soft matter, biophysics, fluid mechanics

Jonas Rønning

# Defect self-propulsion in active nematic films with spatially-varying
activity

Jonas Rønning1 M. Cristina Marchetti 2 and Luiza Angheluta1 1Njord Centre,
Department of Physics, University of Oslo, P. O. Box 1048, 0316 Oslo, Norway
2Department of Physics & Biomolecular Science and Engineering Program,
University of California Santa Barbara, Santa Barbara, CA 93106, USA
<EMAIL_ADDRESS>
###### Abstract

We study the dynamics of topological defects in active nematic films with
spatially-varying activity and consider two setups: i) a constant activity
gradient, and ii) a sharp jump in activity. A constant gradient of extensile
(contractile) activity endows the comet-like $+1/2$ defect with a finite
vorticity that drives the defect to align its nose in the direction of
decreasing (increasing) gradient. A constant gradient does not, however,
affect the known self-propulsion of the $+1/2$ defect and has no effect on the
$-1/2$ that remains a non-motile particle. A sharp jump in activity acts like
a wall that traps the defects, affecting the translational and rotational
motion of both charges. The $+1/2$ defect slows down as it approaches the
interface and the net vorticity tends to reorient the defect polarization so
that it becomes perpendicular to the interface. The $-1/2$ defect acquires a
self-propulsion towards the activity interface, while the vorticity-induced
active torque tends to align the defect to a preferred orientation. This
effective attraction of the negative defects to the wall is consistent with
the observation of an accumulation of negative topological charge at both
active/passive interfaces and physical boundaries.

###### keywords:

active nematics, topological defects, nematic liquid crystals, hydrodynamics

##  1 Introduction

Active nematics are collections of elongated apolar particles that consume
energy from their surroundings to generate dipolar forces that drive self-
sustained flows [1]. Much progress in understanding the rich dynamics of these
active liquid crystals has been achieved through a minimal hydrodynamic theory
that couples orientational order and flow and captures the behavior of
biological systems from subcellular to multicellular scales [2]. Within the
biological realm, the active nematic paradigm describes mixtures of
cytoskeletal filaments and motor proteins [3, 4, 5, 6], bacterial suspensions
[1, 7] and confluent cell monolayers [8, 9].

What distinguishes the hydrodynamics of active nematics from that of their
passive counterparts is the presence of an active stress generated by active
processes, which sets up spontaneous spatio-temporally chaotic flows [7]. The
active stress is given by $\sigma^{a}_{ij}=\alpha Q_{ij}$ with $\mathbf{Q}$
the nematic order parameter and $\alpha$ a scalar activity parameter that
encapsulates the biochemical processes that generate active forces [10, 11,
12, 13]. It can have either sign: $\alpha>0$ corresponds to a system of
"pullers" generating contractile stresses on their surroundings, whereas
$\alpha<0$ reflects a system of "pushers" and their induced extensile active
stresses. With increasing activity, active flows are induced spontaneously and
create large distortions of the nematic order, including the formation of
pairs of topological defects that sustain active turbulence [3, 14].

The lowest energy topological defects in active nematic films have half-
integer charge, corresponding to the comet-shaped $+1/2$ with polarity
$\mathbf{e}_{+}$ defined by a head-tail arrow, and the $-1/2$ which has three-
fold symmetry (see Fig. 1). These defects disrupt the nematic order locally
and induce long-range distortions in the orientation field, generating active
stresses, which in turn lead to spontaneous active flows surrounding the
defects [12, 15, 16]. There is a net active flow through the core of the polar
$+1/2$ defect which makes it intrinsically motile and is referred to as the
defect self-propulsion. For an isolated $+1/2$ defect, the self-propulsion
velocity aligns with the polarity vector and, depending on the
contractile/extensile properties of the active nematic, the defect moves
in/opposite to the direction of its polarization. The $-1/2$ defect does not
create any net flow at the defect position and, thus, is not self-propelled in
systems with uniform activity. The motion of defects in the presence of
spatially inhomogeneous activity is far less understood and explored [17].

There are several approaches to realize experimentally systems with spatially-
dependent activity. In Ref. [18], a varying substrate topography is used to
control the frictional damping in a film of a microtubule-kinesin suspension.
This results in spatial variations of the concentration of active agents, thus
indirectly in the local activity. In Ref. [19] a similar effect is achieved by
manipulating light-sensitive myosin motors that activate the microtubules.
Both studies find that the $-1/2$ defects localize near the interface
separating the region of higher activity from that of lower activity. In Ref.
[19], it was reported that the $+1/2$ defects are deflected by the
active/passive interface. Analytical work based on a hydrodynamic theory of
the defect gas has predicted that a passive/active interface can be used to
separate positive and negative topological charge [17]. Numerical studies of
how the defect dynamics is affected by the spatially-dependent activity show
that the polarity of the $+1/2$ defect tends to align parallel to the activity
gradient [20, 21, 19, 22, 23], and that the confinement and motion of defects
can be manipulated by varying the steepness of the activity gradients [24,
25]. A recent numerical study also shows that the formation of defect dipoles
can be controlled by imprinting special geometries into the activity profile
[22].

