Propulsion driven by self-oscillation via an electrohydrodynamic instability

TL;DR

This paper demonstrates a method to induce self-oscillations in bio-inspired microrobots using a uniform electric field, leveraging an elasto-electro-hydrodynamic instability to mimic natural flagella motion.

Contribution

It introduces a novel approach to achieve self-oscillation in artificial systems through electric field-induced instability, combining theory and simulations.

Findings

Self-oscillations achieved via electric field in microrobots

Identification of three behaviors: stationary, swimming, spinning

Oscillations arise from a Hopf bifurcation mechanism

Abstract

Oscillations of flagella and cilia play an important role in biology, which motivates the idea of functional mimicry as part of bio-inspired applications. Nevertheless, it still remains challenging to drive their artificial counterparts to oscillate via a steady, homogeneous stimulus. Combining theory and simulations, we demonstrate a strategy to achieve this goal by using an elasto-electro-hydrodynamic instability. In particular, we show that applying a uniform DC electric field can produce self-oscillatory motion of a microrobot composed of a dielectric particle and an elastic filament. Upon tuning the electric field and filament elasticity, the microrobot exhibits three distinct behaviors: a stationary state, undulatory swimming and steady spinning, where the swimming behavior stems from an instability emerging through a Hopf bifurcation. Our results imply the feasibility of engineering self-oscillations by leveraging the elasto-viscous response to control the type of bifurcation and the form of instability. We anticipate that our strategy will be useful in a broad range of applications imitating self-oscillatory natural phenomena and biological processes.

Equations

$$\partial \theta /\partial \bar{t}$$

$$\partial {\bar{\mathcal{P}}}_Q/\partial \bar{t}$$

$$\partial {\bar{\mathcal{P}}}_3/\partial \bar{t}$$

Full text

Flagella and cilia exhibit oscillatory movements for locomotion, pumping and fluid mixing. To mimic these functionalities, various approaches have been developed to oscillate their artificial counterparts using magnetic Singh et al. (2005); Evans et al. (2007), electrostatic den Toonder et al. (2008), piezoelectric Kieseok et al. (2009), optical van Oosten et al. (2009) and hydrogel-based actuations Sidorenko et al. (2007); Masuda et al. (2013). In general, a time-dependent stimulus generates oscillations in many biomimetic systems. An apparent exception is the use of the Belousov-Zhabotinsky (BZ) oscillating chemical reaction Masuda et al. (2013) (inspired by Ref Yoshida et al. (1996)) to deform polymer brushes periodically. Nonetheless, time-dependent forcing is not necessary for biological systems which can occasionally generate oscillations by steady stimuli to deliver functionalities such as otoacoustic emissions Gold (1948); Kemp (1979) and glycolysis Sel’kov (1968), etc. These behaviors, namely, the generation and maintenance of a periodic motion powered by a source without a corresponding periodicity is referred to as self-oscillation Jenkins (2013); sc .

Self-oscillation plays a crucial role in some inertia-dominated flow cases, such as the collapse of the Tacoma Narrows Bridge bri and the sound generation of wind musical instruments (including whistling and the human voice), owing to inertia-induced nonlinearity. In this Rapid Communication, we create self-oscillations of artificial structures in a situation with negligible inertia by applying a uniform, time-independent electric field. We exploit an elasto-electro-hydrodynamic (EEH) instability by marrying an electrohydrodynamic instability with an elasto-viscous response. Combining theory and simulations, we investigate a composite microrobot that achieves unidirectional locomotion by self-oscillatory wiggling of an elastic appendage.

It was discovered in 1896 by Quincke Quincke (1896) that a uniform DC electric field can trigger the spontaneous rotation of a dielectric particle immersed in a dielectric solvent with higher conductivity. Quincke rotation (QR) occurs as an electrohydrodynamic instability emerging from an equilibrium configuration where the induced-charge dipole $\bm{\mathcal{P}}$ of the particle is opposite to the applied field. When the field strength $E$ exceeds a threshold value, $E^{\mathrm{cri}}$, the symmetric yet antiparallel configuration is unstable to an infinitesimal disturbance, spontaneously breaking the mirror symmetry through a supercritical pitchfork bifurcation Turcu (1987); Peters et al. (2005); the particle then spins steadily where the electric and viscous torques balance.

