# Superscreening and polarization control in confined ferroelectric nematic
liquids

Federico Caimi  Dept. of Medical Biotechnology and Translational Medicine,
University of Milano, Italy  Giovanni Nava  Dept. of Medical Biotechnology and
Translational Medicine, University of Milano, Italy  Susanna Fuschetto  Dept.
of Medical Biotechnology and Translational Medicine, University of Milano,
Italy  Liana Lucchetti  SIMAU Dept., Università Politecnica delle Marche,
Ancona, Italy  Petra Paiè  Dipartimento di Fisica, Politecnico di Milano,
Milano, Italy  Roberto Osellame  Istituto di Fotonica e Nanotecnologie,
Consiglio Nazionale delle Ricerche (IFN-CNR), Milano, Italy  Xi Chen  Dept. of
Physics, Soft Materials Research Center, University of Colorado, Boulder, CO,
USA  Noel A. Clark  Dept. of Physics, Soft Materials Research Center,
University of Colorado, Boulder, CO, USA  Matthew Glaser
<EMAIL_ADDRESS>Tommaso Bellini<EMAIL_ADDRESS>
###### Abstract

The combination of large spontaneous polarization and fluidity makes the newly
discovered ferroelectric nematic liquid crystalline phase (NF) responsive to
electric fields in ways that have no counterpart in other materials. We probe
this sensitive field response by confining a NF fluid in microchannels that
connect electrodes through straight and curved paths. We find that by applying
electric fields as low as E $\approx$ 0.5 V/mm, the NF phase orders with its
polarization smoothly following the winding paths of the channels even when
oriented antiparallel to the line connecting positive to negative electrodes,
implying analogous behavior of the electric field. Upon inversion of E, the
polar order undergoes a complex multistage switching process dominated by
electrostatic interactions. Multistage polarization switching dynamics is also
found in numerical simulations of a quasi-2D continuum model of NF liquid
crystals in microchannels, which also clarify the conditions under which the
electric field is guided by the microchannels. Experiments and theory indicate
that all observations are direct consequences of the prompt effective
screening of electric field components normal to the channel walls. This
electric “superscreening” effect emerges as a distinctive property of the NF
phase, capable of inducing conditions in which both the polarization and the
electric field are guided by microchannels.

The discovery of molecules capable of ordering into a ferroelectric nematic
(NF) liquid crystal (LC), a state predicted a century ago [1, 2, 3] but only
recently observed [4, 5, 6, 7], has attracted immediate interest because of
the novel properties of this new fluid state, involving its electric behavior,
its elasticity, and its non-linear susceptibilities [8, 9, 10, 11, 12, 13].
Conventional nematic LCs are non-polar anisotropic fluids, in which molecules
partially align along a common axis, the so-called nematic director
$\mathbf{n}$. In the NF phase, a bulk electric polarization density
$\mathbf{P}$ develops parallel to $\mathbf{n}$ through a weak first order
phase transition. Molecules found to give rise to NF ordering have large
permanent electric dipoles ($>10$ D) and a high degree of polar orientational
order, resulting in a large ferroelectric polarization, up to
$P_{0}=\lvert\mathbf{P}\rvert\approx 6\ \mu\mathrm{C/cm}^{2}$ [6, 14, 15].

The presence of a bulk polarization field that is readily reoriented makes the
electric field response of (NF) markedly different from the dielectric
response of conventional nematics, in ways that have yet to be fully explored.
Because of the dielectric anisotropy of conventional nematics, the electric
field ($\mathbf{E}$) has a tensorial coupling to $\mathbf{n}$, producing
torques on the director proportional to $E^{2}$, and of a magnitude comparable
to the orientational elasticity. In the NF phase the presence of a bulk
polarization gives rise to a dipolar coupling to $\mathbf{E}$ and self-
interaction of space charges arising from divergence of the polarization,
producing electric torques orders of magnitude larger. Indeed, because of the
fluidity of the NF phase, the orientation of $\mathbf{P}$ is a Goldstone
variable with energetics dominated by electrostatics, with nematic elasticity
and surface anchoring playing a minor role. The unconstrained rotation of
$\mathbf{P}$ is affected only by the weak elasticity arising from its non-
uniform orientation and by the self-interaction caused by electrostatic forces
arising from polarization space charge at surfaces
($\sigma_{\mathrm{P}}=\mathbf{P\cdot u}$, $\mathbf{u}$ being the unit vector
normal to a limiting surface of NF) and in the bulk
($\rho_{\mathrm{P}}=-\nabla\cdot\mathbf{P}$).

All these factors combine to create a fundamentally new kind of fluid,
markedly different from conventional nematic LCs and from solid ferroelectric
materials, whose polarization, intrinsically constrained by the symmetry of
the crystal unit cell, is typically limited to a set of easy axis
orientations. In this work we explore the interplay of external fields,
geometry, and electrostatic self-interaction NF fluids confined within
straight and bent microchannels. Indeed, because of its spontaneous bulk
polarization, a ferroelectric nematic fluid confined to a channel should, in
the absence of external electric fields and other couplings, align with
$\mathbf{n}$ along the channel to minimize the overall electrostatic energy, a
behavior with intriguing implications in the case of bent channels.

