A crystal's surface atoms have fewer neighbors than its interior — they sit in a higher-energy, more loosely bound state. As a particle shrinks, the fraction of atoms that live on the surface grows as 1/r, so that destabilizing surface energy comes to dominate the total free energy. The Gibbs–Thomson equation quantifies exactly how much this lowers the melting point.
T_m(r) = T_m,bulk · [1 − 4·σ_sl / (ΔH_f·ρ_s·r)]
σ_sl = solid–liquid interfacial energy (J/m²)
ΔH_f = latent heat of fusion, per unit mass (J/kg)
ρ_s = solid density (kg/m³)
r = particle radius (m)
- Material — sets Tm,bulk, σsl, ΔHf and ρs to real literature values for gold, tin and lead nanoparticles.
- Radius r — the curve on the right plots Tm(r) directly from the equation above; drag it down toward a few nanometres to see the depression grow sharply (it diverges as r→0).
- Temperature T — compared against the size-corrected Tm(r), not the bulk value, to decide the nanoparticle's phase below.
- Premelting shell — because surface atoms are the least coordinated of all, they go liquid-like below Tm(r) first; the cross-section shows this quasi-liquid skin thickening inward as T approaches Tm(r), then the whole particle collapsing to liquid right at Tm(r).
Real-world relevance: this effect is why gold nanoparticles a few nanometres across can melt hundreds of degrees below bulk gold's 1337 K, and why nanoscale solder, catalysis and sintering processes all run at temperatures a bulk phase diagram would call impossible.