Evaporation Driven assembly of colloidal particles
Original entry by Joerg Fritz, AP225 Fall 2009
Eric Lauga and Michael P. Brenner: Physical Review Letters, 2004, 93, pp 238301-1 to 238301-4.
On a very general level self-assembly works best if the assembling particles have specific properties. One of them is the property that they are not very symmetrical, they have structure. This is especially true when we talk about the creation of nontrivial constructs that go beyond simple chains. This is not surprising, after all information about the final structure has to be encoded in the particle-environment system.
For this reason cases where spherical particles assemble into complex structures are very interesting from a theoretical point of view. Where does the information in these systems come from? One of the most spectacular of these kinds of systems is an emulsion of oil in water, where the oil drops contain a small number of polystyrene spheres (with a diameter of about 900 nm each). When the oil is evaporated preferentially, these spheres assemble into complex clusters. Very surprisingly, experiments by the Manoharan group at SEAS (see Building Materials by Packing Spheres for details) indicate that these sphere packings are unique, that is, if you know the number of particles in the assembly, you can predict exactly how they will assemble. Considering the immense number of possible arrangements that spheres can assume this observation requires an explanation.
To answer these questions, the paper takes two routes. It first describes precise numerical simulations that can reproduce the shapes exactly. It then investigates how much of these structures can be understood by simple physical arguments and geometric considerations.
The numerical solution is based on one basic assumption. Most forces in the system don't contribute in an essential way to the final packing of the spheres. In fact the configuration for any volume of the oil droplet can be calculated by minimizing the surface energies present in the system, thus ignoring all electrostatic or van der Waals interactions. The total surface energy of the system is the sum of the surface energy of the droplet (D in figure 2a), the droplet-particle interface (DP in figure 2a) and the particle surface itself (P in figure 2a). Note the implicit assumption here that polystyrene spheres can be modeled by the same laws as fluids. These assumptions can be used to determine packings for any number of spheres in the droplet using a small number of additional constraints (spheres do not overlap and they have a much higher surface tension than the drop of oil). The results agree with the experiments over an extremely large parameter regime, from different contact angles, number of spheres to detailed initial conditions. While these results are impressive they do not explain the most puzzling property of this system, why are the packings produced by the experiments (and now the simulations) unique?
The answer to this question can be found at the point where the oil droplet has to deviated from its spherical volume if it wants to continue to shrink and evaporate. The situation at this moment is shown in figure 2b. Below this so called critical volume, there is a significant capillary force acting on all the particles, as depicted quantitatively in figure 2c. The explanation of the uniqueness of the packings is now based on two arguments:
1.) The packing of the spheres at the critical volume is mathematically identical to the packing of circles on a sphere, a problem which has been studied for a long time. For this problem it has been proven that the packings (of circles on spheres) are unique for basically any number of circles (or polystyrene spheres in our problem). Thus we start with an unique circular packing at the critical volume, without regard of the initial conditions
2.) The subsequent evolution of the particles is highly constrained. Simple estimates of the number of degrees of freedom for the interaction between the particles, taking into account capillary forces, indicate that in nearly every case only one final packing exists that is consistent with the constraints.
In this picture the unique packing is actually a result of the mathematical properties of the circles on a sphere packing problem and the particular interactions of the spheres with the surface of the oil droplet, which impose a high number of constraints on the dynamics of the system. So in this case the information we mentioned earlier is stored in specific properties of the particles and its environment that are not immediately visible or obvious.
The soft matter in all of this
Soft matter ideas are employed in several stages of the arguments of this paper. The first interesting idea is to describe the solid polystyrene spheres as liquid droplets with a very high surface tension. The success of the simulations shows that the concept of material properties that lie somewhere between solids and liquids can sometimes even be extended to the boundaries itself, when traditional solids show the behavior of liquids with extreme properties.
This paper also reinforces again that the properties of many systems can be understood on a basic level just by thinking about its geometry and basic physical effects, just as we have seen repeatedly for different complex fluids.