Description of the experiment

We have produced, via femtosecond laser micromachining assisted by chemical
etching [16, 17], microchannels buried in monolithic fused silica that connect
gold wire electrodes. The channels, shown in Fig. 1a-b, are all of length
$\ell\approx 1\ \mathrm{mm}$ and have a rounded square cross-section of width
$w\approx 35\ \mu\mathrm{m}$. Four different shapes are produced: straight
along the x-axis direction connecting the electrodes (I-shaped channel), with
the central part bent with respect to the x-axis of $90^{\circ}$ (L-shaped
channel), $135^{\circ}$ (Z-shaped channel) and $180^{\circ}$ (S-shaped
channel) (see Fig. 1b). The channels were filled with RM734, a liquid crystal
material that exhibits nematic (N) and ferroelectric nematic (NF) phases [4,
6], with the N–NF transition temperature $T_{\mathrm{N-N}_{\mathrm{F}}}\approx
133^{\circ}C$. Prior to filling, the glass channel surfaces were silanized
with hexadecyltrimethoxysilane (HDTMS) to favor planar orientation of RM734
[18]. A detailed description of the fabrication process is given into the SI
section.

Response to static fields

In the absence of an applied voltage difference $\Delta V$ between the
electrodes, the nematic director exhibits partial ordering along the channel
in both the N and NF phases (Fig. 1c), probably because of the residual
roughness of the channels walls. By applying a voltage $\Delta V$, which gives
rise to a “nominal” electric field $E\textsubscript{0}\equiv\Delta V/\ell$,
the NF ordering becomes virtually perfect in all four channels (Fig. 1c-d)
when $E\textsubscript{0}>0.25\ \mathrm{V/mm}$, while no significant effect is
visible in the N phase for fields up to $E\textsubscript{0}\approx 50\
\mathrm{V/mm}$. We closely inspected the field-induced NF alignment by
polarized transmission optical microscopy (PTOM) and found no sign of either
defect lines or defect walls even in the curved sections of the channels that
bend contrary to the naïve electrode-to-electrode direction expected for
$E_{0}$ in uniform dielectrics (Fig. 1e). This finding indicates that under
these conditions both the nematic director $\mathbf{n}$ and the polarization
$\mathbf{P}$ are always nearly parallel to the solid surface and continuously
follow the channel, as sketched in Fig. 1f. The continuity of NF polarization
along the channels also implies a similar continuity of the electric field,
which thus follows the channels rather than being directed along the paths
connecting the electrodes as in homogeneous dielectric media. This behavior is
not dissimilar to what happens in a bent conductive wire between electrodes,
where the surface accumulation of free charges steers the field along the
wire, independently of its path. In NF this same effect must be obtained with
bound charges.

Figure 1: Ferroelectric nematic ordering in microchannels. a: drawing of the
buried microchannels. Yellow cylinders represent the gold wire electrodes. b:
pictures of the 4 channels considered in this work, designed so that the
central part forms an angle of 0°, 90°, 135° and 180° with respect to the x
axis. c: PTOM microscope pictures between cross polarizers in the absence
(left) and in presence (right) of an electric field of the “I” channel filled
with RM734. The fully dark appearance of the channel in the bottom-right
picture indicates that in the presence of a field, the orientation of the N
director is along the channel. d: PTOM microscope pictures between cross
polarizers of the Z, L and S channels, respectively, oriented so that the
lateral portions are at 45° (top panels) or along (bottom panels) the analyzer
while $\Delta V=1V$ is applied to the electrodes. e: enlargement of a curved
portion of the Z channel at various orientations with respect to the
polarizer, as indicated. f: sketch of the continuity of the direction of the
polarization $\mathbf{P}$ (green arrows) within the S channel indicating that,
in its central section, the polarization is pointing in the opposite direction
with respect to the nominal applied electric field $E_{0}$.

Ferroelectric superscreening

We understand the continuity of polarization and electric field along the
channels as the consequence of the orientational freedom of $\mathbf{P}$ in
NF, enabling an immediate deposition of bound charges that readily cancels the
normal components of $\mathbf{E}$ within the channel, which become thus
parallel to the channel axis.

The phenomenon is sketched in Fig. 2. The bare (ferroelectric polarization-
free) microchannel exposed to a transverse field $E_{0\perp}$ behaves like a
capacitor with dielectric constant $\epsilon$ and capacitance per unit length
of channel $C$ (Fig. 2a). In the NF phase an additional a bulk polarization
field $\mathbf{P}$ is present (Fig. 2b), which has the key features of being
reorientable, a result of the symmetry, energy degeneracy, and fluidity of the
LC, and spontaneously spatially uniform, to avoid polarization space charge in
the LC volume [19]. $\mathbf{P}$ is drawn parallel to the channel ($\psi$ = 0)
since $\psi\neq$ 0 results in the deposition of surface polarization
charge/unit length of magnitude $\lambda$ = wP$sin\psi$ on the channel edges
(Fig. 2c). The resulting electric field $E_{P}(\psi)$ within the channel
provides a restoring torque density that opposes the reorientation, a sort of
electrostatic orientational spring that tends to force $\psi$ back to zero.
Application of an external $\psi$-independent torque results in a balance with
this restoring torque at finite psi (Fig. 2c). This electrostatic spring is
exceptionally stiff, as can be estimated from $V_{sat}=Pw/C$, the voltage
generated across the channel when the orientation is saturated at
$\psi=90^{\circ}$ (Fig. 2d.). For our channels we estimate $V_{sat}\approx
10^{6}$V so that $\mathbf{P}$ being perpendicular to the channel is an
idealized condition actually never achievable in practice since it would
require creating a gigantic electric field.