In this paper, we provide a theoretical study of how spatially-varying
activity affects the self-propulsion and reorientation of isolated topological
defects. We consider the representative basic setups where the activity
gradient is either constant or a Dirac delta function (sharp interface). For
constant activity gradients, the $+1/2$ defect rotates due to a vorticity-
induced active torque acting on the defect polarization until the defect
aligns parallel to the activity gradient, and moves in the direction of lower
magnitude of activity. Thus the defect slows down. We show analytically that
the vorticity at the $+1/2$ defect core is proportional to the hydrodynamic
dissipation length $\ell_{d}=\sqrt{\eta/\Gamma}$, which measures the strength
of viscous dissipation $\eta$ relative to friction $\Gamma$. Numerical
simulations of the active flow generated in a disk of radius $R$ show that the
vorticity depends on the system size for small $R$, and crosses over to the
analytically-predicted value for large systems. The vorticity field induced by
an activity gradient parallel to the $+1/2$ defect’s polarization has a
quadruple structure with regions of alternating vorticity. This is confirmed
by numerical simulations for a disk geometry where four vortices are formed
around the $+1/2$ defect. In contrast, the vorticity induced by constant
activity gradients at a $-1/2$ defect has an $8$-fold symmetry that leads to
eight vortices with alternating circulation in a finite domain. We also
calculate the net translational self-propulsion and reorientation that both
$\pm 1/2$ defects acquire near a sharp active/passive interface. The $+1/2$
defect slows down as it moves towards the interface, and the vorticity-induced
torque tends to re-orient it such that its polarization becomes normal to the
interface regardless of the sign of activity. The $-1/2$ defect also acquires
a preferred orientation at the interface, and those that approach the
interface with this stable orientation are then attracted by it.

The structure of the paper is as follows. We start in Section 2 by introducing
the minimal hydrodynamic model active nematic films on a substrate. In Section
3, we derive and discuss the self-propulsion and spontaneous rotation of
$+1/2$ defects in the presence of a constant activity gradient. Section 4
focuses on the analytical derivation of the self-propulsion and vorticity of
$\pm 1/2$ defects close to a sharp active/passive interface. Summary and
concluding remarks are presented in Section 5.

Figure 1: Illustration of (a) the $+1/2$ defect and (b) the $-1/2$ defect and
their corresponding polarizations for $\theta_{0}=0$. The negative defect has
three equivalent polarizations.

##  2 Hydrodynamics of active nematics with spatially-varying activity

We consider the familiar hydrodynamic model of a 2D active nematics that
couples the flow velocity $\mathbf{u}(\mathbf{r})$ to the nematic order
parameter $Q_{ij}=S(\hat{n}_{i}\hat{n}_{j}-\frac{1}{2}\delta_{ij})$, where $S$
quantifies the degree of order and
$\mathbf{\hat{n}}(\mathbf{r})=\left(\cos\theta(\mathbf{r}),\sin\theta(\mathbf{r})\right)$
is the orientational director field with head-tail symmetry. In the simplest
formulation, the Q-tensor is a minimizer of the de Gennes-Landau free energy
[7]

$\displaystyle\mathcal{F}=\int
d\mathbf{r}\left[\frac{K}{2}|\partial_{i}Q_{jk}|^{2}+\frac{g}{4}\textrm{Tr}\left(1-\textrm{Tr}\left(\mathbf{Q}^{2}\right)\right)^{2}\right]\;,$
(1)

with isotropic elastic constant $K>0$ and $g$ the strength of the local
ordering potential. The uniform nematic ordered state corresponds to
$S_{0}=1$. The flow field satisfies a Stokes equation that balances forces on
a fluid element, given by [7]

$\displaystyle\left(\Gamma-\eta\nabla^{2}\right)\mathbf{u}=\bm{\nabla}\cdot[\alpha(\mathbf{r})\mathbf{Q}(\mathbf{r})]-\bm{\nabla}p(\mathbf{r}),\qquad\bm{\nabla}\cdot\mathbf{u}=0\;,$
(2)

where $\Gamma$ is a friction coefficient per unit area, $\eta$ is the shear
viscosity, and $\alpha$ is the activity coefficient, with dimensions of
stress. For simplicity, we neglect additional passive elastic stresses, which
are of higher order in the gradients of $\mathbf{Q}$ compared to the active
stress and a more important contributions for nematic textures with many
defects. Instead we focus on the active flows generated by an isolated $\pm
1/2$ in the presence of non-homogeneous activity $\alpha(\mathbf{r})$.

We rescale the Stokes equation in units of the nematic relaxation time
$\tau=\gamma/g$ (where $\gamma$ is the nematic rotational friction) [26, 16]
and the coherence length $\xi=\sqrt{K/g}$. Different dynamical regimes are
then controlled by one dimensionless number $\zeta=\ell_{d}/\xi$, where
$\ell_{d}=\sqrt{\eta/\Gamma}$, and the rescaled activity
$\alpha(\mathbf{r})\rightarrow\alpha(\mathbf{r})\gamma/(\Gamma K)$. The
dimensionless form of the Stokes equation reads as

$\left(1-\zeta^{2}\nabla^{2}\right)\mathbf{u}=\mathbf{F}_{\pm}-\bm{\nabla}p\;,\qquad\bm{\nabla}\cdot\mathbf{u}=0\;,$
(3)

where the active force field induced by an isolated $\pm 1/2$ defect is given
by

$\mathbf{F}_{\pm}=\mathbf{Q}(\mathbf{r})\cdot\bm{\nabla}\alpha(\mathbf{r})+\alpha(\mathbf{r})\bm{\nabla}\cdot\mathbf{Q}(\mathbf{r})=\mathbf{F}_{\pm}^{I}+\mathbf{F}_{\pm}^{B}\;.$
(4)

