# Evaporation-Driven Assembly of Colloidal Particles

Wiki entry by Emily Gehrels, Fall 2012

Based on the article: Lauga, E., Brenner, M.P. (2004). Evaporation-Driven Assembly of Colloidal Particles. Physical Review Letters, **93**, 238301.

## Background

Vinny Manoharan wrote a paper describing how clusters of small (d=844nm) polystyrene spheres could be formed in reproducible configurations by dispersing them in a toluene-water emulsion and then preferentially evaporating the toluene. The geometries that formed minimized the second moment of the cluster as defined by <math>\mathcal{M}=\sum_{i}||r_i-r_o||^2</math> where <math>r_o</math> is the position of the center of the droplet. This paper examines the theoretical basis for these findings.

## Theories and Simulations

For a droplet with particles attached to the surface, the particles arrange themselves in such a way to minimize the total surface energy between the particles and the drop, the drop and the surrounding medium and the particle and the surrounding medium. A program (Brakke's Surface Evolver) was used to simulate the equilibrium positions of particles in a droplet. The system was initialized with a certain number of particles positioned randomly on the droplet and then the volume of the drop was slowly decreased in steps. At each step the particles were moved to the positions that minimized the energy. The final packings (when the dop volume went to zero) in the simulation agreed with the experimentally observed results as seen in the figure below where in each pair of images the left image is the experimentally observed packing and the right image is the packing obtained from the numerical simulation.

To understand why these particular packings arise, the authors then study the theoretical basis for the packing selection. For large enough volumes, the minimum energy configuration is a spherical droplet where the particles do not interact. However, when the volume of the droplet decreases to the critical volume (<math>V_C</math>), the droplet cannot remain spherical as the particles come close together. At this critical volume the particles reach a critical packing that is determined by the "cone of influence" of each particle. The cone of influence is a cone that starts at the center of the drop and extends out, tangent to the particle, to intersect with the surface of the droplet. The interactions between the particles on the surface of the droplet are directly related with the interactions between the intersections of the cones of influence of the particles with the surface of the droplet. So, the problem of packing particles on the droplet at the critical volume is like packing circles on a sphere. Most circle packings are unique for a given number of circles. However, there are some numbers of circles (5, 19, 20, 23, ...) where there is a degree of freedom that arises from on of the circles being free to 'rattle' around. There are also some numbers of circles (such as 15) where here are two (or more) different configurations that work.

Below <math>V_C</math> we need to find a way of theoretically determining the particle configuration that minimizes the energy If we decrease the volume by a small amount, the forces on each particle are from capillary forces from the droplet interface and from contact forces between the particles. To determine whether a packing will be unique, we will look at the number of degrees of freedom involved in these systems. Each particle as 3 degrees of freedom and the droplet itself has one (arising from the pressure). This means that the system has a total of 3N+1 degrees of freedom. Now we have constraints on the system. The first is that any rotation of the entire droplet does not change the packing. This removes three degrees of freedom. The fact that the particles cannot overlap removes an additional <math>n_c</math> where <math>n_c</math> denotes the number of contacts between different particles when they are critically packed. The final constraint is that the capillary and interparticle forces on have to balance on each particle at equilibrium.

<math>F_i+\sum_{j\in C(i)}f_{ij} =0</math>

This last constraint is not as straight forward as the other two. In order to satisfy this equation, the capillary forces on each of the particles must satisfy compatibility relations. This constrains the rearrangement of the particles such that only <math>n_f</math> of the N capillary forces can be chosen independently. This condition imposes an additional <math>N-n_f</math> constraints to the problem. This leaves a total of

<math>3N+1-3-n_c-(N-n_f)=2N-2+n_f-n_c\equiv n_m</math>

degrees of freedom in the problem.

For most of the cases examined in the experimental study, there is only one final degree of freedom (<math>n_m=1</math>). The only cases where this dof is not one is in the case where a rattler is present. This means that for the cases studied, there is exactly one set of capillary forces that is consistent with the constraints and the final packing is unique.

We can find the exact equation relating the capillary force on each particle as the radius of the droplet changes from R to <math>R+\delta R</math> is given by:

<math>F_i=-2\pi \gamma_D cos \beta (\delta r_i + \frac{Acos(\beta )}{4\pi R^2-NA} \sum_i \delta r_i)</math>

where <math>\alpha</math> is the dry angle of the particle on the critical packing, <math>\theta</math> is the equilibrium contact angle, and <math>\beta=\alpha-\theta</math>.

This formula is valid only for small changed in volume, but can be applied iteratively as the droplet volume is decreased by small amounts. The results for the second moment given by this theoretical analysis (<math>M_m</math>), found in the experiment (<math>M_{exp}</math>) and the minimum second moment (<math>M_2</math>) are given in the table below.

As can be seen, there is excellent agreement between all three values.