Low-Reynolds-number, biflagellated Quincke swimmers with multiple forms of motion

Han, Endao; Zhu, Lailai; Shaevitz, Joshua W.; Stone, Howard A.

doi:10.1073/pnas.2022000118

Low-Reynolds-number, biflagellated Quincke swimmers with multiple forms of motion

Endao Han https://orcid.org/0000-0002-6934-1407 [email protected], Lailai Zhu https://orcid.org/0000-0002-3443-0709, Joshua W. Shaevitz https://orcid.org/0000-0001-8809-4723, and Howard A. Stone https://orcid.org/0000-0002-9670-0639 [email protected]Authors Info & Affiliations

Edited by Michael P. Brenner, Harvard University, Cambridge, MA, and accepted by Editorial Board Member John A. Rogers June 1, 2021 (received for review October 21, 2020)

July 15, 2021

118 (29) e2022000118

https://doi.org/10.1073/pnas.2022000118

PDF/EPUB

Significance

We demonstrate a centimeter-scale swimmer that converts direct current electric power into thrust in the low Reynolds number regime with minimal design and engineering. The swimmers show multiple forms of motion despite their simple geometry, which are reminiscent of some swimming microorganisms in nature. We map the state diagram to reveal how the swimmers translate and rotate and how they switch between these different forms of motion. Finally, our experiments confirm a theoretically predicted self-oscillation in such systems due to an elastohydrodynamic–electrohydrodynamic instability, which is a fundamentally different driving mechanism from the forced oscillation exploited by other artificial swimmers.

Abstract

In the limit of zero Reynolds number (Re), swimmers propel themselves exploiting a series of nonreciprocal body motions. For an artificial swimmer, a proper selection of the power source is required to drive its motion, in cooperation with its geometric and mechanical properties. Although various external fields (magnetic, acoustic, optical, etc.) have been introduced, electric fields are rarely utilized to actuate such swimmers experimentally in unbounded space. Here we use uniform and static electric fields to demonstrate locomotion of a biflagellated sphere at low Re via Quincke rotation. These Quincke swimmers exhibit three different forms of motion, including a self-oscillatory state due to elastohydrodynamic–electrohydrodynamic interactions. Each form of motion follows a distinct trajectory in space. Our experiments and numerical results demonstrate a method to generate, and potentially control, the locomotion of artificial flagellated swimmers.

In a Newtonian fluid, locomotion of microswimmers requires nonreciprocal body motions (1–3). Bacteria or eukaryotic cells achieve this by beating or rotating their slender hair-like organelles, flagella (4, 5) or cilia (6), powered by molecular motors. Mimicking these organisms, artificial swimmers propelled by rotating helices (7, 8) or whipping filaments (9–12) have been fabricated. They are commonly driven by an external power source such as a magnetic field (7–9, 13, 14), sound (15), light (16, 17), and biological materials (12). However, there are very few electrically powered microswimmers (18–20), although electric fields have been exploited to drive other active systems (21–26) via a phenomenon called Quincke rotation (27).

Quincke rotation originates from an electrohydrodynamic instability (28–30). Submerged in a liquid with permittivity

ε_{l}

and conductivity

σ_{l}

, a spherical particle with permittivity

ε_{s}

and electric conductivity

σ_{s}

is polarized under a uniform, steady electric field

\vec{E}

. When the particle is stationary, the induced dipole

\vec{p}

due to the free charges is parallel or antiparallel to

\vec{E}

(Fig. 1A): if the particle’s relaxation time

τ_{s} = ε_{s} / σ_{s}

is shorter than that of the ambient liquid,

τ_{l} = ε_{l} / σ_{l}

,

\vec{p}

points in the same direction as

\vec{E}

; when

τ_{s} > τ_{l}

,

\vec{p}

is opposite to

\vec{E}

, which generates an electric torque

{\vec{Γ}}_{Q} = \vec{p} \times \vec{E}

that amplifies any angular perturbation. However, due to the resisting viscous torque

{\vec{Γ}}_{μ}

, the system becomes unstable only when

E = | \vec{E} |

exceeds a threshold

E_{c}

. This instability causes the particle to rotate with a constant angular velocity

ω

:

ω = \frac{1}{τ} \sqrt{{(\frac{E}{E_{c}})}^{2} - 1},

[1]

where

τ = \frac{ε_{s} + 2 ε_{l}}{σ_{s} + 2 σ_{l}}

is the relaxation time of the system (see SI Appendix, SI Text, or refs. 28, 29, 31 for derivation), and the rotational axis can be in any direction perpendicular to

\vec{E}

. During steady-state Quincke rotation, there is a constant angle between

\vec{p}

and

\vec{E}

(Fig. 1A), which results in a nonzero

{\vec{Γ}}_{Q}

.

Fig. 1.

Quincke rotation and the experimental setup. (A) Distribution of free charge and the corresponding dipole $\vec{p}$ on a sphere in a uniform, steady electric field $\vec{E}$ . The sphere is (*Left*) stationary, (*Middle*) stationary, and (*Right*) rotating with a constant angular velocity $ω$ . (B) A sketch of the biflagellated swimmer. Dashed lines show the roll axis (blue) and pitch axis (green). (C) A schematic illustration of the experimental setup.

Recently, a flagellated swimmer in unbounded space driven by Quincke rotation has been proposed theoretically (32, 33). In light of the theory, we built a laboratory prototype, a biflagellated Quincke swimmer composed of a spherical particle and two attached elastic filaments, as shown in Fig. 1B, and systematically studied its behaviors at low Reynolds number (

R e < 0.3

; Materials and Methods). Varying the electric field

\vec{E}

and the angle between the two filaments, the Quincke swimmers exhibit three distinct forms of motion—two unidirectional rotations, which we call roll and pitch, and a self-oscillatory rotation, due to the balances between the electrical, elastic, and hydrodynamic torques, resulting in distinct trajectories in space. Surprisingly, it was recently reported (34) that spherical bacteria Magnetococcus marinus exhibit a similar pitch motion as our biflagellated artificial swimmers, which is rarely adopted by other microorganisms. Moreover, we found a threshold tail angle that separates the swimmers’ preferred forms of rotation, and within a small range close to this threshold angle, the three forms of motion coexist.

