## Abstract

Biomimetic micro-swimmers can be used for various medical applications, such as targeted drug delivery and micro-object (e.g. biological cells) manipulation, in lab-on-a-chip devices. Bacteria swim using a bundle of flagella (flexible hair-like structures) that form a rotating cork-screw of chiral shape. To mimic bacterial swimming, we employ a computational approach to design a bacterial (chirality-induced) swimmer whose chiral shape and rotational velocity can be controlled by an external magnetic field. In our model, we numerically solve the coupled governing equations that describe the system dynamics (i.e. solid mechanics, fluid dynamics and magnetostatics). We explore the swimming response as a function of the characteristic dimensionless parameters and put special emphasis on controlling the swimming direction. Our results provide fundamental physical insight on the chirality-induced propulsion, and it provides guidelines for the design of magnetic bi-directional micro-swimmers.

## 1. Introduction

In the modern era of medical science where miniaturized means to perform diagnosis and non-invasive surgery are demanded, micro-swimmers and microbots are a key focus of research and development [1–9]. Many miniaturized externally powered propulsive devices have been suggested that have potential applications in targeted drug delivery and biological cell manipulation in lab-on-a-chip devices [1–11]. As the fundamental challenge lies in achieving a non-reciprocal motion at these small length scales [12,13], inspiration can be drawn from nature to design bioinspired artificial micro-swimmers [1,4,5,9–11,14–16]. Bacteria swim using a bundle of flagella (flexible hair-like structures) that form a rotating cork-screw of chiral shape, where each flagellum is actuated by a molecular motor at the bacterial surface, as shown in figure 1. The quest for a simple and viable design of biomimetic bacterial micro-swimmers has attracted considerable attention in the literature [9,13–27]. Various experimental and analytical studies have been performed to analyse the evolution of the chiral shape for an elastic filament or rod-like structure subjected to external actuation [18,19,21–23,25–28]. Also, in an experimental study performed by Garstecki *et al.* [15] a flexible planar structure is deformed into a chiral shape on-the-fly owing to the opposing torques imposed by the externally applied magnetic field and the resisting viscous forces of the surrounding fluid. However, systems that generate chirality ‘on-the-fly’ do not allow for direction reversal, because the chiral polarity is directly linked to the external actuation. To obtain swimming direction reversal, one has to uncouple the chiral polarity from the external actuation, which can be done by resorting to a pre-manufactured chiral shape. Zhang *et al.* [24] have proposed an elegant procedure that allows manufacturing of a helical belt at the micrometre-scale. However, the manufacturing process involves many more steps compared with systems that feature chirality on-the-fly. This brought us to pose the following question: *Can we achieve propulsion by developing chirality on-the-fly as natural bacteria and still be able to achieve direction-reversal through external actuation*? We address this question in this letter and use a computational approach to magnetically control the chirality of an artificial bacterial micro-swimmer. We investigate the swimming response using dimensionless parameters and explore the underlying physics of the chirality-induced propulsion. We exploit the available physical forces (i.e. elastic, magnetic and viscous) to form the chiral shape on-the-fly and relate the swimming dynamics to the intrinsic competition of these forces. The obtained computational results are supported by analytical expressions derived using resistive force theory [29], which provides the first-order approximation of the swimming velocity.

The article is organized as follows. In §2, we describe the computational model and approach used in this study, where the fluid–solid interaction (FSI) problem is solved by properly accounting for the magnetic forces owing to the external magnetic actuation. Additionally, we identify the dimensionless parameters that govern the swimming dynamics of chirality-induced propulsion. Subsequently, in §3, the results associated with the unidirectional (§3*a*) and the bi-directional (§3*b*) swimming are presented and analysed. Finally, conclusions are drawn that summarize the key points of this work in §4.

