Exploring the dynamics of active matter: From a single chiral active Brownian particle to collective thermophoresis of active rods

Abstract

Active matter has emerged as a fascinating field of research over the last few decades. Active matter is driven out of equilibrium on the shortest scale of individual constituents, consuming and dissipating energy from the local environment or internal energy source. It can self-propel by conversion of energy into mechanical motion, breaking the time-reversal symmetry and equilibrium fluctuation-dissipation relation. Active matter can be found in the natural world in a variety of levels ranging from the single particle level (molecular motors, individual cells, and bacteria) to the collective level (bird flocks, fish schools, or human crowds). Taking inspi- ration from such natural active matter systems, researchers have also designed artificial active matter systems in the laboratory environment, e.g., Janus particles (using phoretic force as the source of self-propulsion), vibrated granular matter, and hexbugs. In this thesis, we have explored both the single-particle dynamics and the collective behavior of active-matter sys- tems. First, we investigate the dynamics of a single chiral active Brownian particle to capture the essence of the active matter dynamics at the single particle level. Next, we studied the collective thermophoresis of self-propelled active rods in the presence of a temperature gradient. The motion of self-propelled agents is often described in terms of three related models: the ac- tive Brownian particles (ABP), run-and-tumble particles (RTP), and active Ornstein-Uhlenbeck process (AOUP). Up to the second moment, their dynamics is equivalent and can easily be mapped from one to another. The generation of self-propulsion often utilizes a break in parity in the direction of motion, the heading direction, which undergoes either continuous (ABP, active colloids) or discrete reorientation (RTP, bacteria). In the active phoretic motion of colloids, such asymmetry is inherent to the design of Janus colloids. However, the left-right parity symmetry around the heading direction can also be broken for chiral (or circular) active Brownian particles, forcing them to turn in the broken-symmetry direction while performing self-propulsion. Chirality in active matter is observed in various natural systems, such as bac- teria near walls and interfaces, sperm cells, etc. In synthetic systems, colloidal microswimmers with broken chiral symmetry, motile droplets, granular ellipsoids, and cholesteric droplets also show active chiral motion. In part A, we provide a unified Laplace transform method to generate the exact time-dependent moments associated with the dynamics of a chiral active Brownian particle( cABP) up to any arbitrary order without solving the Fokker-Planck equation directly. We consider three differ- ent situations. In all three problems, we start by writing the coupled Langevin equation for the evolution of the position r(t) and heading direction û(t) of the active force with time t and show the detailed derivation of the corresponding Fokker-Planck equation to capture the time evolution of the probability distribution function. Next, we apply the Laplace transform to that Fokker-Planck equation and multiply it by an arbitrary dynamical moment of the position and orientation vector, ψ[r(t), û(t)], and get a moment-generating equation. Follow- ing all the necessary mathematical details, we obtain a closed-form expression of the momentii in Laplace space. Taking the inverse Laplace transform to that, we finally obtain the exact time-dependent expression of the moments. All analytical expressions are verified with the simulation results obtained by integrating the corresponding Langevin dynamics using the Euler-Majorama scheme. As the first problem, we choose to study the overdamped dynamics of a cABP in two and three dimensions. The circular trajectory in two dimensions and the helical trajectory in three dimensions are caused by the application of a constant deterministic torque in addition to the random torque used to describe the dynamics of ABP. In two dimensions (2d), implementing torque is equivalent to chirality in ABPs, while in three dimensions (3d), this is introduced as an external torque on the particle. We start with the two-dimensional dynamics of a chiral ABP. We obtain time-dependent expressions of the orientational correlation, mean displace- ment, and mean squared displacement (MSD). Our analytical results are exact as verified by our overdamped Langevin simulations. From the long-time diffusive behavior of the MSD, we obtain an effective diffusion coefficient which is a function of both the translational and rotational diffusion coefficients as well as the active speed and constant rotational rate. Our results match the existing analytical and experimental results. We also provide the exact ex- pression of the fourth moment of displacement and utilize it to calculate the excess kurtosis. The excess kurtosis deviates from Gaussian and shows oscillations with multiple zero crossings at intermediate time scales. In the next part of this work, we analyze the dynamics of an ABP in the presence of an external torque along a fixed direction. Although the MSD and fourth moments of displacement in 2d turn out to be independent of initial orientation, the initial orientation, along with the chirality, influences these quantities non-trivially in 3d. The time evolution of the excess kurtosis shows a deviation from the Gaussian nature at the small time limit as a result of active motion. On intermediate time scales, oscillations can be observed because of the chiral rotations of the cABPs. However, unlike in two dimensions, the trajecto- ries in three dimensions extend in a direction perpendicular to the direction of chiral rotation, thus suppressing such oscillations. In the second problem of this single-particle dynamics part, we consider the previously de- scribed cABP in the presence of an external harmonic potential that is spherically symmetric in both two and three dimensions. Following the same prescription as in the earlier case, we first provide the dynamical moments as a function of time that are in perfect agreement with the simulation. As an effect of the external potential, all moments saturate at their steady-state value in a sufficiently long time limit. Then we focus on the steady-state analysis of excess kurtosis. As the stiffness of the trap increases for a fixed active speed, the