Experimental Setup

Each swimmer was composed of a spherical particle and two symmetric tails (Fig. 1B). We tested swimmers with two different sizes: the small swimmers had a particle radius

R = 1.59

mm and a tail radius

a = 13.7 μ

m, while the big swimmers had

R = 3.18

mm and

a = 24.8 μ

m. In this paper, we report results obtained with the big swimmers that had a fixed tail length

L = 13.6 \pm 0.6

mm (

L / R = 4.3 \pm 0.2

and

a / L = (1.8 \pm 0.1) \times 1 0^{- 3}

) and focus on one geometric control parameter—the angle

θ

between the tails and the symmetry axis. The experimental setup is illustrated schematically in Fig. 1C. A constant and uniform electric field was generated by a pair of parallel copper plates attached to the inner walls of an acrylic container filled with a weakly conductive oil. The density of the oil approximately matched that of the spheres, and its viscosity was

μ = 0.225

Pa

\cdot

s. The intensity of the electric field ranged from 0 to

7.7 \times 1 0^{5}

V/m. We captured and reconstructed the three-dimensional (3D) motion of the swimmers with two cameras imaging from perpendicular directions. Details on the experimental setup, material properties, and image analysis techniques are described in Materials and Methods.

Multiple Forms of Motion

Given a sufficiently large

E

, the biflagellated swimmers displayed three forms of motion as shown in Fig. 2. As the sphere rotated, the tails were deformed by the hydrodynamic forces. Their profiles depended on the geometry of the swimmer: when

θ

was below a threshold

θ_{c}

, the swimmer rotated about its axis of symmetry (roll axis in Fig. 1B) with the tails bent inward, forming a helical shape (Fig. 2A) and generating thrust akin to the observations of refs. 35–38. A swimmer with

θ > θ_{c}

rotated about an axis perpendicular to its axis of symmetry (pitch axis in Fig. 1B), and its tails bent outward (Fig. 2D). The corresponding overlaid images of the rotating tails within one period are shown in Fig. 2 B and E, respectively. Furthermore, we conducted simulations based on refs. 32, 33 that reproduced these two motions, as presented in Fig. 2 C and F (see Materials and Methods for details and SI Appendix, Movies S8 and S9, for the simulated rolling and pitching states, respectively).

Movie S8.

A rolling bi-flagellated swimmer in the numerical simulation. Its trajectory of locomotion is labelled by the dashed white line.

Movie S9.

A pitching bi-flagellated swimmer in the numerical simulation.

Fig. 2.

Biflagellated swimmers driven by a uniform, steady electric field $\vec{E}$ in the horizontal direction (labeled by the black arrows in A, D, and G). (A) Rotation (in one period $T$ ) of a swimmer with $θ = 47 °$ about its roll axis in the comoving frame using the center of the sphere as the reference and the overlapped tail shapes obtained from the (B) experiment and (C) numerical simulation. The scale bar indicating 1 cm is for both A and D. (D) Rotation (in $T$ ) of a swimmer with $θ = 58 °$ around its pitch axis in the comoving frame and the corresponding (E) experimental and (F) numerical tail shapes. The tail profiles inside the projection of the sphere are omitted in E due to visualization difficulties. (G) Experimental time lapse shapes in the comoving frame of the swimmer ( $θ = 44 °$ ) undergoing a self-oscillatory motion. (*Left*) A clockwise rotation (viewing from the top) at time $t \in [0, T / 2]$ and (*Right*) a counterclockwise rotation at $t \in [T / 2$ , $T]$ . (H) Instantaneous angular velocity $ω$ as a function of time for the swimmer with $θ = 44 °$ at $E = 1.65 E_{c}$ showing how it reached a steady oscillatory state from being stationary initially.

Within a small range of

θ

and

E

, the swimmer underwent an oscillatory rotation (Fig. 2G) in contrast to the classical, unidirectional Quincke rotation. Fig. 2H shows the particle’s angular velocity

ω

as a function of time when the amplitude of the oscillation increased away from zero toward a plateau. The occurrence of such a cyclic behavior driven by a steady electric field indicates its self-oscillatory nature as opposed to a forced oscillation, where the magnitude or direction of the external field varies periodically in order to drive the swimmer (8, 9, 13). This self-oscillatory rotation emerges from an elastohydrodynamic–electrohydrodynamic instability through a Hopf bifurcation, as identified theoretically for a similar system (32, 33). Transient oscillations also appeared in the experiments during the early stages of continuous rotations, but as the amplitude increased, eventually, the restoring force provided by the tails was not sufficient to sustain the oscillation, and

ω

became a constant. In the simulations, we observed not only this oscillatory state about the roll axis (SI Appendix, Movie S10) but also an oscillation about the pitch axis (SI Appendix, Movie S11), when the swimmer’s axis of symmetry was initially parallel to

\vec{E}

. However, the latter was absent in our experiments.

Movie S10.

A bi-flagellated swimmer oscillating about its axis of symmetry obtained from the numerical simulation.

Movie S11.

A bi-flagellated swimmer oscillating about the pitch axis (as labelled in Fig. 1 of the main text) obtained from the numerical simulation.