In the channel experiments reported here the external torque is produced by an
applied electric field $\mathbf{E}_{0}$ forming an angle $\beta$ with the
channel axis (Fig. 2e), giving rise to a $\mathbf{P}$ rotation and restoring
torque as in Fig. 2c. In this case the torque balance condition is
$E_{\perp}=0$ within the channel, where $E_{P}(\psi)$ cancels $E_{0\perp}$
[20, 21, 22]. For typical experimental $\mathbf{E}_{0}$ magnitudes and a
typical NF polarization $P\approx 6\mu C/cm^{2}$, this cancellation occurs for
sub-microradian reorientations of $\mathbf{P}$, with
$\psi=wE_{0\perp}/V_{sat}$, leaving $\mathbf{P}$ always essentially parallel
to the local channel axis, $E_{\parallel}=E_{0\parallel}$ and $E_{\perp}$ = 0
in the channel. Such immediate and effective screening, by which the normal
component of $E_{0}$ is readily cancelled, is a distinctive property of the NF
phase that we call ferroelectric superscreening to mark the difference between
this ultimately collective fluid dipolar reorientation and conventional
dielectric or conductive screening by dipoles or free charges.

In the microchannel experiments, ferroelectric superscreening locally cancels
the normal electric field component, but also globally affects its
longitudinal components through the long-range effect of the modified bound
charge distribution.

Figure 2: Schematic description of superscreening in a channel. Schematically,
the NF-containing channel can be considered a dielectric slab of dielectric
coefficient $\epsilon$ into which a reorientable polarization field
$\mathbf{P}(r)$ has been introduced and then exposed to external torques. (a)
The bare channel has a capacitance/length of channel $C$, which is the ratio
of voltage $V$ across the channel to $\lambda$, the charge along each edge per
unit length of channel, such that $\lambda=CV$. (b) Being the reorientable
field $\mathbf{P}$ locally along the nematic director, its spatial variation
is controlled by nematic elasticity and by the coupling of P to electric
fields, both external and internal, due to polarization space charge and
polarization surface charge. The ratio $\sqrt{\epsilon K}/P$, where K is a
Frank elastic constant, is very small (less than $1$ nm), indicating that
electrostatics is dominant, a condition that makes $\mathbf{P}$ uniform and
its orientation $\psi$ responsive to external torques as a spatially uniform
field. (c) application of an external $\psi$-independent mechanical torque is
balanced by an electrostatic restoring torque per unit length of channel
$\tau=(P^{2}/C)sin\psi$ due to polarization surface charge at the channel
edges. (d) If $\mathbf{P}$ is reoriented to be normal to the plates, then
$V\equiv V_{sat}=P/C$, and the orientational potential energy/length will be
$P^{2}/C=PV_{sat}$. The corresponding electric field in the channel midplane
will be $E_{sat}\sim 10^{10}V/m$, $i.e.$ $V_{sat}\sim 10^{5}V$, many orders of
magnitude larger that the fields applied in the experiments. (e) microchannels
in the experiments are in electric fields $\mathbf{E}_{0}$ not generally
oriented with the channel axis, with components $E_{0\parallel}$ and
$E_{0\perp}$ along and perpendicular to the channel axis, respectively. While
$E_{0\parallel}$ is continuous across the channel walls, $E_{0\perp}$ is
cancelled by superscreening. Being the applied fields much smaller than
$E_{sat}$, the tilt angle $\psi$ is so small not to be detectable (in the zoom
panel $\psi$ is enhanced to enable representation).

Field reversal experiments

To further explore the role of bound-charge screening in the ‘polarization
following’, we investigated the response to the reversal of the sign of
$\Delta V$. Upon switching $\Delta V$, the nematic ordering becomes suddenly
unstable, with the transmitted light intensity through crossed polarizers
($I_{P}$) and without analyzer ($I_{S}$) transiently changing during a time
interval that depends on $\Delta V$, being of the order of 100 ms for
$E_{0}=1V/mm$. Specifically, we have measured $I_{P}$ vs. time in portions of
the channels oriented as the polarizer (thus dark in stationary condition as
in Fig. 1c lower right panel), and normalized it with $I_{P_{45}}$, the value
of $I_{P}$ measured in equilibrium conditions (i.e. in stationary state with
applied field) with the channel at $45^{\circ}$ with the polarizer (Fig. 1c
upper right panel). Analogously, we normalized $I_{S}$ with $I_{S_{0}}$, the
transmitted intensity in equilibrium conditions. By inspecting the two
transmittances $\tau_{P}(t)=I_{P}(t)/I_{P_{45}}$ and
$\tau_{S}(t)=I_{S}(t)/I_{S_{0}}$ and associated videos, we can recognize three
phases for the polar switching in the channels: a disordering regime, a
reorganization regime, and a defect-annealing regime. Because of the small
section of the microchannels we couldn’t measure experimentally the
polarization current flowing through the electrodes and compare it to
$\tau_{P}(t)$ and $\tau_{S}(t)$. We instead performed an analogous measurement
on a flat cell (see SI).

