Solution formation

The stoichiometry must be exact. Coprecipitation by solid-solution formation, foreign ion entrapment, and adsorption are possible sources of error. 

Free energy of solution formation, consisting of the same amount of solvent and degradated molecules, is presented as follows ... 

The resulting product depends on the precipitation conditions, and in particular, on the over-saturation level of the solution. Formation of ammonium oxyfluorometalate crystalline compounds occurs at a relatively low pH of the solution. From the standpoint of the interactions described in Equation (143), this means that the interaction between NH4F and Me205 (denoted as interaction 1) is stronger than the interaction denoted as interaction 2. In this case, subsequent processes of the hydroxide treatment lead to some defluorination of the product, but the performance of such processes is usually very problematic. Precipitation at high pH values leads to a strong oversaturation of niobium- or tantalum-containing compounds, which in turn... 

The temperature dependence of electrical conductivity has been used [365] to distinguish between the possible structural modifications of the Mn02 yielded by the thermal decomposition of KMn04. In studies involving additives, it is possible to investigate solid-solution formation, since plots of electrical conductivity against concentration of additive have a characteristic V-shape [366]. 

Measurements of photoconductivity and of the Hall potential [367] are accurate and unambiguous methods of detecting electronic conduction in ionic solids. Kabanov [351] emphasizes, however, that the absence of such effects is not conclusive proof to the contrary. From measurements of thermal potential [368], it is possible to detect solid-solution formation, to distinguish between electronic and positive hole conductivity in semi-conductors and between interstitial and vacancy conductivity in ionic conductors. 

Reactions of the general type A + B -> AB may proceed by a nucleation and diffusion-controlled growth process. Welch [111] discusses one possible mechanism whereby A is accepted as solid solution into crystalline B and reacts to precipitate AB product preferentially in the vicinity of the interface with A, since the concentration is expected to be greatest here. There may be an initial induction period during solid solution formation prior to the onset of product phase precipitation. Nuclei of AB are subsequently produced at surfaces of particles of B and growth may occur with or without maintained nucleation. 

To develop and apply assumption-free learning frameworks and methodologies, aimed at uncovering and expressing in adequate solution formats performance improvement. . opportunities, extracted from existing data which were acquired from plants that cannot be described effectively through first-principles quantitative models. 

Thus, a critical departure from previous approaches, common to all our learning methodologies, is the adoption of a solution format that consists of hyperrectangles (not points) defined in the decision space. 

Since a real vector is a degenerate interval vector whose components are null width intervals, previous conventional pointwise solution formats can be considered a particular case of the suggested alternative and more general solution space, obtained when the minimum allowed region size is reduced to zero, thus converting hyperrectangles into single points. 

Solution format, The solution space consists of hyperrectangles ( == X G), instead of points ( = x g, ), defined in the decision space. 

The preceding set of characteristics and properties of the estimators makes our type of mapping procedures, /, particularly appealing for the kinds of systems that we are especially interested to study, i.e., manufacturing systems where considerable amounts of data records are available, with poorly understood behavior, and for which neither accurate first-principles quantitative models exist nor adequate functional form choices for empirical models can be made a priori. In other situations and application contexts that are substantially different from the above, while much can still be gained by adopting the same problem statements, solution formats and performance criteria, other mapping and search procedures (statistical, optimization theory) may be more efficient. 

Fig. 2. (a) Conventional pattern recognition (b) alternative problem statement and solution format. 

All conventional approaches (mathematical and stochastic programming, parametric and nonparametric regression analysis) adopt as a common solution format real vectors, x and as performance criterion,... 

These conceptual changes in both solution formats and performance... 

If one follows the conventional optimization paradigm, adopting point-wise solution formats, the best feasible answer, x, which minimizes fix), is X = (200 17.9), as can be confirmed by examining the contour plots of / shown in Fig. 5a. 

Finally, it should be added that the conventional problem statement and pointwise solution format can be interpreted as a particular degenerate case of our more general formulations. As the minimum acceptable size for zones in the decision space decreases, the different performance criteria converge to each other and X gets closer and closer to x. Both approaches become exactly identical in the extreme limiting case where Ax = 0, m = 1,..., M, which is the particular degenerate case adopted in traditional formulations. 

In the previous paragraphs we defined the solution format f, performance criterion i/r, mapping procedure /, and performance metric y that characterize our learning methodology for systems with a quantitative metric y. Here we will assemble all these pieces together and briefly discuss the search procedure, S (further details can be found in Saraiva... 

As final solution formats, interval vector decision policies, X p, replace their pointwise counterparts, x p. Thus, a decision policy, Xpp, in the context of this section is an interval vector whose components are intervals of decision variables associated with one or more of the infimal decision units. No connection variables or disturbance factors are involved in their definition ... 

Fig. 4.10 Scheme of (a) coUoid solution formation (b) selenides particles deposition onto the electrode surface. (Reproduced from [107], Copyright 2009, with permission from Elsevier)... 

Rawashdeh-Omary, M.A., Omary, M.A., Patterson, H.H. and Fackler, J.P. Jr (2001) Excited-state interactions for [Au(CN)2 ]n and [Ag(CN)2 ]n oligomers in solution. Formation of luminescent gold-gold bonded excimers and exciplexes./oumol of the American Chemical Society, 123, 11237-11247. 

For a range of simple substitutional solid solutions to form, certain requirements must be met. First, the ions that replace each other must be isovalent. If this were not the case, other structural changes (e.g., vacancies or interstitials) would be required to maintain electroneutrality. Second, the ions that replace each other must be fairly similar in size. From a review of the experimental results on metal alloy formation, it has been suggested that 15% size difference can be tolerated for the formation of a substantial range of substitutional solid solutions. For solid solutions in nomnetal-lic systems, the limiting difference in size appears to be somewhat larger than 15%, although it is very difficult to quantify this. To a certain extent, this is because it is difficult to quantify the sizes of the ions themselves, but also because solid solution formation is very temperature dependent. 

Cation Vacancies If the cation of the host structure has a lower charge than the cation that is replacing it, cation vacancies may be introduced for the preservation of electroneutrality. Alternatively, the substitution of an anion by one of lower charge may also achieve this in certain systems. For example, NaCl is able to dissolve a small amount of CaCl2, and the mechanism of solid-solution formation involves the replacement of two Na+ ions by one Ca ion, leaving one vacancy on the Na" sublattice, Nai 2xCa Cl (where x denotes a vacancy). 

A number of models have been proposed to describe the solution formation process [505-509], some of which can be extended to Include chromatographic processes and other solvent-dependent phenomena. In terms of chromatographic aiqplications the most useful are the solubility parameter concept, solvatochromic parameters and Snyder s solvent strength and selectivity... 

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