Solid-state diffusion coefficient temperature dependence

According to the analysis in the previous sections, the primary particle size in flame reactors is determined by the relative rates of particle collision and coalescence. For highly refractory materials, the characterislic coalescence time (12.6) depends on the solid-state diffusion coefficient, which is a very sensitive function of the temperature. The mechanisms of solid-.staie diffusion depend in a complex way on the structure of the solid. For example, a perfect cubic crystal of the substance AB consists of alternating ions A and B. Normally there are many defects in the lattice structure even in a chemically pure single crystal defect types are shown schematically in Fig. 12.8. The mechanism of diffusion in cry.stalline solids depends on the nature of the lattice defects. Three mechanisms predominate in ionic... 

Solid-State Diffusion Coefficient 343 Temperature Dependence 343 Values of D for Lattice Diffusion 345 High Diffitsivify Paths 346... 

Not much is known about the thermophysical properties of liquid metals, especially the transport properties such as chemical and thermal diffusivities. The existing data are sparse and the scatter makes it difficult to make an accurate determination of the temperature dependency of these properties. This situation was the motivation for Froberg s experiment on Space-lab-1 in which he measured the temperature dependence of the self-diffusion of Sn from 240°C to 1250°C. He found that the diffusion coefficients were 30-50% lower than the accepted values and seemed to follow a 7 dependence as opposed to the Arrhenius behavior observed in solid state diffusion. ... 

It follows from this discussion that all of the transport properties can be derived in principle from the simple kinetic dreoty of gases, and their interrelationship tlu ough k and c leads one to expect that they are all characterized by a relatively small temperature coefficient. The simple theory suggests tlrat this should be a dependence on 7 /, but because of intermolecular forces, the experimental results usually indicate a larger temperature dependence even up to for the case of molecular inter-diffusion. The Anhenius equation which would involve an enthalpy of activation does not apply because no activated state is involved in the transport processes. If, however, the temperature dependence of these processes is fitted to such an expression as an algebraic approximation, tlren an activation enthalpy of a few kilojoules is observed. It will thus be found that when tire kinetics of a gas-solid or liquid reaction depends upon the transport properties of the gas phase, the apparent activation entlralpy will be a few kilojoules only (less than 50 kJ). 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

For the solubility of TPA in prepolymer, no data are available and the polymer-solvent interaction parameter X of the Flory-Huggins relationship is not accurately known. No experimental data are available for the vapour pressures of dimer or trimer. The published values for the diffusion coefficient of EG in solid and molten PET vary by orders of magnitude. For the diffusion of water, acetaldehyde and DEG in polymer, no reliable data are available. It is not even agreed upon if the mutual diffusion coefficients depend on the polymer molecular weight or on the melt viscosity, and if they are linear or exponential functions of temperature. Molecular modelling, accompanied by the rapid growth of computer performance, will hopefully help to solve this problem in the near future. The mass-transfer mechanisms for by-products in solid PET are not established, and the dependency of the solid-state polycondensation rate on crystallinity is still a matter of assumptions. 

The concept of the diffusion coefficient, or diffusivity, also is used as the measure of the diffusion of one substance through another. The driving force is the concentration difference, via Pick s law, which can be related to the partial pressure difference by means of the equation of state. It is applicable to mixtures occurring as a common phase but can be applied as well to the case where the second substance is a solid, say, as in dialysis. The dimensions are ordinarily (distance) /time (e.g., cm /sec). Since the equation of state is implicitly involved, the conversion between permeability and diffusivity is more pronouncedly temperature (and pressure) dependent. 

Reaction heat and temperature difference between gas stream and catalyst leads to the heat transfer processes between gas and solid. The mass and heat transfer coefficients depend on Reynold and Prandtl numbers of fluid flow, i.e., state and physical properties of fluid flow. Mass and heat transfer processes in solid cataljret and catal3dic reaction process on internal surface of catalysts take place simultaneously, which relate to the diffusion coefficients of reactants and products as well as the heat conductivity coefficient of catalysts. Therefore, the overall rate of a catal3dic reaction depends not only on the rate of chemical reaction, but also on several physical processes such as flowing state, mass and heat transfer. The kinetics involving physical process effect is usually called as apparent or macrokinetics, while that having no physical processes is called intrinsic or microkinetics. 

Diffusion rates may, in principle, also be determined from any property or reaction which depends on atomic mobility. By way of illustration, ionic conductivity of the anion is directly proportional to the anion diffusion coefficient (see Electrical conductivity). From high temperature solid state reactions, sintering, oxidation of metals etc. diffusion coefficients may be evaluated provided the detailed mechanism of the processes are known. Examples of this will be given in Chapter 7. 

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