Inspection of PTOM pictures and videos shows that, upon potential inversion
($t=0$), the order is replaced by an irregular pattern formed by micron-sized
domains (Fig. 3a-b) coherent with the notion that $\mathbf{P}$ locally rotates
away from the original orientation in a variety of directions. This leads to
an increase in $\tau_{P}(t)$ due to rotated optical axis (Fig. 3 d-g) and in a
decrease in $\tau_{S}(t)$ due to scattering (see SI). The disorder in the NF
structure grows to a maximum and decreases, as shown by appearance of a
minimum in $\tau_{S}(t)$ (see SI) found at about the same time $t\approx
t_{m}$ at which $\tau_{P}(t)$ shows a dent in its growth.

We find that $t_{m}$ marks the transition between two different regimes.
Pulsed field reversals, in which the sign of $\Delta V$ is inverted for a time
$t_{inv}<t_{m}$, give rise to a simple exponential relaxation of
$\tau_{P}(t)$, while for longer pulses ($t_{inv}>t_{m}$), the relaxation
process in the channel becomes much longer and irregular (see SI). We thus
interpret $t_{m}$ as the time it takes to the NF domains to rotate enough to
start reorganizing in the opposite direction, merging and reducing defect
lines and lowering the turbidity.

In the reorganization regime ($t>t_{m}$), $\tau_{P}(t)$ still grows,
reflecting coarsening and smoothing of the defects, to reach a maximum and
decrease back to $0$ when the uniform polar ordering is established in the
reversed direction. Such uniform NF ordering develops through a nucleation
process taking place in the narrowest section of the channel, typically (but
not always) located halfway between electrodes as a natural result of the
etching process or due to design (see SI). This final stage in the reversal of
$\mathbf{P}$ requires major rearrangements of the director that involve
disentangling of topological defects, closure of defect loops and reversal of
the polarity at the surface. Previous observations in silanized flat cells
suggest the latter to be the slowest component of switching after field
reversal [18].

After its nucleation, the defectless uniform region ($\tau_{P}=0$)
progressively expands via the motion of the interface which separates defected
and defect-free NF phase (Fig. 3b, bottom panels, and Fig. 3c). Thus, the
completion of the $\mathbf{P}$ reversal takes place at a time that grows with
the distance from the nucleation site, sooner in proximity of the channel
center (Fig. 3d, red line) and later (even after up to 1 s) close to the
electrodes (Fig. 3d, blue and green lines).

Figure 3: Polarization inversion in the microchannels. a: PTOM microscope
pictures of the Z channel with two different orientations with $\Delta V$ =
1V. b: time sequence of PTOM images of the central part of the Z channel
following the field inversion (t=0). The sequence shows the appearance and
coarsening of domains and the formation of the homogeneous-defected interface,
that moves toward the electrodes with velocity $v_{I}$. c: time sequence of
PTOM images of a curve of the Z channel showing the motion of the homogeneous-
defected interface with velocity $v_{I}$. d: time evolution of the depolarized
transmittance $\tau_{P}(t)$ measured in the central part and in the two side
arms of an S-channel during the inversion of $\Delta V$ from 0.5V to -0.5V. e:
time evolution of the depolarized transmittance $\tau_{P}(t)$ measured in the
central part of the various types of channels during the inversion of $\Delta
V$ from 0.5V to -0.5V. f: time evolution of the depolarized transmittance
$\tau_{P}(t)$ measured in the central part of an S-channel during the
inversion of different $\Delta V$ between the electrodes resulting in
different $E_{0}$, as in the legend. g: Same data of the previous graph
plotted on the rescaled time t·$E_{0}$. h: Velocity $v_{I}$ of the
homogeneous-defected interface vs the the nominal electric field $E_{0}$
measured in the central part of the various channels in the first instants
after its formation.

New insights emerge from the comparison of $\tau_{P}(t)$ in different
positions within the same channel and in different channels. First, the short
time behavior is the same everywhere in all positions of bent channels. This
is shown in Fig. 3d for the S channel, while very analogous plots are obtained
for the L and Z channels. Second, $\tau_{P}(t)$ measured in the central part
of the bent channels, i.e. close to nucleation site, is basically equal in all
regimes provided the applied voltage is the same, as shown in Fig. 3e for
$\Delta V=0.5V$. Remarkably, $\tau_{P}(t)$ is instead different for the simple
straight geometry, which displays a slower rise and a steeper decay.

Fig. 3f shows that the polarization reversal is a process fully controlled by
electrical interactions: the evolution of $\tau_{P}(t)$ becomes faster for
larger $\Delta V$ (and hence $E_{0}$) with its whole kinetics being
proportional to $E_{0}$ in all regimes, as shown in the collapse in Fig. 3g.
The dent at $\tau_{P}(t_{m})$ is more pronounced with larger $E_{0}$, a
condition in which we also find a larger turbidity (see SI). This behavior
reflects the fact that faster switching occurs through the breaking up of the
system in smaller domains, leading to a larger scattering of light.

