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In this work, we study by means of simulations of hard spheres theequilibrium between a spherical solid cluster and the fluid. In the NVTensemble we observe stable/metastable clusters of the solid phase inequilibrium with the fluid, representing configurations that are global/localminima of the Helmholtz free energy. Then, we run NpT simulations of theequilibrated system at the average pressure of the NVT run and observe that theclusters are critical because they grow/shrink with a probability of 1/2.Therefore, a crystal cluster equilibrated in the NVT ensemble corresponds to aGibbs free energy maximum where the nucleus is in unstable equilibrium with thesurrounding fluid, in accordance with what has been recently shown for vaporbubbles in equilibrium with the liquid. Then, within the Seeding framework, weuse Classical Nucleation Theory to obtain both the interfacial free energy{\gamma} and the nucleation rate. The latter is in very good agreement withindependent estimates using techniques that do not rely on Classical NucleationTheory when the mislabeling criterion is used to identify the molecules of thesolid cluster. We therefore argue that the radius obtained from the mislabelingcriterion provides a good approximation for the radius of tension, Rs. Weobtain an estimate of the Tolman length by extrapolating the difference betweenRe (the Gibbs dividing surface) and Rs to infinite radius. We show that suchdefinition of the Tolman length coincides with that obtained by fitting{\gamma} versus 1/Rs to a straight line as recently applied to hard spheres[Montero de Hijes et al., J. Chem. Phys. 151, 155401, 2019]

Interfacial free energy and Tolman length of curved liquid-solid interfaces from equilibrium studies