Excitable patterns in active nematics
Original Entry: Peter Foster, AP 225, Fall 2011 In Progress...
Authors: L. Giomi, L. Mahadevan, B. Chakraborty, and M. F. Hagan
Publication: Giomi et al. Excitable Patterns in Active Nematics. Phys Rev Lett (2011)
This paper consists simulations of the dynamics of a two dimensional system of mutually propelled rods immersed in a solved. The modeling of this system is very similar to traditional models used to describe liquid crystals, but with terms added in order to accommodate the active nature of the mutual propulsion of the rods. It's the active nature of many biological systems (i.e. microtubule/motor systems) that differentiates them from passive liquid crystal systems and characterizes how the energy released by the consumption of fuel (ATP) is converted into mechanical motion.
The dynamics of an active system are governed by several timescales, including the timescale of active forcing (ta), the relaxation time of the nematic degrees of freedom (tp), the diffusive timescale, and the dissipative time scale of the solvent (td). When tp ~ta, a sort of balance can exist between the active forces, elastic distortion, and flow. When ta<<tp, the passive relaxation cannot keep up with the active driving and there will be a dynamic interplay between the active and passive forces.When ta >>tp, the active nature is irrelevant and thus the system behaves as its passive analogue. 1/(ta) is proportional to the system's activity coefficient. A linear stability analysis predicts that there is a critical value of this activity coefficient (the critical value is when ta ~tp). We expect different dynamics when the activity coefficient is changed.
Figure 1 shows results of the simulations. The top two plots show results for an activity coefficient just above the critical value (ta >tp) and the bottom two plots are for an activity coefficient much higher than the critical value. The plots on the left show the velocity field superimposed on a density plot and the plots on the right show the director field superimposed on the local nematic order parameter. When the activity is just above the critical value, the system forms distinct strips with the velocity field parallel in each stripe. A large activity coefficient leads to the formation of vortices.
Interesting things happen if one choses a value for the activity coefficient in between the values chosen in figure 1 (choosing a=1.5 instead of a=.4 and a=3). The dynamics are shown in figure 2. The system in initially such than the velocity field is parallel everywhere (fig2a). However, the system is dynamic and transitions to a banded structure (fig2b) and eventually to a configuration where the velocity field is alined parallel to the the original conformation (fig2c). The cycle then repeats. Reference 2 shows a video of this flipping behavior. The plot on the left of figure 3 shows plots of the average nematic order parameter and the shear stress versus time. One can clearly see the effects of these periodic bursts and define a burst frequency. The experiments are then repeated for different values of the activity coefficient. The right plot of figure 3 shows the burst frequency as a function of the activity coefficient. The authors do not know the cause of the kink in the plot at a=1.35. Part of the reason that the plot only goes to values of a=1.5 is because if the activity becomes too high, the system becomes chaotic (see lower diagrams of figure 1).
This paper shows that excitability can be obtained from a relatively simple model. This is sort of neat because of its utility in biological systems (e.g. in cardiac rhythms). Outside of this paper, I think this area of research is going to be a very nice meeting point for biologists and physicists. Biological materials (e.g. components of the cytoskeleton) are important examples of active matter. These biological systems can be used as a testbed for theories of active matter. The insights gained from a soft condensed matter analysis could also help to further describe some of the regulation of these biological components.
 Giomi et al. Excitable Patterns in Active Nematics. Phys Rev Lett (2011)