By demonstrating that the polarity inversion linearly depends on $E_{0}$, the
results in Fig. 3f-g convey the notion that $\tau_{P}(t)$ can be used as a
proxy for the local field $E_{L}$. Accordingly, the identical rise of
$\tau_{P}(t)$ for $t<t_{m}$ in the various portions of the channels indicates
that, in that time interval, $E_{L}$ is uniform throughout the channels.
Moreover, the identical full shape of $\tau_{P}(t)$ in the central part of the
L, Z and S channels reveals that $E_{L}$ is the same in the central portion of
all bent channels, provided that width, $\Delta V$ and $\ell$ are the same.

In the latest stage of the switching, defect-free NF domains nucleate in the
slimmest portions of the channels, suggesting that those are the places where
$E_{L}$ is largest, and expand via a moving interface. We measured the
propagation velocity $v_{I}$ of such interface for different $\Delta V$ in a
limited region closest to the nucleation spot. We find $v_{I}$ to be constant,
and thus well-defined, at least in the first $100\mu m$ of displacement, away
from electrodes and channel bends (see SI). In Fig. 3h we plot $v_{I}$ vs.
$\Delta V$ for the various channels. We find $v_{I}$ to be approximately
linear in $\Delta V$ and to be very similar in all the bent channels. Again,
the straight channel makes an exception, with a larger, about double, $v_{I}$.

The linear dependence of $v_{I}$ on $\Delta V$ indicates that the motion of
the interface is electrically driven and follows the local electric field,
confirming that the latter is directed from electrode to electrode along the
channels, despite their bends. Such motion also indicates that the interface
carries electric charge. This is coherent with its marking a discontinuity in
polarization between the fully polarized uniform phase and the less polarized
defected structure, a concept that we further validated through measurements
of polarization current and transmittance in flat cells (see SI). The constant
$v_{I}$ for each $\Delta V$ reveals that the interface motion is opposed by a
viscous-type friction force, quite likely reflecting the viscosity arising
from rotation of the domains, necessary to free the volume from defects (see
SI).

Continuum modeling of switching and equilibrium structure

To gain insight into the experimentally observed phenomena, we performed
numerical simulations of the equilibrium structure and switching dynamics of
NF fluids confined to microchannels. We assume that the ferroelectric
polarization $\mathbf{P}$ has fixed magnitude $P_{0}$ but can vary in
direction, i.e.,
$\mathbf{P}=\mathbf{P}(\mathbf{r})=P_{0}\mathbf{p}(\mathbf{r})$, where the
polar director $\mathbf{p}$ is a unit vector, assumed to be parallel to the
nematic director $\mathbf{n}$. We further assume that the $z$ dimension of the
channels is large relative to the $x$ and $y$ dimensions, and that all
quantities ($\mathbf{n}$, $\mathbf{P}$, $\mathbf{E}$, $\rho$, etc.) are
independent of $z$, e.g., $\mathbf{P}(\mathbf{r})=\mathbf{P}(x,y)$. Finally,
we assume that $\mathbf{P}$ and $\mathbf{E}$ are confined to the $x$-$y$
plane, i.e., that $E_{z}=0$ and $P_{z}=0$. These simplifying assumptions
result in a computationally tractable quasi-2D model that captures the
essential physics governing the behavior of NF fluids in microchannels.

The free energy density of the ferroelectric nematic phase is taken to be the
sum of orientational elastic and electrostatic contributions,
$f(x,y)=f_{\mathrm{elast}}(x,y)+f_{\mathrm{elect}}(x,y)$. The former is the
Frank-Oseen orientational elastic free energy density in the one-elastic-
constant approximation,
$f_{\mathrm{elast}}=\frac{1}{2}K\lvert\nabla\theta\rvert^{2}$, where $K$ is
the elastic constant for splay and bend deformations and $\theta(x,y)$ is the
orientation of $\mathbf{p}$ in the $x$-$y$ plane. The electrostatic free
energy density is
$f_{\mathrm{elect}}=-\mathbf{P}\cdot\mathbf{E}=-P_{0}\left(E_{x}\cos\theta+E_{y}\sin\theta\right)$,
where $\mathbf{E}$ is the total electric field arising from both bound and
free charges.

The equilibrium structure is obtained by functional minimization of the total
free energy, which we accomplish using a relaxation method: the system is
evolved to obtain the equilibrium $\theta(x,y)$ by solving the equations of
motion that result from setting the total torque (the sum of elastic,
electrostatic, and viscous torques) to zero.

