In this article we will discuss about:- 1. Introduction to Solid Solution of Metals 2. Types of Solid Solutions 3. Hume Rothery’s Rules for Primary Substitutional Solid Solubility.

Introduction to Solid Solution:

When the liquid solution of two metals crystallises, and if a solid of only single crystal structure forms, then a solid solution has formed. This happens when atoms of two metals are able to share together a given crystal structure (normally of the solvent metal), such that even in a unit cell of this crystalline solid, both type of atoms are present in proportion to their concentration, Fig. 2.2 (c). Thus, a solid solution of two (or more) elements has a single crystal structure and constitutes a single phase.

All metals are mutually soluble, at least to some degree in the solid state. For example, copper (FCC) is able to dissolve up to 38.4 weight percent to zinc (HCP) without destroying the FCC crystal lattice, and up to 5.5 weight percent copper can dissolve in aluminium (FCC).


In general, a solid solution requires a more special solvent-solute relationship than does a liquid solution, because the crystal structure of the solid is less adaptable. However, some metals like copper and nickel show complete solid solubility from 100% copper to 100% nickel, i.e., they are soluble in each other in the solid state in any proportion to give a series of solid solutions (having different proportions of metals), but with the same single FCC crystal structure.

These are called extended solid solutions. Fig. 2.3 illustrates three of them with three different compositions (90 at % Cu, 10 at % Ni; 50 at % Cu, 50 at % Ni; 10 at % Cu, 90 at % Ni). The crystal structure remains FCC in all the three cases with sight decrease in lattice parameter as the nickel content increases (as lattice constant of nickel is smaller).

Alloy systems, Au-Ag (both FCC), Fe-Cr (BCC both), Mg-Cd (HCP both) also show extensive solid solubilities. The solid solutions occur over a range of compositions, which is basically due to the nature of the metallic bond (attraction between position ions and freely moving electrons between them). This bond is indifferent to a large extent both to the precise proportions of the component atoms, and also to their precise distribution in the crystalline array of atomic sites.

Types of Solid Solutions:

There are two main types of solid solutions:


(i) Substitutional solid solutions.

(ii) Interstitial solid solutions.

(i) Substitutional Solid Solutions:


Two elements (or more) form a substitutional solid solution, when atoms of the solute element substitute the atoms of solvent (also called matrix atoms) in its crystal structure. Atoms share a single common array of atomic sites. Fig. 2.2 (c) illustrates a unit cell of Cu-Ni solid solution, in which three copper atoms of the copper unit cell have been substituted by three nickel atoms but randomly. These three copper atoms must have joined the lattice elsewhere, during the freezing itself.

Since the size and the electronic structure of the solvent and the solute atoms always differ, the crystal lattice of the solvent metal always gets distorted whenever a solid solution is formed. The lattice parameter either increases, (if solute atom is larger than solvent atom, just when copper is added in nickel lattice, or solute atom is in interstitial site), or decreases (when solute atom is smaller).

This is illustrated in Fig. 2.4. This distortion interferes with the movement of dislocations on slip planes, and thus, increases the strength of the alloy. This is the primary cause which strengthens a metal by alloying.

In many substitutional solid solutions, at certain fixed ratio of solute and solvent atoms (particularly at low temperatures), a certain order in the arrangement of solute and solvent atoms takes place, that is, an ordered solid solution is formed.


An ordered solid solution is a substitutional solid solution in which the atoms arrange themselves in a preferred manner, that is, the two species are arranged in some regular alternating pattern as illustrated in Fig. 2.3 (b), whereas Fig. 2.3 (a) illustrates a random solid solution in which the substitution of atoms has taken place at random.

This preference to form unlike pair of atoms to form an ordered solid solution is expressed as:

where, EXY is the energy of an unlike-bond, and EXX and EYY are the energies of like-bonds between X – X and Y – Y atoms respectively. If EXY, the energy of the unlike bonds is very much smaller, then a long range order, i.e., an order over large distances, may take place. To illustrate this point in a unit ceil, let us take the case of beta-brass, which is BCC.


There are two effective atoms per unit cell in BCC, one atom due to all the corner atoms and the other is the body centred atom. If the solid solution has equal atomic proportions of copper and zinc atoms, we have a stoichiometric composition, where all the corner point of the BCC unit cells may be occupied by zinc atoms and all the body centred points may be occupied by the copper atoms (ratio is 1:1).

A perfect order is produced as illustrated in Fig. 2.5 (a). Each zinc atom is surrounded by eight copper atoms and each copper atom is surrounded by eight zinc atoms. This alloy undergoes a change from disordered (random) state to ordered state on cooling below a critical ordering temperature (470°C in (his case), when a superlattice is formed by the regular alternation of unlike atoms through the whole crystal, or atleast through a large region of it.

A perfect super lattice is possible only at a critical and simple proportion of two types of atoms, i.e., 1 to 1 or 3 to 1. This is illustrated in Fig. 2.5 (b) and (c). Here, the formula of a chemical compound such as CuAu and Cu3Au respectively can be assigned to such ordered solid solutions. An ordered solid solution is different than a chemical compound because crystal structure of the solvent metal is retained, and above a critical temperature, the solution becomes disordered.

