Substitutional disorder, solid solutions

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the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations.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. 

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... 

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. 

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. 

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... 

The alloys just considered are substitutional solid solutions. Interstitial solid solutions are alloys with small atoms, for example, H, C, N, and O, in the interstitial sites, usually O and T sites. Some alloys have random distribution (disordered) if the melt is quenched but become ordered if heated and annealed or if cooled slowly. An example is the 1 1 alloy CuAu. The disordered structure is ccp, and the ordered structure is also ccp, except alternate layers parallel to a cell face contain Cu or Au. 

As the amount of Fe is increased, the (111) peak shifts to smaller d-spacings, reflecting a contraction of the lattice. The (111) peak positions in Fig. 11.5 show a continuous shift from pure Pt to pure Fe. The Pt-Fe XRD patterns are consistent with a single-phase, substitutional solid solution (disordered alloy) over the entire compositional range. In contrast, Fig. 11.6 clearly displays diffraction from inter-metallic compounds of lower symmetry. Post-deposition annealing has resulted in an ordering of the Pt and Fe atoms, the effect of which is the crystallization of an ordered metal alloy of lower symmetry than 100% Pt. In essence, the applied vacuum deposition method is ideally suited for the preparation of multi-component,... 

For comparison, Al(III) and Sc(III) were substituted for Mn(III) (Goodenough et al., 2002). These ions also perturb the periodic potential of the MO3 array to maintain localized e electrons at the Mn(III) ions where the orbitals become disordered. In these solid-solution... 

The Kirkendall effect arises from the different values of the self-diffusion coefficients of the components of a substitutional solid solution, determined by Matano s method. Matano s interface is defined by the condition that as much of the diffusing atoms have migrated away from the one side as have entered the other. If DA = DB, its position coincides with the initial interface between phases A and B. If I)A f DB, it displaces into the side of a faster diffusant (see Fig. 1.22c). Note that KirkendalFs discovery only relates to disordered phases. It was indeed a discovery since at that time most reseachers considered the relation l)A = DB to hold for any solid solution of substitutional type. KirkendalFs experiments showed that in fact this is not always the case. 

For the present applications it is relevant to ask how accurate is the intersection of the Ewald sphere with the reciprocal lattice point. Each of these points represents a (series of parallel) lattice planes defined by atom positions in the unit cell. The thermal motion of the atoms expands the ideal plane into a slab. Elements of disorder such as microstrain, chemical impurities in lattice positions (substitution, solid solutions), and interstitial atoms producing "chemical microstrain" also expand the lattice planes effectively into lattice slabs of locally varying thickness. 

Substitutional solid solutions can have any composition within the range of miscibility of the metals concerned, and there is random arrangement of the atoms over the sites of the structure of the solvent metal. At particular ratios of the numbers of atoms superstructures may be formed, and an alloy with either of the two extreme structures, the ordered and disordered, but with the same composition in each case, can possess markedly different physical properties. Composition therefore does not completely specify such an alloy. Interstitial solid solutions also have compositions variable within certain ranges. The upper limit to the number of interstitial atoms is set by the number of holes of suitable size, but this limit is not necessarily reached, as we shall see later. When a symmetrical arrangement is possible for a particular ratio of interstitial to parent lattice atoms this is adopted. In intermediate cases the arrangement of the interstitial atoms is random. 

OPCM route of ceria—zirconia solid solutions synthesis appears to be reasonably flexible approach allowing to make rather broad substitution in cation and anion sublattices of fluorite-like oxides without appearance of new phases provided homogeneous mixing of starting salts is ensured. Reasonably high dispersion of samples thus obtained and disordered bulk and surface structure makes them promising candidates as catalysts and supports for red-ox reactions. 

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. 

There is a third form of solid state organization of racemic compounds, called the sohd solution or pseudoracemic compound. This is a rare form in which the arrangement of the enantiomers in the crystal is variable and not defined. The significance of this is that, while the stoechiometry of the two enantiomers in the whole crystal may well be 1 1, it is possible to find homochiral unit cell structures, one enantiomer being substituted for the other. Such disorder arises from the strong similarity of the two enantiomers, particularly in the case of quasispherical molecules. This arrangement is also called a racemic solid solution of the two enantiomers. 

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