The first contribution is an interfacial force $\mathbf{F}_{\pm}^{I}$
originating from activity gradients. The second term is a bulk force
$\mathbf{F}_{\pm}^{B}$ due to nematic distortions. The defect self-propulsion
velocity $\mathbf{v}_{\pm}$ is defined as the net active flow through the
defect core, and thus can be computed from the active flow velocity
$\mathbf{u}$ obtained from the solution of Eq. (3) evaluated at the origin
[26]. The solution of Eq. (3) is given by

$\mathbf{v}_{\pm}=\frac{1}{2\pi\zeta^{2}}\int
d\mathbf{r}K_{0}\left(\frac{r}{\zeta}\right)\left[\mathbf{F}_{\pm}(\mathbf{r})-\bm{\nabla}p(\mathbf{r})\right]=\mathbf{v}_{\pm}^{I}+\mathbf{v}_{\pm}^{B},$
(5)

where $K_{0}(r)$ is the zeroth order Bessel function which is the Green’s
function of Eq. (3) without the incompressibility constraint, and the pressure
field is the solution of the corresponding Poisson’s equation

$\nabla^{2}p=\bm{\nabla}\cdot\mathbf{F}_{\pm}(\mathbf{r})\;.$ (6)

The net vorticity at the defect core is also obtained from measuring the
vorticity of the flow field induced by the defect distortion, given by
$\omega=\partial_{x}u_{y}-\partial_{y}u_{x}=-\nabla^{\perp}\cdot\mathbf{u}$.
Using Eq. (5) and evaluating it at the defect position $\mathbf{r}_{0}=0$, we
obtain an expression for the defect vorticity

$\mathbf{\omega}_{\pm}=-\frac{1}{2\pi\zeta^{2}}\int
d\mathbf{r}K_{0}\left(\frac{r}{\zeta}\right)\bm{\nabla}^{\perp}\cdot\mathbf{F}_{\pm}(\mathbf{r})=\mathbf{\omega}_{\pm}^{I}+\mathbf{\omega}_{\pm}^{B}\;.$
(7)

Both velocity and vorticity are written as sums of interfacial and bulk
contributions which depend on the defect polarization $\mathbf{e}_{\pm}$ and
are computed analytically in the next sections.

For isolated $\pm 1/2$ point-like defects, we can parameterise the
$\mathbf{Q}$-tensor order-parameter in the quasistatic phase approximation as
[26]

$Q^{\pm}_{xx}(\mathbf{r})=\cos(\pm\phi(\mathbf{r})+2\theta_{0})\;,\qquad
Q^{\pm}_{xy}(\mathbf{r})=\sin(\pm\phi(\mathbf{r})+2\theta_{0})\;,$ (8)

where $\phi(\mathbf{r})=\arctan(y/x)$ is the singular part of the nematic
orientation due to a $\pm 1/2$ defect located at the origin, and $\theta_{0}$
is the slowly-varying part of the background orientation of the nematic
director. The $+1/2$ defect has a well-defined polarization which is
determined by the background nematic orientation $\theta_{0}$ as

$\displaystyle\mathbf{e}_{+}=\left(\frac{\bm{\nabla}\cdot\mathbf{Q}}{|\bm{\nabla}\cdot\mathbf{Q}|}\right)_{\mathbf{r}=0}=[\cos(2\theta_{0}),\sin(2\theta_{0})].$
(9)

For the $-1/2$ defect, we can also introduce a polarization vector determined
by $\theta_{0}$ and aligning with one of the principal axes of the three-fold
symmetry [26]

$\displaystyle\mathbf{e}_{-}=[\cos(2\theta_{0}/3),\sin(2\theta_{0}/3)].$ (10)

Both nematic defects and their respective polarizations are illustrated in
Fig. 1.

It can be shown that a net vorticity at the defect core induces an active
torque that tends to rotate the defect polarization. This follows
straightforwardly from taking the time derivative of the polarization in Eqs.
(9) and (10), and using the evolution of the $\mathbf{Q}$-tensor [17, 26] to
account for the change in the background nematic field $\theta_{0}$ due to
vorticity as $\partial_{t}\theta_{0}\approx\omega/2$. Thus, the evolution of
the defect polarization controlled by vorticity is

$\mathbf{\dot{e}}_{\pm}\approx-3^{-1/2+q}\omega_{\pm}\mathbf{e}_{\pm}^{\perp},$
(11)

where the defect charge is $q=\pm 1/2$ and $\mathbf{e}^{\perp}=[e_{y},-e_{x}]$
represents the $90^{0}$ clockwise rotation of the polarization vector. For
motile defects, there are additional torques due to defect interactions, the
elastic stiffness $K$ or the coupling to the flow alignment [17, 26]. Here, we
focus on the active torque induced by a non-zero vorticity which emerges from
spatially varying activity alone. In the subsequent sections, we investigate
how this active torque reorients the defect polarization relative to activity
gradients for two setups: i) a constant activity gradient, and ii) an
interface with a sharp jump in activity.

##  3 Constant activity gradient

We first study the kinematics of an isolated defect in a region where the
activity gradient is locally constant. Without loss of generality, we consider
an activity gradient in the $x$-direction such that the activity has the
linear profile $\alpha(\mathbf{r})=\alpha_{0}+\alpha_{g}x$. The defect
orientation is arbitrary and controlled by the background nematic orientation
$\theta_{0}$. We demonstrate that a constant gradient $\alpha_{g}$ does not
modify the defect self-propulsion velocity as compared to what was obtained
for uniform bulk activity $\alpha_{0}$. An activity gradient across the
texture of a $+1/2$ defect generates, however, a flow that may yield a finite
vorticity at the defect core, which tends to align the defect polarization
according to Eq. (11) in the direction of the gradient. The $-1/2$ defect
remains stationary both in its motion and orientation.

