**Appendix 1.(b). Droplets lying along the x axis**

**4.4 Morphologies transition observations**

**4.4.1** **(I): Stable morphologies**

The morphologies of cluster aggregates increase dramatically with the number of constituent droplets. The complexity of possible organization is directly related to the droplet’s number and size. The enumeration of clusters determines the set of minimally structures that can be formed by self-assembly.

Arkus et al.[239] have conducted an analysis which combines graph theory with
geome-try to analytically solve the problem for clusters satisfying minimal rigidity constraints(≥3
contacts per particle,≥3N-6 total contacts). They have shown that up to N = 9, every
packing has exactly **3N-6** contacts, so that all packing have the same potential energy.

However, for n≥9 the ground stated degeneracy increases exponentially. If we expand the
graph theory to lower dimensions, the finite close packing is**2N-3**for 2D structures, and
**N-1**as for 1D structures.

To begin with, we limit ourselves with clusters of identical constituent droplets (i.e.

no satellite droplet is presented in the cluster). In the case of N = 2, only doublet can be obtained. For N = 3, stable triangle or chain structures are both possible to be produced with our experiments. For larger number of spheres, i.e. N≥4, three dimensional structures are the finite close packings. The Figure 4.21(a) shows the three dimensional, two dimensional and one dimensional close packing structures of droplets having equal size from N = 2 to N = 6. All the corresponding clusters have been observed under different experimental conditions, results are shown in figure 4.21(b). When N ≥6, the clusters develop degeneracy in both 2D and 3D structures with different probabilities, we confirm this result with our experimental observations.

Figure 4.21: (a) The possible closing packing structures of clusters from N = 2 to N = 6.(b)
The closing packing structures of clusters obtained from experiments with Fluorinated
oil-in-water emulsion. The number varies from N = 2 to N = 6. When N=6, the degeneracy
appears: 2 ground states for 3D structures and 3 ground states for 2D structures. The
scale bar is 10*µm.*

Besides those conventional packings, we also observe some unusual structures such as stable symmetrical T-shaped clusters which can also have long-life state. We have discovered many other packings that are hypostatic, namely they have fewer than the 2N-3 contacts required to be linearly rigid in 2D structures. Those morphologies are unfavorable in the terms of the minimization of potential energy, nevertheless, under hydrodynamic conditions, those clusters could be very stable (see figure 4.22(a)). In the presence of satellite droplets, more complex structures are also available. Figure 4.22(b) shows two

4.4. Morphologies transition observations 95 trains of clusters with satellites droplets.

Figure 4.22: (a) Optical micrograph of a train of stable T-shaped clusters. (b) Optical
micrographs of clusters that consist of non-identical droplets (two droplets and a satellite
droplet/ three droplets with a satellite droplet). The scale bar is 10*µm.*

The rearrangement of droplets configurations takes place immediately after they are produced at the step. The evolution towards stable clusters are shown in the figure 4.23.

Figure(a), (b) are triplets and quadruplets of droplets with diameter 50µm and (c), (d) with diameter 5µm.

Similar fast evolutions towards stable morphologies are observed using numerical sim-ulation(see figure 2, self-assembly kinetics in preprint). We reproduce well the rearrange-ment using dipolar interaction and adhesion interaction which suggests that it is the inter-play between the hydrodynamic and physico-chemical forces that mediates the evolution of constituent droplets positions in the cluster. Here we mainly present the observations and the discussion of the physics will be addressed in the paper.

Figure 4.23: (a) Snapshot showing the evolution towards clusters to triangle-like stable
triplet clusters. (b) The evolution towards clusters to rhombus-like stable quadruplet
clusters. (a), (b) Stable clusters obtained with PDMS system with W=50µm, *h*_{1}=10µm,
*h*2=163µm. Scale bar, 100 *µm. (c) Snapshot showing the evolution towards clusters*
to triangle-like stable triplet clusters. (d) The evolution towards clusters to tetrahedron
stable quadruplet clusters. (c), (d) Stable clusters obtained with PDMS system with
W=20µm, *h*1=1µm, *h*2=22µm. Scale bar, 100 *µm. Scale bar, 10* *µm. The formulation*
used is fluorinated oil droplets in water with 2% SDS.

We characterized the polydispersity of stable clusters obtained. As for the anisotropic objects, we use Feret’s diameter to analyse cluster distribution. Feret’s diameter is de-ducted from the projected area of the particles using a slide gauge. It is defined as the distance between two parallel tangents of the particle at an arbitrary angle. We plot the

histogram of the ratio of the Maximum Feret’s diameter *D** _{f max}*, and Minimum Feret’s
diameter

*D*

*f min*of each clusters flowing in the reservoir. Figure 4.24 shows results of the doublets, triplets and quadruplets of droplets with diameter 50µm and figure (c) the pentamers of droplets with diameter 5µm. The clusters produced with our approach are highly monodispersed, with CV≤5 %.

