# Light-induced swirling and locomotion

Ameneh Maghsoodi  Kaushik Bhattacharya  Corresponding author:
<EMAIL_ADDRESS>
### Abstract

Photomechanical liquid crystal elastomers (LCEs) are responsive polymers that
can convert light directly into mechanical deformation. This unique feature
makes these materials an attractive candidate for soft actuators capable of
remote and multi-mode actuation. In this work, we propose a three-dimensional
multi-scale model of the nonlinear and nonlocal dynamics of fibers of
photomechanical LCEs under illumination. We use the model to show that a pre-
stressed helix-like fibers immersed in a fluid can undergo a periodic whirling
motion under steady illumination. We analyze the photo-driven spatiotemporal
pattern and stability of the whirling deformation, and provide a parametric
study. Unlike previous work on photo-driven periodic motion, this whirling
motion does not exploit instabilities in the form of snap-through phenomena,
or gravity as in rolling. We then show that such motion can be exploited in
developing remote controlled bio-inspired microswimmers and novel micromixers.

### Keywords

Photomechanical materials; Liquid crystal elastomers; Whirling actuation;
Propulsion.

### Significance

The design of untethered devices with no or limited onboard power that can be
activated remotely remains a continuing challenge in soft robotics. This work
addresses this challenge by developing a method of generating a whirling
motion in a pre-stressed photomechanical liquid crystal elastomer fiber using
steady illumination, and then showing how this motion may be exploited for
applications such as microswimmers and micromixers. More broadly, this
provides an unusual example of a physical system capable of periodic motion
under steady stimulus that does not exploit instabilities.

##  1 Introduction

Liquid crystal elastomers (LCEs) are rubbery networks composed crosslinked
polymer chains that contain liquid-crystalline mesogens in their main or in
side chains. External stimuli like heat changes the ordering of the liquid-
crystalline (LC) mesogens leading to a change in shape or stimuli-induced
deformation. This phenomenon is reversible. LCEs containing light-sensitive
molecules, such as azobenzene (azo) photochromes, exhibit a reversible
photomechanical behavior: they absorb light energy and convert it into
mechanical energy by changing their shape. This fascinating photomechanical
effect arises from the trans-cis isomerization of azobenzene dyes, a process
in which the azo molecules absorb light energy and change their conformation
from a linear trans isomer to a bent cis isomer. The steric interaction
between the azo and LC molecules consequently changes the LC ordering leading
to a photo-induced shape-change. Upon removing the light, the cis isomers
thermally relax to the trans state leading to a reversal of the photo-induced
shape-change.

Recent works demonstrate the potential of photomechanical materials for soft
robotics because of their remarkable features: 1) properly synthesized
photomechanical materials can undergo large, reversible deformation upon light
irradiation, 2) these materials are lightweight and soft, and hence,
appropriate to generate flexible motion in soft robotics, 3) light is a clean
power source that can actuate a photomechanical object from distance, thereby
eliminating the need for on-board power or tethers, and 4) light-induced
deformations can be controlled by changing light intensity, wavelength, or
polarization [1, 2, 3] thereby providing a platform for multiplexing and
control.

In particular, recent attention has focussed on locomotion in soft-robotics. A
long-standing problem in the use of active materials is the need to reset the
material after each actuation stroke. The ability to rapidly turn on and turn
off light enables such a reset. This has been exploited to build legged
microwalkers [4], leg-free caterpillar-like crawling [5] and swimming
flagellum [6]. Even more promising, the coupling between photo-isomerization,
light propagation and nonlinear mechanics can be exploited to generate cyclic
motion even under steady illumination. Yamada et al. [7] demonstrated a ring
can be made to roll by simultaneously illuminating it with a combination of UV
and natural light at two different but carefully chosen locations. Wie et al.
[8] demonstrated that a flat monolithic polymer films made of azo-containing
liquid crystalline polymer networks changes its conformation to a spiral
ribbon under illumination. The ribbon also translates a large distance with
continuous irradiation. Gelebart et al. [9] created a wave like motion a
doubly clamped azo-LCE strip using steady illumination and exploited this for
a framed walker under steady illumination. In these examples, the motion is
generated by gravity and the reaction from the surface. In this work, we
explore cyclic motion under steady illumination in an immersive environment,
and propose potential applications as bio-inspired micro-swimmers and micro-
mixers as shown in Figure 1.

Figure 1: Potential applications of photo-driven periodic motion of fibers:
microswimmers and micromixers.

The generation of cyclic deformation has also been a subject of interest in
chemo-mechanical systems since the demonstration of oscillatory swelling and
deswelling in hydrogels [10, 11]. In these systems, cyclic chemistry like the
Belousov-Zhabotinsky reactions or Landolt pH oscillators are combined with
hydrogels to obtain a chemo-mechanical oscillations. These have recently been
used to demonstrate spatio-temporal oscillatory motion that can be exploited
for peristaltic motion and walking [12], as well as homeostatic feedback loops
[13]. A theoretical model of these systems show that the system is statically
bistable and the dissipative kinetics leads to a these chemo-mechanical
oscillations. In this way, these systems are conceptually similar to the wave
like motion [9]. In this work, we explore the emergence of periodic behavior
in a stable system.

We begin by developing a three-dimensional theory of photomechanical rods. The
general theory of photomechanical coupling was developed by Corbett and Warner
[14], and used by Warner and Mahadevan [15] to study the development of light-
induced spontaneous curvature in beams and plates under the assumption of
shallow penetration. Corbett et al. [16] studied deep penetration, and the
deformation of beams under stress-free conditions (i.e., without loads).
Korner et al. [17] developed the theory of photomechanical beams under loads
and used it to explain the rolling of rings [7] and wave-like motion of
doubly-clamped beams [9]. All of this work was limited to the two-dimensional
setting of beams. In this paper, we extend these ideas to the three
dimensional setting of rods by combining the photomechanical actuation with
nonlinear Kirchhoff rod theory. We do so in a dynamical setting with inertia.

We then use the theory to exploit photomechanical coupling and pre-stressed
structural configuration to generate periodic whirling motion under steady
illumination; a fascinating photomechanical response that, to our knowledge,
has not been reported in literature. Importantly, we show that such whirling
motion of a fiber immersed in a viscous fluid generates propulsion along its
rotational axis. This can be exploited in developing remote controlled bio-
inspired microswimmers and micromixers (Figure 1). We also show that this
phenomenon does not rely on instabilities of flapping, or on gravity. Finally,
we show that we can control the photoresponse using illumination conditions
(light intensity and direction), pre-stressed configuration of fiber (twist
and bending) and cross-sectional shape (circular and rectangular). These
provide opportunities to design microactuators for different conditions, and
for control. Together, the model and the results open a new avenue for the
development of remote actuated and controlled soft actuators.

