Breaking of symmetry in microfluidic propulsion driven by artificial cilia

Microfluidic mixing by magnetic particles: Progress and prospects

Microfluidic systems have enormous potential for enabling point-of-care diagnostics due to a number of advantages, such as low sample volumes, small footprint, low energy requirements, uncomplicated setup, high surface-to-volume ratios, cost-effectiveness, etc. However, fluid mixing operations are constrained by molecular diffusion since the flow is usually in the laminar regime. The slow nature of molecular diffusion is a technological barrier to implementing fluid transformations in a reasonable time. In this context, magnetically actuated micro-mixers of different sizes, shapes, materials, and actuation techniques provide a way to enhance fluid mixing in microfluidic devices. In this paper, we review the currently existing micro-mixing technologies. From a fundamental perspective, the different magnetization models for permanent and induced dipoles are discussed. The single-particle dynamics in steady and oscillating magnetic fields is studied in order to determine the flow generated and the torque exerted on the fluid due to the magnetic particles. The effect of particle interactions, both magnetic and hydrodynamic, is examined.

Natural Cilia and Pine Needles Combinedly Inspired Asymmetric Pillar Actuators for All-Space Liquid Transport and Self-Regulated Robotic Locomotion

Natural structures and motion behaviors open new avenues for effective small-scale transport, such as the plant-inspired energy-free liquid transport surfaces and cilia-inspired propulsion systems. However, they are restricted by either the fixed structure or nonself-regulating beating modes, making many complex tasks remain challenging, e.g., the controllable multidirectional liquid transport and flexible propulsion. Herein, inspired by pine needles and natural cilia, we report an asymmetric-structured intelligent magnetic pillar actuator (AI-MPA) with both the "passive" and "active" transport features. Under the control of the magnetic field, the AI-MPA shows an all-space liquid transport ability toward arbitrary directions. Moreover, benefiting from the material's magnetoelasticity and asymmetric-structured design, the AI-MPA enables self-regulation of two-dimensional (2D)/three-dimensional (3D) cilia-like beating modes and can be further developed for robotic crawling and self-rotatable motion. The AI-MPA integrates the superiority of static and dynamic systems in nature and exhibits intelligent self-regulation that could not be achieved before. Confirmed theoretically and demonstrated experimentally, this work provides insights into increasingly functional and intelligent miniature biomimetic systems, with applications from directional liquid transport to robotic locomotion.

Microscopic artificial cilia - a review

Cilia are microscopic hair-like external cell organelles that are ubiquitously present in nature, also within the human body. They fulfill crucial biological functions: motile cilia provide transportation of fluids and cells, and immotile cilia sense shear stress and concentrations of chemical species. Inspired by nature, scientists have developed artificial cilia mimicking the functions of biological cilia, aiming at application in microfluidic devices like lab-on-chip or organ-on-chip. By actuating the artificial cilia, for example by a magnetic field, an electric field, or pneumatics, microfluidic flow can be generated and particles can be transported. Other functions that have been explored are anti-biofouling and flow sensing. We provide a critical review of the progress in artificial cilia research and development as well as an evaluation of its future potential. We cover all aspects from fabrication approaches, actuation principles, artificial cilia functions - flow generation, particle transport and flow sensing - to applications. In addition to in-depth analyses of the current state of knowledge, we provide classifications of the different approaches and quantitative comparisons of the results obtained. We conclude that artificial cilia research is very much alive, with some concepts close to industrial implementation, and other developments just starting to open novel scientific opportunities.

Cilia metasurfaces for electronically programmable microfluidic manipulation

Cilial pumping is a powerful strategy used by biological organisms to control and manipulate fluids at the microscale. However, despite numerous recent advances in optically, magnetically and electrically driven actuation, development of an engineered cilial platform with the potential for applications has remained difficult to realize^1-6. Here we report on active metasurfaces of electronically actuated artificial cilia that can create arbitrary flow patterns in liquids near a surface. We first create voltage-actuated cilia that generate non-reciprocal motions to drive surface flows at tens of microns per second at actuation voltages of 1 volt. We then show that a cilia unit cell can locally create a range of elemental flow geometries. By combining these unit cells, we create an active cilia metasurface that can generate and switch between any desired surface flow pattern. Finally, we integrate the cilia with a light-powered complementary metal-oxide-semiconductor (CMOS) clock circuit to demonstrate wireless operation. As a proof of concept, we use this circuit to output voltage pulses with various phase delays to demonstrate improved pumping efficiency using metachronal waves. These powerful results, demonstrated experimentally and confirmed using theoretical computations, illustrate a pathway towards fine-scale microfluidic manipulation, with applications from microfluidic pumping to microrobotic locomotion.

