Diffusion in Solid Phases

Pick s laws also describe diffusion in solid phases. In solids transport properties can be considerably different than in liquid phases. Only one component can mobile diffuse in the matrix of the second component. At higher temperatures the diffusion coefficient can be more similar in size than in liquid phases, but the diffusion coefficient at room temperature can be orders of magnitudes smaller, e.g., D 10 ° cm s k To overcome the time limitation one must make the diffusion length smaller. Ultra-thin layers or nanoparticles provide such small dimensions. Under such conditions the diffusion is not semi-infinite but has a restricted extension. This has to be considered in the boundary conditions. 

For the study of diffusion phenomena in solids it is also possible to work with potential pulses (potentiostatic pulses) or with constant current pulses (galvanostatic pulses). Examples described in the following paragraphs are based on the coulometric titration method described in Chapter 3. Weppner and Huggins reviewed these methods.In a continuous series of pulses the concentration of lithium in a sheet of aluminum is increased. The diffusion in each pulse is followed by either potential or current measurements. 

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S. Kobayashi. Solute redistribution during solidification with diffusion in solid-phase— a theoretical analysis. J. Cryst. Growth, 88(l) 87-96, 1988. 

To address the short comings of continuum models and yet be able to predict the discharge behavior or capacity fade is an important task. The solution suggested is by developing a novel Monte Carlo method that takes into account design properties, for example, thickness of the cathode, and use it in conjunction with microscopic properties, for example, diffusion in solid phase to predict system level properties of interest. The Monte Carlo algorithm can be considered to follow the framework of continuum models. The next section illustrates the usefulness of the Monte Carlo strategy. 

According to the conditions, two main mechanisms of rrass transfer take part in the nitridation transport in gaseous phase and diffusion in solid phase (Gugel et al., 1979). 

In this paper we show how to use a thermodynamic database in a general Calphad approach (3,4) and to understand phases stabilities of diermoelectric materials. Material microstructures after sintering and thermal behavior are checked with the isothennal section at high temperature. Moreover, Gulliver-Scheil simulations (5,6) (infinite diffusion in liquid phase and no diffusion in solid phase) can be used to determine the solid fractions in a solidfication process and the final microstructure. After an example on lead telluride material we will present the features of a general Calphad t proach in thermoelectric antimonides. 

If X > 1, the solid phase is enriched In the microcomponent during the crystallization, otherwise. It Is depleted In the microcomponent. In addition, the logarithmic distribution law assumes that diffusion In solid phase Is negligible. Therefore, the admixture distribution in the solid phase Is inhomogeneous for X > 1, the highest Impurity concentration is In the centre of the crystal and It decreases towards the surface, while for X < 1 this distribution Is inversed. 

Diffusion involves (1) diffusion along grain boundary, and on surface of minerals, (2) diffusion of dissolved species in aqueous solution (volume diffusion, dispersion (convective diffusion), eddy diffusion, turbulent diffusion), and (3) diffusion in solid phase in the presence of aqueous solution. 

Diffusion in pore and crack in rocks and minerals is generally more important as micro scale mass transfer mechanism in water-rock system rather than diffusion in solid phases and in free water. 

If two minerals contact with each other at constant the pressure-temperature condition where two minerals are unstable, reaction occurs between them to form stable mineral. The dominant rate limiting mechanisms are diffusion of aqueous species dissolved from minerals in fluid and dissolution and precipitation reactions. If fluid is not present, diffusion in solid phase occurs. But the rate of diffusion in solid phase is generally very slow. However, at very high temperature and pressure (metamorphic condition) the diffusion in solid phase may control the mass transfer. Reaction-diffusion model is able to be used to obtain the development of reaction zone between two minerals with time. 

Between the two interfaces, diffusion of oxygen occurs in gas phase that ensures the transport of oxygen from b toward c, short-circuiting the diffusion in solid phase ABO2. 

Therefore, the results of the present paper showed, that the notion of reactive medium heterogeneity connected with free volume representations, that was expected for diffusion-controlled solid phase reactions. If free volume microvoids were not connected with one another, then medium is heterogeneous, and in case of formation of overlapping percolation network of such microvoids - homogeneous. To obtain such definition is possible only within the framework of the fractal free volume conception. 

The Elovich model was originally developed to describe the kinetics of heterogeneous chemisorption of gases on solid surfaces [117]. It describes a number of reaction mechanisms including bulk and surface diffusion, as well as activation and deactivation of catalytic surfaces. In solid phase chemistry, the Elovich model has been used to describe the kinetics of sorption/desorption of various chemicals on solid phases [23]. It can be expressed as [118] ... 

Sorption/desorption is the key property for estimating the mobility of organic pollutants in solid phases. There is a real need to predict such mobility at different aqueous-solid phase interfaces. Solid phase sorption influences the extent of pollutant volatilization from the solid phase surface, its lateral or vertical transport, and biotic or abiotic processes (e.g., biodegradation, bioavailability, hydrolysis, and photolysis). For instance, transport through a soil phase includes several processes such as bulk flow, dispersive flow, diffusion through macropores, and molecular diffusion. The transport rate of an organic pollutant depends mainly on the partitioning between the vapor, liquid, and solid phase of an aqueous-solid phase system. 