Discontinuous Transition in Angular Speed and Hysteresis

To understand the dynamics of the system, we look at the swimmer with

θ = 47 °

, which is slightly below

θ_{c}

. We measured the average angular frequency of the particle

Ω \equiv 2 π / T

, with

T

being one period of oscillation or rotation. For unidirectional rotation (roll or pitch), the instantaneous angular speed is

| ω | = Ω

, but for oscillation,

ω \approx ω_{0} \sin (Ω t)

, where

ω_{0}

is the amplitude of

ω

. Since

Ω

is a well-defined parameter (a nonnegative scalar) for both oscillation and rotation, we plot the dimensionless angular frequency

Ω τ

measured using the relaxation time

τ

versus

E

in Fig. 3A for both a bare sphere and the swimmer. The rotation of the sphere is well described by Eq. 1. In comparison, as we increased

E

, the swimmer was stationary until

E

reached about

1.7 E_{c}

, where the oscillatory state occurred within a narrow window. When a stronger electric field was applied, the rotation about the roll axis became unidirectional and steady, and the angular frequency

Ω

increased with

E

. When we reversed this process by decreasing

E

, hysteresis emerged, and no stable oscillatory state was observed. Note that although we use

θ = 47 °

as an example here, the discontinuous transition and hysteresis were observed in a wide range of

θ

below

θ_{c}

, and the discussions below apply when

θ < θ_{c}

.

Fig. 3.

Experimentally measured angular speed of the swimmer and the corresponding torques. (A) Nondimensional angular frequency $Ω τ$ of a particle and a swimmer with $θ = 47 °$ when the field intensity $E$ increased (solid points) or decreased (open points), with data points labeled in the panel. The solid black curve shows Eq. 1. The solid blue curve is a guide to the eye. The rectangular region enclosed by the dotted lines is enlarged and shown in B. The black arrows indicate how the state of the swimmer changes as $E$ increases or decreases. The error bars are labeled by vertical black lines, but they are mostly smaller than the individual data points. (C) The driving torque $Γ_{Q}$ (Eq. 3) at three different $E$ (blue lines) and the resisting torque $Γ_{res} = | Γ_{μ} + Γ_{ec} + 2 Γ_{t} |$ (solid black line; see *SI Appendix* for details) of the swimmer while rolling. The dashed black line shows the slope of $Γ_{res} (ω)$ as $ω \to 0$ . The stable fixed points are labeled with solid stars and the unstable fixed points with the open star. (D) The trajectory of torque-angular velocity when the swimmer oscillated. The trajectory is calculated using Eqs. 2 and 3 (see *SI Appendix* for details). Following ref. 37, we scale $Γ_{t}$ by $Γ_{0} \equiv κ / L$ , where $κ$ is the bending stiffness of the tails. The green arrows label the direction of time.

To explain the discontinuous transition and hysteresis in Fig. 3A, we examine the torques applied on the particle. Besides the driving torque generated by the Quincke effect

Γ_{Q}

and the viscous torque due to the spinning sphere

Γ_{μ} = - 8 π μ R^{3} ω

, each tail exerted the same torque

Γ_{t}

on the particle since they were symmetric. When the swimmer pitched, the drag force generated a torque

Γ_{ec}

due to the sphere’s eccentric rotation, which was negligible for rolling and oscillation. The total torque vanishes in the low Re limit; thus,

Γ_{Q} + Γ_{μ} + Γ_{ec} + 2 Γ_{t} = 0 .

[2]

Given

E

and

ω (t)

, we calculated

Γ_{Q}

and used Eq. 2 to solve for

Γ_{t}

. Note that the electric torque applied on the tails is negligible compared to that applied on the spherical head, so here we only consider the latter (33).

When the swimmer is stationary or rotates steadily, the driving torque is

Γ_{Q} = 8 π μ R^{3} {(\frac{E}{E_{c}})}^{2} \frac{ω}{1 + {(ω τ)}^{2}},

[3]

shown by the blue curves in Fig. 3C, where

ω

is the angular velocity of the swimmer’s spherical head (see SI Appendix, SI Text, or ref. 31). If the tails of the swimmer are rigid, the resisting torque

Γ_{res} = | Γ_{μ} + Γ_{ec} + 2 Γ_{t} |

is a linear function of

ω

following the dashed black line in Fig. 3C. The system evolves toward where

Γ_{Q}

and

Γ_{res}

intersect (fixed points). Consequently, as

E

increases, the transition occurs via a supercritical pitchfork bifurcation, as in the original Quincke rotation. However, since the elastic tails deform under large torques (Fig. 2),

Γ_{res}

deviates more and more from the dashed straight line as

ω

increases (SI Appendix, SI Text), leading to a subcritical bifurcation. At small

E

, the point

(0,0)

in Fig. 3C is a stable fixed point. The swimmer remains stationary until

d Γ_{Q} / d ω

exceeds

d Γ_{res} / d ω

at

ω = 0

as

E

increases, where the origin becomes unstable and the stable fixed point shifts discontinuously to

ω \neq 0

. For an intermediate

E

, e.g.,

E = 1.66 E_{c}

, there are two stable fixed points [one of them is at

(0,0)

] separated by an unstable fixed point. As a result, the swimmer stayed at different states when

E

increased or decreased, hence the hysteresis.

When the swimmer oscillates (Fig. 2G), both

Γ_{Q}

and

Γ_{t}

do not solely depend on the instantaneous

ω

but also on its preceding values. The trajectory of

Γ_{t} (ω)

in Fig. 3D tends to a limit cycle instead of a fixed point (SI Appendix).

Γ_{t}

accelerated the rotation in the I and III quadrants and decelerated it in the II and IV quadrants of the figure. In each period of oscillation,

Γ_{t}

was large enough to hinder the rotation, and then the residual elastic energy stored in the tails drove the sphere to rotate in the opposite direction.

