Understanding Nucleation Gibbs Energy

Nucleation is the initial process where a new thermodynamic phase forms. This requires overcoming an energy barrier, known as the Gibbs energy of nucleation.

The change in Gibbs energy ($\Delta G$) for forming a spherical nucleus involves two main contributions:

• A negative contribution from the bulk free energy change ($\Delta G_v$), which favors the new phase formation.
• A positive contribution from the surface energy ($\gamma$), which increases with the surface area of the nucleus.

Critical Gibbs Energy Barrier

The nucleation process involves reaching a critical size where the nucleus becomes stable. The energy required to reach this point is the critical Gibbs energy barrier, denoted as $\Delta G^*$.

The formula for the critical Gibbs energy barrier is:

$ \Delta G^* = \frac{16 \pi \gamma^3}{3 (\Delta G_v)^2} $

where:

• $ \gamma $ is the surface tension.
• $ \Delta G_v $ is the difference in Gibbs free energy per unit volume between the parent and new phase.

Gibbs Energy at Equilibrium Temperature

The equilibrium temperature ($T_{eq}$) is the specific temperature where the parent phase and the new phase are in equilibrium. At this temperature, there is no net driving force for the phase transformation from the bulk perspective.

Mathematically, this means the bulk free energy change per unit volume is zero:

$ \Delta G_v = 0 $

Substituting this condition into the formula for the critical Gibbs energy barrier:

$ \Delta G^* = \frac{16 \pi \gamma^3}{3 (0)^2} $

As the denominator approaches zero, the value of $\Delta G^*$ tends towards infinity.

Therefore, the critical value of the Gibbs energy of nucleation at the equilibrium temperature is effectively infinite, representing an infinitely large barrier to initiate nucleation under these precise equilibrium conditions.