Biomimetic Liquid Crystal Cilia and Flagella

Cilia and flagella are a vital part of many organisms. Protozoa such as paramecia rely on the collective and coordinated beating of tubular cilia or flagella for their transport, while mammals depend on the ciliated linings of their bronchia and female reproductive tracts for the continuity of breathing and reproduction, respectively. Over the years, man has attempted to mimic these natural cilia using synthetic materials such as elastomers doped with magnetic particles or light responsive liquid crystal networks. In this review, we will focus on the progress that has been made in mimicking natural cilia and flagella using liquid crystal polymers. We will discuss the progress that has been made in mimicking natural cilia and flagella with liquid crystal polymers using techniques such as fibre drawing, additive manufacturing, or replica moulding, where we will put additional focus on the emergence of asymmetrical and out-of-plane motions.

Morphological Control of Cilia-Inspired Asymmetric Movements Using Nonlinear Soft Inflatable Actuators

Soft robotic systems typically follow conventional control schemes, where actuators are supplied with dedicated inputs that are regulated through software. However, in recent years an alternative trend is being explored, where the control architecture can be simplified by harnessing the passive mechanical characteristics of the soft robotic system. This approach is named “morphological control”, and it can be used to decrease the number of components (tubing, valves and regulators) required by the controller. In this paper, we demonstrate morphological control of bio-inspired asymmetric motions for systems of soft bending actuators that are interconnected with passive flow restrictors. We introduce bending actuators consisting out of a cylindrical latex balloon in a flexible PVC shell. By tuning the radii of the tube and the shell, we obtain a nonlinear relation between internal pressure and volume in the actuator with a peak and valley in pressure. Because of the nonlinear characteristics of the actuators, they can be assembled in a system with a single pressure input where they bend in a discrete, preprogrammed sequence. We design and analyze two such systems inspired by the asymmetric movements of biological cilia. The first replicates the swept area of individual cilia, having a different forward and backward stroke, and the second generates a travelling wave across an array of cilia.

Metachronal patterns in artificial cilia for low Reynolds number fluid propulsion

Cilia are hair-like organelles, present in arrays that collectively beat to generate flow. Given their small size and consequent low Reynolds numbers, asymmetric motions are necessary to create a net flow. Here, we developed an array of six soft robotic cilia, which are individually addressable, to both mimic nature's symmetry-breaking mechanisms and control asymmetries to study their influence on fluid propulsion. Our experimental tests are corroborated with fluid dynamics simulations, where we find a good agreement between both and show how the kymographs of the flow are related to the phase shift of the metachronal waves. Compared to synchronous beating, we report a 50% increase of net flow speed when cilia move in an antiplectic wave with phase shift of -π/3 and a decrease for symplectic waves. Furthermore, we observe the formation of traveling vortices in the direction of the wave when metachrony is applied.

Metachronal Actuation of Microscale Magnetic Artificial Cilia

Biological cells often interact with the environment through carpets of microscopic hair-like cilia. These elastic structures are known to beat in a synchronized wavy fashion called metachronal motion to produce fluid transport. Metachronal motion emerges due to a phase difference between beating cycles of neighboring cilia and appears as traveling waves propagating along the ciliary carpet. We demonstrate submerged in water microscale magnetic cilia that are externally actuated to beat in a metachronal fashion. Two approaches are used to induce coordinated phase differences among the beating cilia. In the first case, we fabricate cilia with an imposed gradient of geometrical properties that are subject to a rotating uniform magnetic field. In the second scenario, a ciliary array is composed of identical cilia that experience a magnetic field that varies spatiotemporally. We demonstrate that magnetic cilia can achieve symplectic, antiplectic, and leoplectic metachrony.