##  2 Model

Figure 2: Dynamic model of a photomechanical fiber under steady illumination.
(A) Nonlinear dynamics of the fiber is described using Kirchhof rod theory.
Frame XYZ represents the fixed inertial frame and $\\{\hat{a}_{i}\\}$
represents the body-fixed frame at each cross-section of the fiber. The steady
illumination is defined by the intensity $\Lambda$ and angles $\psi_{1}$ and
$\psi_{2}$ with respect to X axis and plane XY, respectively. (B) Schematic of
the cross-section of fiber under illumination along axis $a_{1}$.

Consider a long, thin azo-LCE fiber subjected to a steady illumination
$\vec{\Lambda}$ that makes angles $\psi_{1}$ and $\psi_{2}$ with fixed frame
XYZ as illustrated in Figure 2A. We describe the centerline of fiber as a
time-dependent space curve $\vec{R}(s,t)$ where $s$ is the arc-length along
the fiber and $t$ is time. Let $\\{\hat{a}_{i}(s,t)\\}$ be a body-fixed
orthonomal frame at point $s$ and time $t$ with unit vectors $\hat{a}_{1}$ and
$\hat{a}_{2}$ coinciding with the principal axes of the cross-section and
$\hat{a}_{3}$ normal to the cross-section. We assume that the fiber is
inextensible and unshearable so that $\hat{a}_{3}$ is always tangent to the
space curve. We define the curvature vector
$\vec{\kappa}=\\{\hat{a}_{3}^{\prime}\cdot a_{1},\hat{a}_{3}^{\prime}\cdot
a_{2},\hat{a}_{1}^{\prime}\cdot\hat{a}_{2}\\}$ in the $\\{\hat{a}_{i}\\}$
frame. Note that the rod may have intrinsic (stress-free, illumination free)
curvature.

We now specify the photomechanical constitutive behavior of the rod building
on the ideas of [14, 17]. Consider a short section of the filament at position
$s$ and time $t$. Suppose for the moment that it is subjected to an
illumination along axis $a_{1}$ with intensity $\Lambda_{1}$ as shown in
Figure 2B. As the light diffracts into the cross-section, it is absorbed by
the photochrome molecules and attenuates. In this work, we assume shallow
penetration governed by Beer’s law. Recall that, according to Beer’s law, when
a light impinges on a flat surface with intensity $\Lambda_{0}$, the intensity
$\Lambda(d)$ at a depth $d$ is given by $\Lambda_{0}\exp(-d/d_{0})$ where
$d_{0}$ is the penetration depth. Generalizing this to our general cross-
section, the intensity at any point $\vec{y}$ on the cross-section is given by
$\Lambda(\vec{y})=\Lambda_{1}f_{1}(\vec{y})$ for some function
$f_{1}:A\to[0,1]$ which quickly decays to zero away from the surface as
illustrated in Figure 2B. Examples of this function are provided in the
supplementary material.

The photochromes in the trans state absorb light and transform into the cis
state which can thermally relax back to the trans state. Therefore, the number
density $n_{c}(\vec{y})$ of cis molecules at position $\vec{y}$ evolves
according to

$\tau\dot{n}_{c}(\vec{y},t)=-n_{c}(\vec{y},t)+\alpha\Lambda(\vec{y})=-n_{c}+\alpha\Lambda_{1}f_{1}(\vec{y}),$
(1)

where the dot represents the material time derivative, $\tau$ is thermal
relaxation time (or $cis$ lifetime), and $\alpha$ is a constant depending on
material properties and forward-backward reaction rates. This in turn
introduces a spontaneous axial strain $\varepsilon_{c}=-\lambda n_{c}$ where
$\lambda$ is a constant of proportionality with $\lambda>0$ corresponds to a
contraction. It follows that the axial normal stress
$\sigma(\vec{y})=E[(\kappa_{2}-\kappa_{2}^{in})\vec{y}\cdot a_{1}+\lambda
n_{c}(\vec{y})]$ where $\kappa_{2}=\vec{\kappa}\cdot a_{2}$, $\kappa_{2}^{in}$
is the intrinsic (stress-free, illumination-free) curvature in the
2-direction, and $E$ is the Young’s modulus. The bending moment along the
$\hat{a}_{2}$ is

$q_{2}=\int_{A}(\vec{y}\cdot
a_{1})\sigma(\vec{y})\mathrm{d}A=B_{2}(\kappa_{2}-\kappa_{2}^{0}),\quad\kappa_{2}^{0}=\kappa_{2}^{in}-\int_{A}\frac{\lambda
n_{c}(\vec{y})}{I_{2}}(\vec{y}\cdot a_{1})\mathrm{d}A$ (2)

where $B_{2}=EI_{2}$ is the bending rigidity along $\hat{a}_{2}$ and $I_{2}$
is the second moment of inertia along $\hat{a}_{2}$. Now, differentiating
$\kappa_{2}^{0}$ with respect to time and using (1), we conclude that

$\tau\dot{\kappa}_{2}^{0}+(\kappa^{0}_{2}-\kappa^{in}_{2})=\gamma_{1}\Lambda_{1},\quad\gamma_{1}=\frac{-\alpha\lambda}{I_{2}}\int_{A}f_{1}(\vec{y})(\vec{y}\cdot
a_{1})\mathrm{d}A.$ (3)

In the case of general illumination, we split it into components along
$\hat{a}_{1}$ and $\hat{a}_{2}$ to conclude that the photomechanical
constitutive behavior of the beam is given by

$\displaystyle\vec{q}={\mathbf{B}}(\vec{\kappa}-\vec{\kappa}^{0}),$ (4)
$\displaystyle\tau\dot{\vec{\kappa}}^{0}-(\vec{\kappa}^{0}-\vec{\kappa}^{in})={\mathbf{\Gamma}}\vec{\Lambda}$
(5)

where $\vec{q}$ is the vector of bending moments and torque, ${\mathbf{B}}$ is
the tensorial rigidity (diag$[EI_{1},EI_{2},GJ]$ in the $\\{\hat{a}_{i}\\}$
basis with bending and torsional rigidities and
${\mathbf{\Gamma}}=\gamma_{1}\hat{a}_{1}\otimes\hat{a}_{1}+\gamma_{2}\hat{a}_{2}\otimes\hat{a}_{2}$
is the photomechanical rate tensor (diag$[\gamma_{1},\gamma_{2},0]$ in the
$\\{\hat{a}_{i}\\}$ basis).

