Filling space with polydisperse spheres in a non-Apollonian way
Résumé
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. High Internal-Phase-Ratio Emulsions (HIPE) are coarse, long-lasting mixtures of non-miscible liquids in which the internal-phase droplets (e.g. oil) occupy a volume fraction higher than 74%, and are separated by an outer continuous phase (e.g. water + surfactant). These extremely concentrated emulsions were described over a century ago, and have since been patented by industrial researchers for applications such as safety and rocket fuel, oil recovery fracturing fluids, and foam and latex production. Until now, HIPEs have typically been made with a very high concentration of surfactant in the continuous phase (14 to 20%, if the surfactant is non-ionic). Under such conditions, it is relatively easy, using processes inspired by the classical mayonnaise recipe, to obtain HIPE in which the inner phase droplets are nearly monodisperse in their diameters. As their volume fraction increases, the inner-phase droplets remain separated by continuous films of the outer phase, but they become non-spherical with beautiful polyhedral shapes such as rhomboedra, dodecahedra and tetrakaidecahedra packed with short-range order. The surface tension of the interconnected surfactant films causes these HIPEs to behave like elastic solids, since any displacement of a droplet requires the stretching of this film network. Mason et al. likewise observed the same rheological / mechanical behavior through the clever use of polymer dialysis to impose very high osmotic pressure on a HIPE in order to attain 97% volume fraction. In the present work, we show that another class of HIPEs exists, which have completely different structures and properties. They are obtained when surfactant availability is reduced. Indeed, in the experiments reported here, the aqueous concentration (w/w) of surfactant in the continuous phase was only 0.6%. Surprisingly, these HIPEs flow under the effect of very weak forces such as gravity. When swollen by excess continuous phase, they simply become regular oil-in-water emulsions, characterized by a power-law droplet-size distribution. Such extreme polydispersity causes the structure of the HIPEs to exhibit scale invariance instead of the short-range order in monodisperse emulsions. According to X-ray scattering, the relative positions of droplets match an Apollonian construction, where the interstice between droplets are occupied by even smaller ones (of the largest size possible). This is particularly curious because it has long been believed in the domain of concentrated colloidal science that 'this peculiar kind of heterodispersity will rarely, if ever, be encountered' due to the need to fabricate a very large number of droplets of different sizes. Each HIPE was made with the non-ionic surfactant, hexaethylene glycol monododecyl ether (C 12 E 6), water and mineral oil. C 12 E 6 was diluted in Millipore Milli-Q water to 0.6% (w/w) to constitute the HIPE's continuous phase. Oil was then added dropwise under constant shearing from a 4-bladed propeller stirrer. To avoid emulsion inversion, a new oil drop was introduced only after.
Domaines
Matière Molle [cond-mat.soft]
Origine : Fichiers produits par l'(les) auteur(s)