Figure 2: Interfacial active force field from Eq. (12) induced by a $+1/2$
defect with a) $\theta_{0}=0$, b) $\theta_{0}=\pi/8$ and c)
$\theta_{0}=\pi/4$. Note that cases b) and c) lead to rotation of the defect
together with the nematic field until the defect polarization aligns with the
direction of the activity gradient. The dark solid lines in a) show the
nematic field around the $+1/2$ defect oriented in the $x$ direction.

###  3.1 $+1/2$ defect

The interfacial active force given in Eq. (4) arising from a constant activity
gradient $\alpha_{g}$ is

$\mathbf{F}_{+}^{I}(\mathbf{r})=\alpha_{g}\left[\cos(2\theta_{0})\mathbf{\hat{r}}-\sin(2\theta_{0})\mathbf{\hat{r}}^{\perp}\right]\;,$
(12)

where $\mathbf{\hat{r}}$ is the radial unit vector and
$\mathbf{\hat{r}}^{\perp}=(\hat{y},-\hat{x})$. This expression corresponding
to the Helmholtz decomposition of $\mathbf{F}_{+}^{I}$ into a curl-free part
($\sim\mathbf{\hat{r}}$) and a divergence-free part
($\sim\mathbf{\hat{r}}^{\perp}$). These two contributions are plotted in Fig.
(2). The divergence-free part gives a net vorticity at the defect core which
tends to rotate its polarization until it aligns with the activity gradient.
This is most easily demonstrated in the friction-dominated limit where the
active flow velocity is $\Gamma\mathbf{u}=\mathbf{F}_{+}^{I}-\bm{\nabla}p$.
The incompressibility constraint thereby removes the curl-free contribution
through the contribution of the interfacial pressure which is radially
symmetric and given by

$p_{+}^{I}(\mathbf{r})=\alpha_{g}\cos(2\theta_{0})(r-L),$ (13)

making the interface flow purely rotational. Here the constant $L$ is a length
comparable with the system size which controls the divergent terms. More
generally, to incorporate viscous dissipation we need to evaluate the integral
expression for the defect velocity given in Eq. (5). In an infinite system,
the symmetry of the integrand leads to no contribution to the defect speed
from the interfacial active force, thus $\mathbf{v}^{I}_{+}=0$. This
contribution may become finite in non-radially symmetric bounded domains.

The contribution from the bulk active force in Eq. (4) reduces to

$\mathbf{F}_{+}^{B}(\mathbf{r})=\mathbf{F}_{+}^{0}(\mathbf{r})+\alpha_{g}x\bm{\nabla}\cdot\mathbf{Q}_{+}=\mathbf{F}_{+}^{0}(\mathbf{r})+\alpha_{g}\frac{x}{r}(\cos(2\theta_{0})\mathbf{\hat{x}}+\sin(2\theta_{0})\mathbf{\hat{y}})\;,$
(14)

where $\mathbf{F}_{+}^{0}(\mathbf{r})$ is the known active force corresponding
to a constant activity $\alpha_{0}$ which leads to a constant self-propulsion
velocity [26, 16]. The contribution due to $\alpha_{g}$ is anti-symmetric
around the defect position, thus its integral over an infinite domain
vanishes. Therefore, there is no contribution from activity gradients to the
defect self-propulsion. This is not changed when we add the gradient of the
bulk pressure which is given as

$p_{+}^{B}(\mathbf{r})=p_{+}^{0}(\mathbf{r})+\frac{\alpha_{g}\cos(2\theta_{0})}{6}\left(\frac{(x^{2}-y^{2})}{r}+3(r-L)\right)-\frac{\alpha_{g}\sin(2\theta_{0})}{3}\frac{xy}{r}$
(15)

Here $p^{0}_{+}$ is the pressure for the constant activity term $\alpha_{0}$
[16].

Figure 3: Magnitude of the angular velocity of the $+1/2$ defect for different
values of dissipation parameters $\eta$ and $\Gamma$, $\alpha_{g}=1$ and
$\theta_{0}=-\pi/4$. The dashed horizontal lines are the analytical prediction
for an unbounded domain. The dotted lines show the linear scaling with $R$ and
with slopes $1/\eta$.

Activity gradients induce, however, a vortical flow that is finite at the
defect core, resulting in an angular velocity of the $+1/2$ defect, given by

$\omega_{+}=\frac{\alpha_{g}}{2\pi\zeta^{2}}\sin(2\theta_{0})\int
d\mathbf{r}~{}K_{0}\left(\frac{r}{\zeta}\right)\left(\frac{1}{r}+\frac{x^{2}}{r^{3}}\right)\;,$
(16)

where the first term in the bracket originates from the interfacial active
force and the second is due to the bulk force. The integral can be carried out
in polar coordinates, with the result

$\omega_{+}=\frac{3\pi\alpha_{g}}{4\zeta}\sin(2\theta_{0})\;.$ (17)

We can rewrite this equivalently in physical units as

$\omega_{+}=\frac{3\pi\alpha_{g}}{4\Gamma\ell_{d}}\sin(2\theta_{0})=\frac{3\pi\alpha_{g}}{4\sqrt{\Gamma\eta}}\sin(2\theta_{0})=\frac{3\pi\alpha_{g}}{4\eta}\ell_{d}\sin(2\theta_{0}),$
(18)

to highlight that the defect angular velocity scales linearly with the
hydrodynamic dissipation length $l_{d}$, similar to the self-propulsion speed
of a defect in a constant activity [16]. The effect of this vorticity is to
align the polarization so that it is pointing opposite to the activity
gradient. This is consistent with recent numerical results, where defects
align normal on soft interfaces separating extensile and contractile regions
[23].