Finally, we complete the system of equations by considering the balance of
linear and angular momenta [18, 19],

$\displaystyle\left(\frac{\partial\vec{f}}{\partial
s}+\vec{\kappa}\times\vec{f}\right)+\vec{f}_{ext}=m\left(\frac{\partial\vec{v}}{\partial
t}+\vec{\omega}\times\vec{v}\right),$ (6)
$\displaystyle\left(\frac{\partial\vec{q}}{\partial
s}+\vec{\kappa}\times\vec{q}\right)+\hat{a}_{3}\times\vec{f}+\vec{q}_{ext}=\left(\mathbf{I}_{m}\frac{\partial\vec{\omega}}{\partial
t}+\vec{\omega}\times\mathbf{I}_{m}\vec{\omega}\right)$ (7)

where $\vec{v}$ is the particle velocity, $\partial\over\partial t$ is the
spatial time derivative, $\vec{\omega}$ the angular velocity of the frame
$\\{\hat{a}_{i}\\}$, and $\vec{f}_{ext}$ and $\vec{q}_{ext}$ are the external
or applied force and moment per unit length respectively; the inextensibility

$\displaystyle\frac{\partial\vec{v}}{\partial
s}+\vec{\kappa}\times\vec{v}=\vec{\omega}\times\hat{a}_{3};$ (8)

the compatibility between curvature and angular velocity

$\displaystyle\frac{\partial\vec{\omega}}{\partial
s}+\vec{\kappa}\times\vec{\omega}=\frac{\partial\vec{\kappa}}{\partial t};$
(9)

and appropriate initial and boundary conditions.

If the fiber is immersed in a fluid, it experiences a hydrodynamic drag force
per unit length that acts as the external force in (6):

$\vec{f}_{ext}=\vec{f}_{drag}=-\mathrm{diag}(C_{1},C_{2},C_{3})\vec{v};$ (10)

where the normal, bi-normal, and tangential drag coefficients are estimated to
be $C_{1}=C_{2}=4\pi\mu/(\mathrm{ln}(L/D)+0.84)$ and
$C_{3}=2\pi\mu/(\mathrm{ln}(L/D)-0.2)$ for a circular cross-section,
respectively, in the limit of low Reynolds number [20]. Above, $\mu$ is the
dynamic viscosity of fluid, and $L$ and $D$ are the length and diameter of
fiber, respectively. The drag force is incorporated into the dynamic model
through the term $\vec{f}_{ext}$ in (6).

To solve the nonlinear initial-boundary-value problem above, we employ a
finite difference method in both space and time to discretize the equations.
Starting with the initial condition of fiber at $t=0$, the discretized
equations are integrated over space at each successive time step. During
spatial integration, the boundary conditions are satisfied using the shooting
method for boundary-value problems. For details on the numerical method, see
[21, 22, 23].

##  3 Photo-driven motion of a buckled and twisted fiber

We now use the model to study the deformation of initially straight
($\kappa^{in}=0$) fibers with circular ($\gamma_{1}=\gamma_{2},I_{1}=I_{2}$ )
cross-section that are buckled and possibly twisted, clamped at the two ends
and illuminated. We use the parameters listed in Table 1 unless otherwise
specified. A detailed parameter study is provided in supplementary
information.

Table 1: Simulation parameters Parameter (symbol) | Value
---|---
Young’s modulus ($E$) | 4 GPa [24]
Fiber diameter ($D$) | 15 $mu$ m [24]
Fiber length ($L$) | 15 mm [24]
Penetration depth ($d$) | 0.56 $\mu$ m [17]
Relaxation time ($\tau$) | 0.1 s
cis-strain proportionality ($\lambda$) | 0.05
Figure 3: Light-induced periodic motion of a twisted-buckled fiber under
steady illumination. (A) Schematic of fiber under illumination. (B) The
buckled-twisted fiber undergoes a periodic whirling; dark blue refers to start
of the cycle and light blue indicates the end of the cycle. (C) Radial
distance of fiber relative to the X axis during one cycle. (D-F) Three
orthogonal views of the photoactivated fiber. (G-I) Evolution of the light-
induced spontaneous curvature $\kappa_{0}$ of the fiber in XYZ frame. Red dots
refer to maximum radius of fiber and black dots refer to mid-point. (J-L)
Evolution of velocity of the fiber in cylindrical coordinate $r\theta X$ shown
in (A). In this simulation, $L_{b}=0.9L$, $\phi=60^{\circ}$,
$\Lambda=300\mathrm{W/m^{2}}$, $\psi_{1}=30^{\circ}$, and
$\psi_{2}=20^{\circ}$.

We begin by defining a pre-stressed doubly clamped conformation as the initial
state of the fiber. Assume that the fiber is initially straight
($\vec{\kappa}_{in}=0$) with length $L$ and the nematic director is along the
axis of the fiber (so $\lambda>0$ in Eq. (2)). We compress and twist the fiber
quasi-statically by applying a compressive force along and a torque about the
axis of fiber till the fiber buckles (the final end to end distance is
$L_{b}<L$ and angle of twist is $\phi$). We then clamp the two ends by
constraining their position and rotation till the fiber is in equilibrium.
This is initial state of fiber which we then illuminate as shown in Figure 3A.

