Brownian Dynamics Simulations

Brownian dynamics is a computer-based simulation technique. The components of a system are allowed to respond to the instantaneous forces present in a given configuration, which causes the system to adopt a new configuration. The forces in this new configuration are then recalculated, the system is allowed to adopt yet another configuration, and a sequence of such steps simulates the evolution of the true system in time. Even in the absence of an obvious force between components the system will still evolve from one configuration to the next due to thermal fluctuations. These thermal fluctuations, manifest as Brownian motion, are represented in a typical simulation of particles suspended in a fluid as random “kicks” acting on each particle. Each individual kick on each particle embodies the average impact of the fluid molecules in which the particles swim.

We have applied Brownian dynamics to a suspension of spheres subject to thermal fluctuation and with the depletion effect represented by a relatively novel effective interaction potential. Brownian dynamics simulations allow us to study the formation and time evolution of aggregated emulsions in a controlled way that can be compared to experiment.

The animated sequence corresponds to 20 seconds of real time for 512 particles of 2 micron diameter, corresponding to a dispersed phase volume fraction of 10%. The time step size (in real time units) is 40 microseconds, so this short movie is the result of 500,000 configuration recalculations starting from an initial random configuration. The periodic boundary conditions mean that droplets that vanish across one face of the simulation box reappear from the opposite face.

The particles jiggle about, displaying thermal Brownian motion, unable to inhabit one another’s space. However they also feel one another’s presence through the depletion interaction, which gives rise to aggregation and the droplet network that forms over time. Despite the presence of randomising thermal effects the particles stay aggregated, though significantly the network is not totally quiescent. By comparing the initial and final frames of the simulation, it is easy to see just how much structure has developed as a result of the depletion attraction:

Initial configuration	Final configuration

To explore the impact of polydispersity, we have also simulated mixtures containing equal numbers of 1, 2, 5 and 10 micron-diameter particles. In the following animation, aggregation is switched on:

The corresponding initial and final configurations clearly show polydisperse aggregation:

Initial configuration	Final configuration

This suggests that polydisperse systems can form relatively strong networks in which the primary network structure is dominated by larger droplets, with added strength arising from smaller droplets populating the interstices.

The simulations capture an era of a dispersion’s evolution, the initial aggregation step, that is hidden from our experiments since the time scale is so short. It is interesting to see what effect gravity has on that initial stage, since in a real system for which there exists a buoyancy mismatch a net external field (in this case gravity) might well be expected to play some role. Structure “baked in” at the formation stage of an aggregated dispersion might reasonably be expected to have an impact on the subsequent properties of the network at much later times. A movie is available [AVI, 9Mb] which shows the same monodisperse system as above but with the disperse phase 50% denser than the continuous phase:

Despite the presence of an external field the system still aggregates into a network. Moreover, there is a hint of anisotropy – compare the initial and final frames of this sequence to those above in which there is no gravity:

Start sequence	Finish sequence