Active Brownian Particles (ABPs) transmit a swim pressure $Π^{swim}=nζD^{swim}$ to the container boundaries, where $ζ$ is the drag coefficient, $D^{swim}$ is the swim diffusivity and $n$ is the uniform bulk number density far from the container walls. In this work we extend the notion of the isotropic swim pressure to the anisotropic tensorial swim stress $\mathbfσ^{swim} = - n ζ\mathbf{D}^{swim}$, which is related to the anisotropic swim diffusivity $\mathbf{D}^{swim}$. We demonstrate this relationship with ABPs that achieve nematic orientational order via a bulk external field. The anisotropic swim stress is obtained analytically for dilute ABPs in both 2D and 3D systems, and the anisotropy is shown to grow exponentially with the strength of the external field. We verify that the normal component of the anisotropic swim stress applies a pressure $Π^{swim}=-(\mathbfσ^{swim}\cdot\mathbf{n})\cdot\mathbf{n}$ on a wall with normal vector $\mathbf{n}$, and, through Brownian dynamics simulations, this pressure is shown to be the force per unit area transmitted by the active particles. Since ABPs have no friction with a wall, the difference between the normal and tangential stress components -- the normal stress difference -- generates a net flow of ABPs along the wall, which is a generic property of active matter systems.
Non-equilibrium dynamics of topological defects can be used as a fundamental propulsion mechanism in microscopic active matter. Here, we demonstrate swimming of topological defect-propelled colloidal particles in (passive) nematic fluids through experiments and numerical simulations. Dynamic swim strokes of the topological defects are driven by colloidal rotation in an external magnetic field, causing periodic defect rearrangement which propels the particles. The swimming velocity is determined by the colloid's angular velocity, sense of rotation and defect polarity. By controlling them we can…
Active (i.e., self-propelled or swimming) particles moving through an isotropic fluid exhibit conventional diffusive behavior. We report anomalous diffusion of an active particle moving in an anisotropic, nematic background. Whilst the translational motion parallel to the nematic director shows ballistic behavior, the long-time transverse motion is super-diffusive, with an anomalous scaling $\propto t \ln t$ of the mean squared displacement with time $t$. This behavior is predicted by an analytical theory that we present here, and is corroborated by numerical simulation of active particle diff…
We study the collective dynamics of elongated swimmers in a very thin fluid layer by devising long, filamentous, non-tumbling bacteria. The strong confinement induces weak nematic alignment upon collision, which, for large enough density of cells, gives rise to global nematic order. This homogeneous but fluctuating phase, observed on the largest experimentally-accessible scale of millimeters, exhibits the properties predicted by standard models for flocking such as the Vicsek-style model of polar particles with nematic alignment: true long-range nematic order and non-trivial giant number fluct…
Scallop Theorem for Swimming in Anisotropic Fluids
University of Pennsylvania（宾夕法尼亚大学）；Kent State University（肯特州立大学）；University of Ljubljana（卢布尔雅那大学）；University of Warwick（华威大学）；Jožef Stefan Institute（约瑟夫·斯特凡研究所）
Active matter comprised of self-propelled interacting units holds a major promise for extraction of useful work from its seemingly chaotic out-of-equilibrium dynamics. Streamlining active matter to produce work is especially important at microscale, where the viscous forces prevail over inertia and the useful modes of transport require very specific non-reciprocal type of motion. Here we report that microscopic active droplets representing aqueous dispersions of swimming bacteria Bacillus subtilis show unidirectional propulsion when placed in an inactive nematic medium. Random motion of bacter…
Swimming bacteria in passive nematics in the form of lyotropic liquid crystals are defined as a new class of active matter known as living liquid crystals in recent studies. It has also been shown that liquid crystal solutions are promising candidates for trapping and detecting bacteria. We ask the question, can a similar class of matter be designed for background nematics which are also active? Hence, we developed a minimal model for the mixture of polar particles in active nematics. It is found that the active nematics in such a mixture are highly sensitive to the presence of polar particles…
We study a swimming undulating sheet in the isotropic phase of an active nematic liquid crystal. Activity changes the effective shear viscosity, reducing it to zero at a critical value of activity. Expanding in the sheet amplitude, we find that the correction to the swimming speed due to activity is inversely proportional to the effective shear viscosity. Our perturbative calculation becomes invalid near the critical value of activity; using numerical methods to probe this regime, we find that activity enhances the swimming speed by an order of magnitude compared to the passive case.
