The platonic pancake: phase change and the ultimate shape of spreading droplets

The spreading of viscous fluids, coupled with phase change, presents a compelling weekend experiment for many families, with important implications for our understanding of magmatic intrusions and many industrial processes. A key question is the extent to which the fluid spreads. Here we show that for many materials, even those with a temperature-dependent rheology, the ultimate shape may be set by the dynamics of phase change (e.g. cooking for pancakes or solidification for many fluid droplets) so that the final radius scales with the volume as $R_\infty \sim V^{5/12}$. Further, we show that with a single, non-dimensional parameter, $\Lambda$, two limiting behaviours in $R_\infty$ may be observed. For very inviscid droplets, $\Lambda << 1$, spreading is first halted by surface tension followed by phase change, while for very viscous droplets $\Lambda >> 1$, spreading is halted by the change of phase itself. We test our simple analytical theory against data from several hundred experiments, from the cooking of a large number of pancakes and crepes, showing good collapse with the variation in measured viscosity, to the solidification of a range of viscosity fluids. Finally, we use our results to speculate on the origin of pancakelike domes observed on the surface of Venus.