Apollonian Packing in Polydisperse Emulsions

TL;DR

This paper reports the discovery of polydisperse high internal-phase emulsions that organize into Apollonian packings, revealing a new geometric structure in emulsions formed during specific emulsification processes.

Contribution

It introduces the existence of polydisperse HIPE with Apollonian packing structure and links their formation to coalescence and fragmentation processes following simple geometric rules.

Findings

Droplets organize in Apollonian packing patterns.

Droplet size distributions follow power laws with Apollonian exponents.

Packing structure confirmed by Small-Angle X-Ray Scattering.

Abstract

We have discovered the existence of polydisperse High Internal-Phase-Ratio Emulsions (HIPE) in which the internal-phase droplets, present at 95% volume fraction, remain spherical and organize themselves in the available space according to Apollonian packing rules. These polydisperse HIPE are formed during emulsification of surfactant-poor compositions of oil-surfactant-water two-phase systems. Their droplet size-distributions evolve spontaneously towards power laws with the Apollonian exponent. Small-Angle X-Ray Scattering performed on aged HIPEs demonstrated that the droplet packing structure coincided with that of a numerically simulated Random Apollonian Packing. We argue that these peculiar, space-filling assemblies are a result of coalescence and fragmentation processes obeying simple geometrical rules of conserving total volume and minimizing surface area.