Electrically kinetic equations

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

In deriving the kinetic equation describing the arrival of various ionic species at the cathode, it is assumed that the primary species N2 + is formed at the central wire at a constant rate, and during its passage in the direction x perpendicular to the axis its concentration is modified by various reactions. In this treatment both ion diffusion and ion-ion or electron-ion recombination processes are neglected because the geometry of the discharge tube and the presence of an electric field would... 

The kinetic equation describing the change in concentration of conductivity electrons and consequently the time dependence of electric conductivity has the following shape... 

Thus, the rigorous solution of kinetic equation describing the change in electric conductivity of a semiconductor during adsorption of radicals enables one to deduce that information on concentration of radicals in ambient volume can be obtained measuring both the stationary values of electric conductivity attained over a certain period of time after activation of the radical source and from the measurements of initial rates in change of electric conductivity during desactivation or activation of the radical flux incident on the surface of adsorbent, i.e. 

The discussion up to this point has been concerned essentially with metal alloys in which the atoms are necessarily electrically neutral. In ionic systems, an electric diffusion potential builds up during the spinodal decomposition process. The local gradient of this potential provides an additional driving force, which acts upon the diffusing species and this has to be taken into account in the derivation of the equivalents of Eqns. (12.28) and (12.30). The formal treatment of this situation has not yet been carried out satisfactorily [A.V. Virkar, M. R. Plichta (1983)]. We can expect that the spinodal process is governed by the slower cation, for example, in a ternary AX-BX crystal. The electrical part of the driving force is generally nonlinear so that linearized kinetic equations cannot immediately be applied. 

Kinetic equations for the recombination luminescence intensity in the presence of a permanent electric field for the arbitrary non-pair initial spatial distribution of the reagents in the case when the concentration of one of them considerably exceeds that of the other, have been obtained in ref. 21. These equations have the form... 

MEISs and macroscopic kinetics. Formalization of constraints on chemical kinetics and transfer processes. Reduction of initial equations determining the limiting rates of processes. Development of the formalization methods of kinetic constraints direct application of kinetics equations, transition from the kinetic to the thermodynamic space, and direct setting of thermodynamic constraints on individual stages of the studied process. Specific features of description of constraints on motion of the ideal and nonideal fluids, heat and mass exchange, transfer of electric charges, radiation, and cross effects. Physicochemical and computational analysis of MEISs with kinetic constraints and the spheres of their effective application. 

This is of course also true if we need to consider the general electrochemical reaction Eq. (92). If the applied driving force (cf. electrical experiment) is an electrical potential gradient, Eq. (97) leads to the well-known non-linear Butler-Volmer equation.79 We will become acquainted with equally important kinetic equations for the cases of the tracer and the chemical experiment.172... 

Here, Ry and ay are the active resistance and the corresponding electric conductance of the circuit fragment between points i and j. Note that in kinetic equation (1.31), thermodynamic rushes fr of the reactant groups behave as electric potentials in the points, while parameter Ey is equivalent to electric conductance ay. 

While comparing the stationary kinetic equations (in their thermodynamic form) for the intermediate concentrations of system (1.34) to the Kirchhoff equation for the electric current inflow and outflow at all junction points of an equivalent electric circuit, one can easily ascertain that the combination of reactions (1.34) will be described by the equivalent electric diagram... 

It has to be mentioned that such equivalent circuits as circuits (Cl) or (C2) above, which can represent the kinetic behavior of electrode reactions in terms of the electrical response to a modulation or discontinuity of potential or current, do not necessarily uniquely represent this behavior that is other equivalent circuits with different arrangements and different values of the components can also represent the frequency-response behavior, especially for the cases of more complex multistep reactions, for example, as represented above in circuit (C2). In such cases, it is preferable to make a mathematical or numerical analysis of the frequency response, based on a supposed mechanism of the reaction and its kinetic equations. This was the basis of the important paper of Armstrong and Henderson (108) and later developments by Bai and Conway (113), and by McDonald (114) and MacDonald (115). In these cases, the real (Z ) and imaginary (Z") components of the overall impedance vector (Z) can be evaluated as a function of frequency and are often plotted against one another in a so-called complex-plane or Argand diagram (110). The procedures follow closely those developed earlier for the representation of dielectric relaxation and dielectric loss in dielectric materials and solutions [e.g., the Cole and Cole plots (116) ]. 

