Substitutional disorder, solid

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

In somewhat earlier work, Vlot et al. [229,230] made calculations of Lennard-Jones binary mixtures in which the pure components are identical but in which the unlike interactions have departures from the Lorentz-Berthelot combining rules. They use this as a model of mixtures of enantiomers. A variety of solid-fluid phase behavior can be obtained from the model. Both substitutionally ordered and substitutionally disordered solid solutions were found to occur. 

So far, the solids that we have studied have been ordered, in the sense that they possess perfect translational symmetry. However, this perfection is really an idealization and, in reality, an actual crystal can be expected to have some sort of disorder, which breaks the long-range periodicity of the lattice. There are a number of ways in which disorder can arise. For instance, interstitial disorder occurs when an impurity atom is placed in the vacant space between two substrate atoms, which remain at their original locations in the lattice. Another situation is that of structural disorder, where the substrate atoms move away from their positions on the perfect lattice. However, the situation of interest in this chapter is that of substitutional disorder. Here, a perfect lattice of one type of atoms (say, A) has some of its members randomly replaced by another type (B). The result is a structurally periodic lattice, but with the constituent atoms A and B randomly placed on the lattice sites. The relative numbers of A and B atoms can be represented by the concentrations ca and cB, with ca + cB = l. The randomness of this type of solid introduces a level of difficulty into the theory, that we have not yet encountered. 

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). 

DISTRIBUTION LAWS AND SUBSTITUTIONAL DISORDER Driessens (2 ) has discussed the consequences of substitutional disorder on component activities in solid solutions. For example, solid solutions of the Formula ... 

In this way and by numerical evaluation, Driessens (2) proved that the experimental activities could be explained on the basis of substitutional disorder, according to Equation (27), within the limits of experimental error. It seems, therefore, that measurements of distribution coefficients and the resulting activities calculated by the method of Kirgintsev and Trushnikova (16) do not distinguish between the regular character of solid solutions and the possibility of substitional disorder. However, the latter can be discerned by X-ray or neutron diffraction or by NMR or magnetic measurements. It can be shown that substitutional disorder always results in negative values of the interaction parameter W due to the fact that... 

This is also valid for the more complex spinel solid solutions of FejO, Mn304 and CO3O4, in which electron exchange occurs in addition to substitutional disorder (2). 

Substitutional Disorder In Regular Solid Solutions. Most simple ionic solutions in which substitution occurs in one sublattice only are not ideal, but regular 2, J3) Most complex ionic solid solutions in which substitution occurs in more than one sublattice are not only regular in the sense of Hildebrand s definition (15) but also exhibit substitutional disorder. The Equations describing the activities of the components as a function of the composition of their solid solutions are rather complex ( 7, V7, 1 ), and these can be evaluated best for each individual case. Both type II and type III distributions can result from these conditions. 

Wolska, E. (1990) Studies on the ordered and disordered aluminium substituted maghemites. Solid State Ionics 44 119-123 Wolska, E. Szajda.W. Piszora, P. (1992) Determination of solid solution limits based on the thermal behaviour of aluminium substituted iron hydroxides and oxides. J. Thermal Analysis 38 2115-2122 Wolski.W. (1985) Das Eisenoxidgelb. Farbe Lack 91 184-189... 

Although most of the studies of this model have focused on the fluid phase in connection with the theory of electrolyte solutions, its solid-fluid phase behavior has been the subject of two recent computer simulation studies in addition to theoretical studies. Smit et al. [272] and Vega et al. [142] have made MC simulation studies to determine the solid-fluid and solid-solid equilibria in this model. Two solid phases are encountered. At low temperature the substitutionally ordered CsCl structure is stable due to the influence of the coulombic interactions under these conditions. At high temperatures where packing of equal-sized hard spheres determines the stability a substitutionally disordered fee structure is stable. There is a triple point where the fluid and two solid phases coexist in addition to a vapor-liquid-solid triple point. This behavior can be qualitatively described by using the cell theory for the solid phase and perturbation theory for the fluid phase [142]. Predictions from density functional theory [273] are less accurate for this system. 

Materials that contain defects and impurities can exhibit some of the most scientifically interesting and economically important phenomena known. The nature of disorder in solids is a vast subject and so our discussion will necessarily be limited. The smallest degree of disorder that can be introduced into a perfect crystal is a point defect. Three common types of point defect are vacancies, interstitials and substitutionals. Vacancies form when an atom is missing from its expected lattice site. A common example is the Schottky defect, which is typically formed when one cation and one anion are removed from fhe bulk and placed on the surface. Schottky defects are common in the alkali halides. Interstitials are due to the presence of an atom in a location that is usually unoccupied. A... 

There are three different substituted benzene compounds with the formula C6H4F2. Assume that the benzene rings pack similarly into their crystal lattices. If the positions of the H and F atoms are statistically disordered in the solid state, which isomer will have the least residual molar entropy ... 

Anion Interstitials The other mechanism by which a cation of higher charge may substitute for one of lower charge creates interstitial anions. This mechanism appears to be favored by the fluorite structure in certain cases. For example, calcium fluoride can dissolve small amounts of yttrium fluoride. The total number of cations remains constant with Ca +, ions disordered over the calcium sites. To retain electroneutrality, fluoride interstitials are created to give the solid solution formula... 

Mutual solid-state solubility a simple structural representation - order/ disorder. In a number of systems such as the previously described V-Mo and Cs-Rb, continuous solid solutions are formed in the whole range of compositions, characteristics and structures of which will be discussed in more detail in Chapter 3. These result from two metals having the same crystal structure, which is maintained for all the intermediate compositions, due to a continuous random substitution of the atoms of one kind for another and vice versa. 

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