State Diagram

The form of motion of a swimmer, or its “state” (stationary, rolling, pitching, or oscillatory), was mainly controlled by its geometry and the applied field. Fixing the ratio of tail length to particle size

L / R = 4.3

, we mapped out the biflagellated Quincke swimmers’ state diagram with respect to the tail angle

θ

and the relative applied field

E / E_{c}

, as shown in Fig. 4. We predict the swimmers’ motion by identifying the axis around which it experiences the least viscous torque, assuming the tails are both rigid. We then numerically calculated the threshold electric field

E_{θ}

for continuous rotation for different

θ

, shown by the black curve in Fig. 4, with no adjustable parameter (SI Appendix, SI Text). The calculations agree reasonably well with the experimental observations* .

Fig. 4.

State diagram for swimmers with $L / R = 4.3$ and different $θ$ . The solid black curve shows the calculated onset field at different $θ$ without any fitting parameters. The dashed black line indicates the boundary of the roll (left) and pitch (right) states at $θ_{c} = 49.8 °$ .

The boundary separating the rolling and pitching motions is at

θ_{c} = 49.8 °

. Swimmers with

θ < θ_{c}

preferred rolling, and those with

θ > θ_{c}

preferred pitching. Within a small region where

θ \in (43 °, θ_{c})

and

E

was slightly below

E_{θ}

, the swimmers exhibited stable oscillatory rotations. When

θ \approx θ_{c}

, the swimmer was almost equally likely to rotate about any axis (SI Appendix), so the three forms of motion, roll, pitch, and oscillation, coexisted. In this case, besides the field intensity

E

, the eventual stable form of motion was significantly affected by the initial orientation of the swimmer relative to

\vec{E}

. As introduced above, Quincke rotation can occur about any axis perpendicular to the external field. Consequently, if the symmetry axis of the swimmer was perpendicular to

\vec{E}

, it tended to roll, while if the axis was parallel to

\vec{E}

, it tended to pitch.

Translational Motion and Propulsive Force

Last, we studied the locomotion of the swimmers while executing different forms of motion. For each swimmer, the translational motion of the particle is determined by three forces: the viscous drag

{\vec{F}}_{μ} = - 6 π μ R \vec{v}

, the gravitational force due to a slight density mismatch

{\vec{F}}_{ρ} = \frac{4}{3} π R^{3} (ρ_{s} - ρ_{l}) \vec{g}

, and the propulsive force

{\vec{F}}_{2 t}

provided by both tails. Here

\vec{v}

is the velocity of the particle,

\vec{g}

is the gravitational acceleration, and

ρ_{s}

and

ρ_{l}

are the densities of the particle and the liquid, respectively. These three forces add up to zero, so

{\vec{F}}_{2 t} = - {\vec{F}}_{μ} - {\vec{F}}_{ρ}

. Typical trajectories corresponding to rolling, oscillation, and pitching are presented in Fig. 5 A–C, respectively, along with the instantaneous propulsive force

F_{2 t}

. While rolling, the swimmer moved along a smooth and relatively straight trajectory (Fig. 5A), and the force almost always pointed in the same direction as

\vec{v}

. When oscillating (Fig. 5B), the trajectory became sinusoidal, and the direction and magnitude of the propulsive force varied with the translational and angular velocities. When pitching, since the rotation was eccentric, the particle followed a helical trajectory resulting from a combination of a fast circular motion and a slow drift (Fig. 5C).

Fig. 5.

Trajectories and propulsive forces generated by different forms of motion. (A–C) Trajectories of the particle and the propulsive force vectors at each time point (red arrows). The forms of motion are (A) roll ( $θ = 40 °$ ), (B) oscillation ( $θ = 44 °$ ), and (C) pitch ( $θ = 58 °$ ). The thick black curves highlight three periods of rotation or oscillation in each plot, with the two ends indicated by black dots. The green curves show the smoothed trajectories, and the directions of motion are labeled by the green arrowheads. The views are perpendicular to the trajectories in every plot, and the scale bar indicating the particle radius $R$ is shared in A–C. The direction of the external field $\vec{E}$ for each panel is labeled by the gray arrows. (D) The propulsive force averaged over each period $F_{T} = ⟨\frac{1}{T} |\int_{t}^{t + T} {\vec{F}}_{2 t} (t) d t|⟩$ as a function of the nondimensional angular speed. Solid symbols represent increasing $E$ , and open symbols represent decreasing $E$ . (E) Relationship between the propulsive force generated and torque applied on each tail. The forces and torques are nondimensionalized with $F_{0} = κ / L^{2}$ and $Γ_{0} = κ / L$ , respectively, where $κ$ is the bending stiffness of the tail (37).

To compare the propulsive features of the swimmers under different conditions, in Fig. 5D we plot the time-averaged propulsive force generated by the tails in each rotation period

F_{T} = ⟨\frac{1}{T} |\int_{t}^{t + T} {\vec{F}}_{2 t} (t) d t|⟩

. Although all three forms of motion were nonreciprocal, only rolling was able to achieve an effective unidirectional translation for the tested swimmers. The propulsion peaked when

E

slightly exceeded

E_{θ}

, then

F_{T}

decreased as

ω

increased. Pitching resulted in poor locomotion because of its symmetry: although the instantaneous force was several times larger compared to the other two forms of motion, they added up to a small value in one cycle of rotation. In comparison, oscillation was able to generate a net force along its drift speed, but it was not as efficient, because the component of the force parallel to the drift direction was relatively small, which can be seen in Fig. 5B.

