steady state, the fluxes of the diffusing species have to be coupled such that electroneutrality is maintained. Hence, in most situations of great practical importance such as creep, sintering, oxidation of metals, efficiency of fuel cells, and solid-state sensors, to name a few, it is the coupled diffusion, or ambipolar diffusion, of two fluxes that is critical. To illustrate, four phenomena that are reasonably well understood and that are related to this coupled diffusion are discussed in some detail in the next subsections. The first deals with the oxidation of metals, the second with ambipolar diffusion in general in a binary oxide, the third with the interdiffusion of two ionic compounds to form a solid solution. The last subsection explores the conditions for which a solid can be used as a potentiometric sensor. 

To best illustrate the notion of ambipolar diffusion, the oxidation of metals will be used as an example following the elegant treatment first developed by 

Wagner. Another reason to go into this model is to appreciate that it is usually the electrochemical potential, rather than the chemical or electric potential, that is responsible for the mobility of charged species in solids. It also allows a link to be made between the notions of chemical stability 

Oxidation rates can be measured by a variety of methods. One of the simplest is to expose the material for which the oxidation resistance is to be measured (typically metal foils) to an oxidizing atmosphere of a given oxygen partial pressure for a given time, cool, and measure the thickness of the oxide layer that forms as a function of time. Long before any atomistic models were put forth, it was empirically fairly well established that for many metals the oxidation rate was parabolic. In other words, the increase in thickness Ax of the oxide layer was related to time by 

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The movement of the fast electrons leads to the fonnation of a space-charge field that impedes the motion of the electrons and increases the velocity of the ions (ambipolar diffusion). The ambipolar diffusion of positive ions and negative electrons is described by the ambipolar diffusion coefficient... 

One may ask why some experiments, for instance those done by microwave-afterglow technique 15,16 and the experiments by Canosa et al., 21,22 gave no indications of an anomalous decay. In part, the answer may be that small variations of the deionization coefficients are not easily detected in the presence of ambipolar diffusion. They were detected in the work of Adams et al. and of Smith and Spanel only because the diffusion losses were unusually slow in their large flow tube. 

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. 

Corrosion reactions of a metal with gaseous species such as oxygen, chlorine, sulfur containing molecules or water vapor to produce a thin layer of product phase are typical of ambipolar diffusion reactions. For example, metal oxidation... 

Some examples ambipolar diffusion, total rate of topochemical reaction, change in the light velocity when transiting from vacuum into the given medium, resultant constant of chemical reaction rate (initial product - intermediary activated complex - final product). 

Polymerization Mechanism in Region III. In region III, all the electrons cannot be transported to the anode in a half cycle of the discharge frequency. A possible charge transportation mechanism is an ambipolar diffusion of ion and electron pairs which will cause polymerization. The diffusion of free radicals may also contribute to the polymerization. In our experiment, the contribution of these two mechanisms cannot be distinguished because the ion and electron pairs behave as neutral gases. 

From photocarrier grating measurements at low laser intensities (100 mW cm-2), the value of the ambipolar diffusion coefficient of meso PS is deduced to be about... 

From the formation reaction of protonic defects in oxides (eq 23), it is evident that protonic defects coexist with oxide ion vacancies, where the ratio of their concentrations is dependent on temperature and water partial pressure. The formation of protonic defects actually requires the uptake of water from the environment and the transport of water within the oxide lattice. Of course, water does not diffuse as such, but rather, as a result of the ambipolar diffusion of protonic defects (OH and oxide ion vacancies (V ). Assuming ideal behavior of the involved defects (an activity coefficient of unity) the chemical (Tick s) diffusion coefficient of water is... 

Figure 24. Models illustrating the source of chemical capacitance for thin film mixed conducting electrodes, (a) Oxygen reduction/oxidation is limited by absorption/de-sorption at the gas-exposed surface, (b) Oxygen reduction/ oxidation is limited by ambipolar diffusion of 0 through the mixed conducting film. The characteristic time constant for these two physical situations is different (as shown) but involves the same chemical capacitance Cl, as explained in the text.

The parabolic-rate law for the growth of thick product layers on metals was first reported by Tammann (1920), and a theoretical interpretation in terms of ambipolar diffusion of reactants through the product layer was advanced later by Wagner (1936, 1975). Wagner s model can be described qualitatively as follows when a metal is... 

We will discuss next the ambipolar diffusion, that is, electro-diffusion of two oppositely charged ions in a solution of a univalent electrolyte with local electro-neutrality. Assume the dimensionless ionic diffusivities are constant. Then the relevant version of (1.9) is... 

Equation (1.57a) implies that in the locally electro-neutral ambipolar diffusion concentration of both ions evolves according to a single linear diffusion equation with an effective diffusivity given by (1.57b). Physically, the role of the electric field, determined from the elliptic current continuity equation... 

Parallel-plate with electric barriers Helium discharge Ambipolar diffusion model 67... 

Parallel-plate radial flow CF4 etching Si Two-dimensional isothermal Navier-Stokes Ambipolar diffusion with assumed electron density and energy 84... 

There is also a potential difference between the positive column and tube wall. This potential difference is created because the electrons are much more mobile than heavy ions and tend to flow rapidly out toward any bounding surface. Since the tube wall is an insulator, they tend to collect there causing the insulator to assume a negative potential relative to the plasma. This creates an electric field close to the tube wall which hinders further electron flow towards it. A deficit of electrons forms in a sheath close to the surface, and this sheath assumes a net positive charge. Ions in the plasma, however, see the tube wall potential which is negative compared to the plasma and are attracted to it. This is the diffusion to the tube walls mentioned in the previous paragraph, and is often referred to as "ambipolar" diffusion. 

Thereafter, combining Equations 5.68 through 5.70 and Equation 5.72, the flux of oxygen can be written with the help of the ambipolar diffusion equation [45]... 

The simultaneous movement of ionic and electronic charge carriers under the driving force of a gradient in the electrochemical potential of oxygen facilitates transport of oxygen in the oxide bulk. The flux density of oxide anions is given (Figure 8.12) [77-79,109] by the ambipolar diffusion equation (see Section 5.7.6) [110,111]... 

In this regard, hydrogen flux by proton transport in a dense oxide membrane, in the short circuit case, is described by the ambipolar diffusion expression (see Figure 10.3a) [40,48,49]... 

When oxygen is transported through a membrane in a steady state, there is no net charge current and the flux of oxygen anions, O2-, can be written using the ambipolar diffusion expression, as follows [49]... 

As a consequence, the electrons which are accelerated in the cathode sheath are forced onto a closed loop drift path parallel to the target surface because of the Lorentz Force. This magnetic trapping of the electrons and the corresponding ambipolar diffusion of the ions raises the plasma density in front of the target. A much higher ion current and therefore deposition rate is possible. Furthermore, the pressure can be decreased, which improves... 

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