Interstitial Solid Solution:

Here, solute atoms are much smaller than the solvent atoms and thus, occupy randomly interstitial space (inter atomic space) in between the solvent atoms in the crystal lattice of the solvent. The solute atoms do not occupy lattice sites as illustrated in Fig. 2.2 (b). For example, carbon atoms dissolve in FCC-iron (gamma-iron) by occupying the interstitial space of FCC-gamma iron structure.

Fig. 2.6 illustrates one such interstitial space-octahedral void-in which carbon atom sits. Small carbon atom fits here better but lattice distortion is still sufficient to make it impossible to locate carbon atoms at all the interstitial sites in FCC-iron structure, as illustrated in 2.6 (b).

In an interstitial solid solution, the lattice parameter of the crystal structure of the solvent always increases. Such solutions are called interstitial solid solutions as illustrated in 2.6 (b). Atoms of only few elements are small enough to act as interstitial solutes in metals. Table 2.1 lists atoms with their atomic radii which form interstitial solid solutions. The atomic radius of all of them is less than 1 A.

In multi components alloys, some atoms of the alloying elements may dissolve substitutionally while other like carbon interstitially. In manganese steel, manganese atoms replace (substitute) iron atoms on the lattice points but carbon atom enters the interstice as illustrated in Fig. 2.7.

Hume Rothery’s Rules for Primary Substitutional Solid Solubility:

The pioneering work of Hume Rothery on a number of alloy systems led to the formulation of conditions that favour extensive primary substitutional solid solubility.

These empirical conditions are called Hume Rothery’s rules (there are numerous exceptions to these rules):

1. Atomic-Size Factor:

The more the difference in the size of the solute atom and the solvent atom, the smaller is the solid solubility. For complete solid solubility, the size factor must be less than 8%. It the atomic diameters differ by more than 15 percent, the size factor is unfavorable and the solid solubility is small.

Metals like silver and gold have a difference of 0.2%; nickel and copper of 2.7%, and show complete solid solubility. But zinc and copper have 4.2% difference with maximum solubility of 38.4 wt.% Zn. (other factors are less favourable); Cadmium in copper with 16.5% size difference shows a solid solubility of 1.7 wt.%.

The size factor comes into play due to the elastic strains caused in the crystal lattice around a misfitting solute atom. The atoms of the solvent are pushed out or pulled in accordingly as the solute atom is larger or smaller than the solvent atom (or crystal site) as illustrated in Fig. 2.4 (a) and (b).

This change in the interatomic spacing from the ideal value increases the energy of the crystal. As the difference in sizes becomes more, more are then the lattice strains, lesser becomes the solubility of the solute.

As this difference in the effect of size factor, when other three factors are favourable increases to 15%, the equilibrium diagram changes from isomorphous system of Cu-Ni type to one of eutectic system of Cu-Ag with partial solid solubility (An unfavourable relative size factor alone). If the size factor is favourable, the other factors should then be considered for deciding about probable degree of solid solubility.

2. Electro-Chemical Factor:

If one solid solution forming element is more electropositive and the other element is more electronegative, there is a greater tendency to form a compound than a solid solution, and the solid solubility becomes smaller. The electro-chemical behaviour can be computed from the Pauling’s electronegativity values.

The typical metallic elements normally have much less difference in values as given in Table 2.2 and thus, form solid solutions. But a non-metallic element like sulphur has a value of 2.5, and thus tries to form sulphides than to dissolve in elements. P, As, Sb and Bi generally form compounds with metals such as Mg and Li. but with lesser electropositive metals, such as Cu and Ag, there is some solubility if size factor is favourable.

3. Crystal-Structure Factor:

The crystal structure of the solute and the solvent metal should be of same type to get complete solid solubility. There shall be a continuous change of crystal lattice parameter of one metal to that of the other metal with the increase of the concentration of the solute metal.

4. Relative-Valency Effect:

Other factors being equal, a metal of lower valency is more likely to dissolve the metal of higher valency than vice versa. It is better that both the elements have same valency. This rule is valid mainly for the alloys of copper, silver or gold. This factor is a reflection of tolerance of metallic bond to changes in electron concentrations.

It is generally found that an excess of electrons is readily tolerated rather than a deficiency of valence electrons. For example, zinc (divalent) dissolves up to 38.4 wt. % in copper (monovalent), whereas copper dissolves up to only about 3 weight % in zinc.

If the mutual solid solubility is restricted (as in Cu-Ag system) to only those portions of the phase diagram that are linked to the pure elements, the solid solutions formed are called as primary (or terminal) solid solutions, which have same crystal structure as of solvent metals. If other phases are present in the system, they are usually called intermediate phases, or intermetallic phases, having different crystal structure than either of the two elements. Such an intermediate phase having a range of solubility is called secondary solid solution.


From the following data, predict whether aluminium, nickel or chromium as solute metal shall form extensive substitutional solid solubility in solvent copper.


All the solute metals are within ± 15% difference radius range, but Al has more than ± 8% difference which reduces chance of 100% solubility. The electronegativity values are not vastly different, though Ni has more similarity than others. The crystal structure of Cr is BCC as compared to FCC of copper, which reduces extent of solubility.

Thus, Ni in copper has same crystal structure, similar electronegativity and just – 2.3% difference in atomic radius. There is good chance of extensive solid solubility. Al in copper has large difference of atomic radius and more difference in electronegativity, and same crystal structure may give around 10% solid solubility. Cr in copper has low% radius difference, little more difference in electronegativity (than Ni) but different crystal structure may lead to low solid solubility.

The observed experimental values are: