  In the study of crystal nucleation via computer simulations, hard spheres are
arguably the most extensively explored model system. Nonetheless, even in this
simple model system, the complex thermodynamics of crystal nuclei can sometimes
give rise to counterintuitive results, such as the recent observation that the
pressure inside a critical nucleus is lower than that of the surrounding fluid,
seemingly clashing with the strictly positive Young--Laplace pressure we would
expect in liquid droplets. Here, we re-derive many of the founding equations
associated with crystal nucleation, and use the hard-sphere model to
demonstrate how they give rise to this negative pressure difference. We exploit
the fact that, in the canonical ensemble, a nucleus can be in a (meta)stable
equilibrium with the fluid, and measure the surface stress for both flat and
curved interfaces. Additionally, we explain the effect of defects on the
chemical potential inside the crystal nucleus. Lastly, we present a simple,
fitted thermodynamic model to capture the properties of the nucleus, including
the work required to form critical nuclei.