To test the validity of these analytical predictions for a bounded system, we
have solved numerically the Stokes Eq. (3) in a disk of radius $R$. In Fig.
(3), we show that the defect angular velocity is proportional to $R$ for radii
smaller than $l_{d}$, and crosses over to the asymptotic value for an infinite
system given by Eq. (18) at large $R$.

We have also computed the vorticity field for $\alpha_{0}=0$ and different
defect orientations relative to the activity gradient, as shown in Fig. (4).
The systems are solved using the finite element package FEniCS [27, 28]. When
the defect polarization is parallel to the activity gradient ($\theta_{0}=0$),
we observe a quadruple structure of the vortical flow. This is consistent with
the analytical prediction in the friction-dominated limit, where the vorticity
field away from the defect is determined by the activity gradient $\alpha_{g}$
as (for $\alpha_{0}=0$)

$\omega_{+}(r,\phi)=\frac{\alpha_{g}\sin(2\phi)}{2\Gamma
r}\cos(2\theta_{0})+\frac{\alpha_{g}}{\Gamma
r}\sin(2\theta_{0})\left(1+\cos^{2}(\phi)\right)\;,$ (19)

where $r$ and $\phi$ are the polar coordinates centered at the defect
position. In contrast, when the defect polarization is normal to the activity
gradient ($\theta_{0}=\pi/2$), we obtain a single vortex centered at the core
of the defect.

Figure 4: Active flow streamlines induced by a uniform activity gradient along
x-direction with $\alpha_{0}=0$ and by a $+1/2$ defect with orientation (a)
$\theta_{0}=0$, (b) $\theta_{0}=-\pi/4$, and (c) $\theta=-\pi/40$. Since
$\alpha_{0}=0$, the flow velocity vanishes at the defect core. The background
colormap represents the vorticity field. When $\theta_{0}\neq 0$, the defect
acquires a non-zero vorticity at its center as predicted theoretically. The
four-fold vortex structure is only visible for small values of $\theta_{0}$,
i.e., when the defect is closely aligned with the direction of the activity
gradient.

###  3.2 $-1/2$ defect

Figure 5: The vorticity field induced by an uniform activity gradient along
the x-direction for $\alpha_{0}=0$ and a $-1/2$ defect with orientation (a)
$\theta_{0}=0$ and (b) $\theta_{0}=-\pi/4$. The black lines shows the director
field.

A similar analytical calculation can be carried out for the $-1/2$ defect
using the parameterization of the $\mathbf{Q}$-tensor in Eq. (8). The
interfacial and bulk components of the active force field are obtained from
Eq. (4) as

$\displaystyle\mathbf{F}_{-}^{I}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{g}}{r}\left[(x\mathbf{\hat{x}}-y\mathbf{\hat{y}})\cos(2\theta_{0})+(y\mathbf{\hat{x}}+x\mathbf{\hat{y}})\sin(2\theta_{0})\right],$
(20) $\displaystyle\mathbf{F}^{B}_{-}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{\alpha_{g}x}{r^{3}}\left[(y^{2}-x^{2})(\cos(2\theta_{0})\mathbf{\hat{x}}+\sin(2\theta_{0})\mathbf{\hat{y}})+2xy(\cos(2\theta_{0})\mathbf{\hat{y}}-\sin(2\theta_{0})\mathbf{\hat{x})}\right]\;.$
(21)

From symmetry considerations these forces as well as their curl vanish upon
integration. This implies that a constant activity gradient alone does not
induce any self-propulsion of the $-1/2$ defect nor a rotation of its
orientation. Including the pressure contributions does not alter this effect.

In the friction-dominated limit and for $\alpha_{0}=0$, we can evaluate the
vorticity field, and show that a constant activity gradient $\alpha_{g}$
induces eight counter rotating vortices, with a vortical flow given by

$\omega_{-}(r,\phi)=-\frac{\alpha_{g}\cos(2\theta_{0})}{2\Gamma
r}(3\sin(4\phi)-\sin(2\phi))+\frac{\alpha_{g}\sin(2\theta_{0})}{2\Gamma
r}\left(3\cos(4\phi)-\cos(2\phi)\right)\;.$ (22)

The same structure is observed in bounded domains where the vorticity form
vortices of alternating circulation, as shown in Fig. (5) for a disk geometry.

##  4 Activity jump at an interface

We now consider an activity profile corresponding to a sharp interface
separating a region of high activity $\alpha_{0}$ from a region of low
activity $\alpha_{1}$. Isolated $\pm 1/2$ defects are situated at a distance
$x_{v}$ from the interface in the region of high activity, $\alpha_{0}$, as
illustrated in Fig. (6). The activity profile across this interface is given
by the Heaviside step function

$\alpha(\mathbf{r})=\alpha_{0}-\Delta\alpha H(x-x_{v}),$

corresponding to a singular activity gradient
$\partial_{x}\alpha=-\Delta\alpha\delta(x-x_{v})$ with
$\Delta\alpha=\alpha_{0}-\alpha_{1}$ the interfacial jump in activity. An
active/passive interface corresponds to $\alpha_{1}=0$ and
$\Delta\alpha=\alpha_{0}$. In this case, we find that the self-propulsion of
the $+1/2$ defect is reduced as the defect approaches the interface. The
vorticity-induced active torque tends to reorient the $\pm 1/2$ defects moving
toward the interface to preferred orientations that depend on
extensile/contractile activity. The $-1/2$ defect that already has the
selected orientation is attracted to the wall, while that with different
polarizations might be repelled.