### Typical result

A typical result
($L_{b}=0.9L,\phi=60^{\circ},\Lambda=300\text{W/m}^{2},\psi_{1}=30^{\circ},\psi_{2}=20^{\circ}$)
is shown in Figure 3B-L. After an initial transient, the fiber undergoes a
steady cyclic motion, snapshots of which over one cycle are displayed in
Figure 3B (in perspective view) and D-F (three normal projections). Note from
Figure 3D that the rotation about the X axis (line joining the end points) is
uniform and from Figure 3E,F that the shape is slightly asymmetric about the
X-axis and about the center. Snapshots of the radial distance (distance to the
line joining the end points) during the cycle is shown in Figure 3C; note that
this is almost invariant. In short, the buckled and twisted fiber rotates
about the axis quite uniformly and with an almost a rigid shape. However, the
rotation is not rigid since the ends are clamped. Figure 3G-I plot the spatio-
temporal evolution of the spontaneous curvature of the fiber (components of
$\vec{\kappa}_{0}$ with respect to the laboratory frame) over first five
cycles, while Figure 3J-L plot the velocity. These confirm that the motion is
indeed periodic after an initial transient. We also note that after an initial
transient, the radial and axial components of velocity remain very small
confirming the almost fixed shape we observed earlier. The angular velocity
shows small periodic fluctuations showing that the rotational motion is
almost, but perfectly, uniform.

As the fiber is illuminated, the photomechanical coupling causes a change in
spontaneous curvature which leads to a change of shape. The overall length of
the fiber endows it with a very small twisting stiffness, and this enables the
fiber to accommodate this change of shape by a rotary motion. This changes the
illumination conditions (the angle between the direction of light propagation
and the tangent to the fiber) leading to a further change of shape and this
cascades into a cyclic motion that we observe. We emphasize that the fiber is
clamped at both ends, so it is not free to rotate about the X-axis. This is
consistent with the periodic fluctuations in the angular velocity. Still, the
small twisting stiffness makes this a low energy manifold of deformation,
thereby enabling the motion. Importantly, there is no instability which we
discuss further in the case with zero twist.

### Zero twist

We now consider the case of zero twist ($\phi=0$). The pre-stressed shape is
planar and we assume that it belongs to the XY plane as shown schematically in
Figure 4A. There are two cases. The first is the planar case where the
illumination is also in the XY plane ($\psi_{2}=0^{\circ}$). This is exactly
the situation studied by Gelebart et al. [9] experimentally (though they used
a strip) and through modeling by Korner et al. [17] (though they used a
quasistatic model with no inertia). When illuminated, we observe that the
fiber goes into a flapping motion as shown in Figure 4B coinciding with the
results of Korner et al. [17]. The second case is the non-planar case where
the illumination is not in the XY plane ($\psi_{2}\neq 0^{\circ}$). In this
case, we see in Figure 4C that the fiber goes into whirling motion similar to
that of the case with twist.

Figure 4: Light-induced periodic motion of a buckled fiber under steady
illumination. (A) Pre-stressed buckled fiber is fully constrained in rotation
and translation at both ends. (B) The fiber subjected to illumination with
$\psi_{1}=0^{\circ}$ and $\psi_{2}=90^{\circ}$ undergoes periodic flapping
motion between two stable states. (C) The fiber subjected to illumination with
$\psi_{1}=15^{\circ}$ and $\psi_{2}=90^{\circ}$ undergoes periodic whirling
motion about its end-to-end (X) axis. (D) The lowest eigenvalue of stiffness
matrix of fiber (B) during periodic flapping motion. (E) The lowest eigenvalue
of stiffness matrix of fiber (C) during periodic whirling motion. (F)
Illumination direction affect the frequency and pattern of motion of initially
buckled fiber. In these simulations, $L_{b}=0.95$ and $\phi=0^{\circ}$.

There is a crucial difference between the flapping motion in the planar case
and the whirling motion in the non-planar case. Korner et al. [17] showed that
the flapping motion in the planar case is enabled by snap-through buckling
instability between two distinct buckled states. This is also shown in Figure
4B where the arrows trace the position of the peak during one cycle. There are
two segments where the evolution is slow as the peak/trough moves to the right
(dashed horizontal arrows), and two segments where the evolution is fast as
the fiber snaps between the up and down buckled states (diagonal arrows). The
snap-through is confirmed by analyzing the stability by examining the smallest
eigenvalue of the stiffness matrix in Figure 4D. We note that this becomes
negative indicating instability. In contrast, the motion in whirling is always
smooth and there is no instability. This is confirmed by analyzing the
stability by examining the smallest eigenvalue of the stiffness matrix in
Figure 4E. This is also in the case of an initially buckled and twisted fiber
($\phi\neq 0^{\circ}$) (see supplemental information). Figure 4F shows he
frequency of flapping is smallter than that of whirling.

##  4 Photo-driven propulsion

Figure 5: Photoactivation of fiber subjected to hydrodynamic drag produces a
steady propulsion along X axis. For a planar fiber ($\phi=0^{\circ}$), (A) net
force along Z axis remains zero, and (B) the trajectory of propulsion is
sinusoidal. For a spatial fiber ($\phi\neq 0^{\circ}$), (C) the average of net
forces along Y and Z axes remain zero, and (D) the trajectory of propulsion is
helical. In these simulations $\Lambda=500\mathrm{W/m^{2}}$,
$\psi_{1}=30^{\circ}$, and $\psi_{2}=0^{\circ}$.

We now examine whether the photo-driven motion discussed in the previous
section can be used to generate propulsion. We consider that the fiber is
immersed in a fluid and therefore subject to a hydrodynamic drag force. The
total drag force $\vec{F}_{pro}$ on the filament in laboratory frame is given
by integrating the drag force per unit length over the length of the filament

$\vec{F}_{pro}=\int_{0}^{L}\vec{f}_{drag}\mathrm{d}s.$ (11)

Further, the average propulsive force over the cycle is

$\vec{\bar{F}}_{pro}=\frac{1}{T}\int_{0}^{T}\vec{F}_{pro}\mathrm{d}t;$ (12)

where $T$ is the period.

We assume that the fiber is attached to a bead, and therefore this force
propels the bead through the fluid with the velocity
$\vec{V}=-C^{-1}\vec{F}_{pro}$ where $C$ is the drag coefficient for the bead
and $\vec{V}$ is the velocity of bead in laboratory frame. We solve the
equations of the fiber to obtain the total force, and then use this force to
study the motion of the bead. We now describe results where the immersing
fluid is water with viscosity of $\mu=0.001$Pa.s, and for simplicity, we
assume $C$ is identity.