## 2. Computational model and dimensional analysis

The artificial flagellar micro-swimmer is designed as a planar partially magnetic flexible film that can be easily manufactured with state-of-the-art polymer processing technologies such as inkjet printing (figure 2). The concept exploits the dynamics of a partially magnetic film in which the chiral shape is induced on-the-fly through an externally applied magnetic field. Continuous rotation of such a chiral micro-swimmer leads to a steady-state propulsion subjected to the magnetic actuation (figure 2). To analyse the swimming dynamics of such a chiral micro-swimmer, a computational framework is developed that accounts for the fluid–structure interaction (FSI) between the deforming swimmer and the surrounding fluid, while properly incorporating the changing magnetic forces imposed by the applied magnetic field. We use a finite-element-based framework in which the solid mechanics, fluid dynamics and magnetostatics equations are solved simultaneously. In the model, the micro-swimmer is represented by an assemblage of shell elements, which act as an internal boundary to the fluid domain. The fluid–structure interaction is considered by implicitly coupling the fluid dynamics and solid mechanics equations, where the Stokeslets method is used to account for the viscous environment (and implemented using a boundary-element method). During the simulations, the application of an external magnetic field leads to the generation of magnetic body couples that attempt to align the magnetic portion of the film with the applied magnetic field vector. These magnetic body couples are considered as input (external force vector) to the FSI model. The approach is summarized in appendix A; full details can be found elsewhere [30].

The swimming velocity of the micro-swimmer will depend on the system parameters, such as the geometry and flexibility of the polymer film, viscosity of the aqueous environment, and the magnitude and angular frequency of the applied magnetic field. To explore the influence of each variable on the swimming response of the micro-swimmer, the governing (virtual work) equations can be used to derive a set of characteristic dimensionless numbers that capture the underlying physics of the chirality-induced propulsion,
2.1in which *L*_{0}/*L* is a normalized measure of the magnetized region in the polymer film [31] (figure 2), *F*_{n} is the fluid number and represents the ratio of viscous forces and elastic forces [31–33] and *M*_{n} is the magnetic number and represents the ratio of magnetic forces and elastic forces [31,32]. The derivation of these characteristic dimensionless parameters is given in appendix B. Now, *F*_{n} can be independently altered by changing the viscosity (*μ*) or frequency (*ω*=2*π*/*t*_{cycle}). Similarly, *M*_{n} can be independently controlled by changing the remanent magnetization (*M*) or the external magnetic field (*B*). Here, we choose the following (fixed) parameters: length (*L*)=100 μm, width (*W*)=20 μm, thickness (*h*)=2 μm, elastic modulus (*E*)=490.6 kPa, Poisson's ratio (*ν*)=0.3 (giving a shear modulus, *G*=188.7 kPa), remanent magnetization (*M*)=62.9 kA m^{−1} and viscosity (*μ*)=1.0 mPa s^{−1}. The period of the rotating magnetic field (*t*_{cycle}), magnetic size *L*_{0} and magnitude of the external magnetic field (*B*) are considered as variables in the analysis.

## 3. Results and discussion

### (a) Unidirectional swimming

Once the partially magnetic flexible film is actuated with a rotating external magnetic field, the chiral shape evolves as a natural consequence of the solid–fluid interaction. The viscous forces of the fluid hinder the motion of the film, while the magnetic portion of the film follows the rotating external magnetic field resulting in opposing torques that forms a twisted ribbon of chiral shape as shown in figure 2. The viscous forces form the chiral shape on-the-fly, which eventually stabilizes leading to a steady-state and unidirectional propulsion subjected to the magnetic actuation (figure 2*b*).

In figure 3*a*, we show the swimming velocity as a function of the angular frequency of the applied magnetic field (*ω*) for three different values of the magnetic field. It can be seen that the velocity increases with frequency until a critical frequency *ω*_{c} is reached. This critical frequency increases with increasing magnetic field. Given the definition of the dimensionless numbers (see equation (2.1)), the critical frequency corresponds to a specific critical value of the fluid number, (*F*_{n})_{c}. Here, we can make connection to several experimental studies showing a critical (step-out) frequency *ω*_{c} at which the velocity suddenly drops [9,15–17,22,26,27]. In general, *ω*_{c} (or, similarly, (*F*_{n})_{c}) quantifies the responsiveness of the system to an externally applied rotating magnetic field. When *F*_{n}>(*F*_{n})_{c}, the magnetic micro-swimmer is not able to follow the external magnetic field and starts lagging behind, which results in a tumbling motion and an associated sudden drop of the swimming velocity (figure 3*a*). The tumbling motion is owing to the fact that when the actuation frequency becomes larger than the critical frequency (i.e. *F*_{n}>(*F*_{n})_{c}), the magnetization (** M**) and external magnetic field (