We exploit this QR instability by grafting an inextensible elastic filament of radius $a$ and length $L$ onto a dielectric spherical particle of radius $A$ (Fig. 1a), where $s$ denotes the arclength of the filament’s centerline with position $\mathbf{r}(s,t)$, and $\alpha=A/L$ is the size ratio. The slenderness of the filament $\epsilon_{\mathrm{sl}}=a/L\ll 1$. The filament base $s=0$ is clamped at the particle surface J, where the tangent vector $\mathbf{e}_{\mathrm{p}}=\partial\mathbf{r}/\partial s|_{s=0}$ at the base always passes through the particle center P and $\mathbf{e}_{\mathrm{p}}$ denotes the orientation of the object. We define an elasto-electro-viscous (EEV) parameter $\bar{\mu}=8\pi\mu L^{4}/D\tau_{\mathrm{s}}$ indicating the ratio of the elasto-viscous time scale $8\pi\mu L^{4}/D$ to the charge relaxation time $\tau_{\mathrm{s}}=\epsilon_{\mathrm{s}}/\sigma_{\mathrm{s}}$ of the solvent, where $\mu$, $\epsilon_{\mathrm{s}}$ and $\sigma_{\mathrm{s}}$ denote respectively, the dynamic viscosity, permittivity and conductivity of the solvent, and $D$ is the bending stiffness of the filament. $\bar{\mu}$ indicates the relative strength of the viscous to the elastic forces, where $\bar{\mu}=0$ corresponds to a rigid filament and increasing $\bar{\mu}$ corresponds to a more compliant filament. To focus on the elasto-viscous response of the filament, we do not consider its polarization. We also do not take into account the hydrodynamic interactions between the particle and the filament.

We adopt the proper Euler angles $[\theta,\phi,\psi]$ to characterize the rotation of the object. The uniform electric field $\mathbf{E}=E\mathbf{e}_{z}$ and the particle dipole $\bm{\mathcal{P}}$ are expressed in the reference coordinate system $[\mathbf{N},\mathbf{Q},\mathbf{e}_{3}]$ rotating and translating with the object, where $\mathbf{e}_{3}$ coincides with the its orientation $\mathbf{e}_{\mathrm{p}}$, $\mathbf{N}$ indicates the nodal line direction and $\mathbf{Q}=\mathbf{e}_{3}\times\mathbf{N}$ (Fig. 1a). We constrain the object’s motion to the $yz$-plane, hence the dipole $\bm{\mathcal{P}}$, the orientation $\mathbf{e}_{\mathrm{p}}$, and the filament lie in the same plane, resulting in $\mathbf{N}=\mathbf{e}_{x}$. Using $\tau_{\mathrm{s}}$, $\tau_{\mathrm{s}}^{-1}$, $D/L$, $E^{\mathrm{cri}}$ and $D/\left(LE^{\mathrm{cri}}\right)$ as the characteristic time, rotation rate, torque, electrical field and polarization dipole strength, respectively, the nondimensional electrohydrodynamic equations are Cēbers et al. (2000) (see Supplemental Material)

[TABLE]

where $\;\bar{}\;$ denotes dimensionless quantities hereinafter, $\bar{\bm{\Gamma}}^{\mathrm{f}\rightarrow\mathrm{p}}$ is the elastic torque exerted by the filament onto the particle with respect to its center; $\bar{\eta}=\alpha^{3}\bar{\mu}$, $\kappa=(R+2)/(S+2)$, $R=\sigma_{\mathrm{p}}/\sigma_{\mathrm{s}}$ and $S=\epsilon_{\mathrm{p}}/\epsilon_{\mathrm{s}}$, where $\epsilon_{\mathrm{p}}$ and $\sigma_{\mathrm{p}}$ are the permittivity and conductivity of the particle, respectively.

Applying a fixed electric field, $\bar{E}=1.5$ for example, we discovered a self-oscillatory response of the composite object by tuning the filament elasticity (Fig. 1). When $\bar{\mu}=635$, the particle wobbles spontaneously rather than to rotate steadily like the classical QR counterpart, as indicated by the rotational velocity $\bar{\Omega}=\mathrm{d}\theta/\mathrm{d}\bar{t}$ (Fig. 1b). The local peak $\bar{\Omega}^{\mathrm{lpk}}$ of $\bar{\Omega}$ increases with $\bar{t}$ rapidly during the initial period and eventually saturates to a constant value $\bar{\Omega}^{\mathrm{mag}}$ corresponding to a time-periodic state.

To understand the initial dynamics, we examine the initial phase $\bar{\Omega}(\bar{t})$, as shown in Fig. 1b (highlighted in the cyan box). This local peak $\bar{\Omega}^{\mathrm{lpk}}$ (Fig. 1c) initially grows exponentially, as confirmed by the inset displaying the linear dependence of $\log{\bar{\Omega}^{\mathrm{lpk}}}$ on $\bar{t}$. Thus, the self-oscillation arises through a linear instability mechanism, similar to other self-oscillation phenomena Jenkins (2013). Furthermore, the system reaches a time-periodic state, namely, the particle oscillates with a fixed amplitude (Fig. 1d highlighting the green box of Fig. 1b). To understand how the grafted filament reacts to the particle, we show in Fig. 1e the particle-filament configurations at six times within a period. We observe that the oscillating particle drives the filament to wiggle, a scenario resembling the locomotion of a flagellated microorganism that acquires thrust by propagating oscillatory bending waves from the head towards the tail. A striking yet natural consequence of this self-oscillation is that the object undulates and translates, hence demonstrates Propulsion by harnessing thrust from the wiggling filament.

