Hume-Rothery rules, named after

William Hume-Rothery William Hume-Rothery OBE FRS (15 May 1899 – 27 September 1968) was an English metallurgist and materials scientist who studied the constitution of alloys. Early life and education Hume-Rothery was born the son of lawyer Joseph Hume-Rother ...

, are a set of basic rules that describe the conditions under which an element could dissolve in a

metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...

, forming a

solid solution A solid solution, a term popularly used for metals, is a homogenous mixture of two different kinds of atoms in solid state and have a single crystal structure. Many examples can be found in metallurgy, geology, and solid-state chemistry. The word ...

. There are two sets of rules; one refers to substitutional solid solutions, and the other refers to

interstitial An interstitial space or interstice is a space between structures or objects. In particular, interstitial may refer to: Biology * Interstitial cell tumor * Interstitial cell, any cell that lies between other cells * Interstitial collagenase, ...

solid solutions.

Substitutional solid solution rules

For substitutional solid solutions, the Hume-Rothery rules are as follows: # The

atomic radius The atomic radius of a chemical element is a measure of the size of its atom, usually the mean or typical distance from the center of the nucleus to the outermost isolated electron. Since the boundary is not a well-defined physical entity, there ...

of the solute and

solvent A solvent (s) (from the Latin '' solvō'', "loosen, untie, solve") is a substance that dissolves a solute, resulting in a solution. A solvent is usually a liquid but can also be a solid, a gas, or a supercritical fluid. Water is a solvent for ...

atoms must differ by no more than 15%: #:

\% \text = \left ( \frac \right ) \times 100\% \le 15\%.

# The

crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...

s of solute and solvent must be similar. # Complete

solubility In chemistry, solubility is the ability of a substance, the solute, to form a solution with another substance, the solvent. Insolubility is the opposite property, the inability of the solute to form such a solution. The extent of the solubil ...

occurs when the solvent and solute have the same valency. A metal is more likely to dissolve a metal of higher valency, than vice versa. # The solute and solvent should have similar

electronegativity Electronegativity, symbolized as , is the tendency for an atom of a given chemical element to attract shared electrons (or electron density) when forming a chemical bond. An atom's electronegativity is affected by both its atomic number and the d ...

. If the electronegativity difference is too great, the metals tend to form

intermetallic compounds An intermetallic (also called an intermetallic compound, intermetallic alloy, ordered intermetallic alloy, and a long-range-ordered alloy) is a type of metallic alloy that forms an ordered solid-state compound between two or more metallic elemen ...

instead of solid solutions.

Interstitial solid solution rules

For

solid solutions, the Hume-Rothery Rules are: # Solute atoms should have a smaller radius than 59% of the radius of solvent atoms. # The solute and solvent should have similar

. # Valency factor: two elements should have the same valence. The greater the difference in valence between solute and solvent atoms, the lower the solubility.

Solid solution rules for multicomponent systems

Fundamentally, the Hume-Rothery rules are restricted to binary systems that form either substitutional or interstitial solid solutions. However, this approach limits assessing advanced alloys which are commonly multicomponent systems. Free energy diagrams (or

phase diagrams A phase diagram in physical chemistry, engineering, mineralogy, and materials science is a type of chart used to show conditions (pressure, temperature, volume, etc.) at which thermodynamically distinct phases (such as solid, liquid or gaseous ...

) offer in-depth knowledge of equilibrium restraints in complex systems. In essence the Hume-Rothery rules (and

Pauling's rules Pauling's rules are five rules published by Linus Pauling in 1929 for predicting and rationalizing the crystal structures of ionic compounds. First rule: the radius ratio rule For typical ionic solids, the cations are smaller than the ani ...

) are based on geometrical restraints. Likewise are the advancements being done to the Hume-Rothery rules. Where they are being considered as critical contact criterion describable with

Voronoi diagrams In mathematics, a Voronoi diagram is a Partition of a set, partition of a plane (geometry), plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, s ...

. This could ease the theoretical phase diagram generation of multicomponent systems.

Substitutional solid solution rules

Interstitial solid solution rules

Solid solution rules for multicomponent systems

See also

References

Further reading