Figure 6: Setup of (a) $+1/2$ defect and (b) $-1/2$ defect at a sharp
interface separating a region with higher activity $\alpha_{0}$ from that with
lower activity $\alpha_{1}$.

###  4.1 $+1/2$ defect

The active force field induced by a $+1/2$ defect located at a distance
$x_{v}$ from a sharp interface is given by

$\displaystyle\mathbf{F}^{I}_{+}(\mathbf{r})$ $\displaystyle=$
$\displaystyle-\Delta\alpha\delta(x-x_{v})\left(\cos(2\theta_{0})\mathbf{\hat{r}}-\sin(2\theta_{0})\mathbf{\hat{r}}^{\perp}\right)\;,$
(23) $\displaystyle\mathbf{F}^{B}_{+}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\left(\frac{\alpha_{0}}{r}-\frac{\Delta\alpha}{r}H(x-x_{v})\right)\mathbf{\hat{e}}^{+}\;.$
(24)

Inserting these expressions in Eq. (5), we obtain the contributions to the
self-propulsion velocity from interfacial and bulk active forces as

$\displaystyle\mathbf{v}^{B}_{+}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{0}\pi}{4\zeta}\mathbf{\hat{e}}^{+}-\frac{\Delta\alpha}{2\pi\zeta^{2}}\mathbf{\hat{e}}^{+}\int
d\mathbf{r}K_{0}\left(\frac{r}{\zeta}\right)H(x-x_{v})\frac{1}{r}\;,$ (25)
$\displaystyle\mathbf{v}_{+}^{I}$ $\displaystyle=$
$\displaystyle-\frac{\Delta\alpha}{2\pi\zeta^{2}}\mathbf{\hat{e}}^{+}\int_{-\infty}^{\infty}dyK_{0}\left(\frac{\sqrt{x_{v}^{2}+y^{2}}}{\zeta}\right)\frac{x_{v}}{\sqrt{x_{v}^{2}+y^{2}}}\;.$
(26)

The first term in the bulk contribution is the well-known constant self-
propulsion velocity from a constant activity $\alpha_{0}$ [16]. The second
term is the additional drift due to the activity jump $\Delta\alpha$ and
depends on the distance $x_{v}$ from the interface. As we will see below, this
contribution suppresses the defect self-propulsion near the interface.

If we now specialize to the case of an active/passive interface, i.e.,
$\Delta\alpha=\alpha_{0}$. In dimensional units, the self-propulsion velocity
of the $+1/2$ defect is then given by

$\mathbf{v}_{+}=\frac{\alpha_{0}}{4\eta}\pi\ell_{d}f_{v}^{+}(x_{v})\mathbf{\hat{e}}^{+}\;,$
(27)

with

$\displaystyle
f^{+}_{v}(x_{v})=1-\frac{2}{\pi^{2}}\int_{-\infty}^{\infty}dy\Bigg{[}K_{0}\left(\sqrt{x_{v}^{2}+y^{2}}\right)\frac{x_{v}}{\sqrt{x_{v}^{2}+y^{2}}}+\int_{x_{v}}^{\infty}dxK_{0}\left(r\right)\frac{1}{r}\Bigg{]}\;.$
(28)

The function $f^{+}_{v}(x_{v})$ is plotted in Fig. (7 a). We note that the
self-propulsion speed vanishes as the defect hits the interface $x_{v}=0$. In
other words, the defect slows down as it approaches the interface, and
eventually remains at rest at the interface. We note that Eq. (27) is obtained
by incorporating the incompressibility constraint only in the
$\mathbf{v}_{+}^{0}$ term. Additional pressure gradients may arise due to
activity jump. These are however, difficult to obtain analytically and are not
included in this study.

We now compute the vorticity at the defect position to investigate how its
contribution to the active torque tends to re-orient the defect as it
approaches the interface to a stable orientation. From Eq. (7), we obtain the
following expressions for interfacial and bulk contributions

$\displaystyle\omega^{I}_{+}=-\frac{\Delta\alpha}{2\pi\zeta^{2}}\sin(2\theta_{0})\int_{-\infty}^{\infty}dy\left[K_{0}\left(\frac{\sqrt{x_{v}^{2}+y^{2}}}{\zeta}\right)\frac{x_{v}^{2}}{(x_{v}^{2}+y^{2})^{3/2}}+K_{1}\left(\frac{\sqrt{x_{v}^{2}+y^{2}}}{\zeta}\right)\frac{x_{v}^{2}}{\zeta(x_{v}^{2}+y^{2})}\right]\;,$
(29)
$\displaystyle\omega^{B}_{+}=-\frac{\Delta\alpha}{2\pi\zeta^{2}}\sin{2\theta_{0}}\int_{-\infty}^{\infty}dy\left[K_{0}\left(\frac{\sqrt{x_{v}^{2}+y^{2}}}{\zeta}\right)\frac{1}{\sqrt{x_{v}^{2}+y^{2}}}-\int_{x_{v}}^{\infty}dxK_{0}\left(\frac{r}{\zeta}\right)\frac{x}{r^{3}}\right]\;,$
(30)

where the bulk vorticity diverges at $x_{v}=0$. The total defect angular
velocity is given by the sum of these two contributions evaluated at the
defect core. In dimensional units, it is given by