An elastic membrane with embedded nematic molecules is considered as a model of anisotropic fluid membrane with internal ordering. By considering the geometric coupling between director field and membrane curvature, the nematic texture is shown to induce anisotropic stresses additional to Canham-Helfrich elasticity. Building upon differential geometry, analytical expressions are found for the membrane stress and torque induced by splaying, twisting and bending of the nematic director as described by the Frank energy of liquid crystals. The forces induced by prototypical nematic textures are vi…
Suspensions of self-propelled bodies generate a unique mechanical stress owing to their motility that impacts their large-scale collective behavior. For microswimmers suspended in a fluid with negligible particle inertia, we have shown that the virial `swim stress' is a useful quantity to understand the rheology and nonequilibrium behaviors of active soft matter systems. For larger self-propelled organisms like fish, it is unclear how particle inertia impacts their stress generation and collective movement. Here, we analyze the effects of finite particle inertia on the mechanical pressure (or …
Ordering configurations of a director field on a curved membrane induce stress. In this work, we present a theoretical framework to calculate the stress tensor and the torque as a consequence of the nematic ordering; we use the variational principle and invariance of the energy under Euclidean motions. Euler-Lagrange equations of the membrane as well as the corresponding boundary conditions also appear as natural results. The stress tensor found includes attraction-repulsion forces between defects; likewise, defects are attracted to patches with the same sign in gaussian curvature. These force…
Several recent experiments investigate the orientational and transport behavior of self-driven bacteria and colloidal particles in nematic liquid crystals. Correspondingly, we study theoretically the dynamics of a minimal model microswimmer in a uniaxially anisotropic fluid. As a first step, the hydrodynamic Green's function providing the resulting fluid flow in response to a localized force acting on the anisotropic fluid is derived analytically. On this basis, the behavior of both puller- and pusher-type microswimmers in the anisotropic fluid is analyzed. Depending on the propulsion mechanis…
We calculate the Casimir stresses in a thin layer of active fluid with nematic order. By using a stochastic hydrodynamic approach for an active fluid layer of finite thickness $L$, we generalize the Casimir stress for nematic liquid crystals in thermal equilibrium to active systems. We show that the active Casimir stress differs significantly from its equilibrium counterpart. For contractile activity, the active Casimir stress, although attractive like its equilibrium counterpart, diverges logarithmically as $L$ approaches a threshold of the spontaneous flow instability from below. In contrast…
Recent experiments in various cell types have shown that two-dimensional tissues often display local nematic order, with evidence of extensile stresses manifest in the dynamics of topological defects. Using a mesoscopic model where tissue flow is generated by fluctuating traction forces coupled to the nematic order parameter, we show that the resulting tissue dynamics can spontaneously produce local nematic order and an extensile internal stress. A key element of the model is the assumption that in the presence of local nematic alignment, cells preferentially crawl along the nematic axis, resu…
We study the flow behaviour of a twist-bend nematic $(N_{TB})$ liquid crystal. It shows three distinct shear stress ($σ$) responses in a certain range of temperatures and shear rates ($\dotγ$). In Region-I, $σ\sim\sqrt{\dotγ}$, in region-II, the stress shows a plateau, characterised by a power law $σ\sim{\dotγ}^α$, where $α\sim0.1-0.4$ and in region-III, $σ\sim\dotγ$. With increasing shear rate, $σ$ changes continuously from region-I to II, whereas it changes discontinuously with a hysteresis from region-II to III. In the plateau (region-II), we observe a dynamic stress fluctuations, exhibitin…
Living active collectives have evolved with remarkable self-patterning ability to meet the physical and biological constraints for growth and survival. However, how complex multicellular patterns emerge from a single founder cell remains elusive. Here, by recourse to an agent-based model, we track the three-dimensional (3D) morphodynamics and cell orientational order of growing bacterial biofilms encased by agarose gels. Confined growth causes spatiotemporally heterogeneous stress buildup in the biofilm. High hydrostatic and low shear stresses at the core of the biofilm promote viscous-to-elas…
A continuum mechanical framework is developed for determining a) the class of stress-free deformed shapes and corresponding director distributions on the undeformed configuration of a nematic glass membrane that has a prescribed spontaneous stretch field and b) the class of undeformed configurations and corresponding director distributions on it resulting in a stress-free \emph{given} deformed shape of a nematic glass sheet with a prescribed spontaneous stretch field. The proposed solution rests on an understanding of how the Lagrangian dyad of a deformation of a membrane maps into the Euleria…
Photoactive nematic elastomers are soft rubbery solids that undergo deformation when illuminated. They are made by incorporating photoactive molecules like azobenzene into nematic liquid crystal elastomers. Since its initial demonstration in 2001, it has received increasing interest with many recent studies of periodic and buckling behavior. However, theoretical models developed have focused on describing specific deformation modes (e.g., beam bending and uniaxial contraction) in the absence of mechanical loads, with only limited attention to the interplay between mechanical stress and light-i…