A series of simulations was performed to examine the influence of the filament elasticity. By varying $\bar{\mu}$, we identify three states of the composite object: an undulatory motion (Fig. 2a,b, where $\bar{\mu}=800$, $1015$ respectively) similar to the $\bar{\mu}=635$ case reported in Fig. 1d,e, though here we observe a larger oscillation amplitude characterized by $\bar{\Omega}^{\mathrm{mag}}$; and a steady spinning motion (Fig. 2c,d, where $\bar{\mu}=1020$, $2000$ respectively), resembling a QR particle towing a passively bent filament that breaks the mirror symmetry about the filament centerline; a stationary state when $\bar{\mu}$ is below a critical value, where the object is stationary ($\bar{\Omega}=0$) and possesses mirror symmetry. The three elasticity-dependent states are identified with a dashed line (stationary), triangles (undulatory) and diamonds (spinning) in Fig. 2e, which represent the bifurcation diagram of a one-parameter ($\bar{\mu}$) dynamical system: the stationary state is a symmetric fixed-point solution, which transits through a supercritical Hopf bifurcation Strogatz (1994) at $\bar{\mu}_{1}^{\mathrm{cri}}\approx 625$ to a limit-cycle solution corresponding to the undulating state. This periodic solution jumps, via a secondary bifurcation at $\bar{\mu}_{2}^{\mathrm{cri}}\approx 1017$, to another asymmetric fixed-point solution representing the spinning state. The Hopf bifurcation is confirmed by the quadratic variation of $\bar{\Omega}^{\mathrm{mag}}$ in $\bar{\mu}$ in the vicinity of $\bar{\mu}_{1}^{\mathrm{cri}}$ shown in Fig. 2b (linear dependence of $\left(\bar{\Omega}^{\mathrm{mag}}\right)^{2}$ on $\bar{\mu}$). It is worth-noting that the bifurcation diagram featured with these two bifurcations remains unchanged when the electric field $1<\bar{E}<\bar{\mathcal{E}}^{\mathrm{cri}}$, where $\bar{\mathcal{E}}^{\mathrm{cri}}$ corresponds to the critical field above which the particle with a rigid filament ($\bar{\mu}=0$) undergoes QR; when $\bar{E}>\bar{\mathcal{E}}^{\mathrm{cri}}$, the object spins steadily regardless of $\bar{\mu}$.

The composite object achieves self-oscillatory propulsion only in the undulating regime $\bar{\mu}\in\left(\bar{\mu}_{1}^{\mathrm{cri}},\bar{\mu}_{2}^{\mathrm{cri}}\right)$, attaining zero net locomotion when $\bar{\mu}\rightarrow\bar{\mu}_{1}^{\mathrm{cri}}+$ and $\bar{\mu}\rightarrow\bar{\mu}_{2}^{\mathrm{cri}}-$. We expect its propulsive performance to exhibit a non-monotonic dependence on $\bar{\mu}$ and peaks at an optimal EEV parameter $\bar{\mu}^{\mathrm{opt}}$. We quantify the performance by the translational velocity $\bar{\mathcal{U}}$ of the swimmer along its effective straight path connecting the most convex points on the wavelike trajectory (Fig. 3a). The trajectory shape depends on $\bar{\mu}$: for the stiffest filament $\bar{\mu}=650$, it matches a sinusoidal wave with a high frequency, almost preserving fore-aft temporal symmetry. Conversely, when $\bar{\mu}=825$, the wavy trajectory is characterized by a larger amplitude and lower frequency. For the most floppy case shown $\bar{\mu}=950$, the trajectory is significantly coiled, exhibiting a pronounced fore-aft asymmetry. Consequently, the swimmer’s backward movement is comparable to the forward movement, leading to a nearly reciprocal motion. The increasing coiled trajectory for $\bar{\mu}$ is closely linked to the more deflected filament shown in Fig. 3b. Fig. 3c confirms our anticipation of the non-monotonically varying $\bar{\mathcal{U}}$ with peak value of $\bar{\mathcal{U}}^{\mathrm{opt}}\approx 6\times 10^{-3}$ at $\bar{\mu}^{\mathrm{opt}}\approx 825$. This velocity $\bar{\mathcal{U}}^{\mathrm{opt}}$ lies in the range $\left(1,15\right)\times 10^{-3}$ of the dimensionless speed of a magnetically-driven flexible flagellum Dreyfus et al. (2005), implying the reasonable efficiency of this self-oscillatory propulsion mechanism. We notice that $\bar{\mathcal{U}}$ oscillates with $\bar{\mu}$ when $\bar{\mu}\rightarrow\bar{\mu}_{2}^{\mathrm{cri}}$. We do not attempt to unravel this peculiar variation here, keeping in mind that the main focus of the current work is on engineering self-oscillation to achieve various functionalities such as locomotion. A thorough analysis on the propulsive features of the microrobot will be conducted in future work.