Why does

F_{T}

decrease with

ω

when the swimmer was rolling? We found that

F_{T}

was approximately proportional to the torque applied on each tail

| Γ_{t} |

(Fig. 5E). However, when a stronger electric field was applied,

| Γ_{t} |

decreased as the rolling speed

ω

increased. In general, for a tilted elastic fiber rotating about one fixed end,

ω (| Γ_{t} |)

is an S-shaped function, and the trend of

| Γ_{t} |

versus

ω

is controlled by a dimensionless bending parameter

B

(also known as the sperm number)

B = \frac{4 π μ L^{4} ω}{κ [\ln (L / a) + 1 / 2]},

[4]

where

κ

is the bending stiffness of the fiber (35–37). We tested swimmers with different geometries (sphere radius, tail radius, tail length, and tail angle) in liquids with different

τ

, and indeed we found a transition at

B \approx 1 0^{2}

:

| Γ_{t} |

decreased with

ω

when

B < 1 0^{2}

and increased when

B > 1 0^{2}

(SI Appendix, Fig. S14). The swimmer shown in Fig. 5 was in the former regime.

Relating the locomotion with the state diagram (Fig. 4), we can see that drastic changes in motion can be achieved by adjusting

E / E_{c}

in a small range, especially around

θ_{c}

. For example, with

θ = 49.4 °

, the swimmer was stationary at

E = 1.71 E_{c}

, oscillated at

E = 1.77 E_{c}

, and rolled at

E = 1.84 E_{c}

with a consistent propulsion. Moreover, since rolling and pitching coexisted for this swimmer, how it rotated depended on its initial orientation relative to

\vec{E}

at the moment when

E

exceeded the threshold. All these features could potentially lead to further questions on controlling the motion of such swimmers.

Conclusions

We created an artificial swimmer driven by constant and uniform external electric fields, exploiting Quincke rotation. The swimmer had a rigid spherical body and two elastic filamentous tails, which allowed it to move at low Reynolds number with three different forms of motion, rolling, pitching, and oscillation, controlled by its geometry and the external field. Among them, rolling allowed the swimmer to generate steady translational locomotion.

Because of the simple structure and driving method of the swimmer, there is a potential to scale down their dimensions exploiting techniques such as photolithography (14, 40) or 3D printing (41) and scale up their numbers so that they become a model system for studying collective motion, swarming (42), or other phenomena in active matter. Increasing the Reynolds number of the system by enlarging the dimension of the swimmer or reducing the viscosity of the ambient liquid can also potentially lead to rich and interesting phenomena (43).

Materials and Methods

Experimental Setup and Sample Preparation.

The experiments were performed with a rectangular acrylic container. A constant and uniform electric field was generated by a pair of parallel copper plates attached to the inner walls of the container. Each copper plate was 6.3 mm thick, with machined and polished surfaces. The top of the container was covered by an acrylic plate to avoid surface flows at the liquid–air interface induced by high electric fields. The distance between the surfaces of the two electrodes was 3.9 cm. When the swimmer was close to the center of the container, its electrical interactions with the electrodes due to the mirror charges were negligible (SI Appendix, SI Text). The other two dimensions of the container were 15 cm (length) and 10 cm (depth). This enclosed volume was fully filled with oil. The voltage applied on the copper plates was provided by a direct current (DC) power supply (Micronta) and a DC voltage amplifier (DCH 3034N1, Dean Technology). The amplifier transformed a 0- to 12-V input voltage into a 0- to 30-kV output voltage. Two single-lens reflex cameras (Nikon D3300 or Nikon D5100) were used to take videos of the swimmer from two perpendicular directions. One camera was placed above the container and the other one on the side, which allowed us to track the motion of the swimmer in three dimensions. In each experiment, the voltage applied on the plates was adjusted to an expected value, then we started recording with both cameras simultaneously.

The spherical heads of the swimmers were made of high-density polyethylene (HDPE), with a radius

R = 3.18

mm and a density

ρ_{s} = 0.94 \pm 0.01

g/

{c m}^{3}

. The tails were No. 8-0 surgical sutures (from S&T, nylon, 24.8

μ

m in radius). We measured the Young’s modulus of the fiber

Y = 2.7 \pm 0.5

GPa. When preparing the swimmer, the nylon fiber was cut to a certain length, and its middle point was attached to the HDPE sphere with a small amount of glue (Loctite 401). After the glue cured, the fiber was folded symmetrically and formed an angle. Colored spots were painted on the surface of the sphere as tracers. Their areas were sufficiently small so that they did not affect the electric or hydrodynamic properties of the sphere surface.

The liquid was an equal mixture by volume of olive oil (Filippo Berio or Spectrum [NF grade]) and castor oil (Alfa Aesar). The viscosity of the mixed oil was 0.225 Pa

\cdot

s measured with a rheometer (Anton Paar MCR 301), and its density was

ρ_{l} = 0.94

g/

{c m}^{3}

, which approximately matched the density of the HDPE sphere

ρ_{s}

. The electric properties of the mixed liquid were characterized by measuring the angular speed

ω

of HDPE spheres (with no tail attached) under different applied fields

E

. Fitting the data with Eq. 1, we obtained

E_{c}

and relaxation time

τ

. In the experiments shown here, the mean threshold electric field was

E_{c} = 382 \pm 8

kV/m, and the mean relaxation time was

τ = 0.28 \pm 0.02

s. To prevent the oil from being contaminated by the dust in the air, the whole setup was placed in a glove box, and the oil was filtered on a daily basis.

Last, we estimate the Reynolds number. The maximum angular speed of the swimmer did not exceed

ω τ = 2

in our experiments, so we can calculate the upper limit of

R e = ρ_{l} R^{2} ω / μ

using

ω_{max} = 7.1

rad/s. Consequently, we get

R e < 0.3

, which is in the low

R e

regime.

Numerical Method.

In this work together with our previous studies (32, 33), we have identified and investigated an elastohydrodynamic–electrohydrodynamic problem that integrates the elastohydrodynamics of flexible filaments in viscous fluids and the electrohydrodynamics of a dielectric particle in dielectric solvents. The numerical method adopted here closely resembles that described in the appendix of ref. 33 despite two differences: first, two filaments are attached to the particle here compared to one in ref. 33; second, full 3D motion of the Quincke swimmer is pursued here, in contrast to the constrained planar motion (33).

