Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2021 / Paper 2 / Section A(i) / 14

Prepare size-characterized gold-particle ensembles, heat them on a calibrated in-situ electron diffraction stage, and follow the loss of sharp fcc diffraction features and appearance of liquid-like diffuse scattering. Measure the particle size distribution rather than treating a polydisperse ensemble as one diameter. Temperature-dependent weakening of a diffraction peak alone is insufficient: thermal vibrations, particle motion, coalescence, evaporation and beam damage can also reduce intensity. Low-dose repeats, heating/cooling runs, size checks before and after heating, a bulk-like reference and support-dependent controls help distinguish melting from those effects. Electron diffraction measurements on gold particles down to about 2 nm are a classic demonstration.^[ref]

The physical origin is the large interface contribution to free energy at small radius. A curved solid–liquid interface raises the solid's thermodynamic equilibrium chemical potential relative to the flat-interface case, so coexistence occurs below the bulk melting temperature. Reduced coordination and surface disorder provide a microscopic interpretation, but the quantitative continuum relation is a free energy balance, not simply a claim that every surface atom has a universal lower melting point.

Let $g_l-g_s$ be the liquid-minus-solid free energy density at the same temperature. Near the bulk coexistence temperature $T_m$, linearization givesFor a sphere, an interface displacement changes area and volume with $dA/dV=2/r$. Equality of the bulk driving force and the capillary cost at a curved solid–liquid interface therefore gives $g_l-g_s=2\gamma /r^{\ast }$. Equivalently, differentiating $4\pi r^2\gamma -(4\pi /3)r^3(g_l-g_s)$ gives the same stationary radius. Combining the expressions yieldsThis form of the Gibbs-Thomson effect assumes isotropic interfacial energy, a spherical interface, bulk-like latent heat per volume and a near-coexistence free energy expansion. More detailed free-particle models must distinguish solid–vapor, liquid–vapor and solid–liquid energies.

The radius is half the diameter, $r^{\ast }=1.5\times 10^{-9}\mathrm{m}$, and the latent heat density is $1.25\times 10^9\mathrm{J}{\mathrm{m}}^{-3}$. HenceUsing the diameter directly in the radius formula would underestimate the depression by a factor of two. The result is the stipulated continuum-model estimate; at only a few nanometres atomistic structure and support effects can matter substantially.

In terms of diameter $d$, the model isThus melting temperature rises toward the bulk value as diameter increases, with a depression proportional to $1/d$. The analytic sketch below includes the calculated point. It deliberately does not extrapolate the continuum expression down to a zero or negative melting temperature.

Facet-dependent energies, non-spherical shapes, size-dependent interfacial energies and latent heat, structural changes or magic cluster sizes can modify the trend. Premelting can broaden the transition. A substrate or encapsulating matrix changes interface energetics and can stabilize or destabilize the solid. Ligands, impurities, strain, particle size distributions, coalescence during heating and nonequilibrium temperature/beam effects can shift the measured transition. These possibilities motivate separate thermal, structural and size controls rather than a fit to diameter alone.