# Equilibrium and Nonequilibrium States in Microﬂuidic Double Emulsions

Original entry: Tony Orth, APPHY 226, Spring 2009

Nicolas Pannacci, Henrik Bruus, Denis Bartolo, Isabelle Etchart, Thibaut Lockhart, Yves Hennequin, Herve Willaime and Patrick Tabeling, Equilibrium and Nonequilibrium States in Microfluidic Double Emulsions., PRL 101, 164502 (2008)[1]

Antony Orth

### Brief Summary

A cascaded microfluidic flow-focusing device was used to generate double emulsions. The stability of the various orderings of the phases was studied as well as predicting the time scale for the transient structures as the double emulsions proceeded to equilibrium states. The authors also demonstrate UV curing of core-shell and Janus droplets, thereby creating interesting colloidal geometries easily attainable with droplet microfluidic devices.

### Surface Tension Phenomena

Figure 1: Possible orderings of phase encapsulation. Depending on the relative values of the interfacial tensions (captured by the "spreading coefficients" $S_{2,3}$), one phase is either completely or only partially engulfed by the other, or the two initially cascaded phases do not share a stable interface so that the inner droplet spontaneously exits the outer droplet. See the appropriate reference in the paper for an explanation of the parameter $\phi$

The dynamics of this interesting study are entirely a result of the relative values of the surface tensions between each of the liquid phases. Suppose we label the continuous (wetting) phase by as phase "2". The other two phases which will form droplets either within phase 2 or each other are phases "1" and "3". Further, let's define the spreading coefficients $S_{i} = \gamma_{jk} - \gamma_{ij} - \gamma_{ik}$. First we note that the region in figure 1 lying above the $S_3 = -S_2$ line is not allowed since this would lead to a negative surface energy. The region below this line is split into three subregions, each of which is occupied by a distinct state of the three-phase emulsion.

Figure 2: Micrographs of transient double emulsion states. (a) A $S_2 <0 ; S_3>0$system which leads to a energetically favourable non-engulfed configuration. (b) A $S_2 <0 ; S_3<0$system which leads to a energetically favourable partially-engulfed configuration.

The meaning of each subregion is more intuitively understood by writing out the inequality explicitly in terms of the interfacial energies. The $S_3 <0 ; S_2>0$ subregion leads to $\gamma_{13}>\gamma_{23} + \gamma_{21}$ and $\gamma_{12}<\gamma_{23} + \gamma_{13}$. This means that a unit of 2,3 + a 1,2 interface is energetically favourable to a 1,3 interface; a 1,2 - a 1,3 interface is favourable to a 2,3 interface. The result being that the most energetically stable state is actually a non-engulfed structure (shown in the lower right corner of figure 1). The same arguments lead to the partially engulfed ad completely engulfed states also pictured in figure 1.

Figure 2 shows a micrograph of the time evolution of the double emulsion state. The stable state of this particular interfacial energy ordering (figure 2 (a) ) lies in the $S_2<0$; $S_3<0$ subregion. Consequently, a 1,2 - 2,3 interface costs less energy than 1,3 interface causing the inner droplet to replace some of the outer droplet's interface with the wetting phase. However, because the 1,3 (inner,outer droplet) interface has less energy than a 2,3 + 2,1 interface, the two droplets remain fused or "partially engulfed". The magnitude of this energy preference (ie. $S_2$) determines the geometry of the dual-phase structure.

Figure 2 shows the time-evolution of (a) $S_2 >0 , S_3 < 0$ and (b) $S_2 , S_3 < 0$ system, wherein once the inner droplet comes in contact with the outer droplet's boundary it (a) is ejected or (b) sticks, because its energy is lowered. Presumably, when the inner droplet and outer droplet boundaries come in contact in a fully engulfing system, the inner droplet simply "bounces off" of the wall formed by the outer droplet's interface. Though the authors did not mention this, it would be interesting to study the "bouncing", "sticking" and "ejecting" dynamics in detail. The author's do estimate a time scale for relaxation of the emulsion to its stable state, which is dictated by the relative speed of the two droplets in the channel. The reader is referred to the paper and references therein [2]. The reader is also referred to a nice discussion of the $S_2$ and $S_3$ space [3].

Figure 3: Polymer droplets irradiated with UV light to produce (a) core-shell and (b) Janus colloids.

As an interesting application of partially- and fully-engulfed emulsions, the authors proceeded to make droplets out of a photocurable polymer. After exposing the drops to UV radiation for 100 ms, they hardened to form microparticles with the nontrivial geometries shown in figure 3. Since the curvature of the Janus indentation (figure 3 (b)) is determined by the interfacial tensions, one could presumably control this parameter to some extent. A microlens mould comes to mind as an application of such a colloid.