As in refs. 32, 33, we do not consider hydrodynamic interactions among the two filaments or those between the particle and the filaments. We use the semi-implicit backward Euler scheme to time-march the nonlinear governing equations in a fully coupled fashion, hence solving for the translational and rotational velocities of the particle, the induced dipole, and the instantaneous profile of the filament simultaneously. This in-house solver has been cross-validated against another finite-element method (FEM) solver developed in the framework of the commercial package COMSOL Multiphysics (I-Math, Singapore). Before the cross-validation, the FEM solver was first validated against the numerical implementation for elastohydrodynamics of filaments (44) and our in-house solver for the Quincke swimmers with one tail (32, 33).

Data Availability

Text files and videos have been deposited in GitHub (https://github.com/endaohan/Data-in-papers). All other study data are included in the article and/or SI Appendix.

Acknowledgments

We thank Janine Nunes and Nan Xue for the help with the experiments. We thank Ellie Acosta, Benjamin Bratton, Yong Dou, Matthias Koch, and Talmo Pereira for useful discussions. E.H. acknowledges the support by the NSF through the Center for the Physics of Biological Function (PHY-1734030). L.Z. thanks the start-up grant provided by the National University of Singapore (R-265-000-696-133). H.A.S. acknowledges the support by the NSF through the Princeton University Materials Research Science and Engineering Center (DMR-2011750). The computational work for this article was performed on resources of the National Supercomputing Center, Singapore (https://www.nscc.sg).

Supporting Information

Appendix (PDF)

Download
4.80 MB

Movie S1.

A swimmer rotating about its axis of symmetry (roll) in the comoving frame with respect to the center of the sphere. The swimmer had a particle radius R = 3.18 mm, tail radius a = 24.8 μm, tail length L = 13.5 mm, and tail angle θ = 46.2°. The applied electric field was E = 1.86E_c. Images obtained from the top camera. The applied electric field is vertically pointing up in this movie.

Download
561.88 KB

Movie S2.

A swimmer rotating about an axis perpendicular to its axis of symmetry (pitch) in the comoving frame with respect to the center of the sphere. The swimmer had a particle radius R = 3.18 mm, tail radius a = 24.8 μm, tail length L = 14.2 mm, and tail angle θ = 79.9°. The applied electric field was E = 1.54E_c. Images obtained from the top camera. The applied electric field is vertically pointing up in this movie.

Download
591.94 KB

Movie S3.

Oscillation of the swimmer in a comoving frame with respect to the center of the sphere. The swimmer had a particle radius R = 3.18 mm, tail radius a = 24.8 μm, tail length L = 14.4 mm, and tail angle θ = 46.9°. The applied electric field was E = 1.78E_c. Images obtained from the side camera. The applied electric field is vertically pointing down in this movie.

Download
832.78 KB

Movie S4.

Trajectory of the swimmer while rolling in the lab frame. The gray circle shows the three dimensional position of the swimmer’s spherical head. The arrows on the sphere represent the driving force F_2t⃗ (green), the viscous drag F_μ⃗ (red), and the force due to density mismatch F_ρ⃗(blue). The trajectory of the sphere is shown by the black curve. The orange arrows indicate the direction of the external electric field. In this video, the swimmer had a particle radius R = 3.18 mm, tail radius a = 24.8 μm, tail length L = 13.5 mm, and tail angle θ = 46°. The applied electric field was E = 1.86E_c.

Download
2.91 MB

Movie S5.

Trajectory of the swimmer while pitching in the lab frame. The notations are identical to Movie 4. In this video, the swimmer had a particle radius R = 3.18 mm, tail radius a = 24.8 μm, tail length L = 13.2 mm, and tail angle θ = 58°. The applied electric field was E = 1.53E_c.

Download
2.25 MB

Movie S6.

Trajectory of the swimmer while oscillating in the lab frame. The notations are identical to Movie 4. In this video, the swimmer had a particle radius R = 3.18 mm, tail radius a = 24.8 μm, tail length L = 13.8 mm, and tail angle θ = 44°. The applied electric field was E = 1.65E_c.

Download
3.03 MB

Movie S7.

Measure instantaneous angular velocity. The swimmer in the video had a particle radius R = 3.18 mm, tail radius a = 24.8 μm, tail length 13.8 mm, and tail angle θ = 44°. The applied electric field was E = 1.65E_c. Fig. S7 is one frame in the video.

Download
5.77 MB

Movie S8.

A rolling bi-flagellated swimmer in the numerical simulation. Its trajectory of locomotion is labelled by the dashed white line.

Download
484.47 KB

Movie S9.

A pitching bi-flagellated swimmer in the numerical simulation.

Download
2.66 MB

Movie S10.

A bi-flagellated swimmer oscillating about its axis of symmetry obtained from the numerical simulation.

Download
1.12 MB

Movie S11.

A bi-flagellated swimmer oscillating about the pitch axis (as labelled in Fig. 1 of the main text) obtained from the numerical simulation.

Download
1.88 MB

References

1

G. I. Taylor, Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. Math. Phys. Sci. 209, 447–461 (1951).

Crossref

Google Scholar

2

E. M. Purcell, Life at low Reynolds number. Am. J. Phys. 45, 3–11 (1977).

Crossref

Google Scholar

3

E. Lauga, Floppy swimming: Viscous locomotion of actuated elastica. Phys. Rev. E 75, 1–16 (2007).

Crossref

Google Scholar

4

H. C. Berg, R. A. Anderson, Bacteria swim by rotating their flagellar filaments. Nature 245, 380–382 (1973).

Crossref

PubMed

Google Scholar

5

H. C. Berg, The rotary motor of bacterial flagella. Annu. Rev. Biochem. 72, 19–54 (2003).

Crossref

PubMed

Google Scholar

6

R. E. Goldstein, Green algae as model organisms for biological fluid dynamics. Annu. Rev. Fluid Mech. 47, 343–375 (2015).