The elastic, electric and viscous torques acting on $\theta(x,y)$ are
$\tau_{\mathrm{elast}}(x,y)=K\nabla^{2}\theta$,
$\tau_{\mathrm{elect}}(x,y)=\left(\mathbf{P}\times\mathbf{E}\right)\cdot\mathbf{z}=P_{0}\left(E_{y}\cos\theta-
E_{x}\sin\theta\right)$, and
$\tau_{\mathrm{visc}}(x,y)=-\gamma\partial\theta/\partial t$, where $\gamma$
is the rotational viscosity. Starting from a given initial condition, the
equations of motion are evolved to reach equilibrium
($\tau_{\mathrm{elastic}}+\tau_{\mathrm{elect}}=0$), and the same equations of
motion are solved to model the switching dynamics upon field reversal. These
equations are solved numerically, with $\theta$ defined on a discrete grid
with grid spacing $a$, using an explicit Euler time-stepping scheme. The
geometry of the simulated channels (S, L and I) matches that of the real
channels in the $x$-$y$ plane. For computational expediency, we perform
simulations with a potential difference between the end of the channels of
$+1000$ V, $i.e.$ about $10^{3}$ times larger than that applied in the
experiments, since with this value the entire switching process is largely
complete within $500\ \mu$s. A grid spacing of $a=1.5\ \mu$m was used in the
calculations presented here, but we have verified that varying grid spacings
give similar results (see SI). In the simulations described below, we have
also introduced homogeneous surface roughness of amplitude $\delta=0.76\ \mu$m
into the channel boundaries, to more faithfully model the experimental
situation (see the SI for a discussion of the effects of surface roughness on
equilibrium structure and switching dynamics). Full details of the
computational method are provided in the Methods section and in the SI.

Figure 4:  Simulated time evolution of polarization $\mathbf{P}$, total (free
+ bound) charge density $\rho=\sigma_{P}+\rho_{P}+\rho_{f}$, and electric
field $\mathbf{E}$, following sign reversal of the potential on the electrodes
(at $t=0=t_{0}$). The starting condition is the equilibrium configuration for
the opposite sign of potential difference. All channels are of width $40\
\mu$m and contour length $\ell\approx 1$ mm. (a): Snapshots showing
$\mathbf{P}$, $\rho$ and $\mathbf{E}$ in the S-shaped channel at five specific
times marked by color frames. The arrows denoting $\mathbf{P}$ and
$\mathbf{E}$ are of fixed size, with the field amplitude indicated by the
color of the arrows. For clarity, $\mathbf{P}$ and $\mathbf{E}$ are shown only
at a subset of grid points (1 in 196). (b) and (c): $P_{\parallel}(t)$ and
$E_{\parallel}(t)$ averaged over the central section of an S-shaped channel.
Vertical colored lines mark the times relative to the snapshots in panel a of
matching color. (d) and (e): $P_{\parallel}(t)$ and $E_{\parallel}(t)$ from
different regions of S-, L-, and I-shaped channels, with the color code
indicated in the schematic diagram. The color shading in panels (b)-(e)
highlights the time intervals during which the electric field reverses (red
shading), and during which the polarization reverses and $E_{\parallel}\approx
1V/\mu m=\Delta V/\ell$, indicating electric field guiding (blue shading).

Simulation results

The general features of the equilibrium structure and switching dynamics of
ferroelectric nematics confined to microchannels can be understood from the
simulated switching dynamics upon voltage reversal for the S-shaped channel,
shown in Fig. 4. Fig. 4a shows the time evolution of polarization density,
total charge density, and electric field upon voltage reversal. Figs. 4b and
4c show the $t$ dependence of $P_{\parallel}$ and $E_{\parallel}$, the
components of $\mathbf{P}$ and $\mathbf{E}$ parallel to the local channel
direction, averaged over the central linear section of the S-shaped channel,
where the colored vertical lines indicate the times corresponding to the
snapshots in Fig. 4a with matching colored frames. Immediately after voltage
reversal ($t=0$) the electric field in the channel is nonuniform in magnitude
and has a substantial component transverse to the channel. This transverse
component of field is rapidly screened out by a slight reorientation of the
polarization at the channel surfaces, and by $t=15\ \mu$s (red vertical line)
the electric field in the channel follows the channel contour and is nearly
uniform in magnitude ($\sim 1\ \mathrm{V}/\mu$m) (see Fig. 3e, red frame). In
this time interval, the field outside the channel switches direction, with a
corresponding change in sign of the bound charge density at the channel
surfaces. Fig. 4d-e shows that the same field-reversal time interval during
which $\mathbf{E}$ changes direction while $\mathbf{P}$ remains nearly
constant is observed in different regions within the L, S and I channels,
identified by the color code in the schematic diagram between Fig. 4d and 4e.

At the end of the field-reversal process, the polarization field in the
channel is antiparallel to the electric field and thus in a state of unstable
equilibrium, leading to reorientation and disordering of the polarization
field at later times. The configuration at time $t=75\ \mu s$ (corresponding
to $P_{\parallel}(t_{2})\approx 0$) is shown in the blue-framed snapshots of
Fig. 3e, where it appears that the inversion of $P_{\parallel}$ takes place
through a disordered state, indicating that reduction of the free energy
occurs via breaking of the polar director field into small domains, enabling
local minimization of the energy cost due to bulk and surface charge
accumulation. The orientation of the electric field within the channel becomes
somewhat disordered during this process, but overall $E_{\parallel}$ and
$\rho$ remain nearly constant in the interval $t=25$–$125\ \mu s$ (blue
shading of Fig 4b-e), the time interval during which $P_{\parallel}$ nearly
completes its inversion. Thus, most of the $\mathbf{P}$ inversion process
takes place while $E_{\parallel}$ is roughly constant and uniform. Very
similar behavior of $P_{\parallel}$ and $E_{\parallel}$ is found over this
time interval for different regions of the various channels (Fig. 4d-e).