Magnetic Actuation of Surface Walkers: The Effects of Confinement and Inertia

Driven by a magnetic field, the rotation of a particle near a wall can be rectified into a net translation. The particles thus actuated, or surface walkers, are a kind of active colloid that finds application in biology and microfluidics. Here, we investigate the motion of spherical surface walkers confined between two walls using simulations based on the immersed-boundary lattice Boltzmann method. The degree of confinement and the nature of the confining walls (slip vs no-slip) significantly affect a particle's translational speed and can even reverse its translational direction. When the rotational Reynolds number Re_ω is larger than 1, inertia effects reduce the critical frequency of the magnetic field, beyond which the sphere can no longer follow the external rotating field. The reduction of the critical frequency is especially pronounced when the sphere is confined near a no-slip wall. As Re_ω increases beyond 1, even when the sphere can still rotate in the synchronous regime, its translational Reynolds number Re_T no longer increases linearly with Re_ω and even decreases when Re_ω exceeds ∼10.

Anti-Biofouling and Self-Cleaning Surfaces Featured with Magnetic Artificial Cilia

The fouling of surfaces submerged in a liquid is a serious problem for many applications including lab-on-a-chip devices and marine sensors. Inspired by the versatility of cilia in manipulating fluids and particles, it is experimentally demonstrated that surfaces partially covered with magnetic artificial cilia (MAC) have the capacity to efficiently prevent attachment and adhesion of real biofouling agents-microalgae Scenedesmus sp. Actuation of the MAC resulted in over 99% removal of the algae for two different scenarios: (1) actuating the MAC immediately after injecting the algae into a microfluidic chip, demonstrating antifouling and (2) starting to actuate the MAC 1 week after injecting the algae into the chip and leaving them to grow in static conditions, showing self-cleaning. It is shown that the local and global flows generated by the actuated MAC are substantial, resulting in hydrodynamic shear forces acting on the algae, which are likely to be key to efficient antifouling and self-cleaning. These findings and insights will potentially lead to novel types of self-cleaning and antifouling strategies, which may have a relevant practical impact on different fields and applications including lab-on-a-chip devices and water quality analyzers.

Why antiplectic metachronal cilia waves are optimal to transport bronchial mucus

The coordinated beating of epithelial cilia in human lungs is a fascinating problem from the hydrodynamics perspective. The phase lag between neighboring cilia is able to generate collective cilia motions, known as metachronal waves. Different kinds of waves can occur, antiplectic or symplectic, depending on the direction of the wave with respect to the flow direction. It is shown here, using a coupled lattice Boltzmann-immersed boundary solver, that the key mechanism responsible for their transport efficiency is a blowing-suction effect that displaces the interface between the periciliary liquid and the mucus phase. The contribution of this mechanism on the average flow generated by the cilia is compared to the contribution of the lubrication effect. The results reveal that the interface displacement is the main mechanism responsible for the better efficiency of antiplectic metachronal waves over symplectic ones to transport bronchial mucus. The conclusions drawn here can be extended to any two-layer fluid configuration having different viscosities, and put into motion by cilia-shaped or comb-plate structures, having a back-and-forth motion with phase lags.

Microfluidic Magnetic Mixing at Low Reynolds Numbers and in Stagnant Fluids

Microfluidic mixing becomes a necessity when thorough sample homogenization is required in small volumes of fluid, such as in lab-on-a-chip devices. For example, efficient mixing is extraordinarily challenging in capillary-filling microfluidic devices and in microchambers with stagnant fluids. To address this issue, specifically designed geometrical features can enhance the effect of diffusion and provide efficient mixing by inducing chaotic fluid flow. This scheme is known as "passive" mixing. In addition, when rapid and global mixing is essential, "active" mixing can be applied by exploiting an external source. In particular, magnetic mixing (where a magnetic field acts to stimulate mixing) shows great potential for high mixing efficiency. This method generally involves magnetic beads and external (or integrated) magnets for the creation of chaotic motion in the device. However, there is still plenty of room for exploiting the potential of magnetic beads for mixing applications. Therefore, this review article focuses on the advantages of magnetic bead mixing along with recommendations on improving mixing in low Reynolds number flows (Re ≤ 1) and in stagnant fluids.

