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 ∼ tⁿ 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.