Theory of Shape-Shifting Droplets

Recent studies of cooled oil emulsion droplets uncovered transformations into a host of flattened shapes with straight edges and sharp corners, driven by a partial phase transition of the bulk liquid phase. Here, we explore theoretically the simplest geometric competition between this phase transition and surface tension in planar polygons and recover the observed sequence of shapes and their statistics in qualitative agreement with experiments. Extending the model to capture some of the three-dimensional structure of the droplets, we analyze the evolution of protrusions sprouting from the vertices of the platelets and the topological transition of a puncturing planar polygon.

Figure 1

Shape-shifting droplets. (a) Main stages of droplet shape evolution, following Ref. [3]. Initially spherical droplets flatten, first into hexagons, then into quadrilaterals or triangles, before growing thin protrusions. (b) Energy diagram, showing a range of temperatures close to the melting temperature where the formation of a rotator phase is energetically favored. (c) Cross section of flattened droplet. A layer of plastic rotator phase has formed at the edges of the droplet.

Figure 2

Model of shape-shifting droplets. (a) Classification of polygonal shapes with interior angles of 60° or 120°, and transitions between shapes. (b) Experimental image of pentagon state. (c) Simplest droplet model: a polygonal frame of area $\mathcal{A}$ and perimeter $\mathcal{P}$ defines a flat droplet of surface area $\mathcal{S}=2\mathcal{A}$. (d) Extended droplet model: conservation of the droplet volume $V$ and the finite height $H$ of the rim of rotator phase modify the relation between $\mathcal{S}$ and $\mathcal{A}$. (e) Droplet energy: effect of competing surface tension and tendency to form rotator phase illustrated for the simplest, triangular droplet shape.

Figure 3

Droplet modes and shape statistics. (a) Triangle mode ($\mathrm{T}$), with eigenvalue 1. (b) Parallelogram mode ($\mathrm{P}$), with eigenvalue $2/3$. There are two linearly independent modes of this ilk; others are obtained by rotation by 60° or 120°. (c) Shape regions for perturbations of fixed magnitude $ϵ$ around the hexagon state, with eigendirections $\mathrm{T}$ and $\mathrm{P}$ indicated. Linearly independent $\mathrm{P}$ modes span the equatorial plane. Parameter values are $\alpha =1$, $ϵ=0.04$. (d) Proportion of initial hexagons that evolve into triangles in $(ϵ,\alpha )$ space. (e) Proportion of initial hexagons that evolve into parallelograms in $(ϵ,\alpha )$ space.

Figure 4

Droplet puncture. (a) Punctured parallelogram as outcome of droplet shape evolution. Scale bar $20\mu \mathrm{m}$. (b) Idealized cross section of an inverted droplet. (c) Trajectories of Eq. (6) indicated by solid lines, overlaid with the puncture condition Eq. (5), represented by dashed lines. Inset: Definition of geometric variables. Thick solid line shows trajectory through saddle. Parameter values are $\alpha =1$, $H/V^{1/3}=0.25$, 0.047.

Figure 5

Growth of protrusions. (a) Geometry: bending of sides removes defects and allows protrusions to grow. (b) Parallelograms and trapezoids sprout protrusions from 60° angles, but not from 120° angles. Scale bar $20\mu \mathrm{m}$.