The characteristics of heterogeneous nucleation on concave spherical surfaces were analysed using a novel analytical approach and compared with nucleation on both convex and planar surfaces. The complex expressions of partial derivative Delta G/partial derivative r (Delta G: free energy change and r: embryo radius) for nucleation on concave spherical surfaces can be reduced to the simple form for nucleation on planar surfaces by introducing an appropriate pseudo-contact angle, providing a new fundamental link. The advantages of heterogeneous nucleation on a concave spherical surface of radius R over a planar surface occur primarily when 2R < 10r* (r*: nucleus radius) and diminish rapidly when 2R > 10r*. This is similar to nucleation on convex spherical surfaces, where the disadvantages of nucleation over a planar surface occur primarily when 2R < 10r* and diminish rapidly when 2R > 10r*. The substrate size 10r* thus provides an approximate borderline that distinguishes between curved and planar surfaces for nucleation. The advantages or disadvantages of nucleation over a planar surface are most outstanding when the concave (advantageous) or convex (disadvantageous) surface shows a specific contact angle with the nucleus. The nanoscale nature of the threshold size 10r* presents a clear fundamental support to surface nanopatterning for directed nucleation. A minimum depth is required for a nanoscale crater to be a favourable nucleation site. (C) 2012 Elsevier B.V. All rights reserved.