Engineering & Science
Gibbs-Thomson Nanoparticle Melting-Point Depression & LSW Ostwald Ripening
Calculate curvature-induced melting-point depression Delta_Tm = (2 * gamma_SL * Tm) / (Delta_Hf * r) in Kelvin for nanoparticles/dendrite tips and Lifshitz-Slyozov-Wagner coarsening radius r(t) = (r0^3 + K * t)^(1/3).
Why 5 nm metal nanoparticles sinter and melt tens of degrees below bulk Tm!
Atoms on the sharply curved surface of a spherical nanoparticle or dendrite tip of radius r have fewer nearest-neighbor bonds and higher chemical potential than a flat bulk crystal! The Gibbs-Thomson equation predicts an inverse-radius melting-point depression (undercooling): Delta_Tm = Tm_bulk - Tm(r) = (2 * gamma_SL * Tm_bulk) / (Delta_Hf * r), where gamma_SL is solid-liquid interfacial energy (J/m^2), Tm_bulk is bulk absolute melting point (K), Delta_Hf is volumetric latent heat of fusion (J/m^3), and r is particle radius in meters (r_nm * 10^-9)!
For copper nanoparticles (gamma_SL = 0.20 J/m^2, Tm_bulk = 1,358 K, Delta_Hf = 1.8e9 J/m^3), a 50 nm particle drops only 6.04 K, whereas a 5 nm nano-ink particle melts 60.36 K lower (at 1,297.64 K)! That exact same capillary potential drives big precipitates in superalloys and emulsions to devour small ones—Lifshitz-Slyozov-Wagner (LSW) Ostwald ripening, where mean radius cubed grows linearly with time: r(t) = ((r0^3) + (K_LSW * t)) ^ (1/3)!
| Copper Particle Radius r | Gibbs-Thomson Depression Delta_Tm | Depressed Melting Point Tm(r) | Capillary Scaling |
|---|---|---|---|
| r = 5 nm (Printed Electronics Ink) | 60.36 K | 1,297.64 K (1,024.49 °C) | 10x smaller r -> 10x larger drop |
| r = 50 nm (Submicron Powder) | 6.04 K | 1,351.96 K | Approaches bulk 1,358 K |
| LSW Ostwald Ripening (r0=10 nm, K*t=7000) | r(t) = (1000 + 7000)^(1/3) | 20.00 nm (Doubled radius!) | 8x volume increase per particle |
Gibbs-Thomson Melting-Point Depression Delta_Tm (Kelvin, Radius in nm)
Converts nanoparticle radius B5 from nanometers to meters (* 10^-9) and computes (2 * gamma_SL * Tm) / (Delta_Hf * r).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Gibbs-Thomson Inputs", ROUND((2 * B2 * B3) / (B4 * B5 * 0.000000001), 2))Depressed Nanoparticle Melting Temperature Tm(r) = Tm_bulk - Delta_Tm (Kelvin)
Subtracts Gibbs-Thomson curvature undercooling from bulk melting temperature B3.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Gibbs-Thomson Inputs", ROUND(B3 - ((2 * B2 * B3) / (B4 * B5 * 0.000000001)), 2))Try the example
Use the original values or change them to test your understanding.