Thus rate laws for precipitation reactions tend to be complicated, even in pure solutions. Mixed precipitates can be inhomogeneous solids with one component restricted to a thin outer layer because of slow diffusion. New solid phases can precipitate homogeneously onto the surfaces of existing solid phases. Weathering solids may provide host surfaces onto which more stable phases may precipitate. 

Many factors will prevent the reaction from completion and lower the yield of solid-phase synthesis. Besides chemical issues, factors hke solvent [8], reagent diffusion [9], mixing method [10], site interaction [11], and others may all influence the synthesis. In this section, discussion will focus on the reagent diffusion issue in solid-phase synthesis. 

As a reaction medium, solvents are used to bring reactants at suitable concentrations. For endothermic reactions, heat can be supplied readily by heating the solvent for exothermic reactions, the solvent can act as a heat sink. If necessary, surplus heat can be removed by allowing the solvent to boil. After the reaction, products have to be freed of solvent. As diffusion in solids is slow, solid-solid reactions are slow at room temperature. To get reactants to mix at a molecular level requires use of the gas phase. 

Wen CJ, Huggins RA. Chemical diffusion in intermediate phase in the lithium-silicon system. J Solid State Chem 1981 37 271-278. 

In the solid-state reaction, nucleation and growth have a fundamental role, because, in essence, the solid-state reaction is a phase transformation. In this type of reactions, nucleation and growth follow similar principles as those previously analyzed in Section 3.1 the principal difference being the increased role of diffusion in solid-state reactions [30],... 

In this book we are concerned only with mass transport, or diffusion, in solids. Self-diffusion refers to atoms diffusing among others of the same type (e.g., in pure metals). Interdiffusion is the diffusion of two dissimilar substances (a diffusion couple) into one another. Impurity diffusion refers to the transport of dilute solute atoms in a host solvent. In solids, diffusion is several orders of magnitude slower than in liquids or gases. Nonetheless, diffusional processes are important to study because they are basic to our understanding of how solid-liquid, solid-vapor, and solid-solid reactions proceed, as well as [solid-solid] phase transformations in single-phase materials. 

As an explanation of the process of macroradicai decay the mechanism of radical state migration by means of hydrogen atoms elimination from adjacent chains has been suggested and confirmed. The activation energies of various macroradicai reactions have been measured and found to be very high. This is due to a specifity of the kinetic process in solid phase and to the participation of the diffusion phenomenon. Comparative investigation of the reactivity of various macroradicals reveals a great importance of the steric factor, i. e. the entropy ot the transition state for the kinetics of macroradicai reactions. 

The fact that the reaction rates in solid phase synthesis are not drastically reduced, compared to the homogeneous reactions, indicates that the diffusion of the reagent into the polymeric matrix is not a limiting factor for the method. This has been confirmed by Andreatta and Rinkll9) in kinetic studies on both cross-linked and linear polystyrenes. This means that the intrinsic problems of solid phase synthesis arise from deviations in the linear kinetic course in the final stages of reaction due to non-equivalence of functional groups. 

For lakes which have undergone significant acidification, it has been suggested that heavy metals could be released from surface sections by pH-dependent dissolution, resulting in sub-surface maxima in sedimentary heavy metal concentrations. In two Canadian acid lakes, however, Carignan and Tessier found that downward diffusive fluxes of dissolved zinc from overlying waters into anoxic pore waters were responsible for the pronounced sub-surface sediment maxima in solid phase zinc, presumably as the insoluble sulfide. 

Many such studies of sedimentary phosphorus profiles, also incorporating pore water measurement of soluble reactive phosphate, have demonstrated that redox-controlled dissolution of iron (hydr)oxides under reducing conditions at depth releases orthophosphate to solution. This then diffuses upwards (and downwards) from the pore water maximum to be re-adsorbed or co-precipitated with oxidized Fe in near-surface oxic sections. The downwards decrease in solid phase organic phosphorus indicates increasing release of phosphorus from deposited organic matter with depth, some of which will become associated with hydrous iron and other metal oxides, added to the pool of mobile phosphorus in pore water or contribute to soluble unreactive phosphorus . The characteristic reactions involving inorganic phosphorus in the sediments of Toolik Lake, Alaska, are shown in... 

Substantial differences between solid-phase reactions and hydrothermal synthesis reactions have been stated in numerous investigations. In solid-phase processes, the sequence of intermediate products formation does not depend on reagents ratio in the initial mixture, and the excess product appears to be a compound with the highest crystallization temperature. On contrary, for the formation of a definite product by hydrothermal synthesis, the initial mixture should contain reagents at an exact stoichiometric ratio [19,20]. In solid-phase reactions, the interaction rate is determined by the rate of diffusion processes, while in hydrothermal processes the determining factor is the rate of dissolution of the initial products in the water. Water simplifies diffusion transport of particles in the system the formation of nuclei and crystal growth occur faster than in solids. 

In the case of a sediment devoid of iron oxide, such that none of the reduced sulfate is incorporated in solid phases such as FeS, the concentration of total dissolved HoS equals (SO )q - iS04) providing the diffusion coefficients for H2S and SO4 are equivalent. 

W. A, Steele, Computer Simulation of Surface Diffusion in Adsorbed Phases, in Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces, ed. W. Rudzinski, W. A. Steele and G. Zgrablich (Elsevier, Amsterdam, 1996), 451-486. 

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