Crossref

PubMed

Google Scholar

7

A. Ghost, P. Fischer, Controlled propulsion of artificial magnetic nanostructured propellers. Nano Lett. 9, 2243–2245 (2009).

Crossref

PubMed

Google Scholar

8

L. Zhang et al., Artificial bacterial flagella: Fabrication and magnetic control. Appl. Phys. Lett. 94, 2007–2010 (2009).

Google Scholar

9

R. Dreyfus et al., Microscopic artificial swimmers. Nature 437, 862–865 (2005).

Crossref

PubMed

Google Scholar

10

C. H. Wiggins, D. Riveline, A. Ott, R. E. Goldstein, Trapping and wiggling: Elastohydrodynamics of driven microfilaments. Biophys. J. 74, 1043–1060 (1998).

Crossref

PubMed

Google Scholar

11

T. S. Yu, E. Lauga, A. E. Hosoi, Experimental investigations of elastic tail propulsion at low Reynolds number. Phys. Fluids 18, 2–6 (2006).

Crossref

Google Scholar

12

B. J. Williams, S. V. Anand, J. Rajagopalan, M. T. A. Saif, A self-propelled biohybrid swimmer at low Reynolds number. Nat. Commun. 5, 3081 (2014).

Crossref

PubMed

Google Scholar

13

E. F Box Han, C. R. Tipton, T. Mullin, On the motion of linked spheres in a Stokes flow. Exp. Fluid 58, 1–10 (2017).

Google Scholar

14

H. W. Huang et al., Adaptive locomotion of artificial microswimmers. Sci. Adv. 5, eaau1532 (2019).

Crossref

PubMed

Google Scholar

15

D. Ahmed et al., Artificial swimmers propelled by acoustically activated flagella. Nano Lett. 16, 4968–4974 (2016).

Crossref

PubMed

Google Scholar

16

C. L. Van Oosten, C. W. M. Bastiaansen, D. J. Broer, Printed artificial cilia from liquid-crystal network actuators modularly driven by light. Nat. Mater. 8, 677–682 (2009).

Crossref

PubMed

Google Scholar

17

S. Palagi et al., Structured light enables biomimetic swimming and versatile locomotion of photoresponsive soft microrobots. Nat. Mater. 15, 647–653 (2016).

Crossref

PubMed

Google Scholar

18

S. Gangwal, O. J. Cayre, M. Z. Bazant, O. D. Velev, Induced-charge electrophoresis of metallodielectric particles. Phys. Rev. Lett. 100, 058302 (2008).

Crossref

PubMed

Google Scholar

19

G. Loget, A. Kuhn, Electric field-induced chemical locomotion of conducting objects. Nat. Commun. 2, 535 (2011).

Crossref

PubMed

Google Scholar

20

F. Ma, X. Yang, H. Zhao, N. Wu, Inducing propulsion of colloidal dimers by breaking the symmetry in electrohydrodynamic flow. Phys. Rev. Lett. 115, 208302 (2015).

Crossref

PubMed

Google Scholar

21

L. Lobry, E. Lemaire, Viscosity decrease induced by a DC electric field in a suspension. J. Electrost. 47, 61–69 (1999).

Crossref

Google Scholar

22

N. Pannacci, L. Lobry, E. Lemaire, How insulating particles increase the conductivity of a suspension. Phys. Rev. Lett. 99, 2–5 (2007).

Crossref

Google Scholar

23

A. Bricard, J. B. Caussin, N. Desreumaux, O. Dauchot, D. Bartolo, Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98 (2013).

Crossref

PubMed

Google Scholar

24

Q. Brosseau, G. Hickey, P. M. Vlahovska, Electrohydrodynamic Quincke rotation of a prolate ellipsoid. Phys. Rev. Fluid. 2, 014101 (2017).

Crossref

Google Scholar

25

H. Karani, G. E. Pradillo, P. M. Vlahovska, Tuning the random walk of active colloids: From individual run-and-tumble to dynamic clustering. Phys. Rev. Lett. 123, 208002 (2019).

Crossref

PubMed

Google Scholar

26

D. Das, E. Lauga, Active particles powered by Quincke rotation in a bulk fluid. Phys. Rev. Lett. 122, 194503 (2019).

Crossref

PubMed

Google Scholar

27

G. Quincke, Ueber rotationen im constanten electrischen felde. Ann. Phys. Chem. 295, 417 (1896).

Crossref

Google Scholar

28

A. O. Tsebers, Internal rotation in the hydrodynamics of weakly conducting dielectric suspensions. Fluid Dyn. 15, 245–251 (1980).

Crossref

Google Scholar

29

T. B. Jones, Quincke rotation of spheres. IEEE Trans. Indus. Appl. 20, 845–849 (1984).

Crossref

Google Scholar

30

A. Cebers, E. Lemaire, L. Lobry, Electrohydrodynamic instabilities and orientation of dielectric ellipsoids in low-conducting fluids. Phys. Rev. E 63, 016301 (2001).

Crossref

Google Scholar

31

I. Turcu, Electric field induced rotation of spheres. J. Phys. Math. Gen. 20, 3301–3307 (1987).

Crossref

Google Scholar

32

L. Zhu, H. A. Stone, Propulsion driven by self-oscillation via an electrohydrodynamic instability. Phys. Rev. Fluid. 4, 1–7 (2019).