**) vectors are not parallel anymore. Obviously, there is a competition between the magnetic (driving) forces and the viscous (opposing) forces that governs the responsiveness of a magnetic micro-swimmer. In terms of the dimensionless numbers, the ratio of the magnetic forces and viscous forces is represented by**

*B**M**=

*M*

_{n}/

*F*

_{n}∝

*MB*/

*μω*—the inverse of the Mason number [34]—which represent the responsiveness of the magneto-mechanical system. The micro-swimmer will be fully responsive (to the externally applied rotating magnetic field) when the magnetic forces will fully overcome the viscous resistance of the fluid (i.e. when

*M**>

*M**

_{c}).

To further explore the critical responsiveness, we investigate the swimming response of the chiral micro-swimmer solely in terms of the dimensionless numbers *F*_{n} and *M*_{n}. The normalized swimming velocity as a function of responsiveness of the system (*M**=*M*_{n}/*F*_{n}) for various values of *F*_{n} and for *L*_{0}=0.1*L* is shown in figure 3*b*. The swimming velocity (*U*) is normalized by the factor *ωW*^{2}/*L* that has been obtained by the first-order calculation based on resistive force theory, [29], which gives
3.1where is the maximum twist angle present in the film, and *C _{dL}* and

*C*are the local drag-coefficients for a chiral micro-swimmer in the length and thickness direction, respectively (see appendix C for the derivation). For all

_{dh}*F*

_{n}values, the swimming velocity (

*U**) initially increases with

*M** followed by a saturation phase and eventually starts decreasing slightly at higher values of

*M** (figure 3

*b*). The swimming velocity will initially increase as the film starts responding to the external magnetic field (

*M** regime between 0 and 0.5), and once the film becomes fully responsive at

*M**≥

*M**

_{c}=0.5 (see above in the discussion of figure 3

*a*), any further increase in

*M** will not get reflected into the corresponding increase in the swimming velocity, and a saturation will be observed.

The analytical expression for the velocity based on the resistive force theory (equation (3.1)) holds for a constant helicity along the swimmer and shows that the swimming velocity is directly proportional to the maximum twist angle () accommodated by the film. In the numerical simulations, however, owing to the elastic and magnetic interactions with the viscous fluid, the twist angle distribution becomes nonlinear. We therefore did not include in the normalization for *U*. To explore the relationship between twist angle and applied magnetic field, we analyse the twist angle variation along the length of the film as a function of *M*_{n} for *F*_{n}=10, *L*_{0}=0.1*L* (figure 4*a*). It can be clearly seen that the twist angle for lower values of *M*_{n} (*M**<*M**_{c}=0.5) is limited as the film is not able to overcome the fluid's resistance. The twist angle increases with *M*_{n} and eventually attains a saturation at higher values of *M*_{n} when the film becomes fully responsive (i.e. *M**≥*M**_{c}=0.5). Note that the maximum twist angle () accommodated by the film depends on the magnitude of the viscous forces (related to *F*_{n}) and will be discussed later. Interestingly, higher values of *M*_{n} result into reducing the twist angle variation inside the magnetic portion of the film and eventually reduce , which explains the slight decrease in swimming velocity for large *M*_{n} values as shown in figure 3*b*. This is a natural consequence of the external magnetic torque acting only on the magnetic portion of the film, which ultimately restrains the deformation of the magnetic part of the film at higher *M*_{n} values, and causes the magnetic portion to behave as rigid. Note that this effect becomes more prominent for higher values of *L*_{0}/*L* as shown in figure 4*a*. For a fully responsive micro-swimmer (*M**≥0.5), the influence of the magnetic portion *L*_{0}/*L* on the swimming velocity is shown in the inset of figure 4*a*. As *L*_{0}/*L* also quantifies the availability of the micro-swimmer's length to form a chiral shape, the swimming velocity is linearly dependent on *L*_{0}/*L*. *L*_{0}/*L*=0 and 1 represent the extreme cases for the magnetic portion and in both cases the swimming velocity will be zero owing to either the absence of external actuation (*L*_{0}=0) or reciprocal motion of the film (*L*_{0}=*L*). Next, we study the effect of viscous forces in the limit of fully responsive micro-swimmers (*M**=1). The response of the twist angle variation along the micro-swimmer length as a function of *F*_{n} (for *M**=1) is shown in figure 4*b*. The maximum twist angle increases with *F*_{n} which leads to an increase in the swimming velocity, as shown in the inset. Increasing *F*_{n} at constant *M** physically corresponds to decreasing the stiffness of the micro-swimmer. This leads to an increasing twist angle, and thus an increasing velocity (inset), in correspondence with equation (3.1).