Figure 5 shows the results for a planar fiber ($\phi=\psi_{2}=0^{\circ}$) with
a flapping motion and a spatial fiber ($\phi\neq 0^{\circ}$) with a whirling
motion. We first consider the planar case. Figure 5A shows that the propulsive
force is periodic after an initial transient; this is expected since the
motion and the velocity $\vec{v}$ of fiber are periodic as noted earlier.
There is a large propulsive force in the $Y-direction$ that is normal to the
axis of the fiber; however the average of this force is zero. The propulsive
force in the Z direction normal to the plane is zero as expected. Finally, we
have spikes in the propulsive force in the X direction during the snap-through
phase of the cycle; importantly, this is always in the same direction, and
thus the net propulsive force (Eq. (12)) in the X direction is non-zero. All
of this leads to the motion of the bead shown in Figure 5B. Note that it
follows a sinusoidal path of cyclic excursions in the Y direction that produce
no net translation, but a quasi-steady motion in the X direction. This
sinusoidal trail is reminiscent of the crawling of the nematode and C.
elegance.

We then turn to the spatial case. Figure 5C shows that the propulsive force is
periodic after an initial transient as expected. Once again we see large
forces in the Y and Z directions, but they average to zero. However, there is
a small but steady force in the X direction. This leads to the helical or
spiral motion as shown in Figure 5D that is reminiscent of swimming of sperm
and flagellated bacteria.

Figure 6: The effect of illumination and pre-stressed conformation of fiber on
propulsion. (A) Propulsive force as a function of light intensity, where
$\psi_{1}=30^{\circ}$ and $\psi_{2}=0^{\circ}$. (B) Propulsive force as a
function of illumination direction for various illumination angles $\psi_{1}$
and $\psi_{2}$, where $L_{b}=0.95L$ and $\phi=120^{\circ}$,
$\Lambda=200\mathrm{W/m^{2}}$. (C) Propulsive force as a function of the
initial twist
$\phi=\\{0^{\circ},15^{\circ},30^{\circ},45^{\circ},60^{\circ},120^{\circ},180^{\circ}\\}$
and buckling $L_{b}=\\{0.9,0.925,0.95,0.975\\}L$. In these simulations
$\Lambda=500\mathrm{W/m^{2}}$, $\psi_{1}=30^{\circ}$, and
$\psi_{2}=0^{\circ}$. (D) Propulsive force vs. frequency at different
illumination intensities $\Lambda=[50-500]\mathrm{W/m^{2}}$.

Figure 6 presents a parameter study. Figure 6A shows that the average
propulsive force over the cycle increases with illumination. This is
consistent with our finding that the frequency increases with illumination
(cf. supplementary information). Figure 6B shows that the role of illumination
direction, and this reflects the dependance of the frequency since the twist
and the buckled length remain the same. Similarly, Figure 6C shows that
propulsive force increases with increasing twist, and that this increase is
greater than that can be expected from the increase of frequency alone (cf.
supplementary information). Also, the propulsive force increases with
decreasing $L_{b}$, and that this dependance is greater than that of the
frequency.

Figure 6D shows that the propulsive force and the frequency are proportionally
correlated. This is consistent with the resistive force theory and
experimental data [25, 26] for a rotating rigid cylindrical helix at low
Reynolds number that the propulsive thrust increases linearly with respect to
rotation frequency. We note that the slight deviation from a perfect linear
function can correspond to the slight deformation of fiber during rotation
while its overall shape remains almost constant (cf. Figure 3).

### Acknoweledgment

We gratefully acknowledge the support of the US Office of Naval Research
through Multi-investigator University Research Initiative Grant ONR
N00014-18-1-2624.

## References

  * [1] K. Kumar, C. Knie, D. Bléger, M.A. Peletier, H. Friedrich, S. Hecht, D.J. Broer, M.G. Debije, and A.P.H.J. Schenning.  A chaotic self-oscillating sunlight-driven polymer actuator.  Nature Communications, 7:11975, 2016.
  * [2] H. Zeng, P. Wasylczyk, D.S. Wiersma, and A. Priimagi.  Light robots: bridging the gap between microrobotics and photomechanics in soft materials.  Advanced Materials, 30:1703554, 2018.
  * [3] N. Tabiryan, S. Serak, X-M. Dai, and T. Bunning.  Polymer film with optically controlled form and actuation.  Optics Express, 13:7442–7448, 2005.
  * [4] H. Zeng, P. Wasylczyk, C. Parmeggiani, D. Martella, M. Burresi, and D.S. Wiersma.  Light-fueled microscopic walkers.  Advanced Materials, 27:3883–3887, 2015.
  * [5] H. Zeng, O.M. Wani, and A. Wasylczyk, P.and Priimagi.  Light-driven, caterpillar-inspired miniature inching robot.  Macromolecular Rapid Communications, 391:1700224, 2018.
  * [6] C. Huang, J. Lv, X. Tian, Y/ Wang, Y. Yu, and J. Liu.  Miniaturized swimming soft robot with complex movement actuated and controlled by remote light signals.  Scientific Reports, 5:17414, 2015.
  * [7] M. Yamada, M. Kondo, J. Mamiya, Y. Yu, M. Kinoshita, C.J. Barrett, and T. Ikeda.  Photomobile polymer materials: towards light-driven plastic motors.  Angewandte Chemie, 120:5064–5066, 2008.
  * [8] J.J. Wie, M.R. Shankar, and T.J. White.  Photomotility of polymers.  Nature Communications, 7:13260, 2016.
  * [9] A.H. Gelebart, D. Jan Mulder, M. Varga, A. Konya, G. Vantomme, E.W. Meijer, R.L.B. Selinger, and D.J. Broer.  Making waves in a photoactive polymer film.  Nature, 546:632–636, 2017.
  * [10] R. Yoshida, T. Takahashi, T. Yamaguchi, and H. Ichijo.  Self-oscillating gel.  Journal of the American Chemical Society, 118:5134–5135, 1996.
  * [11] C.J. Crook, A. Smith, R.A. Jones, and A.J. Ryan.  Chemically induced oscillations in a ph-responsive hydrogel.  Physical Chemistry and Chemical Physics, 4:1367–1369, 2002.
  * [12] R. Kim, Y.S. an Tamate, A.M. Akimoto, and R. Yoshida.  Recent developments in self-oscillating polymeric systems as smart materials: from polymers to bulk hydrogels.  Materials Horizons, pages 38–54, 2017.
  * [13] X. He, M. Aizenberg, O. Kuksenok, L.D. Zarzar, A. Shastri, A.C. Balazs, and J. Aizenberg.  Synthetic homeostatic materials with chemo-mechano-chemical self-regulation.  Nature, 487:214–218, 2012.
  * [14] D. Corbett and M. Warner.  Nonlinear Photoresponse of Disordered Elastomers.  Physical Review Letters, 96:237802, 2006.
  * [15] M. Warner and L. Mahadevan.  Photoinduced deformations of beams, plates, and films.  Physical Review Letters, 92:134302, 2004.
  * [16] D. Corbett, C. Xuan, and M. Warner.  Deep optical penetration dynamics in photobending.  Physical Review E, 92:013206, 2015.
  * [17] K. Korner, A.S. Kuenstler, R.C. Hayward, B. Audoly, and K. Bhattacharya.  A nonlinear beam model of photomotile structures.  Proceedings of the National Academy of Sciences, 117:9762–9770, 2020.
  * [18] E.H. Dill.  Kirchhoff’s theory of rods.  Archive for History of Exact Sciences, 44:1–23, 1992.
  * [19] S. Goyal, N.C. Perkins, and C.L. Lee.  Nonlinear dynamics and loop formation in kirchhoff rods with implications to the mechanics of dna and cables.  Journal of Computational Physics, 209:371–389, 2005.
  * [20] J. Howard.  Mechanics of motor proteins and the cytoskeleton: Sinauer assoc.  Sunderland, MA, 2001.
  * [21] C. Gatti-Bono and N.C. Perkins.  Physical and numerical modelling of the dynamic behavior of a fly line.  Journal of sound and vibration, 255:555–577, 2002.
  * [22] A. Maghsoodi, A. Chatterjee, I. Andricioaei, and N.C. Perkins.  A first model of the dynamics of the bacteriophage t4 injection machinery.  Journal of Computational and Nonlinear Dynamics, 11::041026, 2016\.
  * [23] J.I. Gobat, M.A. Grosenbaugh, and M.S. Triantofyllou.  Generalized-alpha time integration solutions for hanging chain dynamics.  Technical report, Woods Hole Oceanographic Instituteion, 2002.
  * [24] M.L. Smith, M.R. Shankar, R. Backman, V.P. Tondiglia, K.M. Lee, M.E. McConney, D.H. Wang, L-S. Tan, and T.J. White.  Designing light responsive bistable arches for rapid, remotely triggered actuation.  In Behavior and Mechanics of Multifunctional Materials and Composites 2014, volume 9058, pages 121–130. SPIE, 2014.
  * [25] B. Rodenborn, C-H Chen, H.L Swinney, B. Liu, and H.P. Zhang.  Propulsion of microorganisms by a helical flagellum.  Proceedings of the National Academy of Sciences, 110:E338–E347, 2013.
  * [26] X. Zhong, K.W. Moored, V. Pinedo, J. Garcia-Gonzalez, and A.J. Smits.  The flow field and axial thrust generated by a rotating rigid helix at low reynolds numbers.  Experimental Thermal and Fluid Science, 46:1–7, 2013.