Crossref

Google Scholar

33

L. Zhu, H. A. Stone, Harnessing elasticity to generate self-oscillation via an electrohydrodynamic instability. J. Fluid Mech. 888, A311–A3135 (2020).

Crossref

Google Scholar

34

K. Bente et al., High-speed motility originates from cooperatively pushing and pulling flagella bundles in bilophotrichous bacteria. eLife 9, e47551 (2020).

Crossref

PubMed

Google Scholar

35

M. Manghi, X. Schlagberger, R. R. Netz, Propulsion with a rotating elastic nanorod. Phys. Rev. Lett. 96, 068101 (2006).

Crossref

PubMed

Google Scholar

36

B. Qian, T. R. Powers, K. S. Breuer, Shape transition and propulsive force of an elastic rod rotating in a viscous fluid. Phys. Rev. Lett. 100, 078101 (2008).

Crossref

PubMed

Google Scholar

37

N. Coq, O. du Roure, J. Marthelot, D. Bartolo, M. Fermigier, Rotational dynamics of a soft filament: Wrapping transition and propulsive forces. Phys. Fluids 20, 051703 (2008).

Crossref

Google Scholar

38

M. K. Jawed, N. K. Khouri, F. Da, E. Grinspun, P. M. Reis, Propulsion and instability of a flexible helical rod rotating in a viscous fluid. Phys. Rev. Lett. 115, 168101 (2015).

Crossref

PubMed

Google Scholar

39

G. E. Pradillo, H. Karani, P. M. Vlahovska, Quincke rotor dynamics in confinement: Rolling and hovering. Soft Matter 15, 6564–6570 (2019).

Crossref

PubMed

Google Scholar

40

H. W. Huang, M. S. Sakar, A. J. Petruska, S. Pane, B. J. Nelson, Soft micromachines with programmable motility and morphology. Nat. Commun. 7, 12263 (2016).

Crossref

PubMed

Google Scholar

41

I. Liashenko, J. Rosell-Llompart, A. Cabot, Ultrafast 3d printing with submicrometer features using electrostatic jet deflection. Nat. Commun. 11, 753 (2020).

Crossref

PubMed

Google Scholar

42

D. B. Kearns, A field guide to bacterial swarming motility. Nat. Rev. Microbiol. 8, 634–644 (2010).

Crossref

PubMed

Google Scholar

43

F. Peters, L. Lobry, E. Lemaire, Experimental observation of Lorenz chaos in the Quincke rotor dynamics. Chaos 15, 13102 (2005).

Crossref

PubMed

Google Scholar

44

A. K. Tornberg, M. J. Shelley, Simulating the dynamics and interactions of flexible fibers in Stokes flows. J. Comput. Phys. 196, 8–40 (2004).

Crossref

Google Scholar

Information & Authors

Information

Published in

Proceedings of the National Academy of Sciences

Vol. 118 | No. 29
July 20, 2021

PubMed: 34266946

Classifications

Copyright

Data Availability

Text files and videos have been deposited in GitHub (https://github.com/endaohan/Data-in-papers). All other study data are included in the article and/or SI Appendix.

Submission history

Published online: July 15, 2021

Published in issue: July 20, 2021

Keywords

Acknowledgments

We thank Janine Nunes and Nan Xue for the help with the experiments. We thank Ellie Acosta, Benjamin Bratton, Yong Dou, Matthias Koch, and Talmo Pereira for useful discussions. E.H. acknowledges the support by the NSF through the Center for the Physics of Biological Function (PHY-1734030). L.Z. thanks the start-up grant provided by the National University of Singapore (R-265-000-696-133). H.A.S. acknowledges the support by the NSF through the Princeton University Materials Research Science and Engineering Center (DMR-2011750). The computational work for this article was performed on resources of the National Supercomputing Center, Singapore (https://www.nscc.sg).

Notes

*One reason why the predicted onset field deviates from experimental measurements (e.g., at

θ = 0.3 π

) could be that we assumed the bases of the two tails were next to each other in the calculation to keep the model simple, while in actual experiments there was, on average, an approximately 1.8 mm gap between them. Confinement could also increase the onset electric field (39), although the effect should be weak in our case.

This article is a PNAS Direct Submission. M.P.B. is a guest editor invited by the Editorial Board.

Authors

Affiliations

Endao Han¹ https://orcid.org/0000-0002-6934-1407 [email protected]

Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ, 08544;

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Lailai Zhu https://orcid.org/0000-0002-3443-0709

Department of Mechanical Engineering, National University of Singapore, Singapore 117575;

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Joshua W. Shaevitz https://orcid.org/0000-0001-8809-4723

Joseph Henry Laboratories of Physics, Princeton University, Princeton, NJ, 08544;

Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, 08544;

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Howard A. Stone¹ https://orcid.org/0000-0002-9670-0639 [email protected]

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ, 08544

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Notes

1

To whom correspondence may be addressed. Email: [email protected] or [email protected].

Author contributions: E.H., L.Z., and H.A.S. designed research; E.H. and L.Z. performed research; E.H. and L.Z. contributed new reagents/analytic tools; E.H., L.Z., J.W.S., and H.A.S. analyzed data; and E.H., L.Z., J.W.S., and H.A.S. wrote the paper.

Competing Interests

The authors declare no competing interest.

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E. Han,
L. Zhu,
J.W. Shaevitz,
& H.A. Stone,

Low-Reynolds-number, biflagellated Quincke swimmers with multiple forms of motion, Proc. Natl. Acad. Sci. U.S.A. 118 (29) e2022000118, https://doi.org/10.1073/pnas.2022000118 (2021).

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Experimental Setup

Multiple Forms of Motion

Discontinuous Transition in Angular Speed and Hysteresis

State Diagram

Translational Motion and Propulsive Force

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Experimental Setup and Sample Preparation.

Numerical Method.

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