### (b) Bi-directional swimming

Direction reversibility is a desirable feature for object manipulation in confined flow geometries, such as in the human arterial system and in miniaturized lab-on-a-chip devices. The approach studied in the previous section of forming a chiral shape on-the-fly by exploiting the viscous forces, severely reduces the manufacturing demands for a bacterial micro-swimmer. However, it does not have the possibility to change/reverse swimming direction. To achieve swimming bi-directionality, we suggest to magnetically control the initial chirality through two magnetic sections with opposite remanent magnetization (figure 5). Here, (controlled or non-controlled) manufacturing imperfections will serve as a perturbation to the system resulting in an initial chiral shape when the film is brought under the influence of an external magnetic field (figure 5) and swimming can be achieved by subsequently rotating the magnetic field. As the chiral shape is now independent of the rotation direction of the magnetic field, forward and backward swimming can be triggered by inverting the rotation direction. Note that the twist angle (chirality) is now explicitly induced through the initial application of the external magnetic field and has an upper limit of 180° (figure 5).

The dependence of the swimming velocity (*U**) on the responsiveness of the system (*M**=*M*_{n}/*F*_{n}) for various values of *F*_{n} and for *L*_{0}/*L*=0.05*L* is shown in figure 6*a*. For all *F*_{n} values, the swimming velocity increases initially with *M*_{n} as the system responds to the externally applied magnetic field and eventually saturates at higher *M** values. Again, any increase in the *M*_{n} value directly increases the initial twist angle in the film (as the film becomes more responsive). The upper limit for the twist angle (180°) is reached at relatively high *M** values leading to a saturation in the swimming velocity (as , see equation (3.1)). Note that the lower velocities for *M**<100 are not owing to the ‘slipping’ or ‘tumbling’ motion observed for the previous system (figure 3), but is entirely owing to the elastic resistance that limits the twist angle. The influence of *L*_{0}/*L* on the swimming velocity is linear (see inset of figure 6*a*), which is owing to the fact that the magnetized portion of the film does not contribute to the chiral shape formation as it behaves as rigid under the influence of the large external magnetic field. It is worth mentioning that there are two significant differences in the new approach; first, the swimming velocity saturates at relatively high *M** values (*M**>100^{1}), and second the swimming velocity does not go to zero when the film becomes fully magnetic (*L*_{0}/*L*=0.5) owing to the novel design of the micro-swimmer, where both (end) magnetic portions are used to form the chiral shape.

We further explore the influence of *F*_{n} on the swimming velocity for a fully responsive system (*M*_{n}∝*F*_{n}) as shown in figure 6*b*. The swimming velocity increases with *F*_{n} initially and then saturates, which is followed by small dip at large *F*_{n} values. The initial trend can be explained by the corresponding increase in *M*_{n} that increases the induced initial chirality () in the micro-swimmer. However, the observed dip at higher *F*_{n} values is related to the change in the twist angle distribution along the micro-swimmer length as shown in the inset of figure 6*b* for various values of *F*_{n}. It can be clearly seen that at higher *F*_{n} values the steady-state chirality distribution is different from what is initially induced. The large viscous forces of the fluid at higher *F*_{n} values start deforming the magnetically induced (initial) chiral shape, which results in a nonlinear distribution of chirality along the micro-swimmer length instead of a linear distribution induced initially. We have analysed the effect of nonlinearity by assuming a power-law relationship for the twist angle variation, (see appendix C). Analytical results based on resistive force theory show that the swimming velocity is maximal for a linear distribution of twist angle along the micro-swimmer length (*n*=1), see appendix C.

