Solid solution theory

When other elements dissolve in a metal to form a solid solution they make the metal harder. The solute atoms differ in size, stiffness and charge from the solvent atoms. Because of this the randomly distributed solute atoms interact with dislocations and make it harder for them to move. The theory of solution hardening is rather complicated, but it predicts the following result for the yield strength... 

Mott played a major part, with his collaborator Frank Nabarro (b. 1917) and in consultation with Orowan, in working out the dynamics of dislocations in stressed crystals. A particularly important early paper was by Mott and Nabarro (1941), on the flow stress of a crystal hardened by solid solution or a coherent precipitate, followed by other key papers by Koehler (1941) and by Seitz and Read (1941). Nabarro has published a lively sequential account of their collaboration in the early days (Nabarro 1980). Nabarro originated many of the important concepts in dislocation theory, such as the idea that the contribution of grain boundaries to the flow stress is inversely proportional to the square root of the grain diameter, which was later experimentally confirmed by Norman Fetch and Eric Hall. 

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... 

A dependence of w upon composition must also be adduced in the case of the Fe-Ni solid solutions. Over the range from 0 to 56 at. per cent Ni, these solid solutions exhibit essentially ideal behavior,39 so that w 0. Since the FeNi3 superlattice appears at lower temperatures, either w is markedly different at compositions about 75 at. per cent Ni than at lower Ni contents, or w 0 for the solid solutions about the superlattice. Either possibility represents a deviation from the requirements of the quasi-chemical theories. 

Other ordering systems show striking discrepancies with the predictions of the quasi-chemical theories. Cu-Pt,67 Co-Pt,38 and Pb-Tl36 are binaries the solid solutions of which exhibit a positive partial excess free energy for one of their components, as well as positive excess entropies of solution. Co-Pt goes even further in deviating from theory in that it has a positive enthalpy of solution,... 

Table II also demonstrates the discrepancy existing between E0/RTe calculated by the Yang-Li quasi-chemical theory and the experimental ratio. E0 is the energy difference between a fully ordered superlattice and the corresponding solid solution with an ideally random atom species distribution. It is a quantity that can only be estimated from existing experimental information, but the disparity between theory and experiment is beyond question.

Whereas the quasi-chemical theory has been eminently successful in describing the broad outlines, and even some of the details, of the order-disorder phenomenon in metallic solid solutions, several of its assumptions have been shown to be invalid. The manner of its failure, as well as the failure of the average-potential model to describe metallic solutions, indicates that metal atom interactions change radically in going from the pure state to the solution state. It is clear that little further progress may be expected in the formulation of statistical models for metallic solutions until the electronic interactions between solute and solvent species are better understood. In the area of solvent-solute interactions, the elastic model is unfruitful. Better understanding also is needed of the vibrational characteristics of metallic solutions, with respect to the changes in harmonic force constants and those in the anharmonicity of the vibrations. 

The theory of dilute solid solutions, in which one component is present in small amount only, has already been considered... 

Alloys of lead and thallium have a structure based upon cubic closest packing from 0 to about 87-5 atomic percent thallium. The variation of the lattice constant with composition gives strong indication that ordered structures PbTl, and PbTl, exist. In the intermediate ranges, solid solutions of the types Pb(Pb,Tl)a and Pb(Pb,Tl)TlB exist. Interpretation of interatomic distances indicates that thallium atoms present in low concentration in lead assume the same valence as lead, about 2-14, and that the valence of thallium increases with increase in the mole fraction of thallium present, having the same value, about 2-50, in PbTls and PbTl, as in pure thallium. A theory of the structure of the alloys is presented which explains the observed phase diagram,... 

Between 1865 and 1887, Dmitri 1. Mendeleev developed the chemical theory of solutions. According to this theory, the dissolution process is the chemical interaction between the solutes and the solvent. Upon dissolution of salts, dissolved hydrates are formed in the aqueous solution which are analogous to the solid crystal hydrates. In 1889, Mendeleev criticized Arrhenius s theory of electrolytic dissociation. Arrhenius, in turn, did not accept the idea that hydrates exist in solutions. 

The loops around the precipitates act as stress concentrators. They exert shearing stresses in addition to the applied stress on the particles. When enough of them have accumulated, the precipitates will be plastically sheared as the loops disappear one by one. This is the basis of a theory of precipitation hardening in an aluminum-copper alloy by Fisher, Hart, and Pry (1953). The precipitate in this case is CuA12, and the precipitates cause an increment of hardening added to the hardness of the solid-solution (Al-Cu) matrix. Quantitative agreement with experimental measurements is fair. 

Gold forms a continuous series of solid solutions with palladium, and there is no evidence for the existence of a miscibility gap. Also, the catalytic properties of the component metals are very different, and for these reasons the Pd-Au alloys have been popular in studies of the electronic factor in catalysis. The well-known paper by Couper and Eley (127) remains the most clearly defined example of a correlation between catalytic activity and the filling of d-band vacancies. The apparent activation energy for the ortho-parahydrogen conversion over Pd-Au wires wras constant on Pd and the Pd-rich alloys, but increased abruptly at 60% Au, at which composition d-band vacancies were considered to be just filled. Subsequently, Eley, with various collaborators, has studied a number of other reactions over the same alloy wires, e.g., formic acid decomposition 128), CO oxidation 129), and N20 decomposition ISO). These results, and the extent to which they support the d-band theory, have been reviewed by Eley (1). We shall confine our attention here to the chemisorption of oxygen and the decomposition of formic acid, winch have been studied on Pd-Au alloy films. 

