Interstitial | Substitutional Solid Solution

Materials are transformed into various phases under the effects of temperature and pressure. The iron is available in BCC form (α-iron), FCC form (y-iron) and BCC form again (δ-iron) at certain temperatures.

The development of alloys, whether non-ferrous brass or ferrous high speed steel (H.S.S.); magnetic alnico or antifriction babbits, require the knowledge of solid solutions. The mixing of atoms of two or more elements requires study of different phases. Solids and their blending follow certain rules in their formation.

It may be Hume-Rothery rules, Gibb’s phase rule or lever rule; a sound knowledge of systems, components and degrees of freedom has to be borne in mind. In this article we are going to equip ourselves with all these information.

Solid Phases in Alloys

Pure metals generally suffer from one or the other deficiency in their mechanical and other properties. They are, therefore, not suitable for most engineering applications. These metals, in their alloy forms, show remarkable improvement in properties. That is why, alloys find wider use.

As an illustration consider aluminum. Its use in pure form is very rare. But its alloys such as alclad, hindalium, duralumin etc. are vastly used.

Composition of alloy: An alloy is composed of two or more elements of which at least one element is a metal. Constitution of an alloy mainly consists of a

  • Base metal, and
  • Alloying elements

The metal constituent present in an alloy with highest proportion is known as the base metal. Other constituents, metallic or nonmetallic, present in the alloy are called alloying elements.

Nichrome, an alloy of nickel and chromium, contains 80% Ni and 20% Cr. Here nickel is base metal while chromium is an alloying element. We may call this alloy as nickel based alloy.

Effects of alloying elements on properties of base metal: Marked changes in the properties of base metal are noticed on adding different percentage of alloying elements. The quantity of alloying element may be small or high. Constantan is chromium based alloy. Percentage of nickel as an alloying element in it is as high as 45% whereas proportion of silicon in silicon steel is only 4%.

The nature and extent of changes in properties of base metals are influenced by the alloying elements. It depends on whether they dissolve or do not dissolve in the base metal, or whether they form a new phase or not.

Forms of solid phases: Solid phases in most alloy systems occur in the following forms.

  1. Solid solution
  2. Intermetallic compound
  3. Intermediate compound

Solid Solution

A homogeneous mixture of atoms of two or more elements in solid state is called as a solid solution. It is a single phase system. Atoms of different elements in it cannot be either mechanically separated or physically distinguished. Solid solution may be classified as follows.

  • Interstitial solid solution, and
  • Substitutional solid solution

Interstitial Solid Solution

Small solute atoms dissolve in the interstitial space available amongst big solvent atoms, and form an interstitial solid solution.

Carbon atoms of atomic radius 0.7 oA form solid solution of this kind in a-iron of atomic radius 1.24 oA. The arrangement is shown in Figure 1(a). Solubility is generally random and limited in nature.

Small atoms of hydrogen, carbon, nitrogen and boron etc. more readily form the interstitial solution in transition metals. This is because the transition metals such as Fe, Ni, Zr, Ti etc. have incomplete inner orbits.

substitutional solid solution, interstitial solid solution

Substitutional Solid Solution

Exactly on the lines of substitutional impurities, the solute atoms occupy positions governed by Hume-Rothery’s rule. The binary Au-Ag, Ni-Cu, Si-Ge systems, or the ternary Ag-Au-Ni system showing complete solid solubility are the examples of substitutional solid solution.

Such solid solutions may be of following types viz.

  • Random or disordered, and
  • Preferably arranged or ordered.

Figures 1 (b and c) explain the two kinds of arrangements. Random solid solution is shown in Figure 1 (b), and the preferred substitutional solid solution in Figure 1(c). Temperature is deciding factor in the random and preferably arranged solid solutions. Brass, an alloy of Cu-Zn system has ‘ordered arrangement’ below 723 K in (3-phase (see Figure 2).

