  Heterogeneous nucleation of a new bulk phase on a flat substrate can be
associated with the surface phase transition called wetting transition. When
this bulk heterogeneous nucleation occurs on a completely-wettable flat
substrate with a zero contact angle, the classical nucleation theory predicts
that the free energy barrier of nucleation vanishes. In fact, there always
exist a critical nucleus and a free energy barrier as the first-order
pre-wetting transition will occur even when the contact angle is zero.
Furthermore, the critical nucleus changes its character from the critical
nucleus of surface phase transition below bulk coexistence (undersaturation) to
the critical nucleus of bulk heterogeneous nucleation above the coexistence
(oversaturation) when it crosses the coexistence. Recently, Sear [J.Chem.Phys
{\bf 129}, 164510 (2008)] has shown by a direct numerical calculation of
nucleation rate that the nucleus does not notice this change when it crosses
the coexistence. In our work the morphology and the work of formation of
critical nucleus on a completely-wettable substrate are re-examined across the
coexistence using the interface-displacement model. Indeed, the morphology and
the work of formation changes continuously at the coexistence. Our results
support the prediction of Sear and will rekindle the interest on heterogeneous
nucleation on a completely-wettable substrate.