The mechanics of a flexible membrane decorated with a nematic liquid-crystal texture is considered in a variational framework. The variations on the splay, twist and the bend energy of the nematics are obtained from the local deformations leading to changes in the shape membrane. The Euler-Lagrange derivatives and the Noether charges are identified from the variational equations. The nematic stress tensor is obtained as a consequence of translational invariance. Likewise, the rotational invariance implies the torque nematic tensor. The corresponding boundary conditions are obtained for free ed…
One objective of active matter science is to unveil principles by which chaotic microscale dynamics could be transformed into useful work. A nematic liquid crystal environment offers a number of possibilities, one of which is a directional motion of an active droplet filled with an aqueous dispersion of swimming bacteria. In this work, using the responsiveness of the nematic to the electric field and light, we demonstrate how to control the direction and speed of active droplets. The dielectric response of nematic to the electric field causes two effects: (i) reorientation of the overall direc…
We investigate the effect of an anisotropic substrate on the turbulent dynamics of a simulated two dimensional active nematic. This is introduced as an anisotropic friction and an effective anisotropic viscosity, with the orientation of the anisotropy being defined by the substrate. In this system we observe the emergence of global nematic order of topological defects that is controlled by the degree of anisotropy in the viscosity and the magnitude of the active stress. No global defect alignment is seen in passive liquid crystals with anisotropic viscosity or friction confirming that ordering…
The topological properties of many materials are central to their behavior, with the dynamics of topological defects being particularly important to intrinsically out-of-equilibrium, active materials. In this paper, local manipulation of the ordering, dynamics, and topological properties of microtubule-based extensile active nematic films is demonstrated in a joint experimental and simulation study. Hydrodynamic stresses created by magnetically actuated rotation of disk-shaped colloids in proximity to the films compete with internal stresses in the active nematic, enabling local control of the…
We numerically investigate how spatial variations of extensile or contractile active stress affect bulk active nematic systems in two and three dimensions. In the absence of defects, activity gradients drive flows which re-orient the nematic director field and thus act as an effective anchoring force. At high activity, defects are created and the system transitions into active turbulence, a chaotic flow state characterized by strong vorticity. We find that in two-dimensional (2D) systems active torques robustly align $+1/2$ defects parallel to activity gradients, with defect heads pointing tow…
Topological defects are increasingly being identified in various biological systems, where their characteristic flow fields and stress patterns are associated with continuous active stress generation by biological entities. Here, using numerical simulations of continuum fluctuating nematohydrodynamics we show that even in the absence of any specific form of active stresses associated with self-propulsion, mesoscopic fluctuations in either orientational alignment or hydrodynamics can independently result in flow patterns around topological defects that resemble the ones observed in active syste…
We investigate the effects of extrinsic curvature on the turbulent behavior of a 2D active nematic confined to the surface of a cylinder. The surface of a cylinder has no intrinsic curvatrue and only extrinsic curvature. A nematic field reacts to the extrinsic curvature by trying to align with the lowest principle curvature, in this case parallel to the long axis of the cylinder. When nematics are sufficiently active, there is a proliferation of defects arising from a bend or splay instability depending on the nature of the active stress. The extrinsic curvature of the cylinder beaks the rotat…
Growing tissue and bacterial colonies are active matter systems where cell divisions and cellular motion generate active stress. Although they operate in the non-equilibrium regime, these biological systems can form large-scale ordered structures such as nematically aligned cells, topological defects, and fingerings. Mechanical instabilities also play an essential role during growth by generating large structural folding. How active matter dynamics and mechanical instabilities together develop large-scale order in growing tissue is not well understood. Here, we use chain forming Bacillus subti…
I put forward a continuum theory for active nematic gels, defined as fluids or suspensions of orientable rodlike objects endowed with active dynamics, that is based on symmetry arguments and compatibility with thermodynamics. The starting point is our recent theory that models (passive) nematic liquid crystals as relaxing nematic elastomers. The interplay between viscoelastic response and active dynamics of the microscopic constituents is naturally taken into account. By contrast with standard theories, activity is not introduced as an additional term of the stress tensor, but it is added as a…
We develop a formal analogy between configurational stresses in two distinct physical systems, and study the flows that they induce when the configurations of interest include topological de- fects. The two systems in question are electrokinetic flows in a nematic fluid under an applied electrostatic field, and the motion of self propelling or active particles in a nematic matrix (a living liquid crystal). The mapping allows the extension, within certain limits, of existing results on trans- port in electrokinetic systems to self propelled transport. We study motion induced by a pair of point …
Point-like motile topological defects control the universal dynamics of diverse two-dimensional active nematics ranging from shaken granular rods to cellular monolayers. A comparable understanding in higher dimensions has yet to emerge. We report the creation of three-dimensional active nematics by dispersing extensile microtubule bundles in a passive colloidal liquid crystal. Light-sheet microscopy reveals the millimeter-scale structure of active nematics with a single bundle resolution and the temporal evolution of the associated nematic director field. The dominant excitations of three-dime…