Since the interplay of theory and experiment is central to nearly all the material covered in this chapter, it is appropriate to start by defining the various concepts and laws needed for a quantitative theoretical description of the thermodynamic properties of a dilute solid solution and of the various rate processes that occur when such a solution departs from equilibrium. This is the subject matter of Section II to follow. There Section 1 deals with equilibrium thermodynamics and develops expressions for the equilibrium concentrations of various hydrogen species and hydrogen-containing complexes in terms of the chemical potential of hy-... 

A calculation procedure could, in theory, predict at once the distribution of mass within a system and the equilibrium mineral assemblage. Brown and Skinner (1974) undertook such a calculation for petrologic systems. For an -component system, they calculated the shape of the free energy surface for each possible solid solution in a rock. They then raised an n -dimensional hyperplane upward, allowing it to rotate against the free energy surfaces. The hyperplane s resting position identified the stable minerals and their equilibrium compositions. Inevitably, the technique became known as the crane plane method. 

Another defect problem to which the ion-pair theory of electrolyte solutions has been applied is that of interactions to acceptor and donor impurities in solid solution in germanium and silicon. Reiss73>74 pointed out certain difficulties in the Fuoss formulation. His kinetic approach to the problem gave results numerically very similar to that of the Fuoss theory. A novel aspect of this method was that the negative ions were treated as randomly distributed but immobile while the positive ions could move freely. 

Although the theory of solutions has been widely used in formulating problems of defects in solids the problems encountered differ in certain respects. The most obvious point is that defects are restricted to discrete lattice sites, whereas the ions in a solution can occupy any position in the fluid. Sometimes no allowance is made for this fact. For example, it has not been demonstrated that at very low concentrations, in the absence of ion-pair effects, the activity coefficients are identical with those of the Debye-Hiickel theory. It can be plausibly argued51 that at sufficiently low concentrations the effect of discreteness is likely to be negligible, but clearly in developing a theory for any but the lowest concentrations the effect should be investigated. A second point... 

But many computations of phase-formation based on the application of pseudo-potential, quantum-mechanical techniques, statistic-thermodynamic theories are carried out now only for comparatively small number of systems, for instance [1-3], A lot of papers dedicated to the phenomenon of isomorphic replacement, arrangement of an adequate model of solids, energy theories of solid solutions, for instance [4-7], But for the majority of actual systems many problems of theoretical and prognostic assessment of phase-formation, solubility and stable phase formation are still unsolved. 

Does Surface Precipitation occur at Concentrations lower than those calculated from the Solubility Product As the theory of solid solutions (see Appendix 6.2) explains, the solubility of a constituent is greatly reduced when it becomes a minor constituent of a solid solution phase (curve b in Fig. 6.10).Thus, a solid species, e.g., M(OH)2 can precipitate at lower pH values in the presence of a hydrous oxide (as a solid solvent), than in its absence. 

Ihe present paper is intended to review the most important literature in this field and to extend the theory from the widely accepted ideal solid solutions to the more general models of regular solid solutions ( 5), with and without ordering (6 ) or substitutional disorder (2, b, 1). 

The distribution of componentsof binary solid solutions over the solid phase and the aqueous phase has been studied for a number of systems. Table I contains a summary of some of these systems with references. This literature review is not complete more data are available especially for rare earth and actinide compounds, which primarily obey type I Equations to a good approximation. In the following sections, the theory above will be applied to some special systems which are relevant to the fields of analytical chemistry, inorganic chemistry, mineralogy, oceanography and biominerals. 

As described in Section 15.2.1, eutectic systems can be purified in theory by single-stage crystallisation, whereas solid solutions always require multistage operations. Countercurrent fractional crystallisation processes in column crystallisers are described in Section 15.4.3. 

Investigations of the vibrational spectra of spinels of the II—III type 177) and L1YX408 type 178), which are not built up of discrete units, have also been reported. In normal cubic II—III spinels four bands are observed in the IR spectrum, as predicted by theory 177). The analysis of the spectra for solid solutions as well as isotopic data indicate that vi and V2 correspond roughly to the vibrations of the octahedrally co-ordinated trivalent cation. In comparison, vs and Vi cor-... 

The fibrils are produced as the result of the partial coalescence of a disperse medium consisting of a highly viscous liquid or a solid solution. According to the solid-liquid theory the fibrils are produced by the coalescence of the dispersed sol in the form of... 

The mechanism of dyeing (see First Report on Colloid Chemistry, B. Ass. 1917, for the complete rdsum6 of the various theories) has received a great deal of attention and has led to the formulation of a variety of hypotheses such as the solid solution, the chemical, the adsorption and electrical hypothesis. A critical examination of the data on which these various alternative hypotheses rests would lead one to suppose that they are by no means irreconcilable with one another. 

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