Light-induced swirling and locomotion

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.

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.

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].

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.

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.

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 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).

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 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.

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.