| dc.description.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. | en_US |