Finally, the two suggested concepts to obtain a chiral shape (viscosity-induced or magnetically induced) can be compared. The swimming response associated with these concepts as a function of the fluid number *F*_{n} is shown in figure 7 for the fully responsive systems. Note that the magnetically induced chirality concept should be preferred as (i) it facilitates bi-directionality in the swimming direction and (ii) the swimming velocity remains independent of the fluid number (*F*_{n}), which allows operational accommodation of different fluid viscosities (*μ*) and allows tuning the applied frequency *ω*, given that the magnetic number (*M*_{n}) is large enough to overcome the elastic resistance of the film and form a chiral shape. It is advisable to increase the length of the magnetized portion (*L*_{0}/*L*) to reduce the deformations owing to the viscous force at higher *F*_{n} values. The micro-swimmers can be fabricated with ink-jet printing or photolithography [35–38] by using magnetic nanoparticles for the two magnetic domains. However, fabricating a film with two ends magnetized in opposite directions is not straightforward. A two-step manufacturing procedure would be required using two opposing directions of the applied magnetization field in each step. The opposing polarities are established by shielding one end in the first step and the other end in the second.

## 4. Summary and conclusion

Artificial micro-swimmers can potentially be used for various bio-medical applications, such as micro-object manipulation on lab-on-a-chip devices and for targeted drug delivery. A bioinspired approach has been taken by mimicking bacterial swimming through magnetically driven chirality-induced propulsion. We have used a computational approach based on a functional polymer film whose chiral shape and rotational velocity are controlled by a rotating magnetic field. Furthermore, we have identified the key dimensionless parameters of the chirality-induced propulsion and investigated the competition between elastic, viscous and magnetic forces that are responsible for the physical response. The suggested bacterial micro-swimmers form a chiral shape on-the-fly (from an initial planar geometry) by exploiting the available physical forces (i.e. elastic, magnetic and viscous). In this on-the-fly system, viscous forces are used to induce a chiral shape during swimming. However, this does not permit bi-directionality. To overcome this, a new concept is proposed that uncouples the chiral-polarity from the external actuation by forming an initial chiral shape through two magnetic sections with opposite remanent magnetization. Forward and backward swimming can be achieved by tuning the rotation direction of the magnetic field. The micro-swimmer—a planar functional polymer film—can be easily manufactured with state-of-the-art polymer processing technologies such as inkjet printing, and opens possibilities for various bio-medical applications such as micro-object manipulation in lab-on-a-chip devices.

## Funding statement

We acknowledge the Dutch Polymer Institute (DPI) for funding under project code DPI-699 (ARTFLAG).

## Appendix A. Computational model

To solve the coupled FSI involved in the swimming of bacterial flagella, we use a three-dimensional computational framework in which the solid mechanics, fluid-dynamics and magneto-static equations are solved simultaneously. By implicitly coupling the solid mechanics and fluid dynamics equations, we incorporate the equivalent drag matrix owing to the fluid into the stiffness matrix. The magnetic body couples acting on the micro-swimmer owing to external magnetic-actuation is computed on-the-fly and supplied to the FSI model as an external force vector. Here, we briefly summarize the computational framework; for full details of the approach, the reader is referred to Khaderi & Onck [30].

**(a) Magnetostatics**

The magneto-responsive micro-swimmers are subjected to magnetic body couples (** N**) owing to the externally applied magnetic field (

**), which can be obtained from the cross-product of the film remanent magnetization (**

*B***) and the external magnetic field,**

*M***=**

*N***×**

*M***. The magnetic body couples (**

*B***) are considered as input to the FSI model and supplied as an external force vector [30], see equation (A 12). Magnetic interactions between multiple magnetic domains are assumed to be small and are therefore not accounted for.**

*N***(b) Finite-element formulation for the solid mechanics equations**

The micro-swimmer is represented by an assemblage of shell elements, which can undergo large deformation. For the shell elements, the displacement of any point on the normal can be written in terms of the displacements on the mid-surface (*u*_{0},*v*_{0},*w*_{0})
A1where *β _{x}* and

