Radius (nm) ΔG (J) x 10 16 100 80 60 40 20 0 -1.0 -0.6 -0.2 0.2 0.6 1.0 Radius (nm) ΔG (J) x 10 17 10 8 6 4 2 0 -1.0 -0.6 -0.2 0.2 0.6 1.0 Radius (nm) ΔG (J) x 10 18 1.0 0.8 0.6 0.4 0.2 0 -1.0 -0.6 -0.2 0.2 0.6 1.0 criticalradius
0 (K)
dΔGrxn / d|ΔT|
-10.0 (J/[mole.K])
0.015 (J.m-2)
100 (cm3/mol)
ΔGrxn = 0.0 (J/mole)
ΔGv = 0.0 (J/cm3)
Critical Volume = 0.0 (nm3)
Critical Volume = 0.0 (formula units)
Critical Radius = 0.0 (nm)

Figure 2.06. Critical Radius for Nucleation. This diagram shows the Gibbs energy change ΔG associated with a spherical nucleus as a function of the radius of the nucleus. Sliders are provided to change various parameters used to calculate ΔG. To see the graph, move the ΔT slider to change the temperature away from the equilibrium temperature for the reaction that produces the new phase. The scale of the graph can be changed by clicking the "Change Scale" button in the upper right.

For a small sphere of the new phase, the Gibbs energy of the interface (σA) between the new phase and its surrounding phase may cause the net Gibbs energy change to be positive, even though the change of Gibbs energy due to replacing the volume of the reactant phase(s) with the volume of the product phase (ΔGV) is negative. On the graph, the volumetric Gibbs energy is shown in red, the interfacial Gibbs energy is shown in blue, and the sum of the two Gibbs energy values is shown in black, all as a function of the radius of the spherical nucleus.

A spherical nucleus can coninue to grow spontaneously only if the sum of the Gibbs energy changes (volumetric and interfacial) declines as the radius increases. This decline of Gibbs energy occurs for radii larger than a "critical radius," which is marked on the diagram.

The critical radius (nm) and the associated critical volume (nm3) of a spherical nucleus are shown. The critical volume is also given in terms of formula units for the chemical formula used to define the volumetric Gibbs energy change in units of J/mole. The number of atoms in the critical volume of the nucleus is found by multipying the number of forumla units by the number of atoms in the formula.

The ΔT can be positive or negative as long as the volumetric Gibbs energy is decreasing, so the absolute value of the temperature change |ΔT| is used. Therefore, this critical nucleus diagram can be used to think about rising T reactions, such a garnet nucleation during prograde metamorphism of a shale, and falling T reactions, such as olivine nucleation during the cooling of a basalt magma.

This simplified model does not change the interfacial Gibbs energy with temperature. Also, there may be strain energy term due to growth of the new phase, but this Gibbs energy is not included here.