Figure 4.24: (a) Definition of Feret’s diameters, the maximun diameter is illustrated in red
and minimun in green. (b) Size distribution of a population of 152 doublets, 382 triplets
and 217 quadruplets with d= 50*µm in diameter, showing a coefficient of variation CV (i.e.,*
the quantity 100 *σ/ (D*_{f max}*/D** _{f min}*)) below 5%, where

*σ*is the standard deviation and

*D*

*f max*

*/D*

*f min*is the ratio between the Feret’s maxium diameter and the Feret’s minimum of clusters obtained with image processing. Scale bar, 100µm. (c) Size distribution of a population of 2134 pentamers, with d= 5

*µm in diameter. Scale bar, 10µm.*

**4.4.2** **(II): Oscillatory morphologies**

So far we have presented the stable morphologies. We observe that the clusters could develop permanent 2D configurational chaotic oscillations too, a phenomenon never re-ported in the literature. In the following part, we will present qualitatively this dynamical oscillation behavior.

The chaotic oscillations are observed as the velocity field increases and the adhesion between the droplets decreases. Those clusters develop 2D configurational chaotic oscil-lations. Figure 4.25 shows time sequences of oscillation events. Clusters composed of 3, 4, 5 droplets show similar oscillatory behaviors. Droplet may glide and rotate from one

4.4. Morphologies transition observations 97 position to another without detaching from the each other.

Figure 4.25: Snapshot showing the oscillation of clusters (N = 3, 4, 5). Clusters are flowing from left to right. The scale bar is 5µm. Fluorinated oil droplets in water with 0.4% SDS.

With the image treatment, we measure the average of droplet distances inside a cluster:

*r*=

*N*

P

*i,j*

*r**ij*

*C*^{2}_{N}

We plot the average of droplet distances *r* normalized by the droplet radius R as a
function of the time (see figure 4.26). In the finite close packing state, the
dimension-less value *r/R* is the lowest value corresponding to the most compact form. When the
cluster undergoes oscillations, this dimensionless value will fluctuate according to the
con-formations. The oscillatory time scale under this experiment condition is in the order of
seconds.

Figure 4.26: Average of droplet distances *r* normalized by the droplet radius R as a
function of the time. (a) Oscillatory cluster composed of three droplets. (b) Oscillatory
cluster composed of four droplets.

**4.4.3** **Adhesion force effect**

For a given number of droplets, the cluster aggregates can adopt stable configurations or show dynamical oscillatory states. Whether the morphologies of clusters adopt stable or oscillatory states strongly depends on the experimental conditions. We identify that the adhesive interactions between droplets play an essential role in the clustering process.

If we reduce the adhesion between the droplets, the clusters tends to move from stable state to oscillatory state (which is the case that in figure4.23 (with 2% SDS) and figure 4.25 (with 0.4% SDS)). If the adhesion is weak enough, the clusters cannot be obtained whatever the flow condition is; the droplets don’t aggregate to be transported as an entity but as individual objects. In the figure 4.27, we show one example of such non-adhesion behaviors (with 0.2% SDS). The analysis concerning adhesion interactions will be discussed in details later in Chapter 5.

The figure 4.27 shows the experimental result of fluorinated oil in water stabilized with 0.2% (1 CMC) SDS. Three droplets are produced successively from one plug-like droplet. Then they get dispersed following the streamlines separately once entering into the reservoir.

Figure 4.27: Snapshot showing the separation of non-adhesive droplets in the reservoir channel, the clustering process cannot be achieved using Fluorinated oil droplets in water with 0.2% SDS. The scale bar is 20µm

We compare behaviors of ‘adhesive’ and ‘non adhesive’ droplets in different flow con-ditions, the experimental observations are shown in figure 4.28.

Figure 4.28: Comparison behaviors of ‘adhesive’ and ‘non adhesive’ droplets in different flow conditions. ‘Adhesive’ emulsion is Fluorinated oil in water with 2% (10 CMC) SDS and ‘non-adhesive’ emulsion is with 0.33% (10 CMC) copolymer Pluronic F-68.

**4.4.4** **Flow condition effect**

With the same formulation of emulsion (dispersed and continuous phases, surfactant etc.) and experimental conditions (droplet size, temperature etc.), we observe that the flow field can dramatically alter cluster morphologies too. In order to screen the controlling factors which govern the behaviors of the cluster self-assembly, we adopt the emulsion

4.5. Numerical simulation 99