$\displaystyle\omega_{+}(x_{v},\theta_{0})=-\frac{\Delta\alpha}{2\pi\eta}\sin(2\theta_{0})f_{\omega}^{+}(x_{v})\;,$
(31)

where the wall-dependence function $f_{\omega}^{+}(x_{v})$ is plotted in Fig.
(7 b). Hence, near an active/passive interface, the vorticity-induced rotation
is at a rate $\dot{\theta}_{0}=\frac{1}{2}\omega_{+}(x_{v},\theta_{0})$ until
$\omega_{+}(x_{v},\theta_{0})=0$. It is clear from Fig. (7 b) that re-
orientation only occurs within a distance of order $\ell_{d}$ from the wall.
As the $+1/2$ defect approaches the wall, $f_{\omega}^{+}$ increases and
eventually diverges at $x_{v}\rightarrow 0$. This means that the defect tends
to re-orient its polarization until $\sin(2\theta_{0})=0$. From the stability
criterion that $\frac{d\omega_{+}}{d\theta_{0}}<0$, this corresponds to the
stable orientation $2\theta_{0}=\pi$ for $\alpha_{0}<0$ (extensile) and
$2\theta_{0}=0$ for $\alpha_{0}>0$ (contractile). In both cases, the defect
polarization is normal to the interface $\mathbf{e}_{+}=[\mp 1,0]$ and points
away from the interface for extensile systems and into the interface for
contractile systems, respectively. Numerical simulations [19] report that
$+1/2$ defects tend to reorient and drift parallel to the boundary when the
angle between the interface and the incoming velocity is below a critical
value that depends on activity. Above this critical angle, i.e more head-on
collisions, the defect hits the wall and tunnels through it. This effect is
likely coming from the additional contributions to the active torque that are
not considered here, namely the interactions between defects, deformations in
the nematic order parameter due to the wall and the coupling to flow
alignment. It is likely that these terms are important close to the interface,
both for determining the defect orientation and the tunneling effect observed
both experimentally and numerically [19].

Figure 7: Plot of (a) $f_{v}^{+}$ and (b) $f_{\omega}^{+}$ as functions of the
distance $x_{v}$ of the $+1/2$ defect from the interface. Notice that
$f_{\omega}^{+}$ diverges at $x_{v}=0$ due to the bulk terms.

###  4.2 $-1/2$ defect

The components of the interfacial active force due to a $-1/2$ defect at a
distance $x_{v}$ from the activity jump are given by

$\displaystyle
F^{I}_{x-}=-\frac{\Delta\alpha}{r}\delta(x-x_{v})(x\cos{2\theta_{0}}+y\sin{2\theta_{0}})\;,$
(32) $\displaystyle
F^{I}_{y-}=-\frac{\Delta\alpha}{r}\delta(x-x_{v})(-y\cos{2\theta_{0}}+x\sin{2\theta_{0}})\;.$
(33)

The corresponding bulk active force is

$\displaystyle F^{B}_{x-}=(\alpha_{0}-\Delta\alpha
H(x-x_{v}))\frac{1}{r^{3}}[(y^{2}-x^{2})\cos{2\theta_{0}}-2xy\sin{2\theta_{0}}]\;,$
(34) $\displaystyle F^{B}_{y-}=(\alpha_{0}-\Delta\alpha
H(x-x_{v}))\frac{1}{r^{3}}[(y^{2}-x^{2})\sin{2\theta_{0}}+2xy\cos{2\theta_{0}}]\;.$
(35)

Using these expressions, and neglecting the contribution from the pressure
gradient, the net drift velocity of the defect can be written as

$\displaystyle\mathbf{v}_{-}(x_{v})=-\frac{\Delta\alpha}{2\pi\eta}\ell_{d}f_{v}^{-}(x_{v})\mathbf{\hat{n}}^{-},$
(36)

where
$\mathbf{\hat{n}}^{-}=\cos(2\theta_{0})\mathbf{\hat{x}}+\sin(2\theta_{0})\mathbf{\hat{y}}$.
The function $f_{v}^{-}(x_{v})$ describes the dependence on the distance
$x_{v}$ to the interface and is given by

$f_{v}^{-}(x_{v})=\int_{\infty}^{\infty}dyK_{0}(\sqrt{x_{v}^{2}+y^{2}})\frac{x_{v}}{\sqrt{x_{v}^{2}+y^{2}}}-\int_{x_{v}}^{\infty}\int_{-\infty}^{\infty}dxdyK_{0}(r)\frac{x^{2}-y^{2}}{r^{3}}\;.$

It has been be evaluated numerically and is plotted in Fig. (8 a). The $-1/2$
defect acquires a finite self-propulsion close to the wall in a region of
thickness of order $\ell_{d}$ near the activity jump. Its motion is either
towards or away from the boundary, depending on the defect’s orientation and
the sign of the activity.

Figure 8: Profile of (a) $f_{v}^{-}(x_{v})$ and (b) $f_{\omega}^{-}(x_{v})$ as
function of $x_{v}$. Note that the function $f_{\omega}^{-}(x_{v})$ diverges
at $x_{v}=0$.