In all reactions given above, we have oxygen ions O on the right part of the equations. Corresponding to these reactions, the current discharge density 7 can be determined by the kinetic equations (2.12)-(2.16). Furthermore, the capacitance of the electric double layer at the TPB among gas-oxide-SE-YSZ is different from that of the electric double layer at the TPB among gas-metal-SE-YSZ [38, 61]. In fact, for SEs, based on metal Pt or Au, the capacitance of the electric double layer is about 50-150 juC/cm and basically independent of temperature and Pq2 fluctuations. However, for... 

The above analysis shows that in the simple case of one adsorbed intermediate (according to Langmuirian adsorption), various complex plane plots may be obtained, depending on the relative values of the system parameters. These plots are described by various equivalent circuits, which are only the electrical representations of the interfacial phenomena. In fact, there are no real capacitances, inductances, or resistances in the circuit (faradaic process). These parameters originate from the behavior of the kinetic equations and are functions of the rate constants, transfer coefficients, potential, diffusion coefficients, concentrations, etc. In addition, all these parameters are highly nonlinear, that is, they depend on the electrode potential. It seems that the electrical representation of the faradaic impedance, however useful it may sound, is not necessary in the description of the system. The systen may be described in a simpler way directly by the equations describing impedances or admittances (see also Section IV). In... 

A first attempt to consider the role of the Debye counterion atmosphere on the transport of a surfactant ion through the DL was made by Mikhailovskij (1976, 1980) (cf. Kortilm 1966, Lyklema 1991). In contrast to a macro-kinetic model, Mikhailovskij derived kinetic equations for a multi-component system under the influence of an external electric field. The basis of this derivation was the set of Bogolubow equations for the partial distribution functions. As the result of the model derivation the following set of electro-diffusion equations is obtained. 

The electrical breakdown results from the tunneling of the charge carriers from electrodes or from the valence band or from the impurity levels into the conduction band, or by the impact ionization. The tunnel effect breakdown happens mainly in thin layers, e.g., in thin p-n junctions. Otherwise, the impact ionization mechanism dominates. For this mechanism, the dielectric strength of an insulator can be estimated using Boltzmann s kinetic equation for electrons in a crystal. 

Valuable information about the physics involved in the kinetic treatment of a specific problem can be obtained by considering the consistent macroscopic balance equations of the electrons, Eqs. (31) to (33), adapted to the specific kinetic problem. On the right side of the power and momentum balance, Eqs. (32) and (33), a difference between the corresponding gain fi-om the electric field and the total loss in collisions occurs. Gain and loss terms arise on the right side of the particle balance equation, Eq. (31), too if nonconservative electron collision processes (for instance, ionization and attachment) are additionally taken into account in the kinetic equation, Eq. (8), and thus in the equation system (12). 

Another important point, closely connected with electron kinetics, concerns the self-consistent treatment of electron kinetics, of the particle and/or power balance equations for heavy particles (such as different ions and excited atoms or molecules), and of the Maxwell equations (or a reduced version such as the Poisson equation or appropriate electric circuit equations) to obtain a more complete description of all important plasma components as well as of the internal electric field. This self-consistent treatment is usually tricky and is based on an iterative approach to the solution of the various types of equations involved (Loffhagen and Winkler, 1994 Uhrlandt and Winkler, 1996 Yang and Wu, 1996). To integrate the electron kinetic equation in such an approach adequately, a very effective solution procedure for this equation is of particular importance, although remarkable progress with respect to the speed of computation has been achieved in recent years. 

First, consider the dynamics of integration of drops with fully retarded surfaces in the absence of electric field. If drops are uncharged, the kernel of the kinetic equation is given by the relation (13.100), namely... 

Consider now the dynamics of integration of conducting drops in the external electric field of strength Eo, provided that Eo does not exceed the critical value at which the approaching drops may split. In this case the kinetic equation kernel is determined by the expression (13.134)... 

CAT+ would vary periodically in both the phases along with the ratio r= [CAT ]org/[CAT+]aq, which would be responsible for electric potential oscillations. The formation of reverse micelles in the non-aqueous phase would be responsible for the decrease in concentration of [CAT ] ,g in the non-aqueous phase. On the other hand, it would be compensated by the transfer from the aqueous phase. But this would require more H in the aqueous phase to interact with CTAB to generate CTA. The required amount of H+ would be produced by transfer of HP from the non-aqueous phase and the cycle would be repeated over and over again, giving rise to oscillations in electric potential and PH. Using relevant kinetic equations [27] and linear stability analyses, mathematical formalism has been developed. 

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