Hume-Rothery’s Rule

Desire of a strong and useful alloy is natural to engineers and materials scientists. Its development necessitates certain conditions. Hume-Rothery has given some criteria in this regard for substitutional solid solution. These are known as Hume-Rothery’s rules, and are given below.

1. Similar crystal structure: The two or more metals should have similar crystal structures such as FCC and FCC, or BCC and BCC. Tungsten alloy steel has Fe and W both of BCC structures, while platinum-silver (Pt-Ag) has FCC structures.

2. Relative atomic Size: Atoms of the two metals should have their sizes within 15% of each other. If the size difference is more than 15%, only limited solid solubility will be obtained.

3. Valency criteria: The valency of base metal and the alloying element should be the same.

4. Electronegativity: The solid solubility will be limited if the two metals possess greater electronegativity. If electro-negativity is too high, the two metals will form intermediate phase instead of solid solution.

Intermetallic Compounds and Intermediate Compounds

Two or more metallic constituents under specific conditions form an intermetallic compound. They are ionically or covalently bonded. Copper aluminide (CuAl2) and magnesium suicide (Mg2Si) are the examples. Based on the specific conditions of their formation, the inter-metallic compounds are sub-classified as:

  1. Valence intermetallic compounds.
  2. Electron intermetallic compounds.
  3. Definite radii ratio intermetallic compounds.
  4. Intermediate compounds.

Valency Intermetallic Compounds

A metal with strong metallic nature combines with another metal of weak metallic properties to form a valency intermetallic compound. In doing so, they obey normal rules of valency. The carbides, fluorides, oxides, hydrides, and combination of metals and metalloids e.g. S, Te, Sb, Bi, Se, As etc. are the examples.

Electron Intermetallic Compounds

The electron intermetallic compounds are formed between two metals having atoms of comparable size but different valency. The CuZn compound has a valency of 3 that includes valencies of copper and zinc as 1 and 2 respectively. Only two atoms are associated in the formation of compound, hence ratio of

valence electrons/number of atoms = 3/2

Electron compounds, generally exist with this ratio of 3: 2 (i.e. 21: 14); 21: 13 and 21: 12 (i.e. 7: 4).

Definite Radii Ratio Intermetallic Compound

Compounds such as Cu2 Mg, LiZn and CaMg etc. bear a definite atomic radii ratio of 1/25,1/15 and 1/24 in their formation. Such compounds are known as definite radii ratio intermediate compounds. A favorable atomic packing is possible with them.

Intermediate Compounds (or Phases)

These compounds are also known as intermediate phases. Similar sounding terms viz. intermediate phases and intermetallic compounds are different from each other.

Intermediate phases are formed in binary alloy systems when their mutual solubility is limited, and the chemical affinity is high. The intermediate phase formed in Cu-Zn system is shown in Figure 2.

solid solution phase diagram

α, β, y Phase of Brass

Various solid phases are α, β, y, δ, e and ἠ. A zinc-rich β-phase appears when solubility of copper exceeds in zinc. This copper-zinc alloy is called as brass. This diagram shows a series of alloys (brass) viz. α-brass, β-brass, y-brass etc. Transformation of α-brass takes place into β-brass when proportion of zinc exceeds 38%. β-brass is harder than α-brass. The intermediate phases have narrow to wider ranges of homogeneity. They are brittle, and poor in electrical conduction.


Physically-distinct, mechanically-separable, and chemically-homogeneous portion of a system is called a phase. A substance can have different phases such as given below.

  1. Gaseous phase,
  2. Liquid phase,
  3. Solid phase,
  4. Solid-liquid phase, and others.

Illustrations: The gaseous state or a liquid solution are single phases in themselves. The atoms or molecules of chimney exhaust gases mix readily in air. Sugar dissolves in water and a liquid solution is formed. In the above cases, each system has a single phase.

Contrary to these, a liquid mixture of water and petrol has two phases. These two liquids do not form a chemically homogeneous mixture as they do not mix together.