To see how the $-1/2$ reorients as it approaches the interface, we evaluate
the flow vorticity at the defect core as a function to the wall distance.
Again, there are contributions to the vorticity from both flows driven by
interfacial and bulk forces, given by

$\displaystyle\omega^{I}_{-}=$
$\displaystyle-\frac{\Delta\alpha}{2\pi\zeta^{2}}\sin{2\theta_{0}}\int_{-\infty}^{\infty}dy\left[K_{1}\left(\frac{r}{\zeta}\right)\frac{x_{v}^{2}}{\zeta(x_{v}^{2}+y^{2})}-K_{0}\left(\frac{\sqrt{x_{v}^{2}+y^{2}}}{\zeta}\right)\frac{x_{v}^{2}}{\sqrt{x_{v}^{2}+y^{2}}^{3}}\right]\;,$
(37) $\displaystyle\omega^{B}_{-}=$
$\displaystyle-\frac{\Delta\alpha}{2\pi\zeta^{2}}\sin{2\theta_{0}}\int
d\mathbf{r}K_{0}\left(\frac{r}{\zeta}\right)\Bigg{(}\delta(x-x_{v})\frac{y^{2}-x^{2}}{r^{3}}+H(x-x_{v})\frac{3x(x^{2}-3y^{2})}{r^{5}}\Bigg{)}\;.$
(38)

Note that the bulk term diverge when $x_{v}\rightarrow 0$. The total angular
velocity of the $-1/2$ defect can then be written as

$\displaystyle\omega_{-}(x_{v},\theta_{0})=-\frac{\alpha_{0}}{2\pi\eta}\sin(2\theta_{0})f^{-}_{\omega}(x_{v})\;.$
(39)

The function $f_{\omega}(x_{v})$ has been calculated numerically and is shown
in Fig. (8 b)). The dependence on the wall distance $x_{v}$ changes sign near
the wall, indicating that the vorticity tends to rotate the defect to a
preferred orientation at the wall. The preferred orientation is determined by
the stationary condition $\sin{2\theta_{0}}=0$, and the stability criterion
$\frac{d\omega_{-}}{d\theta_{0}}<0$, which implies that $\theta_{0}=0$ for
$\alpha_{0}<0$ and $2\theta_{0}=\pi$ for $\alpha_{0}>0$. In other words for
extensile activity the stable orientation of a $-1/2$ defect at a sharp
active/passive interface corresponds to a polarization $\mathbf{e}_{-}=[1,0]$.
Therefore, as a result of both their self-induced translational and rotational
motion, in an extensile system $-1/2$ defects are attracted to a sharp
active/passive interface and orient themselves with one of the three axis
normal to the interface. This is consistent with the accumulation of negative
topological charge observed in experiments at active/passive interfaces [18,
19] and near physical walls [29], as well as in simulations [24, 25].

##  5 Conclusion

Activity gradients or sharp jumps can guide the motion and orientation of
nematic defects. In a constant activity gradient, $+1/2$ defects acquire an
angular velocity that may rotate their orientation such that the defect
polarization aligns parallel to the activity gradient. The defects then self-
propels in the direction of the gradient, always moving towards regions of
lower magnitude of activity, where it is less motile. Thus, we expect that
activity gradients will introduce more circular motion in the trajectories of
the $+1/2$ defects. In contrast, a constant activity gradient yields no net
vorticity or active force at the core of the $-1/2$ defect, which remains
stationary.

We find that the self-propulsion velocity of $+1/2$ defects moving towards a
sharp active/passive interface is also reduced, and that the defect will
eventually stagnate at the wall. In contrast, $-1/2$ defects acquire a finite
propulsion speed in the interfacial region and can overcome the positive
defects, explaining the observation of negative charge accumulation in
experiments and simulations [18, 19, 29, 24, 25]. We also predict that the
active torque acting on a $+1/2$ defect that reaches the interface tends to
reorient it toward a preferred polarization that is perpendicular to the
interface and points away/toward it depending on extensile/contractile
activity. The vorticity-induced active torque also acts on the orientation of
a $-1/2$ defect migrating toward interface, by rotating the defect until it
reaches the stable orientation which minimizes the net vorticity at the defect
position. We show that a $-1/2$ defect with a stable orientation gets
attracted to a sharp interface. This stable orientation is selected by the
sign of activity, i.e whether the system is contractile or extensile.
Tunneling across the interface observed numerically may be due to soft
interfaces where the activity gradients are not sufficiently steep, as well as
due to defect interactions and other hydrodynamic effects. Here, we have
neglected additional contributions of pressure gradients induced by activity
gradients, as well as elastic stresses, flow alignment, nematic distortions
due to the active/passive interface and defect interactions, which may change
qualitatively the defect dynamics.

Our results offer a simple understanding of the dynamics of nematic defects in
the presence of spatially-varying activity. They can provide the starting
point for designing structures capable of controlling defect dynamics and
associated active flows.

No ethical dilemmas where encountered in the preparation of this paper.

This is primarily theoretical work and does not have any experimental data.
The computational data and codes for FEniCS are available on GitHub:
https://github.com/jonasron/Defect-Flows

J.R. derived the analytical results and performed finite-element simulations
for finite domains. L.A. verified all analytical calculations. All authors
contributed to a critical discussion of the analytical and numerical results
and participated in writing the manuscript.

We declare we have no competing interests.

M.C.M. was supported by the US National Science Foundation Grant No.
DMR-2041459. J.R. and L.A. acknowledge support from the Research Council of
Norway through the Center of Excellence funding scheme, Project No. 262644
(PoreLab).

M.C.M. thanks Mark Bowick, Luca Scharrer and Suraj Shankar for illuminating
discussions. L.A. and J.R. are thankful to Jorge Viñals for simulating
discussions.

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