Supplementary Material

Light-induced swirling and locomotion

Ameneh Maghsoodi and Kaushik Bhattacharya

##  1 Photomechanical model of Azo-LCE fibers

Consider a long, thin azo-LCE fiber possessing circular cross-section. Under
illumination, Azobenzene molecules absorb photons and transform between linear
$trans$ to bent $cis$ configurations due to light absorption and thermal
decay. Suppose for the moment that the fiber is subjected to an illumination
with intensity $\Lambda_{0}$. As a simple case, assume that illumination is
along axis Y as shown in Figure 1A. As the light diffracts into the cross-
section, it is absorbed by the photochrome molecules and attenuates. In this
work, we assume shallow penetration governed by Beer’s law. Recall that,
according to Beer’s law, when a light $\Lambda_{0}$ impinges on the flat
surface of a strip with thickness $H$, the intensity decreases with depth such
that the intensity at any point $y$ on the cross-section is given by

$\Lambda(s,t,y)=\Lambda_{0}(s,t)\mathrm{exp}\left(-\frac{H/2-y}{d}\right)=\Lambda_{0}(s,t)f(y),y\in[{0},H/2].$
(1)

where $d$ is the penetration depth. Similarly, for a fiber with circular
cross-section shown in Figure 1A

$\Lambda(s,t,r,\theta)=\Lambda_{0}(s,t)sin\theta\mathrm{exp}\left(-\frac{R-r}{d}\right)=\Lambda_{0}(s,t)f(r,\theta),r\in[{0},R],\theta\in[{0},\pi].$
(2)

where function $f\in[{0},1]$.

Figure 1: Illumination of a circular cross-sectional fiber where illumination
(A) is along principal axis, and (B) makes arbitrary angle $\beta$ with
principal axis Y. (C) Profile of light intensity at a rectangular cross-
section. (D) Profile of light intensity at a circular cross-section. (E) The
light intensity penetrating a circular cross-section varies with both radial
depth and angle.

Figure 1C-E compares the light intensity profile for a circular and a
rectangular cross-sections using (1) and (2). As shown in Figure 1C, the light
intensity passing through a rectangular cross section decreases
(exponentially) with depth $H$. However, Figure 1D-E for a circular cross-
section show that the light intensity depends on the depth $r$ as well as the
angle between incident light and radial direction; where the maximum light
intensity corresponds to $\theta=\pi/2$ and $r=R$. Note that the light
intensity at any angle $\theta$ decreases exponentially with radial depth (cf.
(2)). This demonstrates that the fiber cross sectional shape affects the
profile of light intensity.

The photochromes in the trans state absorb light and transform into the cis
state which can thermally relax back to the trans state. Therefore, the number
density $n_{c}$ of cis molecules along cross-section evolves according to

$\tau\dot{n}_{c}=-n_{c}+\alpha\Lambda=-n_{c}+\alpha\Lambda_{0}f,$ (3)

where the dot represents the material time derivative, $\tau$ is thermal
relaxation time (or $cis$ lifetime), and $\alpha$ is a constant depending on
material properties and forward-backward reaction rates. This in turn
introduces a spontaneous axial strain $\varepsilon_{c}=-\lambda n_{c}$ where
$\lambda$ is a constant for proportionality with $\lambda>0$ corresponds to a
contraction. Consequently, the light-induced bending at the circular cross-
section of the fiber

$q(s,t)=B(\kappa(s,t)-\kappa_{0}(s,t)),$ (4)

where

$\kappa_{0}(s,t)=\kappa_{in}(s)+\int_{0}^{2\pi}\int_{0}^{R}\left(\frac{-\lambda}{I_{c}}\right)n_{c}(s,t,r,\theta)r^{2}sin\theta\mathrm{d}r\mathrm{d}\theta,$
(5)

