Macro-kinetic equation

The electrostatic retardation of the adsorption kinetics of ionic siufactants is one of these nonequilibrium surface phenomena to be described on the basis of this physical model, consisting of the electrochemical macro-kinetic equations used in theoretical and colloid electrochemistry. This approach describes the flux of ions in terms of their spatial distribution. The equations were first developed by Overbeek (1943) and later proved to be valid for the theory of different... 

The kinetics of polyurethane curing in adiabatic conditions was studied by the thermometric method (see Section 2.2) for a composition based on macro(diisocyanate) and diamine.48 It was proved that in this case, the second-order kinetic equation was inapplicable for the rangep > 0.7. By analyzing the time dependencies of temperature of the reactive medium, it was established that the... 

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 fundamental approaches to definition of turbulent flows macro-kinetics and macro-mixing processes are considered in [136-139]. Special attention was focused on micro-mixing models in the context of method based on equation for density of random variables probabilities distribution. Advantage of this method is that we can calculate average rate of chemical reaction if know the corresponding density of concentration and temperature possibility distribution. 

TABLE 9.8 Macro Kinetic Rate Expressions for WGSR [25, 51, 52, 56, 73-78] —Cont d Sample Reaction conditions Rate equation... 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

Note here that the relation between mesoscopic and microscopic approaches is not trivial. In fact, the former is closer to the macroscopic treatment (Section 2.1.1) which neglects the structural characteristics of a system. Passing from the micro- to meso- and, finally, to macroscopic level we loose also the initial statement of a stochastic model of the Markov process. Indeed, the disadvantages of deterministic equations used for rather simplified treatment of bimolecular kinetics (Section 2.1) lead to the macro- and mesoscopic models (Section 2.2) where the stochasticity is kept either by adding the stochastic external forces (Section 2.2.1) or by postulating the master equation itself for the relevant Markov process (Section 2.2.2). In the former case the fluctuation source is assumed to be external, whereas in the latter kinetics of bimolecular reaction and fluctuations are coupled and mutually related. Section 2.3.1.2 is aimed to consider the relation between these three levels as well as to discuss problem of how determinicity and stochasticity can coexist. 

For ideal radical polymerization to occur, three prerequisites must be fulfilled for both macro- and primary radicals, a stationary state must exist primary radicals have to be for initiation only and termination of macroradicals only occur by their mutual combination or disproportionation. The rate equation for an ideal polymerization is simple (see Chap. 8, Sect. 1.2) it reflects the simple course of this chain reaction. When the primary radicals are deactivated either mutually or with macroradicals, kinetic complications arise. Deviations from ideality are logically expected to be larger the higher the concentration of initiator and the lower the concentration of monomer. Today termination by primary radicals is an exclusively kinetic problem. Almost nothing has been published on the mechanism of radical liberation from the aggregation of other initiator fragments and from the cage of the... 

The reaction occurs at the electrode/electrolyte interface (sol-id/liquid interface at the surface of the particle). This reaction occurs as a source term in the equations for the macro scale. In the model equations, accounts for the electrochemical kinetics, (intercalation reaction from the electrolyte phase into the solid matrix and vice-versa). It is a modified form of the Butler-Volmer kinetics, and is given by the following expression ... 

Aqueous-phase Ni concentration, however, was found to have a pronounced effect on ion exchange kinetics, which is not consistent with prediction from the IPDC model based on the Nernst-Planck equation. Experimental results could, however, be accounted for with the aid of a macroporous model, originally developed by Yoshida and Kataoka based on the assumption of parallel diffusion of counterions in the solid-gel phase and in the macro-... 

In chemical kinetics the concept of the order of a reaction forms the basis of a kinematics which constitutes a frame for most of the molecular theories of chemical reactions. The fundamental magnitudes of this kinematics are the concentrations and the specific rate constants. In simple cases only the time enters as an independent variable, whereas in a diffusion process both time and space are involved. Diffusion processes are generally described in terms of diffusion coefficients, volume concentrations and thermodynamic potential or activity factors. Partial volume factors and friction coefficients associated with the components of the diffusing mixture are also essential in the description. A feature of the macro-dynamical theory is that it covers any region of concentration. Especially simple equations are connected with the differential diffusion process (diffusion with small concentration differences), for which the different coefficients or factors mentioned above are practically constant. 

From kinetic investigation of the curing of macro-triisocyanates (MTI) based on L-3003, P-2200, and their mixtures, kinetic curves of the polymerization process were obt uned at 298 K. Kinetic curves for the curing of polyurethane prepolymer mixtures obtained by reaction of TDI with L-3003, P-2200, and various mixtures with a double excess of NCO groups with respect to OH groups are linear. These dependences show that NCO group reactions continue until high fractional conversions comply with the second-order equation... 

As far as the present work is concerned, the relevance of numerical stochastic methods for polymer dynamics in micro/macro calculations resides in their ability to yield (within error bars) exact numerical solutions to dynamic models which are insoluble in the framework of polymer kinetic theory. In addition, and mainly as a consequence of the correspondence between Fokker Planck and stochastic differential equations, complex polymer dynamics can be mapped onto extremely efficient computational schemes. Another reason for the efficiency of stochastic dynamic models for polymer melts stems from the reduction of a many-chain problem to a single-chain or two-chain representation, i.e., to linear computational complexity in the number of particles. This circumstance permits the treatment of global ensembles consisting of several tens of millions of particles on current hardware, corresponding to local ensemble sizes of O(IO ) particles per element. 

For these reasons we may attribute the faster kinetics to the temperature change of the dye distribution along the macro= ion chain as done for the system AO-PSS. The process can be described by the following phenomenological equation ... 

Recent progress in experimental methods for observation and characterization of the particle system were presented in Chapter 5 of Volume 2 of this series. As for design methods, it has been pointed out, in Chapter 6 of Volume 1 of this series, that population balance equations can be used. However, such equations contain kinetic terms for birth, growth and death of particles, which must be reliably known for practical application. Moreover, they are continuous, macro-scale representations that - by definition - cannot directly account for micro-scale physics. 

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