Polarization reversal is not the final event in the voltage reversal response.
Indeed, a complex subsequent dynamics takes place, involving a more subtle
reorientation of the polarization field and an associated reorganization of
bound charge in bands of positive and negative sign transverse to the channel,
resulting in nonuniformity in the magnitude of the electric field along the
channel, evident by $t=250\ \mu$s (Fig. 4a, purple frame). In this last stage,
$E_{\parallel}(t)$ evolves differently in different regions of the various
channels (see Fig. 4e for $t>125\ \mu$s) and depend sensitively on details of
channel roughness, as can be seen by comparison with simulations in channels
in which the linear portions of the channels are smooth (see data and
discussion in SI), making impossible to identify an equilibrium state to
compare with experiments.

Switching in the three simulated microchannels thus takes place in three
distinct stages: (i) fast screening of the normal component of the electric
field at the boundaries of the channel (‘superscreening’) leading to uniform
following of the channel contour by the electric field, and accompanied by
reversal of the field direction outside the channel and an associated change
in the sign of bound charge at the surface of the channel; (ii) reversal of
the polarization direction via a disordered intermediate state, culminating in
uniform following of the channel contour by the polarization field while
$\mathbf{E}$ within the channel remains approximately uniform in magnitude and
guided; (iii) roughness-dependent reorganization (‘annealing’) of the
polarization field and associated bound charges within the channel.

Finally, to clarify the role of channel width variation, we performed test
simulations on S and I channels tapered in the central region, finding that
indeed polarization reversal is faster and the local field larger where the
channel has the minimum width (see SI).

Discussion

Despite the strong simplifications adopted in the computation, the simulation
results clearly confirm that the major experimental observations are direct
consequence of the unique features of the NF fluid state. The polarization
continuously follows the channels from electrode to electrode even in bent-
shape geometries, in full agreement with the experimental observations.

The simulations confirm the essential role of NF superscreening, through the
prompt accumulation of surface polarization charges that screen the component
of $\mathbf{E}$ normal to the channel boundaries, causing the electric field
inside each channel to become directed along the channel, with a magnitude
that, during the $\mathbf{P}$ switching, is roughly constant and uniform in
all channels. This notion agrees with the observation that the electrically
controlled switching kinetics (initial part) is indeed similar in the
different positions of the bent channels.

Simulations also confirm that the $\mathbf{P}$ switching taking place through
the observed disruption of the nematic ordering into small domains is also a
consequence of the bulk polarization, a strategy by which the system minimizes
the surface and bulk bound charge during inversion. Of course, missing the
third dimension, this result remains a qualitative indication only.

On longer timescales, simulation predicts variations in electric field
magnitude that depend on the detailed shape of the channel. Experiments
indicate that in the latest stages of $\mathbf{P}$ inversion, the defect-free
volume nucleates in the narrowest sections of the channels where simulation
suggest the field is largest, while the expansion velocity of such domains
($v_{I}$) is equal in the bent channels, a sign that the electric field has
remained equal in the central part of the S, Z and L channels. To assess the
quantitative coherence of these observations a much more detailed level of
modeling would be required.

Experiments show that straight channels behave a bit differently than the bent
ones, with a slower onset of $\mathbf{P}$ disordering and a larger $v_{I}$.
This is not in simple agreement with simulations, which predict a generally
uniform behavior. Even in simulation, however, the straight channel has a
singular behavior: when built with smooth walls shows a much slower
$\mathbf{P}$ inversion; tapering and/or introducing wall roughness has a more
dramatic effect than in bent channels. We understand the peculiarity of the
straight channel as a consequence of its symmetry, by which no easy point of
charge accumulation are available, as in the bends of the other channels.
Elimination of the normal component of $\mathbf{E}$ requires in the straight
channel the buildup of a nonzero $\nabla\cdot\mathbf{P}$, while bending,
tapering and wall noise provide immediate accumulation points for polarization
charges, which speed up the charge distribution process. We thus speculate
that the quantitative features of the straight channel switching behavior
depend on its construction details.
Conclusions

The electric field response of ferroelectric fluid material confined in bent
microchannels, by which polarization and electric field follow the channels
along their paths, brings to light the capacity of these materials of promptly
cancelling any electric field components normal to dielectric surfaces. This
unique and defining property of NF, which we have called ‘superscreening’, is
achieved through minor local changes in the orientation of $\mathbf{P}$ and is
the leading and fastest component of the multiscale dynamic response of NF to
electric fields, determining all the following evolution.

Superscreening mandates that the electric field within the channel ‘follows’
the channel, which further implies that the ferroelectric polarization field
follows the contour of the channel under equilibrium conditions because the
electrostatic energy is minimized when the polarization field is locally
parallel to the electric field everywhere. This behavior strictly depends on
the electrostatic energy being the dominant free energy contribution (in our
conditions about $10^{4}$ times larger than the elastic energy, see SI), a
feature making ferroelectric nematics strikingly different to conventional
nonpolar nematics.

