Active matter: chirality, translational order, and interfaces

Abstract

My PhD work is on chiral active matter with solid and fluid directions, dynamics of the interface of a nonconserved chiral order parameter in an active system, flocking on curved manifolds and field-driven colloids in confined nematics.

We start by formulating theories of layered active chiral matter. We start with constructing the active model H* in two and three dimensions – the chiral and active variant of model H. This theory describes the coupled dynamics of a conserved scalar order parameter and a conserved momentum density field. At thermal equilibrium, chiral molecules form a range of liquid-crystalline phases such as cholesteric, with a helical structure of the molecular orientation. It turns that at long length scales, the mechanics of a cholesteric is precisely the same as that of a smectic A which has an achiral one-dimensional density modulation. It is curious that though microscopic chirality leads to a one-dimensional periodic structure, its asymptotic long-wavelength elasticity and hydrodynamics show no signature of chirality. In this chapter we show that this equivalence does not carry over to active cholesteric and smectic A phases. Thanks to the presence of a mix of solid- and liquid-like directions, we predict that chiral active stresses create a force density tangent to contours of constant mean curvature of the layers. This non-dissipative force in a fluid direction – odder than odd elasticity – leads, in the presence of an undulational instability created by non-chiral active stresses normal to the layers, to spontaneous vortical flows arranged in a two-dimensional array with vorticity aligned along the pitch axis and alternating in sign in the plane. This vortex-lattice state can be switched on or off by means of an externally imposed uniaxial stress. We also show that a two-dimensional active cholesteric is unstable with an activity threshold that goes to zero for an infinite system.

We then move on to formulating the active hydrodynamics of columnar phases, those with two solid and one fluid direction. We show that a bulk active columnar phase is spontaneously unstable to an extensile activity along the column direction via a buckling instability. We predict singular stiffening or softening – depending on whether the active achiral stress is contractile or extensile – of the buckling of fluid columns in all active columnar materials, irrespective of whether they are chiral or polar. Further, we demonstrate that the effect of the active achiral stress in columnars is exactly equivalent to an externally imposed in-plane, isotropic stress; therefore, the instability induced by a singular softening of the column buckling mode – in extensile systems – is exactly equivalent to a columnar Helfrich-Hurault instability under an external stress. This allows us to exactly calculate the threshold activity for this instability in a finite columnar liquid crystal. The instability is mediated by a twist-bend mode resulting in helical columns – same as those that arise from a Helfrich-Hurault instability of passive columnar material. If the active units composing the columnar state are in addition chiral, the buckled and twisted state beyond the spontaneous Helfrich-Hurault instability in an apolar system hosts large-scale shear flows due to a new form of odd elasticity. For polar and chiral columnar systems, we show that two-dimensional solid odd elasticity is naturally realised in this three-dimensional material. The interplay of this odd elasticity with viscous, Stokesian hydrodynamics leads to an optical mode with a frequency that depends on the direction of the relative angle between the wavevector direction and polarity but, crucially, not on the wavenumber. The frequency of this vibrational mode is set by the ratio of the coefficient of chiral and polar active stress and the viscosity. The damping of this mode is also wavenumber-independent. The oscillation is due to the two in-plane displacement fields acting effectively as a position-momentum pair.

In chapter 4, we move to investigating the interfacial dynamics in the bulk chiral active models. In particular, we derive the stochastic partial differential equation (SPDE) describing the dynamics of a fluctuating chiral interface with up-down symmetry in one-dimension. The obtained SPDE has been studied before, and the result on logarithmic corrections to scaling is stated in the Comment by Paczuski et al. However, the derivation from a bulk field theory is new, and the renormalisation group calculation is unavailable in the literature. We derive the SPDE from an active field theory for a non-conserved pseudoscalar field in a uniaxial medium equation, governing the interfacial dynamics separating regions of opposite chirality. This dynamics turns out to be equivalent to that of a steadily forced polymer. The steady-state probability distribution of the one-dimensional shape of the domain wall is the same as the passive Edwards-Wilkinson model. However, surprisingly, the dynamical behaviour of the domain wall shape reveals its activity. A nonlinearity – which by scaling arguments is marginal in one dimension – turns out to lead to anomalous growth. We examine this numerically and analytically using a two-loop RG calculation to obtain the exponent of the logarithmic correction to diffusivity.

The chapter 5 is on analytical modelling of experiments on AC field-driven Janus colloids in confined nematics. Electrokinetics involves study of electrically driven fluid flow (electro-osmosis) and particle motion (electrophoresis). The use of electric fields to transport tiny particles through fluids, is an important technology for macro-molecular sorting, colloidal assembly and a challenging area of soft-matter research. Traditional studies on Electro-osmosis have been on colloids sus- pended in isotropic electrolytes. Janus colloids in an isotropic electrolyte – with dielectric and conducting hemispheres – show unidirectional motility (dielectric forward) – thanks to the contrasting polarisability on the either hemispheres. However, this phenomenon is unsuitable for self-assembly and micro-botic applications for its unidirec- tional motion and eventual sedimentation in an isotropic electrolyte. In a striking depar- ture from conventional electrophoresis, the experiments show that metal-dielectric Janus particles can be piloted at will through a nematic liquid crystal film, in the plane spanned by the axes of the particle and the nematic, and perpendicular to an imposed AC electric field. A complete command over particle trajectories can be achieved by varying field amplitude and frequency, exploiting the sensitivity of electro-osmotic flow to the asymmetries of particle and defect structure. To understand the multi-directional motility of janus particles in the experiments, we calculate the dipolar force density pro- duced by the interplay of the electric field with director anchoring and the contrasting electrostatic boundary conditions on the two hemispheres of the janus colloid to account for the dielectric-forward (metal-forward) motion of the colloids due to induced puller (pusher) force dipoles.

In the final chapter, we study Toner-Tu flocking on curved substrates. We study the dynamics of density and polarisation fields on toroidal substrates and analytically calculate the steady-state profile. The Euler characteristic of the torus is zero and hence defect-free states can exist which are well defined globally. We observe the density profile to be inhomogeneous due to the presence of curvature. We also find delocalisa- tion of extremas of density and polarity field. Further, the active flow allows the system to have long-wavelength propagating sound modes which are gapped by the curvature, while the gapless modes get localized to two special geodesics located on the positively and negatively curved faces.