Solid solutions, rather than chemical compounds, are more likely to form the more similar are the chemical properties of the components. Gold dissolves silver in all proportions and NaCl dissolves KCl. Interstitial solid solutions are often formed when elements that are expected to form small atoms or ions (e.g. H, C, B, O, N) dissolve in a crystal. The two types of solution may in all cases be distinguished by density measurements and measurements of the volume of the unit cell of the solid solution. The density r of the crystal is given by:
where M is the molecular weight (in Mg), V is the volume of the unit cell (in m3 ) and n is the number of formula units per unit cell. In the pure material, n is an integer. In a substitutional solid solution, n is the same integer, but M alters to the average molecular weight M – given by the chemical composition of the solution.5 In an interstitial solution, n is again the same for the solute but the density is increased. In a binary interstitial solution (two components), the value of r is:
where nsi/ns is the ratio of the mole fraction of solute interstitial to that of solvent and Msi is the molecular weight of the solute interstitial. An example of the determination of the type of solid solution is given in Problem 3.9.
The various component atoms of a solid solution are usually randomly distributed among the sites available for them, but in some, below a certain temperature the distribution ceases to be random and what is called ordering occurs. Ordering is most easily described for a metallic solid solution. Part of the (111) plane of a disordered alloy of copper, shown as circles, containing 25 at% of gold, shown as shaded spheres, is shown in Figure 3.28a. Above a temperature of about 390 °C, the copper and gold atoms occur in any of the positions at the lattice points of a c.c.p. lattice. There is no preferred position for gold or copper. Therefore, while on average a (111) plane will have 25 at% gold and 75 at% copper, small regions of the plane chosen at random may have more, or less, than 25 at% gold. The part of the plane of the disordered form of Cu3Au chosen at random in Figure 3.28a happens to be deficient in Au and richer in Cu.
In equilibrium below 375 °C, a (111) plane would appear as in Figure 3.28b – the gold atoms are all surrounded by copper atoms and there is a regular arrangement of both the gold and the copper atoms. Such a structure is called an ordered solid solution. It is apparent from Figure 3.28c that in the fully ordered state for Cu3Au a conventional unit cell can be chosen with the gold atoms at cube corners (0, 0, 0) and the copper atoms at the midpoints of all of the faces. This cell is the primitive unit cell of a simple cubic superlattice. In this particular case, the superlattice is known as the L12 superlattice. The space group symmetry of the crystal structure changes from Fm3 – m in the high-temperature disordered phase to Pm3 – m in the low-temperature ordered phase. In this example, the conventional unit cells of the high-temperature disordered phase and low-temperature ordered phase are identical in volume, but the primitive unit cell of the ordered phase has a volume four times that of the primitive unit cell of the disordered phase.
Order–disorder changes occur in many solid solutions. The fully ordered state is always of lower symmetry than the disordered one, and usually possesses a lattice with larger cell dimensions, which is called a superlattice.
The essential grouping in the ordered state of this alloy, Cu3Au, is one with all the gold atoms surrounded by copper atoms, as shown in Figure 3.28c. When ordering starts in a large crystal it may be ‘out of step’ in the various parts of the crystal. If this occurs, the ordering may be perfect within various regions of the crystal (as in Figures 3.28b and c). These regions are called domains. Where the domains are in contact, the requirement that gold atoms are surrounded by copper atoms is not met, as is shown in Figure 3.28d. The dotted line in Figure 3.28d indicates the trace in (111) of what are called antiphase domain boundaries. In three dimensions, the boundaries are walls separating neighbouring domains. Since the neighbouring atoms are not fully ordered at the domain boundaries, the boundaries represent a source of extra energy in an ordered crystal. Heating a crystal for a long time near to the ordering temperature can lead to their complete removal.
The order we have just described is called long-range order because within any domain one type of lattice site is preferred for a particular atom. Many solid solutions, while not showing long-range order, do not show a truly random distribution of the atoms; unlike atoms occur more frequently as near neighbours than they would by pure chance. Such a state of affairs is called short-range order. This is very common and is shown by some ordered solutions when they are heated above their ordering temperatures.
The structures of some ordered solid solutions are illustrated in Figure 3.29. Examples of materials showing these structures are given in Table 3.5. The B2 (or L20) order– disorder change is characterized by a b.c.c. structure in the disordered state which changes to the caesium chloride structure on ordering. This occurs in CuZn. The perfectly ordered structure has the composition AB (Figure 3.29a). The superlattice is then simple cubic.
The D03 superlattice type with perfectly ordered composition AB3 also has a b.c.c. structure in the disordered state. This superlattice type occurs in Fe3Al. The ordered state is shown in Figure 3.29b. It is most easily described by saying that the superlattice is composed of four equal interpenetrating face-centred cubic lattices with the origins at (0, 0, 0) for lattice 1, ( ) 1 2 , 0, 0 for lattice 3, ( ) 111 , , 444 for lattice 4 and ( ) 3 1 1 , , 444 for lattice 2. The ordered state consists of B atoms occupying the sites of lattices 2, 3 and 4 with A atoms at type 1 lattice sites.
The L12 superlattice has already been described (Figure 3.28c). The fully ordered condition requires the composition AB3, as in Cu3Au. A related superlattice type, also of ideal composition AB3, is called D019 and is typified by Mg3Cd. The disordered state is the closepacked hexagonal structure. The ordered structure (Figure 3.29c) can be described as four interpenetrating h.c.p. structures in parallel orientation, with the c-axis the same length as in the ordered alloy but the lattice parameter aos twice that of the corresponding disordered alloy. The origins of the sublattices in the ordered state are at (0, 0, 0) for sublattice 1, ( ) 1 1, 2 2 , 0 for sublattice 2, ( ) 1 2 , 0, 0 for sublattice 3 and ( ) 1 2 0, , 0 for sublattice 4. B atoms occupy the sites of sublattices 2, 3 and 4 and A atoms are found at the points of sublattice 1.
Reference: https://crystal-algerien1970.blogspot.com/2020/03/crystallography-and-crystal-defects.html
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