Curvature correction to the surface tension of small nuclei
In plain words
Classical nucleation theory uses the surface tension of a flat interface, but critical nuclei are only a few molecules across, where the curvature should change the tension. The size and even the sign of this correction is not agreed on.
Precise statement
For a nucleus of radius $R$, gamma(R) = gamma_inf (1 - 2 $\delta$ / R + ...) with Tolman length delta. Determine the sign and magnitude of delta for the Lennard-Jones liquid-vapor interface and for crystal-melt interfaces, and whether the expansion in $1/R$ is valid at critical-nucleus sizes of about 10 to 30 Angstrom.
What would settle it
Consistent determinations of $\delta$ from independent simulation methods (droplet and bubble free energies, virial route), and a test of the $1/R$ expansion against measured nucleation barriers.
Status in the literature
Simulations for the Lennard-Jones liquid-vapor case point to a small negative $\delta$ of order $-0.1$ particle diameters (approximately), but crystal-melt values and the validity of the expansion are unsettled.