Solid Solutions and the Hume-Rotherty Criteria

5 Phase Diagram for Ideal (Isomorphic) Systems 12,5.1 Solid Solutions and the Hume-Rotherty Criteria 

Complete solid solubility for a binary system means that atoms of one constituent can be substituted for the other over the entire range of compositions in both the solid and liquid state without introducing a different phase. Such alloys are said to be isomorphous. Hume-Rothery developed a set of criteria that must be met in order for a system to exhibit complete solid solubility  

Given the similarity of atomic radii between the 4d and 5d transition metals, one would expect those pairs in the same column to be isomoiphous. This is the case for Hf-Zr, Mo-W, Ru-Os, and Au-Ag. Si-Ge is also isomorphous as are several compoimd systems such as HgTe-CdTe, Al203-Cr203 that exhibit pseudobinary isomorphous phase diagrams. 

Some mixed valency systems such as Cu-Ni and Ag-Pd are isomorphic. Other systems meet all of the Hume-Rothery criteria but are immiscible in the liquid state (e.g., Nb-Ta) or are completely miscible in the melt but decompose spinodally in the solid state (e.g., Pd-Pt). One must conclude that there is nothing hard and fast about the Hmne-Rothery rules and they should be used only as a guide to estimating which systems may be isomorphous. 

We will now set out to compute the free energies for the solid and liquid phases of an ideal isomorphous binary system and the resulting phase diagram using a very simple model. If we add the entropy of mixing (Equation 12.3) to the dhemical potentials of pure A and pure B solid and we can write the free energy of the solid as 

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