Topological mixing in 2D active nematic liquid crystals

Recent years have seen a surge of interest in active materials, in which energy injected at the microscale gives rise to larger-scale coherent motion. One prominent example is an active 2D liquid crystal composed of microtubules in the nematic phase. The activity is generated by molecular motors that consume ATP to generate local shearing between the microtubules. The resulting 2D fluid flow exhibits self-generated mesoscale chaotic dynamics with a characteristic folding and stretching pattern. We analyse this dynamics in the context of chaotic advection, in which the fluid can be viewed as "stirred" by the topological defects in the nematic order parameter. We compute the topological entropy from the braiding of these defects and show that all of the entropy arises from the positive one-half defects; the negative one-half defects, which are also present, contribute nothing to the entropy. We also show that the topological entropy generated by this stirring can be understood as a direct consequence of the micro-scale stretching quantified by the Lyapunov exponent, which is computed from PIV data. Our work is based on experimental fluorescence images of the microtubule structure.