In (4), $\kappa$ and $\kappa_{0}$ denote the curvature and spontaneous stress-
free curvature of the fiber, respectively. The constant $B=EI_{c}$ is the
bending stiffness with the second moments of area $I_{c}=\pi R^{4}/4$ for a
circular cross-section. Combining (3)-(5),

$\tau\dot{\kappa}_{0}(s,t)+(\kappa_{0}(s,t)-\kappa_{in}(s))=\gamma\Lambda_{0}(s,t),$
(6)

where

$\gamma=\left(\frac{-\alpha\lambda}{I_{c}}\right)\int_{0}^{\pi}\int_{0}^{R}(rsin\theta)^{2}\mathrm{exp}\left(-\frac{R-r}{d}\right)\mathrm{d}r\mathrm{d}\theta.$
(7)

Now, in a general case, assume that a steady illumination $\Lambda_{0}$ makes
angle $\beta$ with principal axis Y as illustrated in Figure 1B. Illumination
$\Lambda_{0}$ makes bending about axis $\delta$ such that

$\kappa_{0X}(s,t)=\kappa_{0\delta}(s,t)cos(\beta),$ (8)
$\kappa_{0Y}(s,t)=\kappa_{0\delta}(s,t)sin(\beta),$ (9)

On the other hand, illumination
$\vec{\Lambda}_{0}=\begin{pmatrix}\Lambda_{0}sin(\beta)\\\
\Lambda_{0}cos(\beta)\end{pmatrix}$. Therefore, using (6)

$\tau\frac{\partial\kappa_{0X}(s,t)}{\partial
t}+(\kappa_{0X}(s,t)-\kappa_{inX}(s))=\gamma\Lambda_{0}(s,t)cos(\beta),$ (10)
$\tau\frac{\partial\kappa_{0Y}(s,t)}{\partial
t}+(\kappa_{0Y}(s,t)-\kappa_{inY}(s))=\gamma\Lambda_{0}(s,t)sin(\beta),$ (11)

Combining (6-11) concludes that photoactuaction by the steady illumination
$\Lambda_{0}$ can be written as the sum of two photoactuactions
$\Lambda_{0}sin(\beta)$ and $\Lambda_{0}cos(\beta)$ along principal axes X and
Y, respectively.

##  2 Stability analysis

To examine stability of fiber for the dynamic equilibrium configurations, we
calculate the second variation of total potential energy, $\delta^{2}\Pi$. If
$\delta^{2}\Pi>0$, the potential energy is minimized at the equilibrium
configuration, indicating that the configuration is stable. To derive the
stability criteria, consider the potential energy function in the form

$\displaystyle\Pi=\int_{0}^{1}f\left(s,Y(s),Y^{\prime}(s)\right)ds$ (12)

where $Y=[y_{1},y_{2},...,y_{N}]^{T}$ and
$Y^{\prime}=[y^{\prime}_{1},y^{\prime}_{2},...,y^{\prime}_{N}]^{T}$. To
calculate the second variation of energy, we consider small variations around
equilibrium configuration $Y^{*}$ in the form

$\displaystyle Y(s)=Y^{*}(s)+\epsilon\eta(s)$ (13)

where $\epsilon$ is an arbitrary constant independent of $s$ and $Y$, and
$\eta(s)=[\eta_{1},\eta_{2},...,\eta_{N}]^{T}$ contains arbitrary functions
which are independent of $\epsilon$ and satisfy boundary conditions. We
differentiate $\Pi$ twice with respect to $\epsilon$ and set $\epsilon$ to
zero

$\displaystyle\delta^{2}\Pi=\int_{0}^{1}\left(\eta^{T}\frac{\partial^{2}f}{\partial
Y\partial Y}\eta+2\eta^{T}\frac{\partial^{2}f}{\partial Y\partial
Y^{\prime}}\eta^{\prime}+\eta^{\prime T}\frac{\partial^{2}f}{\partial
Y^{\prime}\partial Y^{\prime}}\eta^{\prime}\right)ds$ (14)

From Rayleigh-Ritz approach, we assume
$\eta_{i}=\Sigma^{n}_{j=1}a_{ij}\phi_{j}$, $i=1,2,..,N$. In doing so, finding
the minimum of second variation of energy subject to the constraint that
$\int_{0}^{1}\eta^{T}\eta ds=1$ becomes (by Lagrange multiplier $\lambda$) an
eigenvalue problem of finding $\lambda$. The positive eigenvalues imply that
the configuration is stable. For a twisted-buckled rod that is clamped at one
end and subjects to force $\vec{F}$ and moment $\vec{M}$ at the other end

$\displaystyle\Pi=\int_{0}^{1}\left(\frac{\mathbf{B}}{2}\left(\frac{\partial\vec{\theta}}{\partial
s}-\vec{\kappa}_{0}\right)^{2}-\vec{F}.\vec{r}^{\prime}-\vec{M}.\frac{\partial\vec{\theta}}{\partial
s}\right)ds$ (15)

Where $Y=\vec{\theta}(s)=[\theta_{1}(s),\theta_{2}(s),\theta_{3}(s)]^{T}$, and
$\vec{r}^{\prime}=\mathbf{R}(\vec{\theta}).\vec{a}_{3}$ in which $\mathbf{R}$
is the rotation matrix from local frame to inertial frame.

As shown in Figure 2A, a pre-stressed buckled-twisted fiber undergoes a
periodic whirling motion under steady illumination. The motion in whirling is
always smooth and there is no instability. This is confirmed by analyzing the
stability by examining the smallest eigenvalue of the stiffness matrix in
Figure 2B. The lowest eigenvalue is always positive.

Figure 2: (A) Whirling motion of pre-stressed buckled-twisted fiber about its
helical axis. (B) The lowest eigenvalue of stiffness matrix during periodic
whirling motion remains positive.

##  3 Parameter Study

### The effect of illumination intensity and direction

Figure 3: Influence of illumination on the frequency and pattern of
oscillations. (A) Frequency of oscillations as a function of light intensity,
where $\psi_{1}=30^{\circ}$ and $\psi_{2}=0^{\circ}$. (B) Frequency of
oscillations as a function of illumination direction for various illumination
angles $\psi_{1}$ and $\psi_{2}$, where $L_{b}=0.95L$ and $\phi=120^{\circ}$,
$\Lambda=200\mathrm{W/m^{2}}$. (C) Comparison of conformational change of
fiber during periodic oscillations at three different illumination directions.
Red dots refer to maximum radius of fiber and black dots refer to mid-point.

