The effects of interfacial energy on nucleation can be evaluated using a sphere as the form of the new phase. A sphere has the smallest surface to volume ratio of any form, so interfacial energy effects from other forms would be larger. Figure 2.05 shows representative cross sections of two tiny spherical nucleii, one with a radius of one nanometer (nm) and one with a radius of two nanometers. Volumes (V) and surface areas (A) of the two spheres are listed in the figure.

**Click on the figure**to see a larger version. Doubling the radius from 1 to 2 nm, doubles the ratio (V/A) of the volume of the sphere (4/3 π R

^{3}) to the area of the sphere (4 π R

^{2}):

When calculating the Gibbs energy change to grow the nucleus, the volume term will become more important as the radius increases.

In Figure 2.05, example values of interfacial Gibbs energy per nm

^{2}(σ

_{A}) and reaction Gibbs energy per nm

^{3}(ΔG

_{V}) are used to calculate the total Gibbs energy change (ΔG

_{T}) for a nucleus of radius R. A graph of the total Gibbs energy change as a function of radius in Figure 05 shows that ΔG

_{T}begins to decline when the radius of a spherical nucleus reaches a critical value (the maximum of ΔG

_{T}on the graph). It is at this radius that the Gibbs energy decreases with nucleus growth. Declining ΔG

_{T}with growth of the product phase is necessary for the reaction to proceed spontaneously.

Thus, if the size of the nucleus of a new phase produced by a reaction is large enough, the phase will grow and the reaction will proceed. However, if the new phase is too small, interfacial energy will prevent it from growing further. Every nucleus must begin very small.

**What circumstances will lead to the appearance of a nucleus that is large enough to grow spontaneously?**