The S-shaped channel, in which there are three parallel three straight
sections, is particularly revealing: in principle, the polarization field
could point in the same direction (say to the right) in all three sections,
with polarization reversal walls in the two curved portions of the channel.
However, such polarization reversal walls are highly charged and thus
inherently unstable, their bound surface charge density $\sim\pm 2P_{0}$
giving rise to fields on the order of $10^{10}$ V/m. Should such reversal
walls transiently form, the large electric field they generate would
immediately reorient the polarization in the central portion of the S-shaped
channel to produce uniform ‘polarization following’ along the channel contour,
as observed.

Superscreening operates at the NF-dielectric interface, but not when NF
material is in direct, high capacitance contact with electrodes, such as those
at the ends of the channels in the experiments described here, where bound
polarization charges are effectively compensated by free charges, a condition
enabling strong electro-optical response, as the one following the potential
reversal.

The last step in the switching involves annealing of defects in the bulk and
polarity inversion at the surfaces. This occurs through a nucleation process
that starts where the field is largest and propagates through the entire
channel via an electrophoretic-type motion of the interface between uniform
and disordered regions. We find the whole behavior to be electrically driven,
and thus specific to this fluid ferroelectric phase in which electrostatic
interactions dominate.

Our findings open to the possibility of exploiting the propagation of order
within channels to design microconfinement geometries that provide a simple
means to control the optical axis in multiple positions in electro-optic
devices. More broadly, this study establishes a conceptual framework for
understanding the behavior of NF materials in a variety of settings in which
geometrical confinement effects are relevant, e.g., in composite structures,
in porous and/or disordered media with quenched or annealed disorder, and in
the presence of topographically patterned substrates.

## Materials and methods

Materials
RM734 (4-[(4-nitrophenoxy)carbonyl]phenyl 2,4-dimethoxybenzoate) has been
synthesized as reported in [6], all the other reagents and materials were
obtained from different commercial sources.

Channel fabrication

Microchannels have been fabricated by femtosecond laser irradiation followed
by chemical etching (FLICE) [16, 17] in 1 mm thick fused silica substrates.
Details of the whole process are provided in the Supplementary Information.
Following laser irradiation, we have exposed the substrate to two different
etchant solutions: first an aqueous solution of hydrofluoric acid (HF) to etch
the access holes, and then a solution of potassium hydroxide (KOH) to etch the
channels. We have used this dual process, to benefit from the advantages of
both the etchants. Indeed, HF guarantees high etching rate for large volume
removal, i.e. for the creation of electrode access holes, while KOH is slow
but provides high etching selectivity for controlled and uniform microchannel
cross-section over the entire length [23]. To smooth the microchannel
surfaces, we annealed the samples in a temperature-controlled furnace [24].
Channel surfaces have then been treated using a modified version of the
procedure reported in [18].

Transmittance measurements

Transmitted light intensities trough crossed polarizers ($I_{P}$,
$I_{P_{4}5}$) and without analyzer ($I_{S}$, $I_{P_{4}5}$) are measured
recording 400FPS videos with a CMOS microscope camera. Analysis are performed
on a $30\times 30\mu m^{2}$ region, aligned along the polarizers (thus dark in
stationary conditions) and centred on the interface nucleation point for
measurements for the central branches and at halfway for the lateral branches.
Transmittance trough crossed polarizers $\tau_{P}(t)$ is calculated as the
ratio between $I_{P}(t)$ and the intensity measured in stationary condition at
$45^{\circ}$ with the polarizer. Transmittance without analyzer $\tau_{S}(t)$
is calculated as the ratio between $I_{S}(t)$ and the intensity measured in
stationary condition at the instant $t_{0}$ ($I_{S_{0}}$). All the
measurements are obtained as an average of at least three negative to positive
switches and three positive to negative switches.

$v_{i}$ measurements

To perform $v_{i}$ measurements 500FPS videos are recorded with a CMOS
microscope camera and analyzed with a tracking-edge MATLAB script whose
functioning is better described in S.I.. Measurements are referred to a single
interface and recorded in the first instants after its formation. All the
measurements are obtained as an average of at least two negative to positive
switches and two positive to negative switches. Error bars are calculated
considering both errors on edge position extimation and averaging between
switches.

Simulations

The dynamical evolution and equilibrium structure observed in our simulations
depend on initial conditions and other details of the simulations, and we have
carried out a variety of numerical experiments to assess these dependencies.
For the results presented here, we adopt the following simulation protocol,
designed to minimize the dependence on initial conditions: (1) initialize the
system with a completely random polar director field (the polar angle $\theta$
at each grid point is assigned a random value in the range $-\pi,\pi$); (2)
evolve the system forward in time for $510\ \mu$s while maintaining a constant
potential difference of $-1000$ V between the two ends of the channel; (3)
change the sign of the voltage between the ends of the channel to $+1000$ V
and evolve the system forward in time for $500\ \mu$s. Here, we present
results from stage (3) of these simulations, under the assumption that this
models the experimental switching dynamics at a qualitative level, and that
the final configuration is a reasonable model of the equilibrium state of the
system for a fixed applied voltage. In our model, the free charge on the
electrodes is represented by a uniform charge density located at and spanning
the channel ends, with equal and opposite magnitude at the two ends of the
channel. A constant electrostatic potential difference $\Delta V$ between the
two ends of the channel is maintained by appropriately scaling the free charge
density at each timestep. Full details of the theoretical model and
computational methods can be found in the SI.

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