  The Gibbs-Thomson (GT) equation describes the shift of the crystallization
temperature for a confined fluid with respect to the bulk as a function of pore
size. While this century old relation is successfully used to analyze
experiments, its derivations found in the literature often rely on nucleation
theory arguments (i.e. kinetics instead of thermodynamics) or fail to state
their assumptions, therefore leading to similar but different expressions.
Here, we revisit the derivation of the GT equation to clarify the system
definition, corresponding thermodynamic ensemble, and assumptions made along
the way. We also discuss the role of the thermodynamic conditions in the
external reservoir on the final result. We then turn to numerical simulations
of a model system to compute independently the various terms entering in the GT
equation, and compare the predictions of the latter with the melting
temperatures determined under confinement by means of hyper-parallel tempering
grand canonical Monte Carlo simulations. We highlight some difficulties related
to the sampling of crystallization under confinement in simulations. Overall,
despite its limitations, the GT equation may provide an interesting alternative
route to predict the melting temperature in large pores, using molecular
simulations to evaluate the relevant quantities entering in this equation. This
approach could for example be used to investigate the nanoscale capillary
freezing of ionic liquids recently observed experimentally between the tip of
an Atomic Force Microscope and a substrate.
