# Self Assembly of spherical particles on an evaporating sessile droplet

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All images taken from the paper (see references).

## Summary

This paper investigates the various packing configurations that result when spherical particles are adsorbed on the surface of a liquid spherical drop, which then evaporates while sitting on a flat surface. These results could lead to new techniques in self-assembly. The authors show that the final packing configuration of the particles exhibits different behaviors depending on the number of particles in the drop and the contact angle of the drop with the surface on which it sits.

### Setup

The liquid drop is assumed to be spherical in nature and sits on a flat surface. The particles themselves are hard spheres of radius $a$ and are adsorbed to the inner surface of the liquid drop, forming a contact angle $\theta$ with the liquid surface. The liquid forms a contact angle $\alpha$ with the flat surface; it is kept fixed during evaporation, meaning that as the radius of the spherical drop decreases, its center may shift vertically.

Diagram of parameters for initial configuration of simulation.

### Initial Packing

When $\alpha < \alpha_c\,$, ring and pyramid initial packings occur. They are shown here for $N=6,7,8\,$.
When $\alpha > \alpha_c\,$, asymmetric initial packings occur. They are shown here for $N=6,7,8\,$.

When the liquid drop is sufficiently large, since that particles are adsorbed to its surface, they will not be touching. As the drop evaporates, however, the radius of the drop decreases and eventually there comes a point where the particles will start to touch. These means that our configuration as a whole is not longer spherical. The configuration of the particles at this critical radius is called the "initial packing". This is in contrast to the configuration of the particles at the very end of the simulation when the liquid drop has completely evaporated, called the "final packing". The authors have shown that there is a one-to-one mapping between the initial packings and final packings.

There can many different possible initial packings for a given number of particles $n\,$ and contact angle $\alpha\,$, so the trouble is finding which ones actually occur. To find these, a "Markov Chain Monte Carlo" algorithm was used. The algorithm is as follows: 1) randomly move a particle by a step, if it an invalid move, undo it, 2) after completing 1 ten times, decrease the radius of the drop, and if it loses its spherical shape, undo it, 3) repeat both steps 10,000 more times. The initial packings that occur for spherical particles on a free sphere are already know (i.e. a liquid drop in midair, not on a surface). The authors were able to use their algorithm to determine all the expected initial packings for up to $n=20\,$, which also happen to be the densest packings. Running the algorithm on for a liquid drop on a surface, however, did not produce merely the densest initial packings even at low $n\,$. The algorithm was run for $n\in[1,20]$ and $\alpha\in\left[\frac{3\pi}{16},\pi\right]$, producing some interesting results. Most notably, below a critical contact angle $\alpha_c\,$, most of the initial packings were either ring or pyramid configurations. Above $\alpha_c\,$, asymmetric configurations arose.

### Final Packings

A program called "Surface Evolver" was used to simulate the final packing that occurred when the liquid drop completely evaporated. The authors only simulated evaporation for $n=5\,$ because the program is very computationally intensive. Some results are given in the two figures below. Depending on the initial configuration of the particles, there can be different final packings. Note that the mapping from initial packing to final packing is still unique, but that there can be more than one possible initial packing for a given $n\,$ and $\alpha\,$.

The final packing for $n=5\,\alpha=\pi\,$, resulting in a stacked final packing.
The final packing for $n=5\,\alpha=\pi\,$, along with figures of the intermediate processes, resulting in a planar final packing.

## Relevance to Soft Matter

Interfacial tensions play an important role in soft matter physics. By making use of how this force changes with decreasing surface area through a process like evaporation, we may be able to create useful self-assembled structures without having to do much templating and electrical/mechanical tuning. There are many interfaces at work in this experiment: the liquid is in contact with both the particles and the flat surface, while the particles are also constrained by the flat surface. The liquid-air interface was not considered much in this simulation, but in physical experimentation, there will interaction between the air and liquid through evaporation.

### Related Papers

Assembly of Colloidal Particles by Evaporation on Surfaces with Patterned Hydrophobicity -- Fengui Fan, et. al., Langmuir, 2004, 20(8)

Evaporation Driven assembly of colloidal particles -- Eric Lauga and Michael P. Brenner: Physical Review Letters, 2004, 93, pp 238301-1 to 238301-4

## References

M. Schnall-Levin, E. Lauga, and M.P. Brenner. "Self-Assembly of Spherical Particles on an Evaporating Sessile Droplet." Langmuir 22, 4547-4551 (2006).