Solids mixing random

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

A fundamental study of solids mixing is given, with experimental data for a rotating horizontal cylinder containing salt and sand. Theoretical random equilibrium mixture, chi-square test, and sampling considerations are discussed. Also rate equations and segregating effects are covered. 

An expression is developed for computing the mean size z, of randomly dispersed agglomerates, from the variance S2, obtained with a certain sample size, R. The author postulates a rate equation for solids mixing, which he feels may also be applicable to the homogenizing process in glass melting. 

The distribution of spot sample compositions of a certain size, taken from a randomly mixed batch of A and B, can be calculated theoretically. The methods of calculation are standard statistical techniques, and several papers have shown how various aspects of these basic ideas can be applied to solids mixing. Most of the calculations and discussion center around three distributions binomial, normal, and Poisson. 

When substances in liquids or solids mix completely on an atomic scale, that is, when they mix homogeneously and randomly, the mixing usually decreases the es-... 

In a part of the mixing process of particulate solids in addition to the stochastic nature of solid mixing at the micro-level, important macro-level random effects are present too. This paper suggests a theoretical approach to model these large-scale random variations and to calculate the residence probability of the particles. A simulation method is also presented on the basis of the stochastic model. The stationary state is also investigated and a sufficient condition for the existence of the stationary state is given. 

Mixing Mechanisms There are several basic mechanisms by which solid particles are mixed. These include small-scale random motion (diffusion), large-scale random motion (convec tion), and shear. 

On the other hand, upon closer examination even the copper-gold solid solutions evince serious discrepancies with the quasichemical theory. There is a composition range where the entropy of solution is larger than that for random mixing (see Fig. 1) where... 

Different metals can very frequently be mixed with each other in the molten state, i.e. they form homogeneous solutions. A solid solution is obtained by quenching the liquid in the disordered alloy obtained this way, the atoms are distributed randomly. When cooled slowly, in some cases solid solutions can also be obtained. However, it is more common that a segregation takes place, in one of the following ways ... 

Two metals that are chemically related and that have atoms of nearly the same size form disordered alloys with each other. Silver and gold, both crystallizing with cubic closest-packing, have atoms of nearly equal size (radii 144.4 and 144.2 pm). They form solid solutions (mixed crystals) of arbitrary composition in which the silver and the gold atoms randomly occupy the positions of the sphere packing. Related metals, especially from the same group of the periodic table, generally form solid solutions which have any composition if their atomic radii do not differ by more than approximately 15% for example Mo +W, K + Rb, K + Cs, but not Na + Cs. If the elements are less similar, there may be a limited miscibility as in the case of, for example, Zn in Cu (amount-of-substance fraction of Zn maximally 38.4%) and Cu in Zn (maximally 2.3% Cu) copper and zinc additionally form intermetallic compounds (cf. Section 15.4). 

While the mole fraction is a natural measure of composition for solutions of metallic elements or alloys, the mole fraction of each molecule is chosen as the measure of composition in the case of solid or liquid mixtures of molecules.1 In ionic solutions cations and anions are not randomly mixed but occupy different sub-lattices. The mole fractions of the atoms are thus an inconvenient measure of composition for ionic substances. Since cations are mixed with cations and anions are mixed with anions, it is convenient for such materials to define composition in terms of ionic fractions rather than mole fractions. In a mixture of the salts AB and AC, where A is a cation and B and C are anions, the ionic fractions of B and C are defined through... 

The amorphous phase is not usually a desirable state for the API because the formation process is more random and difficult to control than a crystallization. A second dispersed liquid phase is usually formed just prior to freezing and may coalesce or disperse under the influence of hydrodynamic forces in the crystallizer, making the process sensitive to micro-mixing effects on scale up. Amorphous solids also have significantly lower thermodynamic stability than related crystalline material and may subsequently crystallize during formulation and storage. Because of the non-uniformity of the amorphous solid it can more easily incorporate molecules other than the API, making purification less effective. 

Almost all the crystalline materials discussed earlier involve only one molecular species. The ramifications for chemical reactions are thereby limited to intramolecular and homomolecular intermolecular reactions. Clearly the scope of solid-state chemistry would be vastly increased if it were possible to incorporate any desired foreign molecule into the crystal of a given substance. Unfortunately, the mutual solubilities of most pairs of molecules in the solid are severely limited (6), and few well-defined solid solutions or mixed crystals have been studied. Such one-phase systems are characterized by a variable composition and by a more or less random occupation of the crystallographic sites by the two components, and are generally based on the crystal structure of one component (or of both, if they are isomorphous). 

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