*β*are the rotations of the normal with respect to the

_{y}*x*- and

*y*-axes, respectively, as indicated in figure 8. The associated non-zero components of the Green–Lagrange strain are A2The variation of these Green–Lagrange strains can be obtained in terms of the axial strain and the curvature

*κ*using the displacement definitions given in equation (A 1) A3Similarly, A4The internal virtual work can be written as A5

where *σ*_{x}, *σ*_{y} and *σ*_{xy} are the components of the second Piola–Kichhoff stress tensor and d*V* represents an elemental volume in the undeformed configuration. Assuming the shell elements of uniform cross section with thickness *h*, we can write the above equation using an elemental area d*A* in the undeformed configuration as
A6where , ** κ**=[

*κ*

_{x}*κ*

_{y}*κ*]

_{xy}^{T}, and

**and**

*P***are the associated membrane forces and bending moments, respectively [30]. The internal virtual work at time**

*M**t*+Δ

*t*can be written as A7which can be expanded linearly in time by assuming

*Q*

^{t+Δt}=

*Q*

^{t}+Δ

*Q*, and can be simplified by neglecting the higher order terms, leading to A8We use the finite-element formulation to discretize the system in terms of the nodal displacements (

*u*,

*v*,

*w*) and rotations (

*θ*,

_{x}*θ*,

_{y}*θ*) of three-noded triangular elements with A9where the subscripts 1, 2 and 3 denote the respective three nodes of the element.

_{z}To capture the deformation of the films, both the bending and membrane stiffnesses (stiffness associated with the in-plane stretching) have to be accurately modelled. To do so, we adopt the approach proposed by Bathe & Ho [39] and model the bending of the films using discrete Kirchhoff triangles, while the membrane behaviour is accounted for using constant strain triangles (CSTs) [30]. To improve the accuracy during in-plane bending, we add drilling degrees of freedom to the CSTs [40], and adopt an updated Lagrangian framework to arrive at the final set of equations. The resulting stiffness matrix includes the geometric nonlinearity, which accounts for large deformations but small strains [30], see below.
A10where *k*_{M} is the material stiffness matrix, *k*_{G} is the geometric stiffness matrix and *f*_{int} is the internal force vector [30]. After performing the standard finite-element assembly and coordinates transformation (local-to-global), we obtain
A11

The external virtual work at time *t*+Δ*t* can be written as
A12where *N _{x}* and

*N*are the externally applied magnetic body couples (see appendix A

_{y}*a*), and

*V*

_{m}is the magnetic portion of the film. After discretization and the standard finite-element assembly procedure, the external virtual work can be written as [30] A13

**(c) Boundary-element formulation for the fluid dynamics equations**

For the fluid we make use of the Stokeslet approach in which the force exerted on the fluid at the surface of the solid structure is approximated by the distribution of Stokeslets along the length of the structure. The velocity and force fields are related by a Green's function that has a singularity proportional to 1/*r* in three dimensions [41]. The expression of Green's function (** G**) for the Stokeslets relates the velocity at location

**, , to the forces at location**

*r***′,**

*r***, through A14where**

*f***=**

*R***−**

*r***′, is the distance between the two locations**

*r***and**

*r***′, and**

*r**δ*is the Kronecker delta. By assuming the point force

_{ij}**to be represented by the traction**

*f***=**

*f***d**

*T**S*over the solid surface, the boundary-integral equation can be written as A15where

**are the tractions imposed on the fluid [33]. In equation (A 15), the boundary-integral equation has been discretized using boundary elements (three-noded shell elements), and the tractions are linearly interpolated using**

*T***=**

*T*

*N***with**

*t***being the tractions at the nodes. When equation (A 15) is used to evaluate the velocity in all nodes of the micro-swimmer, we obtain a system of equations that relates the traction**

*t***exerted by the micro-swimmer on the fluid to its velocity . The integration procedure is adopted from the literature, where the singular integrals are evaluated using the method of change of variables [41] and the non-singular integrals are evaluated using standard two-dimensional Gaussian Quadrature [30]. Once the velocity of the solid surface is known, this relation can be inverted to obtain the nodal tractions [33]: .**

