❄️ Nucleation & Growth — Birth of a New Phase

r* is the cluster radius at which the free energy ΔG(r) is at its maximum. The simulation computes it live as r* = 2γ/ΔGv from the Surface energy and Undercooling sliders. Clusters smaller than r* are shown shrinking and fading (unstable); clusters at or above it are shown growing with a bright gradient (stable). The dashed circle drawn near the top-right of the canvas is this exact r* value, updated every frame.

Undercooling ΔT feeds directly into the driving force ΔGv used by the code (dGv = 0.25 + 3.0 × undercool). Since r* = 2γ/ΔGv, turning ΔT up shrinks r*, so far more randomly-sized embryos land above the critical size and survive — you will see nucleation become much more frequent and the "Surviving nuclei" counter climb faster.

γ is the energy cost of creating new surface area between the old and new phase. Raising γ increases both r* (= 2γ/ΔGv) and the nucleation barrier ΔG* (= 16πγ³/(3ΔGv²)) computed by the simulation, meaning clusters need to reach a larger size before they stop shrinking — fewer random embryos will be large enough to survive.

Each spawned embryo gets a random starting radius drawn from a rough thermal distribution around r*. In the step() function, any cluster with radius below the current r* has its radius reduced every frame until it is dissolved (added to the Dissolved counter); any cluster at or above r* has its radius increased at a rate based on Growth speed and Undercooling, with growth slowing as it approaches roughly half the canvas size.

Clicking or tapping calls the seedAt() function, which places a new cluster at that exact point with radius set to 1.4 × the current r* — guaranteed to be super-critical — so it immediately starts growing regardless of the random thermal distribution used for automatic embryos.