Figure 3 compares the frequency $f$ and pattern of oscillations over a range
of values of illumination intensity $\Lambda$ and angles $\psi_{1}$ and
$\psi_{2}$. One needs a critical intensity before one has periodic motion
(Figure 3A); below this critical intensity, the fiber deforms and attains a
photo-stationary state. Beyond this critical intensity, the frequency
increases with the light intensity: this is consistent with the fact that
greater intensity leads to a higher rate of increase in the photo-induced
strain and spontaneous curvature.

The angle $\psi_{1}$ between the direction of light propagation and the X-axis
(the line joining the end-points) has a significant effect on the motion. When
the illumination is perpendicular to the X-axis, then the fiber reaches a
photo-stationary state (Figure 3B). The frequency increases as $\psi_{1}$
decreases and reaches a peak when the illumination is parallel to X-axis.
There is symmetry about $\psi_{1}=90^{\circ}$; the illumination at angle
$-\psi_{1}$ drives fiber with the same velocity but opposite direction of
illumination at angle $\psi_{1}$.

Figure 3C shows that this angle also affects the shape of the rod, and the
rotary motion becomes less steady as the illumination is perpendicular and the
more steady as it is parallel to the X-axis. The angle $\psi_{2}$ which
describes orientation in the YZ plane has no impact, consistent with almost
rotary motion.

### The effect of pre-stressed conformation of fiber

Figure 4 examines the frequency as a function of the pre-stressed
conformation, both the buckled length $L_{b}$ and the angle of twist $\phi$
(with $\Lambda=500\mathrm{W/m^{2}}$ with $\psi_{1}=30^{\circ}$ and
$\psi_{2}=0^{\circ}$). The figure shows that the case of no twist, $\phi=0$,
is distinct and so we defer its discussion. Once twisted, Figure 4A shows that
the frequency increases almost linearly but only slightly with increasing
twist. Increasing the buckling (i.e., reducing $L_{b}$) also increases the
frequency. Figure 4B shows that the shape changes with a larger bulge with
reduced $L_{b}$.

Figure 4: The effect of pre-stressed conformation of fiber on the
oscillations. (A) Frequency of periodic oscillations as a function of the
initial twist
$\phi=\\{0^{\circ},15^{\circ},30^{\circ},45^{\circ},60^{\circ},120^{\circ},180^{\circ}\\}$
and buckling $L_{b}=\\{0.9,0.925,0.95,0.975\\}L$. The inset compares the
conformation of planar ($\phi=0^{\circ}$) fiber and spatial ($\phi\neq
0^{\circ}$) fiber. (B) Comparison of conformational change of fiber during
periodic oscillations under four different pre-stressed conformations. Red
dots refer to maximum radius of fiber and black dots refer to mid-point. In
these simulations, $\Lambda=500\mathrm{W/m^{2}}$ with $\psi_{1}=30^{\circ}$
and $\psi_{2}=0^{\circ}$.

### Influence of illumination direction on the frequency and propulsion

As shown in Figure 5, the whirling motion of a pre-stressed twisted-buckled
fiber is symmetric with respect to the YZ plane perpendicular to the helical
axis if fiber. Note that the amplitude of frequency and propulsive force are
the same but the directions are the opposite with respect to YZ plane.

Figure 5: (A) Frequency of oscillations as a function of illumination
direction. (B) Propulsion as a function of illumination direction. In this
simulations, $L_{b}=0.95L$ and $\phi=120^{\circ}$, and
$\Lambda=200\mathrm{W/m^{2}}$. The frequency and propulsive force are
symmetric with respect to YZ plane ($\psi_{1}=90^{\circ}$)

### The effect of cross-sectional geometry

We now briefly examine whether the cross-section of fiber affects the photo-
driven motion. In photomechanical model of fiber, the function
$f_{1}(\vec{y})$ depends on both geometrical and material properties of fiber.
Examples of function $f_{1}(\vec{y})$ are presented and compared for circular
and rectangular cross section in the supplementary material. Figure 6 compares
the photomechanical response of the fibers with square and rectangular cross
sections where the illumination is along X-axis from left to right. From
Figure 6A-C for a square cross-sectional fiber, the motion is periodic. The
fiber rotates uniformly about its helical axis with an almost constant shape;
this is similar to that of circular cross-sectional fiber explained in
previous sections.

Figure 6D-F shows the photomechanical response of a rectangular cross-
sectional fiber. The photo-driven motion is periodic; however, the pattern of
deformation is significantly different from circular cross-sectional fiber.
The fiber rotates about its axis, however, the rotation velocity is largely
nonuniform during the cycle as observed in Figure 6D. From Figure 6E-F, the
radial distance of fiber relative to rotation axis significantly changes. Also
the maximum radius (Red dots) translates along X-direction while rotating
about X-axis. These results concludes that the fiber undergoes a coupled
flapping-whirling motion about its axis. Future analysis and parametric study
on rectangular cross-section can reveal the contribution of bending and
torsional elasticity and aspect ratio on the coupled flapping-whirling motion.

Figure 6: The effect of fiber cross-section on photo-driven motion. Comparison
of conformational change of fiber with (A-C) square cross-section with width
and thickness of 15 $mu$ m, and (D-F) rectangular cross-section with width of
15 $mu$ m and thickness of 22 $mu$ m. Note that (C) and (F) present the radial
distance of fiber from its end-to-end axis. Also, red arrows indicate the
direction of rotation, black arrows in (E) shows the flapping trajectory, the
red dots refer to maximum radial distance, and black dots indicate the middle
of fiber at each time. In these simulations $L_{b}=0.95L$,
$\phi=\\{120^{\circ}$, $\Lambda=500\mathrm{W/m^{2}}$, $\psi_{1}=0^{\circ}$,
and $\psi_{2}=0^{\circ}$.

##  4 Animations

### Planarpropulsion.mp4

Shows the motion of a bead attached to a buckled (but not twisted) fiber
illuminated along the buckled plane.

### Spatialpropulsion.mp4

Shows the motion of a bead attached to a buckled (but not twisted) fiber
illuminated at an angle to the buckled plane.