*t***(d) Fluid–solid interaction and implicit coupling**

Coupling of the solid mechanics and fluid dynamics equations will be done in an implicit manner by incorporating the equivalent drag matrix owing to the fluid into the stiffness matrix. The external virtual work owing to the fluid's drag forces (*T*_{d}) can be given as
A16where , ** u** is the displacement vector, and

*d*_{e}is the local nodal displacement vector [30]. Note that the minus sign appears because of the change of reference (from fluid to the structure,

*T*_{d}=−

**). After performing the standard finite-element assembly procedure, we obtain A17where the matrix**

*T*

*G*_{f}relates the velocity of the solid structure to the traction, see the end of the previous subsection. Now, using the no-slip boundary condition it follows that A18where is an equivalent drag matrix and is the stiffness contribution owing to the presence of the fluid [30,33].

**is a matrix that eliminates the rotational degrees of freedom from the global displacement vector Δ**

*A***. Finally, invoking the arbitrariness of the virtual displacements and equating the internal (equation (A 11)) and the external virtual work (the sum of equations (A 13) and (A 18)) we obtain the following equation of motion for the FSI problem, which will be solved to obtain the incremental displacements (Δ**

*d***) after incorporating the appropriate boundary conditions [30]. A19**

*d*## Appendix B. Dimensional analysis for the chiral micro-swimmers

Starting point for the dimensional analysis will be the virtual work expression, given by *δW*_{int}=*δW*_{ext}, see equations (A 6), (A 12) and (A 16), for a film of thickness *h*:
B1Assuming the evolved chiral shape is mainly twisting dominated and the in-plane deformations are negligible, the above equation can be simplified into
B2where *M _{xy}*=(

*Gh*

^{3}/12)

*κ*,

_{xy}*κ*=(∂/∂

_{xy}*y*)(∂

*w*/∂

*x*)−(∂/∂

*x*)(∂

*w*/∂

*y*),

*G*being the shear modulus, and

*L*and

*W*are the length and width of the film, respectively. As the film is partially magnetic, the magnetic body couple will only act on the magnetic portion (

*L*

_{0}) of the film. We now introduce dimensionless variables (.)* such that

*x*=

*X**

*L*,

*y*=

*Y**

*W*and

*A*=

*A**

*LW*(or

*A*=

*A**

*L*

_{0}

*W*for the magnetic part). From the standard torsion formula, the twist angle follows to be

*θ*

_{twist}=

*TL*/

*GJ*, with

*T*and

*J*being the applied torque and the second polar moment of area, respectively. As

*J*∝

*Wh*

^{3}for rectangular cross sections, [42],

*θ*

_{twist}∝

*L*/

*W*for a given value of the film thickness, which leads to

*w*∝

*θ*

_{twist}

*W*≈

*L*or

*w*=

*W**

*L*. Substitution yields the following pre-factors for the elastic, viscous and magnetic parts B3It can be noted that the choice of the characteristic length in the

*x*-direction is not the same for all integrals in the above equation, which leads to a governing dimensionless (length) parameter,

*L*

_{0}/

*L*, defining the normalized length of the magnetic portion in the elastica. Normalization with the elastic term reveals the following governing dimensionless (force) parameters [32,33]: (i) the magnetic number,

*M*

_{n}=12

*N*

_{x}WL_{0}/

*Gh*

^{2}, i.e. the ratio of magnetic to elastic forces, and (ii) the fluid number,

*F*

_{n}=12

*LW*

^{2}

*T*/

_{d}*Gh*

^{3}, i.e. the ratio of fluid to elastic forces.

^{2}From dimensional considerations,

*T*should scale with or

_{d}*μωL*/

*W*, where

*μ*is viscosity of the fluid and

*ω*=2

*π*/

*t*

_{cycle}is the angular frequency of the magnetic field actuation. Thus, the final form of the three governing dimensionless parameters is as follows:

(i) the fraction of film that is magnetic, B4

(ii) the magnetic number, B5

(iii) the fluid number, B6

Note that *F*_{n} can be independently altered by changing the viscosity (*μ*) or frequency (*ω*). Similarly, *M*_{n} can be independently controlled with the remanent magnetization (*M*) or with the external magnetic field (*B*), which eventually influences the magnetic body couple (*N _{x}*).

