Kinetic models, for solid-state

KINETIC MODELS FOR SOLID STATE REACTIONS 3.1. INTRODUCTION... 

Use of Kinetic Models for Solid State Reactions in Combustion Simulations... 

Carter RE (1961) Kinetic model for solid-state reactions. J Chem Phys 34 2010-2015... 

R. E. Carter, Addendum Kinetic Model for Solid State Reactions, J. Chem. Phys. 35(3) 1137-38 (1961). 

Since the traditional kinetic models of solid-state reactions are often based on a formal description of geometrically well defined bodies treated under strictly isothermal conditions, they are evidently not appropriate to describe the real process, which requires accoimt to be taken of irregularity of shape, polydispersity, shielding and overlapping, unequal mixing anisotropy and so on, for sample particles under reaction. One of the measures which has been taken to solve the problem is to introduce an accommodation function a a) [32]. The discrepancy between the idealized /(a) and the actual kinetic model function h a) can be expressed as... 

Chapter 10 begins a more detailed treatment of heterogeneous reactors. This chapter continues the use of pseudohomogeneous models for steady-state, packed-bed reactors, but derives expressions for the reaction rate that reflect the underlying kinetics of surface-catalyzed reactions. The kinetic models are site-competition models that apply to a variety of catalytic systems, including the enzymatic reactions treated in Chapter 12. Here in Chapter 10, the example system is a solid-catalyzed gas reaction that is typical of the traditional chemical industry. A few important examples are listed here ... 

Experimental determination of Ay for a reaction requires the rate constant k to be determined at different pressures, k is obtained as a fit parameter by the reproduction of the experimental kinetic data with a suitable model. The data are the concentration of the reactants or of the products, or any other coordinate representing their concentration, as a function of time. The choice of a kinetic model for a solid-state chemical reaction is not trivial because many steps, having comparable rates, may be involved in making the kinetic law the superposition of the kinetics of all the different, and often unknown, processes. The evolution of the reaction should be analyzed considering all the fundamental aspects of condensed phase reactions and, in particular, beside the strictly chemical transformations, also the diffusion (transport of matter to and from the reaction center) and the nucleation processes. 

A kinetic model for single-phase polymerizations— that is, reactions where because of the similarity of structure the polymer grows as a solid-state solution in the monomer crystal without phase separation—has been proposed by Baughman [294] to explain the experimental behavior observed in the temperature- or light-induced polymerization of substimted diacetylenes R—C=C—C=C—R. The basic feature of the model is that the rate constant for nucleation is assumed to depend on the fraction of converted monomer x(f) and is not constant like it is assumed in the Avrami model discussed above. The rate of the solid-state polymerization is given by... 

The main purpose of quantum-chemical modeling in materials simulation is to obtain necessary input data for the subsequent calculations of thermodynamic and kinetic parameters required for the next steps of multiscale techniques. Quantum-chemical calculations can also be used to predict various physical and chemical properties of the material in hand (the growing film in our case). Under quantum-chemical, we mean here both molecular and solid-state techniques, which are now implemented in numerous computer codes (such as Gaussian [25], GAMESS [26], or NWCHEM [27] for molecular applications and VASP [28], CASTEP [29], or ABINIT [30] for solid-state applications). 

FIGURE 54 Refinement of a solid-state kinetics model for three-dimensional solid-state diffusion (a(t) — 1 — [1 — (kt) /2]3) to the experimental extent of reoxidation of activated H5[PV2Mo1o04o],13l-l20 at 698 K (Ressler and Timpe, 2007). Reprinted from (Ressler and Timpe, 2007), Copyright 2007, with permission from Elsevier. 

Heterogeneously catalyzed reactions are usually studied under steady-state conditions. There are some disadvantages to this method. Kinetic equations found in steady-state experiments may be inappropriate for a quantitative description of the dynamic reactor behavior with a characteristic time of the order of or lower than the chemical response time (l/kA for a first-order reaction). For rapid transient processes the relationship between the concentrations in the fluid and solid phases is different from those in the steady-state, due to the finite rate of the adsorption-desorption processes. A second disadvantage is that these experiments do not provide information on adsorption-desorption processes and on the formation of intermediates on the surface, which is needed for the validation of kinetic models. For complex reaction systems, where a large number of rival reaction models and potential model candidates exist, this give rise to difficulties in model discrimination. 

