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Thermal crossing of a nucleation barrier $F^{\ast }$ has rate proportional to $e^{-F^{\ast }/(k_{B}T)}$ and mean waiting time proportional to $e^{F^{\ast }/(k_{B}T)}$, up to a kinetic prefactor.

Boiling is vaporization through the formation and growth of vapor bubbles within a liquid. It differs from surface evaporation, which can occur without bubbles in the bulk. At equilibrium, the boiling point of a pure liquid is set by equality of its saturation vapor pressure and the external pressure.

A real bubble must also overcome local hydrostatic pressure and the capillary pressure of its curved surface. Creating the interface costs energy, giving a nucleation barrier; a smooth clean vessel can therefore permit superheating. Surface cavities and boiling chips supply sites where bubbles form more readily. The observed onset depends on these sites and heating conditions, while the equilibrium boiling point remains a property specified at the chosen pressure.

Condensation is conversion of a vapor into a liquid. Cooling vapor-bearing air below its dew point is a familiar example: droplets form on a cold surface as the saturation vapor pressure falls. In a mixture, the relevant comparison uses the condensable component's partial pressure, rather than the mixture's total pressure alone.

Creating a new droplet costs interfacial free energy, while conversion to the stable liquid lowers bulk free energy. A nucleation barrier can therefore permit supersaturated vapor before droplets appear. A pre-existing surface can lower that barrier through heterogeneous nucleation. Condensation also releases latent heat, so a cold plate or condenser must remove heat as liquid accumulates. At vapor-liquid equilibrium, evaporation and condensation continue microscopically with equal average opposing rates.

Ice nucleates on an inclusion already immersed in supercooled water. The interfacial nucleation barrier, inclusion properties, droplet volume and observation time affect freezing probability.

A metastable state is locally stable against small disturbances but has a higher thermodynamic potential than another accessible phase. In a free-energy landscape it occupies a local minimum separated from the stable state by a barrier.

A liquid undergoing supercooling illustrates the distinction: small fluctuations can relax without freezing, but a sufficiently large crystal nucleus can grow. The nucleation barrier depends on the competition between interfacial cost and bulk free-energy gain. Waiting time is therefore affected by sample volume, impurities, surfaces and observation duration. A spinodal point is the mean-field limit where the metastable minimum loses local stability; it is distinct from the coexistence condition where the stable phases have equal appropriate intensive potentials. Metastability explains hysteresis without making the equilibrium phase boundary path dependent.

A droplet can remain liquid below its equilibrium melting temperature through supercooling: a microscopic ice embryo must first pay an interfacial free-energy cost. The nucleation barrier can therefore delay freezing even when bulk ice is thermodynamically favoured. Solutes, available surfaces, temperature, droplet volume and exposure time affect the probability; zero degrees Celsius is not an automatic freezing threshold for every atmospheric droplet.

At least four physical routes are relevant:

Mineral particles provide nonbiological examples of heterogeneous nuclei. A particle's presence alone does not guarantee activity at a given temperature. The cloud ice nucleation versus ice growth distinction matters: the processes listed concern the formation of new ice; subsequent depositional growth, riming and aggregation are separate from initiating that first crystal.

Droplets remain supercooled because creating an ice embryo costs interfacial free energy: freezing has a nucleation barrier even below the equilibrium melting point. Small droplets also have a low probability of containing an effective ice-nucleating inclusion.

Three heterogeneous routes are immersion freezing around a particle already inside the droplet, contact freezing following a collision with a suitable mineral particle, and deposition nucleation from ice-supersaturated vapour onto a suitable surface. At sufficiently low temperature, homogeneous ice nucleation occurs without a foreign surface. Its onset depends on volume and observation time rather than being a universal exact temperature. No biological ice-nucleating agents are needed for these mechanisms.

Diamond can be metastable: its Gibbs free energy is higher than graphite's under ambient conditions, but transformation is kinetically inhibited. Conversion requires substantial rearrangement of the strong carbon-bond network and the formation and growth of a new thermodynamic phase, with an activation and nucleation barrier.

At room temperature the rate can be extremely small even though the free-energy change favors graphite. Thermodynamic stability specifies the preferred equilibrium state, not the time needed to reach it. Persistence of diamond therefore does not contradict graphite's ambient equilibrium stability.

Along the stipulated fixed-aspect-ratio pathway, substitute $h=d/k$ before differentiating:The nonzero stationary diameter isComparison with the stated diameter gives $k{\gamma }_f+2{\gamma }_e=6{\gamma }_f+2{\gamma }_e$, hence $k=d/h=6$, provided ${\gamma }_f$ is nonzero. This is a plate-like shape, qualitatively consistent with the cheaper broad coherent faces. For negative $\Delta G$, substitution shows the stationary point is a maximum along this size pathway, as required for a nucleation barrier.

A useful distinction is that this calculation holds shape fixed. Minimizing the interface energy at fixed volume separately gives $d/h={\gamma }_e/{\gamma }_f$ for the cylindrical model; if the selected aspect ratio is also an equilibrium shape, it would require ${\gamma }_e=6{\gamma }_f$. The given critical-diameter relation alone does not otherwise determine that energy ratio. In the degenerate zero-face-energy limit the coefficient comparison cannot determine $k$.

Supercooling is persistence of a phase below the equilibrium temperature where a lower-temperature phase becomes favorable. A liquid remaining unfrozen below its equilibrium melting temperature is the standard example.

Creating a crystal nucleus gains bulk free energy but costs interfacial energy. Small embryos can disappear, while one exceeding the critical size can grow; this nucleation barrier permits a metastable liquid. A surface or suspended inclusion can lower the barrier through heterogeneous nucleation, whereas homogeneous nucleation occurs within the bulk. Measured freezing onset consequently depends on droplet size, purity and waiting time. Growth releases latent heat and can temporarily warm the remaining material toward the coexistence temperature, separating the onset of nucleation from the later heat-removal problem.

Superheating is persistence of a phase above the equilibrium temperature at which a higher-temperature phase becomes favorable. The familiar case is a liquid above its boiling point at the imposed pressure, although delayed melting can also be discussed as superheating.

The liquid can remain in a metastable state because producing a vapor bubble costs interfacial energy. A sufficiently large bubble overcomes the nucleation barrier and grows, converting the liquid and consuming latent heat. Smooth clean containers can delay nucleation, while cavities and boiling chips provide heterogeneous nucleation sites. The observed temperature of sudden bubble formation therefore need not equal the equilibrium boiling point. The amount of superheating is a kinetic and preparation-dependent observation, rather than a new equilibrium phase boundary.