## Appendix C. Analytical study for chirality-induced propulsion

Resistive force theory suggested by Gray & Hancock [29] can be used to get the first-order approximation of the swimming velocity associated with the chirality-induced propulsion. As the chirality is induced owing to the twisting of the film, it can be assumed that the twist angle (*θ*) varies along the length of the micro-swimmer leading to a twist gradient of (*θ*_{2}−*θ*_{1})/d*x* as shown in figure 9. Assuming that the local drag-coefficients for a chiral micro-swimmer are *C _{dL}*,

*C*and

_{dW}*C*in the length, width and thickness direction, respectively, the

_{dh}*x*-component of total force on the chiral micro-swimmer of length

*L*and width

*W*can be written as C1where, C2with d

*F*being the force acting on a segment of length d

_{x}*x*and width d

*y*, and

*θ*=

_{y}*y*d

*θ*/d

*x*is the developed helix or the pitch angle. Note that the propulsion force in the direction perpendicular to the plane of actuation (

*F*) results from the difference of the viscous drag in the tangential and normal directions,

_{x}*C*and

_{dL}*C*, respectively (see equation (C 2)). The above expression is derived assuming that the micro-swimmer is swimming with a velocity

_{dh}*U*and is externally actuated with

*ω*being the angular frequency of actuation (the angular frequency of the external magnetic field in this case; figure 9). In addition, the total force consists of a propulsive part owing to the actuation forces and a retarding part owing to the drag forces opposing the horizontal swimming velocity (the part associated with

*U*). When the micro-swimmer reaches a steady-state swimming velocity

*U*, the propulsive and retarding forces are in (dynamic) equilibrium, so that the total force must be zero [33].

For the general case of chirality-induced propulsion, we account for the nonlinear variation of twist angle by assuming a power-law relationship, , where the power-law-exponent (*n*) quantifies the nonlinearity of the twist angle variation. We begin with obtaining a closed-form expression for the swimming velocity assuming a linear variation for the twist angle (i.e. *n*=1) and considering only the first-order terms (i.e. higher order terms are neglected), which leads to
C3The above expression suggests that the swimming velocity (*U*) scales linearly with the frequency of actuation *ω* and quadratically with the width of the film *W*. Additionally, the swimming velocity is directly proportional to the maximum twist gradient in the film . Next, we explore the influence of nonlinearity in the twist angle variation (i.e. *n*≠1) on the swimming velocity. Here, we consider the full form of the total force without neglecting the higher order terms and solve the equation numerically using `Mathematica`.^{3} The results are shown in figure 10 for *U*_{normalized}=*U*/*U*_{n=1}, where we explicitly assume *C _{dh}*=2

*C*. It can be clearly seen that the swimming velocity will be maximal when

_{dL}*n*≈1. In other words, the linear variation of the twist angle along the micro-swimmer length gives the maximal velocity for a given set of parameters ( and

*W*). This can be interpreted in the following way. For the cases of

*n*≈1, the whole length of the micro-swimmer will be effectively used to generate propulsion as the twist gradient is uniform along the length. However, once we deviate from linearity (

*n*≠0), the portion of the film having the maximum twist gradient is responsible for propulsion, while the rest of the film with a lower twist gradient (mainly) contributes to the opposing drag forces, which leads to a decrease in the swimming velocity. Note that the swimming velocity will be zero for

*n*=0 owing to reciprocal motion of the film in the absence of any twist gradient along the length.

Note that the analytical study does not provide a complete picture for the physical phenomenon. For instance, the elasticity of the film is not accounted for and the role of viscous and magnetic forces are not clearly distinguished. Nevertheless, it is interesting that such a simple analysis can give useful insights on the physics of the problem and can provide an accurate measure of the scaling .

## Footnotes

↵1 For this study, aspect ratio (

*L*/*W*) of the film is 5. For larger aspect ratios of the film, the*M** value required for the saturation in the swimming velocity will be less as the twist angle (for a given external torque)*θ*∝*L*/*W*.↵2 Note that the factor 12 can be removed without loss of generality. Here, we decide to keep it in for historical reasons.

↵3 Mathematica is a trademark of Wolfram Research, Inc.

- Received August 12, 2013.
- Accepted November 26, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.