Model overview
The model used here is based on the Diffusion Limited Aggregation (DLA) model introduced by Witten and Sanders (1981), extended with a mechanistic simulation to drive the system towards an equilibrium state. Each model step consists of two parts: first the DLA step in which new particles are attached to the aggregate and second the simulation of the internal dynamics of the aggregate. In this model, particles are not considered to be of the same type (unlike the standard DLA model), but to be of different types that are distinguished by their surface features. The different particle types get concentrations assigned at the beginning of a simulation which determine the probabilities for generating the different particle types, when they are inserted. The particles are modeled as hard spheres with a diameter Dp and with circular patches on the surface, called binding sites, through which the particles can bind to each other. A binding site has three properties: the position p on the surface of a particle, the opening angle a, and a set of other binding sites S to which it can bind.
Figure 1: Schematic view of a particle with a binding site. The green area marks the surface area at which the particle can bind to other particles. a is the opening angle of the binding site, and p is position vector of the binding site. The position is defined by the center point of the binding site on the particle’s surface.
Figure 1 shows a schematic view of a particle with a binding site. Using the binding site model, two particles i and j can bind if the conditions given in figure 2 are fullfilled. During the DLA step, new particles attach to the aggregate which leads to a growth of the aggregate. In the second step, the internal dynamics of the aggregate is simulated. This is done by simulating the Newtonian trajectories for each particle. During this phase, binding sites can exhibit a mutual attraction force if the binding conditions concerning binding site positions and matching properties as well as the additional condition L ≥ dist(i, j) > Dp are fullfilled, where L determines the distance where bonds break. Further, if the necessary conditions for binding are fullfilled for an unbounded pair of particles a bond is established between the two particles. If any of the three binding conditions is violated for a bond, the bond breaks. During the dynamics simulation the bonds are represented as stiff, damped springs. Collisions between particles that do not result in new bonds are handled as momentum and energy conserving collisions. Before each step of the dynamics simulation, the velocity of each particle gets assigned a random value from a normal distribution. Also, the velocity of each particle gets periodically random contributions. For details, see Schroer & Lindgren (2006) and Lindgren et al (2007).
Illustration of simulation results
In order to explore some of the possible structures, randomized simulations were executed. Particle and binding site types were created at random and then used in simulations. Each simulation had five different particle types with at most eight different binding sites, while three binding site types were created. The diffusion step was executed until ten particles had attached to the aggregate. Then the physics simulation was executed to allow for rearrangements in the cluster. The final aggregate size was 300 particles for each of the aggregates. The results show examples of different structural properties. First, some simulations were encountered in which several small instead of one big aggregate were formed. Further, cyclic bindings can be observed in most obtained structures. Examples of these structures are given in the figure below. In addition to the above described observations, the influence of the particle type concentration on the aggregate growth was examined. To study this topic, an engineered example was created. In this example only one binding site type existed with an opening angle of 20 degrees with only itself as binding partner. In addition, two particle types were defined. The first one (type P0) had six binding sites. Each of them was placed in such a way, that its position vector pointed along a local coordinate frame axis in the positive or negative direction. The second particle type (type P1) had two binding sites. Both of these were placed on opposite positions on the particle surface. Two experiments were conducted of which the first one was about the effect of a high P0 concentration and a low P1 concentration, while the second one examined the effect of a high P1 and a low P0 concentration. The results are shown in figure (d) and (e). These experiments were conducted using an older version of the model in which the internal dynamics of the aggregate were not simulated.
Three movies illustrating the self-assembly process are available for 2, 3, and 6 binding sites: movie-2, movie-3, movie-6.
References:
- Schroer, J.P., and K. Lindgren (2006). Simulation of higher-order self-assembly. Proceedings of ECCS06.
- Lindgren, K., J. Nyström, J.P. Schroer (2007). Prototype simulation package for physically self-assembling units. PACE report for Deliverable D13.
- Witten, T., & L. Sander (1981). Diffusion-limited aggregation, a kinetic critical phenomenon, Phys. Rev. Lett. 47, 1400.
Figure 2 (a)
Example of aggregate formed by the simulation.
Figure 2 (b)
The self-assembly may result in the formation of separate aggregates. Note that there is a "ring" shaped structure in the bottom left aggregate.
Figure 2 (c)
Aggregates may appear in various sizes as illustrated by this simulation.
Figure 2 (d)
The experiment with different concentrations of the two particle types P0 and P1. Here P0 has high concentration; cf. next Figure.
Figure 2 (e)
Here P1 is of high concentration instead, resulting in a structure with long chains.