Nanoparticle Surface-to-Volume Scaling & Melting-Point Depression Simulator

Material

Particle radius

Radius r 10 nm

Log scale: 1 nm → 1 µm (1000 nm). Drag toward the left edge to enter the nanoscale.

Live properties

Surface / volume

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Atoms at surface

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Melting point

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ΔT vs bulk

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Total atoms

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Surface atoms

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How it works

Model the nanoparticle as a sphere of radius r built from atoms of radius a. Shrinking r toward a forces a growing share of those atoms to sit at the surface, and that single geometric fact drives everything else on this page.

SA/V = 4πr² / (4/3 πr³) = 3/r
f_surface ≈ 1 − (1 − a/r)³
ΔT_melt = 2·σ_sl·V_m·T_bulk / (ΔH_f · r)   (Gibbs–Thomson)
N_total ≈ η·(r/a)³,  N_surface = f_surface · N_total

Surface/volume (3/r) — exact for a sphere; it diverges as r→0, which is the geometric root cause of every nanoscale effect below.
Surface-atom fraction — a shell one atomic radius thick holds a near-constant number of atoms while the total atom count falls as r³, so the surface fraction rises sharply below ~10 nm.
Melting-point depression — surface atoms are under-coordinated and less tightly bound, so they can be dislodged at a lower temperature than bulk atoms. The Gibbs–Thomson relation makes this quantitative: ΔT scales as 1/r, using each material's real solid–liquid interfacial energy σ_sl, molar volume V_m and latent heat of fusion ΔH_f.
Surface atom count — the fraction shrinks with r, but the particle's total atom count grows with r³, so the absolute number of catalytically exposed surface atoms per particle still increases with size even as the specific (per-atom) activity falls — the two curves genuinely diverge.

Real-world relevance: this is why gold nanoparticles melt hundreds of degrees below bulk gold (Buffat & Borel, 1976), why nanoparticle catalysts (Pt, Pd, Au) are prized for high specific activity per gram despite using less material, and why sintering and Ostwald ripening accelerate once particles drop below ~10 nm.

Bulk interior Surface shell (1 atomic radius) Current radius marker