Microfluidic pumping using artificial magnetic cilia

One of the vital functions of naturally occurring cilia is fluid transport. Biological cilia use spatially asymmetric strokes to generate a net fluid flow that can be utilized for feeding, swimming, and other functions. Biomimetic synthetic cilia with similar asymmetric beating can be useful for fluid manipulations in lab-on-chip devices. In this paper, we demonstrate the microfluidic pumping by magnetically actuated synthetic cilia arranged in multi-row arrays. We use a microchannel loop to visualize flow created by the ciliary array and to examine pumping for a range of cilia and microchannel parameters. We show that magnetic cilia can achieve flow rates of up to 11 μl/min with the pressure drop of ~1 Pa. Such magnetic ciliary array can be useful in microfluidic applications requiring rapid and controlled fluid transport.

Transport and Mixing Induced by Beating Cilia in Human Airways

The fluid transport and mixing induced by beating cilia, present in the bronchial airways, are studied using a coupled lattice Boltzmann-Immersed Boundary solver. This solver allows the simulation of both single and multi-component fluid flows around moving solid boundaries. The cilia are modeled by a set of Lagrangian points, and Immersed Boundary forces are computed onto these points in order to ensure the no-slip velocity conditions between the cilia and the fluids. The cilia are immersed in a two-layer environment: the periciliary layer (PCL) and the mucus above it. The motion of the cilia is prescribed, as well as the phase lag between two cilia in order to obtain a typical collective motion of cilia, known as metachronal waves. The results obtained from a parametric study show that antiplectic metachronal waves are the most efficient regarding the fluid transport. A specific value of phase lag, which generates the larger mucus transport, is identified. The mixing is studied using several populations of tracers initially seeded into the pericilary liquid, in the mucus just above the PCL-mucus interface, and in the mucus far away from the interface. We observe that each zone exhibits different chaotic mixing properties. The larger mixing is obtained in the PCL layer where only a few beating cycles of the cilia are required to obtain a full mixing, while above the interface, the mixing is weaker and takes more time. Almost no mixing is observed within the mucus, and almost all the tracers do not penetrate the PCL layer. Lyapunov exponents are also computed for specific locations to assess how the mixing is performed locally. Two time scales are introduced to allow a comparison between mixing induced by fluid advection and by molecular diffusion. These results are relevant in the context of respiratory flows to investigate the transport of drugs for patients suffering from chronic respiratory diseases.

Asymmetric motion of magnetically actuated artificial cilia

Most microorganisms use hair-like cilia with asymmetric beating to perform vital bio-physical processes. In this paper, we demonstrate a novel fabrication method for creating magnetic artificial cilia capable of such a biologically inspired asymmetric beating pattern essential for inducing microfluidic transport at low Reynolds number. The cilia are fabricated using a lithographic process in conjunction with deposition of magnetic nickel-iron permalloy to create flexible filaments that can be manipulated by varying an external magnetic field. A rotating permanent magnet is used to actuate the cilia. We examine the kinematics of a cilium and demonstrate that the cilium motion is defined by an interplay among elastic, magnetic, and viscous forces. Specifically, the forward stroke is induced by the rotation of the magnet which bends the cilium, whereas the recovery stroke is defined by the straightening of the deformed cilium, releasing accumulated elastic potential energy. This difference in dominating forces acting during the forward stroke and the recovery stroke leads to an asymmetric beating pattern of the cilium. Such magnetic cilia can find applications in microfluidic pumping, mixing, and other fluid handling processes.

Individually Controllable Magnetic Cilia: Mixing Application

This paper introduces a new design for individually controlled magnetic artificial cilia for use in fluid devices and specifically intended to improve the mixing in DNA microarray experiments. The design has been implemented using a low-cost prototype that can be fabricated using polydimethylsiloxane (PDMS) and off-the-shelf parts and achieves large cilium deflections (59% of the cilium length). The device's performance is measured via a series of mixing experiments using different actuation patterns inspired by the blinking vortex theory. The experimental results, quantified using the relative standard deviation of the color when mixing two colored inks, show that exploiting the individual control leads to faster mixing (38% reduction in mixing time) than when operating the device in a simultaneous-actuation mode with the same average cilium beat frequency. Furthermore, the experimental results show an optimal beating pattern that minimizes the mixing time. The existence and character of this optimum is predicted by simulations using a blinking-vortex approach for 2D ideal flow, suggesting that the blinking-vortex model can be used to predict the effect of parameter variation on the experimental system.