magnitude of the negative kurtosis keeps on increasing, suggesting that the particle gets trapped inside a circle in two dimensions (or a sphere in three dimensions) with its center at the origin and radius related to the magnitude of the active speed and trap stiffness. The probability distribution function shows an off-centered non-Gaussian behavior. Keeping the active velocity fixed, ifiii we further increase the magnitude of the trap stiffness, we again obtain excess kurtosis zero. The Gaussian probability distribution function in this case has a very sharp peak, suggesting that the particles become superlocalized near the origin. This passive(equilibrium)-active(non- equilibrium)-passive reentrant phase transition remains intact in the presence of chirality as well. However, in two dimensions, the radius of the confinement of the particle decreases as the effect of chirality, showing a very stiff Gaussian probability distribution function for a higher magnitude of chirality. We have also observed positive kurtosis for small magnitudes of trap stiffness and intermediate chirality, although the magnitude of positive kurtosis is very small. As for the three-dimensional system, the chirality shows an insignificant effect on the passive-active-passive re-entrant phase transition in contrast to the two-dimensional case, as the circular trajectory does not appear along the direction of the external torque. In the last problem of this part, we consider a point-like cABP with significant mass. Now, the coupled Langevin dynamics is underdamped in the translational degree of freedom, while its rotational motion is still in the overdamped limit. The Fokker-Planck equation is modified accordingly. We show that the circular traces in the trajectory get lost if we keep on increas- ing the mass of the particle. Following the framework, we first get a modified Fokker-Planck equation and then obtain all the exact time-dependent dynamical moments like mean-squared displacement (MSD), mean-squared velocity (MSV), fourth moment of the velocity and the displacement vector. The velocity field gets to its steady-state value at a sufficiently long time limit. In doing so, it imitates the behavior of the position vector in the previous problem, with the harmonic trap replaced by the inverse of the mass of the particle. From the steady state MSV we obtain the kinetic temperature. Using that and the long-time effective diffusion coefficient (mass-independent), we get a modified fluctuation-dissipation relation that vanishes as we consider the particle mass to be infinitely high, thus retaining equilibrium-like behavior for very heavy particles. The excess kurtosis of the velocity field in steady state also confirms that the mass has the opposite effect to the trap stiffness in the previous problem. In part B, we explore the collective behavior of rod-shaped particles in the presence of a temperature gradient. Thermophoresis or the Soret effect deals with the drift or motion of particles of a fluid system along temperature gradients. It plays an important role in various applications, such as microfluidics, convection in ferrofluids, crude oil characterization, and the fabrication of synthetic microswimmers. For example, an experiment with silica beads half- coated with gold showed propulsion when irradiated with a defocused laser beam. The gold caps act as heat sources when they absorb light. Theoretical studies to understand linear and non-linear effects in thermophoresis use either a hydrodynamic or a thermodynamic viewpoint. Most studies on thermophoresis look at spherical colloidal particles. However, filaments such as RNA or microtubules are elongated objects and can be modeled as passive rigid rods that interact via volume exclusion. Naturally, such systems show a rich variety of liquid crystallineiv phases. In a temperature-gradient field, an anisotropic shape should be characterized by mul- tiple thermal diffusivities leading to anisotropic thermophoresis. In a simulation study, it was shown that the director of the prolate ellipsoids aligns perpendicularly to the temperature gradient, whereas the director of the oblate ellipsoids aligns parallel to the gradient. How- ever, simulations of colloidal rods in a fluid indicated that thermophoretic anisotropy due to the shape does not induce the orientation of the rod. As noted earlier, the importance of thermophoresis is not limited to passive particles but in the broader context of particles that self-propel. Experimental realizations of self-propelled rods include rod-shaped bacteria, cy- toskeletal filaments such as actin, and microtubules in vitro motility assays where the filaments are driven by molecular motors, shaken granular particles, and chemically driven rod-shaped Janus particles. We study the collective behaviour of self-propelled hard rods with excluded volume inter- actions, moving on a substrate in two dimensions, and subjected to a small but finite external temperature gradient. We first establish that a rigid rod in a constant external temperature gradient will not experience torque. To show the collective behaviour, we consider coarse- grained hydrodynamic equations involving the time evolution of the fields, local density, local polarization, and local nematic order parameters or the alignment tensor. We show that an ensemble of rigid rods can align either perpendicular or parallel to a constant temperature gradient irrespective of self-propulsion activity. The density field is inhomogenous and (in- creases) decreases as we move towards the hotter end for (thermophilic) thermophobic rigid rods. Our analysis also shows that even at a density lower than the critical density required for the isotropic-nematic phase transition, it is possible to obtain a spatially varying nematic-order parameter. The self-propulsion activity induces a finite polarization. The collective behaviour of thermally active spherical Janus particles has been studied both theoretically and experimentally. Each particle dissipates heat through its surface, creating a spherically symmetric temperature profile. The temperature profile surrounding the sphere is anisotropic and decays sharply with the distance from the centre of the particle, creating a local temperature gradient. Interesting results such as finite local polarization and inhomoge- neous density profiles are reported in such studies. We have tried to extend our work on the collective behaviour of active rods to the ensemble of anisotropic rod-shaped particles, which are thermally active.