This part demonstrates how deterministic models of impedance response can be developed from physical and kinetic descriptions. When possible, correspondence is drawn between hypothesized models and electrical circuit analogues. The treatment includes electrode kinetics, mass transfer, solid-state systems, time-constant dispersion, models accounting for two- and three-dimensional interfaces, generalized transfer functions, and a more specific example of a transfer-function tech-nique.in which the rotation speed of a disk electrode is modulated. 

Nonradiative energy transfer with the simultaneous involvement of different mechanisms has been modelled by Rotman for solid state systems. Although his treatment is particularly directed towards inorganic systems it is generally relevant to organic systems also. The complex kinetics of sequential energy processes involving four species each with a dependence of rate coefficients has also been analyzed. ... 

Because the radiation in most cases will not penetrate the entire sample, the concentration of the reactant is unlikely to approach zero at infinite time. A plot of remaining concentration vs. time will therefore level off at a value greater than zero. This should be taken into account when selecting the kinetic model for studies of solid-state degradation (Sande, 1996). The solid-state degradation will in some cases appear to consist of a series of consecutive processes with different mechanisms and rates (Carstensen, 1974). Such a stepwise change in reaction rate is most likely caused by an alteration in sample surface and fading of subsequent layers. The concept of reaction order may not be useful for photodecomposition in the solid state (De VUliers et al 1992). 

According to the kinetic model for adsorption, the surface of the solid is assumed to serve as effective adsorption sites [39, 40]. When the rate of condensation of the gas molecules on the unoccupied sites on the surface is equal to the rate of evaporation of the gas molecules from the occupied sites, a state of dynamic equilibrium is reached, which leads to the following equation ... 

Dimitrov N, Vasilic R, Vasiljevic N (2(X)7) A kinetic model for redox replacement of UPD layers. Electrochem Solid State Lett 10(7) D79-D83... 

Duan, Y, Wang, L., and Chen, H. (2012) Digital image analysis and fractal-based kinetic modelling for fungal biomass determination in solid-state fermentation. Biochem. Eng. /., 67, 60-67. 

It is clear that the experimental curves, measured for solid-state reactions under thermoanalytical study, cannot be perfectly tied with the conventionally derived kinetic model functions (cf. previous table lO.I.), thus making impossible the full specification of any real process due to the complexity involved. The resultant description based on the so-called apparent kinetic parameters, deviates from the true portrayal and the associated true kinetic values, which is also a trivial mathematical consequence of the straight application of basic kinetic equation. Therefore, it was found useful to introduce a kind of pervasive des-cription by means of a simple empirical function, h(a), containing the smallest possible number of constant. It provides some flexibility, sufficient to match mathematically the real course of a process as closely as possible. In such case, the kinetic model of a heterogeneous reaction is assumed as a distorted case of a simpler (ideal) instance of homogeneous kinetic prototype f(a) (1-a)" [3,523,524]. It is mathematically treated by the introduction of a multiplying function a(a), i.e., h(a) =f(a) a(a), for which we coined the term [523] accommodation function and which is accountable for a certain defect state (imperfection, nonideality, error in the same sense as was treated the role of interface, e.g., during the new phase formation). 

The overall simulation of high-temperature corrosion processes under near-service conditions requires both a thermodynamic model to predict phase stabilities for given conditions and a mathematical description of the process kinetics, i.e. solid state diffusion. Such a simulation has been developed by integrating the thermodynamic program library, ChemApp, into a numerical finite-difference diffusion calculation, InCorr, to treat internal oxidation and nitridation of Ni-base alloys [10]. This simulation was intended to serve as a basis for an advanced computer model for internal oxidation and sulfidation of low-alloy boiler steels. 

---