Magnetically actuated artificial cilia: the effect of fluid inertia

Natural cilia are hairlike microtubule-based structures that are able to move fluid on the micrometer scale using asymmetric motion. In this article, we follow a biomimetic approach to design artificial cilia lining the inner surfaces of microfluidic channels with the goal of propelling fluid. The artificial cilia consist of polymer films filled with superparamagnetic nanoparticles, which can mimic the motion of natural cilia when subjected to a rotating magnetic field. To obtain the magnetic field and associated magnetization local to the cilia, we solve the Maxwell equations, from which the magnetic body moments and forces can be deduced. To obtain the ciliary motion, we solve the dynamic equations of motion, which are then fully coupled to the Navier-Stokes equations that describe the fluid flow around the cilia, thus taking full account of fluid inertial forces. The dimensionless parameters that govern the deformation behavior of the cilia and the associated fluid flow are arrived at using the principle of virtual work. The physical response of the cilia and the fluid flow for different combinations of elastic, fluid viscous, and inertia forces are identified.

Fluid flow due to collective non-reciprocal motion of symmetrically-beating artificial cilia

Using a magneto-mechanical solid-fluid numerical model for permanently magnetic artificial cilia, we show that the metachronal motion of symmetrically beating cilia establishes a net pressure gradient in the direction of the metachronal wave, which creates a unidirectional flow. The flow generated is characterised as a function of the cilia spacing, the length of the metachronal wave, and a dimensionless parameter that characterises the relative importance of the viscous forces over the elastic forces in the cilia.

Propulsion and trapping of microparticles by active cilia arrays

We model the transport of a microscopic particle via a regular array of beating elastic cilia, whose tips experience an adhesive interaction with the particle's surface. At optimal adhesion strength, the average particle velocity is maximized. Using simulations spanning a range of cilia stiffness and cilia-particle adhesion strength, we explore the parameter space over which the particle can be "released", "propelled", or "trapped" by the cilia. We use a lower-order model to predict parameters for which the cilia are able to "propel" the particle. This is the first study that shows how both stiffness and adhesion strength are crucial for manipulation of particles by active cilia arrays. These results can facilitate the design of synthetic cilia that integrate adhesive and hydrodynamic interactions to selectively repel or trap particulates. Surfaces that are effective at repelling particulates are valuable for antifouling applications, while surfaces that can trap and, thus, remove particulates from the solution are useful for efficient filtration systems.

Unperturbing a non-helically perturbed bacterial flagellar filament: Salmonella typhimurium SJW23

Salmonella typhimurium SJW23 has a right-handed, non-helically perturbed filament of serotype gt with a unique surface pattern. Non-helical perturbations involve symmetry reduction along the five-start helical lines resulting in layer lines of fractional Bessel orders and a consequent seam. The flagellin gene, fliC(23), which we sequenced, differs from the sequence of the canonic, plain SJW1655 flagellin, fliC(1655). We modified discrete components of fliC(23) in order to localize, in the expressed filament, the submolecular site responsible for the non-helical perturbation. These modifications include (i) deleting the outermost domain D3(23), (ii) replacing D3(23) with D3(1655), (iii) substituting a hydrophilic α-helix at the interface between the neighboring domains D1 and D2 with a hydrophobic one from fliC(1655), and (iv) substituting a serine/glycine pair in the loop connecting the modified α-helix to its neighbor; these modifications were made in the presence and absence of D3(23). We used S. typhimurium SJW1655 both as a reference and as a source for 'spare parts'. The symmetry of the constructs was assessed from the power spectra through changes in the layer lines at a height of 1/105 and 1/35 Å(-1), unique to the non-helical perturbation. Deleting D3(23), either alone or in combination with various substitutions, or replacing it with D3(1655) transforms the non-helically perturbed filament into a plain one as judged by the disappearance of the typical layer lines from the power spectra. We conclude that the non-helical perturbation is a product of unique interactions in the D3(23) density shell. Whereas other minor structural changes may occur at the filaments interior, they are all helically symmetric.