Single-phase and Multi-phase Solids: Solids may have one or many phases. Hence, they are grouped as

  1. Single phase solids, and
  2. Multi phase solids

Normally single crystal materials and pure metals have single phase. Quartz, mono-crystal titanium and pure copper are such examples. Polycrystalline materials and alloys may have single or multiphase.

Multiphase solids are rocks, ceramics, polymers, wood; iron and steel, and alloys. Multi-phases for single solid and two solid systems are explained in this article later on.

Gibb’s Phase Rule

Composition and phases of materials may be understood well if the phase relations are known. Gibb has enunciated a rule, better known as Gibbs phase rule, which is given by

D = C – P + λ            (λ = 2)……….. Equation (1)

Where D is degrees of freedom; C represents number of components, P the number of phases in equilibrium, and X the system variables. The system variables are only 2 viz. pressure and temperature, therefore λ = 2 in this equation.

Total number of variables in a system: In a system, the number of degrees of freedom D which is equal to the number of independent variables, cannot exceed total number of variables (dependent and independent). Total number of variables in a system is equal to

P(C – 1) + 2 ……….Equation (2)

Therefore, D = C – P + 2 < P (C – 1) + 2

For C = 2 in a phase, the number of variables is (C – 1) = (2 – 1). For a system having P phases, total number of variables in the composition will be P (C – 1). By adding 2 external variables p and T, total number of variables become P(C – 1) + 2.

Limitation of degree of freedom: The degree of freedom cannot be negative. At the most it can be zero. It is evident from Equation 1 that the degree of freedom decreases when number of phases increases. For example, in a two phase system like Cu and Ni, the degree of freedom is two.

Definition of Frequently Used Terms

The definition of frequently used terms in the studies of phases and phase diagrams are as under:

System: The substances that are isolated and unaffected by their surrounding are called as system. A system may be a solid, liquid, gas, or a combination of these three. A system is capable of changing its composition, pressure, density or temperature if such a requirement arises.

A system is expressed by the unit of 1 kg or 1 mol or 1 m3. A small piece of vanadium or polystyrene, an alloy, and gallium-silicon compound are the examples of systems.

State: It is a physical condition of the system, and is governed by the quantities such as pressure, temperature, mass etc. A system may be in the steady or unsteady states depending on whether it changes with time or not. For example the state of the atoms in diffusion process may be steady or unsteady.

Property: The quantities that govern the state of a system are called as properties. The properties of a system may be either (i) intensive, or (ii) extensive.

The intensive properties such as pressure, temperature, velocity etc. are mass independent while the extensive properties such as volume, density, enthalpy, entropy etc. are mass dependent.

Components: These are elements or the compounds that constitute system. The number of components in a system may be two or more. Pb-Sn, Au-Ag and Si-Al are two component systems; Cu-Ni-Zn in German silver (an alloy) and Ag-Au-Pt are three component systems; and duralumin, an alloy of Al, Cu, Mn and Mg is an example of a four component system.

Equilibrium: This is the state of a system at any specified condition when the system possesses minimum free energy. The specified conditions may be temperature, pressure, composition or their combination. Two or more phases can exist in equilibrium in a single, two or more materials system.

Phase: Physically and chemically homogeneous composition of a substance is known as phase in which the minute adjacent regions are indistinguishable from one another. A system may be made up of one, two or more phases. Several phases may coexist in a substance. A phase change is accompanied with a change in the properties of material.

Degree of Freedom: The number of independent variables required to describe the state of a system are called as the degrees of freedom. Temperature, pressure and the composition of phases, control the constitution of a material system.

These are independent variables, and each one is referred as a degree of freedom. We cannot produce an alloy (a system) of any random choice as the number of components and the overall compositions are dependent variables.

Constraint: A system that has zero degree of freedom is said to be constraint. A system may be

  • fully constraint, or
  • partially constraint.

A system is fully constraint at triple points but elsewhere it is partially constraint in the same figure.

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