Why solidification has an S-shaped history

Here we show theoretically that the history of solid growth during “rapid” solidification must be S-shaped, in accord with the constructal law of design in nature. In the beginning the rate of solidification increases and after reaching a maximum it decreases monotonically as the volume of solid tends toward a plateau. The S-history is a consequence of four configurations for the flow of heat from the solidification front to the subcooled surroundings, in this chronological order: solid spheres centered at nucleation sites, needles that invade longitudinally, radial growth by conduction, and finally radial lateral conduction to interstices that are warming up. The solid volume (B(s)) vs time (t) is an S-curve because it is a power law of type B(s) ~ t(n) where the exponent n first increases and then decreases in time (n = 3/2, 2, 1, …). The initial portion of the S curve is not an exponential.