Using the depletion interaction to control self assembly.
As synthesis techniques become evermore sophisticated the availability of colloidal particles of various different shapes and sizes has increased dramatically. From spheres and rods to multi-faceted, or even concave particles, these new building blocks offer many exciting opportunities in the development of advanced new materials and devices.
In order to achieve such goals it is important to understand, and precisely control, the interactions between the particles. A useful tool to this end is the addition of non-adsorbing polymers or nanoparticles to induce an attractive depletion force between the colloids. Depletion is a purely entropic effect, which can be easily tuned by altering the size and quantity of the depletant.
Complementary shapes aiding assembly.
Starting from a colloidal sphere, by creating a pressure imbalance you can make it buckle. Particles made in NYU, and now Oxford, can have very well control dimples that are surprisingly uniform. Combined with the depletion interaction these dimples can form highly directional bonds with another particle that has a complementary shape.
By varying the dimple depth I am looking at the different structures that can form. For deep dimples the particles can chain (see opposite) and for shallower dimples the chains can branch and form more complex aggregates. This work has lead to interesting questions about the effect of packing in the assembly of shapes in general.
In colloabration with Marjolein Dijkstra in Utrecht I am looking at the assembly of colloidal cubes. These particles were made in Utrecht by Laura Rossi and Albert Philipse. Starting from a hematite core a layer of silica is grown around the outside and then the core is washed away. Adding depletants causes the cubes to assemble into crystal structures. We've found that the structure formed is extremely sensitve to the size of depletant and the precise shape of the cube.'
The projects listed here present demanding problems for simulation. Monte Carlo has long been used as a powerful tool in statistical physics to answer questions that are too complicated for theory alone. Many techniques exist to overcome problems such as large free energy barriers, critical slowing down, rare events and optimisation problems. A large part of my work is devoted to developing new methods to overcome such problems. Below are listed some of my principle approaches.
All moving together.
The depletion projects have the common problem of simulating large particles alongside small particles. For this we use the geometric cluster algorithm, or GCA, (see opposite) to move large particles around. The GCA is an iterative scheme that reflects particles in a pivot or a plane until there are no more overlaps. I've extended this algorithm for the anisotropic shapes above. With Nigel Wilding I have also developed statistical tools to study phase transitions in systems with large size asymmetry.
Other cluster algorithms exist that can operate when systems are close to their critical point. I used one such algorithm, the Wolff algorithm, to generate large Ising model configurations in the critical point demonstrations.
Another way around.
Suppose you're walking along a path and the way is blocked. If you can only move on the path then you're stuck. If you can leave the path and find a way around, then you can rejoin the path on the other side. In statistical physics by allowing your system to break free of its normal constraints, even the normal laws of physics, you can often find a way around a block and connect different states on your path. In the example to the left we would like to insert the large particle into a space occupied by small particles. To do this we allow the large particle to exist in a ghost state, gradually pushing the small ones away. For some network problems I use this technique to sample networks that have very strict constraints. By relaxing the constraints one can smoothly move between different solutions.
Statistical Mechanics of Graphs.
Networks are everywhere you look. From social networks, collaboration networks, power networks, the web and the internet, to biological networks such as metabolic networks and gene regulatory networks. Once simplified to their basic structure of nodes and connections many of these networks begin to look very similar. From a statistical physics perspective these networks are big systems with many interacting components, and this is right up our street.
Coping with incomplete data.
It is quite rare in the real world to have completely accurate data for any given network. It can be difficult to know how to fill in the gaps in a fair way. By applying the principle of maximum entropy statistical mechanics provides a way to generate the most likely networks given what information you have. From here one can deal with the ensemble of likely networks to find what features are common.
Through Bath's recently launched Centre for Networks and Collective Behaviour (CNCB) I'm working on problems across several departments that I look forward to updating here soon.
Stuck in traffic.
One of the first features of common networks to be discovered is the "small world" effect. As neatly demonstrated by Stanley Milgram's experiment one can get from one person to another in a reasonably small number of hand shakes. Also discovered was that certain people were more important than others, so-called "hubs". In fact it is easy to think of many situations where the main connection route is through a centralised location. This leads to the question of what happens if this hub becomes congested? Is it better to wait and use the central hub, or is it better to take the long way around? Is it better to have an organisation run everything through a central decision maker, or is it better to bypass the top level?
Along with Neil Johnson (then Oxford) I looked a simple models whereby the more connections that went through a central hub the more congested it became. Depending on the level of congestion different optimal network topologies emerge with a sharp transition between a sparse network and a well connected network. We compared these results to optimal solutions in nature of slime moulds that route resources placed in different locations.