Alloys Hume-Rothery rules

If a Brillouin zone is nearly full or nearly empty, the electron concentration may become an important factor in determining alloy stability. This factor forms the basis of the Hume-Rothery rules (8) and is of great importance in intermetallie semiconductors. 

In general, for a mixture of two or more pure elements, there are two types of solid-solution alloys that may be obtained. Type I alloys are completely miscible with one another in both liquid and solid states. As long as the Hume-Rothery rules are satisfied, a random substitutional alloy will be produced. We will see many examples of these alloys in this section for a variety of metal dopants in stainless steels. By comparison, type II alloys are only miscible in the molten state, and will separate from... 

Certain other alloys also with the body-centred cubic structure, LiHg, MgTl, etc., which have already been mentioned, have sometimes been regarded as exceptions to the Hume-Rothery rules. They fall into line with the other P structures only if we assume that Hg or T1 provides one valence electron. Since, however, the radii of the metal atoms in these alloys, as in those with the NaTl structure, are smaller than the normal values, it is probably preferable not to regard these as 3 electron compounds. 

It is not easy to find other data which show that Eg plays a role in the reactions of solids. There is one case for metals which can be used as a test the formation of alloys, which represents, at least in a sense, the reaction of two solid metals with each other. The stability of alloys of varying composition is influenced by factors such as the relative atom sizes, and the number of valence electrons per atom (the Hume-Rothery rules). 

The simpler of these two structures is the caesium chloride arrangement, found in the phases LiHg, LiTl, MgTl, CaTl and SrTl. This is, of course, also the structure of the / phase in the silver-cadmium system and in other electron compounds (fig. 13.11), and for this reason the systems just mentioned are sometimes quoted as exceptions to Hume-Rothery s rule. Apart from this geometrical resemblance, however, these systems have little in common with the electron compounds, and it seems preferable to regard the Hume-Rothery rule as applicable only to alloys of the T2-B1 type. 

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. 

The formation of an alloy follows three simple rules that are known as the Hume-Rothery rules. 

As a more stringent application of the Hume-Rothery rules that govern the alloying of metals, if the difference in radii is less than 8%, the metals will be soluble throughout the full range of compositions. This is the case for nickel and copper, whose radii are 1.49 and 1.45 A, respectively. Hence, there are over 20 different alloys that are used in industry based on the mutual solubility of copper and nickel in... 

Bla67] Blandin, AE, Theoiy of the Hume-Rothery Rules, in Phase Stabdity in Metals and Alloys, P.S. Rudman, J. Stringer, and R.I. Jaffee, Ed., McGraw-HiU, 1967, p. 115-124... 

M.C. Troparevsky, J.R. Morris, M. Daene, Y. Wang, A.R. Lupini, G.M. Stocks, Beyond atomic sizes and Hume-Rothery rules understanding and predicting high-entropy alloys, JOM 67 (2015) 2350-2363. 

An explanation for the oocurroice of structural vacancies has been given from the Hume-Rothery rules. One knows that the ordered B2 compounds exist only if the number of conduction electrons per atom is near to and does not exceed 1.5. This is the case for sttndiiometric NiAI, if one admits that Ni, a transition metal, contributes zero electrons. Then, for an aluminum-rich alloy Ni, if non-stoichiometry occurs via Al,., antisites,... 

The effect of relative atomic size on alloy structures was discussed on p. 134. It also influences the extent of terminal solid solubility in metallic alloys as expressed in the Hume-Rothery 15% rule " and its extension by Darken and Gurry to also take account of the effect of electronegativity difference in limiting solid solubility. The Hume-Rothery rule has now come to be stated that if the sizes of the solute and solvent atoms differ by less than 15%, extended terminal solid solutions may form, whereas if they differ by more, it is very unlikely that extended solid solutions will form. The rule is only permissive for the formation of extended solid solutions at radii differing by less than 15%, since other factors such as, for example, large electronegativity difference may still prevent their forming. 

Hume-Rothery s rule The statement that the phase of many alloys is determined by the ratio.s of total valency electrons to the number of atoms in the empirical formula. See electron compounds. 

Pt from laboratory wastes have been described. Platinum can be successfully electroplated, at least in thin films Many of the Pt alloys follow the Hume-Rothery structural rules, in which the structure adopted can be related to the average valence electron number for the alloy. Pt-Ir alloy (80-20) is especially useful for its ductility. 

Apart from a few general rules, the alloying behaviour of metals is rather empirical. The classical rules of Hume-Rothery [220] explain this behaviour reasonably well. Such factors as size, electronegativity, valency, electron concentration, free energy, formation of intermediate phases and isomorphism are found to influence the alloying tendency of metals. However, size and electronegativity are the two most important factors, and they profoundly influence the solubility of the solute atoms and greatly affect the crystal structures of the alloys. 

The limit of solubility of B in A, to form a substitutional solution, is governed - particularly with metal alloys - by the empirical rules devised by Hume and Rothery. According to these rules, there are four factors which determine the degree of solubility of a substance B in a substance A ... 

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