In this article we will discuss about the shapes of phases with its diagram.

Any surface or an interface such as a grain boundary has energy related with it, which acts like a surface tension. A solid phase does not easily change its shape in response to the surface tension (due to rigid bonds between atoms in a solid), unless given a long time to adjust, and particularly at high temperatures due to greater mobility of atoms (a lamellar pearlite takes more than 20 hours at 700°C to change to globular pearlite). However, the surface tension does effect the shapes, when phases (or one of the phase) are liquids before solidification occurs.

If a liquid and a solid surface are involved, then depending on the interfacial energies, three cases arise – (a) complete wetting of solid (b) Partial wetting (c) Non-wetting of the solid as illustrated in Fig. 2.23.

If there are two different liquid phases completely insoluble in one another, and if one of them is in smaller amounts, then it assumes a spherical shape because this shape minimises the interfacial area (thus, the interfacial energy), i.e., it is like non-wettability. It is illustrated in Fig. 2.24 (a).

If there are three immiscible liquids as illustrated in Fig. 2.24 (b) where the density of liquid 2 is less than liquid 3, but more than liquid 1. If ϒ (gamma) is used to represent the surface tension, and the subscripts are used to indicate the two liquids on either side of the surface tension force vector, such that ϒ12 is the surface tension between liquids 1 and 2; ϒ31 is the surface tension between liquids 1 and 3, etc.

The shape of the droplet (liquid 2) is determined by the magnitude of three interfacial tensions, ϒ12, ϒ23 and ϒ31 between the pairs of different liquid phases, which exert forces on the surface of the lens-shaped volume of liquid 2. The angles θ1, θ2 and θ3 are the angles between the three pairs of interfaces. The situation is similar to three strings with weights of magnitudes ϒ12, ϒ23 and ϒ31 being tied together at a point as in the triangle law of forces.

Under equilibrium conditions:

where, the magnitude of ϒ12 depends on the chemical nature of liquid 1 and liquid 2, and so on. If the surface tensions between the grains is the same as illustrated in Fig. 2.20 (being grains of the same phase), the angles θ1 = θ2 = θ3 = 120°.

In alloys of two phases, there can be two types of boundaries:

(i) Boundaries separating grains of the same phase, and


(ii) Boundaries separating grains of the two different phases.

Let us take a junction where two grains of one phase meet a grain of different phase at a common intersection as illustrated in Fig. 2.25, where, ϒ11 is the surface tension in the single (gamma one one) phase boundary; ϒ12 is the surface tension in the two phase boundary; and 8, the dihedral angle which the second phase makes between two grains of phase I.

Under the static equilibrium:


The horizontal components give-

ϒ11 = 2ϒ12 cos θ/2, …(2.4)

as the vertical components balance each other, or

The equation (2.5) is used to plot a graph between dihedral angle θ and the ratio ϒ1211, as illustrated in Fig. 2.26.


When this ratio is one (i.e., ϒ12 = ϒ11) then θ, has a value of 120°, and also the following conditions are true:

ϒ12 > ϒ11, θ > 120°

ϒ12 < ϒ11, θ < 120°

and for the two extreme cases,

ϒ12 > > ϒ11, θ → 180° ‘No-wetting’ occurs.

ϒ12 ≤ 0.5 ϒ11, θ → 0° ‘Complete-wetting’ occurs

The physical appearance of phase 2 trapped at either a boundary between two grains, or at grain corner (between three grains) is illustrated in Fig. 2.27. It is clear from this Fig. that the shape of the second phase is largely determined by dihedral angle θ. When θ = 180°, (ϒ12/ ϒ11 >> 2), the liquid region takes the shape of a sphere. It is a condition of non-wettability. Spheroidal graphite probably lakes the shape of spheres in S.G. iron due to this. For, θ = 120°, the second phase takes lenticular shapes.

A significant result appears, when the ratio, ϒ12/ ϒ11 attains a value of 0.5, that is, the dihedral angle θ = 0 (approaches zero). The liquid or the second phase tends to spread as a thin layer (film) along all the grain boundaries, and separates the grains of the first phase from each other, even when the amount of the second phase (even as liquid) is very small.

Fig. 2.28 illustrates a picture as to how the second phase changes when dihedral angle changes form 10° to almost zero degree. This type of development of thin film at the grain boundaries can have lots of important consequences such as hot shortness in steels.