Analysis of fluid flow around a beating artificial cilium

Biological cilia are found on surfaces of some microorganisms and on surfaces of many eukaryotic cells where they interact with the surrounding fluid. The periodic beating of the cilia is asymmetric, resulting in directed swimming of unicellular organisms or in generation of a fluid flow above a ciliated surface in multicellular ones. Following the biological example, externally driven artificial cilia have recently been successfully implemented as micropumps and mixers. However, biomimetic systems are useful not only in microfluidic applications, but can also serve as model systems for the study of fundamental hydrodynamic phenomena in biological samples. To gain insight into the basic principles governing propulsion and fluid pumping on a micron level, we investigated hydrodynamics around one beating artificial cilium. The cilium was composed of superparamagnetic particles and driven along a tilted cone by a varying external magnetic field. Nonmagnetic tracer particles were used for monitoring the fluid flow generated by the cilium. The average flow velocity in the pumping direction was obtained as a function of different parameters, such as the rotation frequency, the asymmetry of the beat pattern, and the cilium length. We also calculated the velocity field around the beating cilium by using the analytical far-field expansion. The measured average flow velocity and the theoretical prediction show an excellent agreement.

Swimming direction reversal of flagella through ciliary motion of mastigonemes

Bio-inspired designs can provide an answer to engineering problems such as swimming strategies at the micron or nano-scale. Scientists are now designing artificial micro-swimmers that can mimic flagella-powered swimming of micro-organisms. In an application such as lab-on-a-chip in which micro-object manipulation in small flow geometries could be achieved by micro-swimmers, control of the swimming direction becomes an important aspect for retrieval and control of the micro-swimmer. A bio-inspired approach for swimming direction reversal (a flagellum bearing mastigonemes) can be used to design such a system and is being explored in the present work. We analyze the system using a computational framework in which the equations of solid mechanics and fluid dynamics are solved simultaneously. The fluid dynamics of Stokes flow is represented by a 2D Stokeslets approach while the solid mechanics behavior is realized using Euler-Bernoulli beam elements. The working principle of a flagellum bearing mastigonemes can be broken up into two parts: (1) the contribution of the base flagellum and (2) the contribution of mastigonemes, which act like cilia. These contributions are counteractive, and the net motion (velocity and direction) is a superposition of the two. In the present work, we also perform a dimensional analysis to understand the underlying physics associated with the system parameters such as the height of the mastigonemes, the number of mastigonemes, the flagellar wave length and amplitude, the flagellum length, and mastigonemes rigidity. Our results provide fundamental physical insight on the swimming of a flagellum with mastigonemes, and it provides guidelines for the design of artificial flagellar systems.

Measurement of fluid flow generated by artificial cilia

We observed and measured the fluid flow that was generated by an artificial cilium. The cilium was composed of superparamagnetic microspheres, in which magnetic dipole moments were induced by an external magnetic field. The interaction between the dipole moments resulted in formation of long chains-cilia, and the same external magnetic field was also used to drive the cilia in a periodic manner. Asymmetric periodic motion of the cilium resulted in generation of fluid flow and net pumping of the surrounding fluid. The flow and pumping performance were closely monitored by introducing small fluorescent tracer particles into the system. By detecting their motion, the fluid flow around an individual cilium was mapped and the flow velocities measured. We confirm that symmetric periodic beating of one cilium results in vortical motion only, whereas asymmetry is required for additional translational motion. We determine the effect of asymmetry on the pumping performance of a cilium, verify the theoretically predicted optimal pumping conditions, and determine the fluid behaviour around a linear array of three neighbouring cilia. In this case, the contributions of neighbouring cilia enhance the maximal flow velocity compared with a single cilium and contribute to a more uniform translational flow above the surface.

Magnetically-actuated artificial cilia for microfluidic propulsion

In this paper we quantitatively analyse the performance of magnetically-driven artificial cilia for lab-on-a-chip applications. The artificial cilia are fabricated using thin polymer films with embedded magnetic nano-particles and their deformation is studied under different external magnetic fields and flows. A coupled magneto-mechanical solid-fluid model that accurately captures the interaction between the magnetic field, cilia and fluid is used to simulate the cilia motion. The elastic and magnetic properties of the cilia are obtained by fitting the results of the computational model to the experimental data. The performance of the artificial cilia with a non-uniform cross-section is characterised using the numerical model for two channel configurations that are of practical importance: an open-loop and a closed-loop channel. We predict that the flow and pressure head generated by the artificial cilia can be as high as 18 microlitres per minute and 3 mm of water, respectively. We also study the effect of metachronal waves on the flow generated and show that the fluid propelled increases drastically compared to synchronously beating cilia, and is unidirectional. This increase is significant even when the phase difference between adjacent cilia is small. The obtained results provide guidelines for the optimal design of magnetically-driven artificial cilia for microfluidic propulsion.