By varying $\bar{E}$, we present a bifurcation diagram in Fig. 4a for the composite object with different $\bar{\mu}$ values. The diagram shares the same feature with Fig. 2e considering $\bar{\mu}$ as the control parameter: a Hopf and a secondary bifurcation occur at $\bar{E}_{1}^{\mathrm{cri}}$ and $\bar{E}_{2}^{\mathrm{cri}}$, respectively. $\bar{E}=1$ (hollow square) indicates the pitchfork bifurcation resulting in the original QR instability. This graph has highlighted the role of the filament in transforming the pitchfork bifurcation into the Hopf bifurcation that leads to self-oscillation. We further conduct a linear stability analysis (LSA) Zhu and Stone (2019) around an equilibrium base solution $[\theta,\bar{\mathcal{P}}_{Q},\bar{\mathcal{P}}_{3}]=[\hat{\theta},-\kappa\bar{\eta}\bar{E}\sin{\hat{\theta}},-\kappa\bar{\eta}\bar{E}\cos{\hat{\theta}}]$ of Equ. 1 ($\hat{\theta}$ can be an arbitrary value without loss of generality), when the composite system is stationary and the filament is undeformed. Without the filament, $\bar{\Gamma}^{\mathrm{f}\rightarrow\mathrm{p}}_{N}=0$, the LSA indeed predicts a critical electrical field of $\bar{E}=1$ corresponding to that of the original QR instability. In the presence of filament, realizing that the filament undergoes weak deformation near the onset of instability, we are able to model the elastic torque $\bar{\Gamma}^{\mathrm{f}\rightarrow\mathrm{p}}_{N}$ following Refs. Wiggins and Goldstein (1998); Wiggins et al. (1998). The critical electric field $\bar{E}_{1}^{\mathrm{cri}}$ predicted by LSA above which self-oscillatory instability occurs is shown as a function of $\bar{\mu}$ and $\alpha$ (Fig. 4b), so as the critical EEV number $\bar{\mu}_{1}^{\mathrm{cri}}$ versus $\bar{E}$ and $\alpha$ (Fig. 4c). These theoretical predictions agree well with their numerical counterparts, especially when $\bar{\mu}$ is large.

In this work, we have uncovered an EEH instability and demonstrated a strategy for engineering self-oscillations based on a steady, uniform electric field. This idea is illustrated by driving the motion of a dielectric particle connected to an elastic filament. By tuning the filament elasticity and electric field strength, the object achieves propulsion enabled by the dual functionalities of the filament: manipulating the bifurcation through its elasto-viscous response, which causes the particle to oscillate; providing thrust by wiggling motion actuated by the oscillating particle. Besides offering the possibility as a swimming microrobot, the object can also transform into a stationary, soft obstacle or spinner when the electric field strength is tuned, respectively, below or above the critical values we have identified.

The key idea we have recognized is to introduce an elastic element to trigger the Hopf bifurcation and consequently the self-oscillatory instability. Therefore, this might be one path for engineering biomimetic oscillatory processes using a time-independent power source. More generally, our results imply the potential for incorporating elastic media in other unstable systems to manipulate and diversify the bifurcations Chen et al. (2000), which possibly can be employed for different functionalities. This concept is different from, but complementary to, taming a structure’s mechanical failures to achieve functionalities Reis et al. (2018). We believe that our ideas offer opportunities to develop a new generation of soft, reconfigurable machines that can morph and adapt to the environment. Experiments to test and further explore the EEH instability introduced in this work are in progress.

We thank Drs. Y. Man, L. Li and G. Balestra, and Profs. O. S. Pak, B. Rallabandi, E. Nazockdast, Y. N. Young and F. Gallaire for useful discussions. Prof. T. Götz is acknowledged for sharing us with his Phd thesis. L.Z. thanks the Swedish Research Council for a VR International Postdoc Grant (2015-06334). We thank the NSF for support via the Princeton University Material Research Science and Engineering Center (DMR-1420541). The computer time was provided by SNIC (Swedish National Infrastructure for Computing).