Designing Bioinspired Artificial Cilia to Regulate Particle–Surface

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Designing Bioinspired Artificial Cilia to Regulate Particle−Surface Interactions Anna C. Balazs,*,† Amitabh Bhattacharya,‡ Anurag Tripathi,§ and Henry Shum† †

Department of Chemical and Petroleum Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15261, United States Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India § Department of Chemical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India ‡

ABSTRACT: Biological cilia play a critical role in a stunning array of vital functions, from enabling marine organisms to trap food and expel fouling agents to facilitating the effective transport of egg cells in mammals. Inspired by the performance of these microscopic, hair-like filaments, researchers are synthesizing artificial cilia for use in lab-on-a-chip devices. There have, however, been few attempts to harness the artificial cilia to regulate the movement of particulates in these devices. Here, we review recent computational studies on the interactions between actuated artificial cilia and microscopic particles, showing that these cilia are effective at transporting both rigid and deformable particles in microchannels. The findings also reveal that these beating filaments can be used to separate microparticles based on their size and stiffness. Importantly, these studies indicate that artificial cilia can be used to prevent fouling by a wide variety of agents because they can expel both passive particulates and active swimmers from the underlying surface. These results can help guide experimental efforts to fully exploit artificial cilia in controlling particle motion within fluid environments.

A

large variety of biological organisms utilize the coordinated motion of microscopic hair-like filaments, called cilia, to perform a range of essential tasks. For example, paramecia harness undulating cilia to propel themselves through water,1 and marine suspension feeders use the beating cilia to capture food particles and transport these particles to their mouths.2−8 Marine organisms also employ the actively beating cilia to prevent settlement of various fouling agents onto their surfaces.9 In humans, motile cilia in the fallopian tubes transport egg cells to the uterus,10 and cilia lining the lungs and trachea are critical in expelling particulates out of the respiratory tract.10 These biological cilia operate at low Reynolds number, and their ability to produce net motion under these conditions has inspired researchers to design artificial cilia that could be used to control fluid flow in microchambers and thus enhance the performance of microfluidic devices. Typically, these artificial cilia11,12 are long, flexible microstructures that are anchored to the walls of a microchannel and actuated by external stimuli, such as electromagnetic fields. Such actuated, synthetic cilia can generate net flows and enhance the mixing of components within these confined geometries. Consequently, these artificial cilia are being utilized in lab-on-chip devices for pumping and mixing a range of complex fluids.11,12

Notably, numerous lab-on-chip applications involve microscale analyses of particulates, such as cells, bacteria, or microbeads, which are suspended in the fluids.13 Hence, it would be useful to harness the artificial cilia to not only pump the fluids but also regulate the transport of the suspended microscopic particles in these microfluidic devices. The latter development requires a fundamental understanding of the interactions among the actuated cilia, the particles, and the surrounding fluid. To date, however, there have been relatively few studies on cilia−particle interactions.14−21 Such studies would be useful for not only employing the artificial cilia to manipulate the movement of microparticles but also could provide insight into the physicochemical factors that enable biological cilia on marine organisms to capture and transport food particles2−8 and actively prevent settlement of fouling agents.9 Importantly, the studies could also provide guidelines for using synthetic cilia to prevent the fouling of microfluidic devices, which is a critical problem that limits the performance of these systems.22 Fouling also affects a broad range of macroscale operations. For example, the fouling of sensors and heat exchangers in many chemical and food processing industries reduces efficiency and increases the maintenance cost; the buildup of biofouling layers on ship hulls is a global problem for the marine industry, leading to larger fuel consumption due to increased frictional drag. Hence, insights obtained from controlling cilia−particle interactions at the

Cilia can be effective at repelling microparticles and cells away from the substrate and thus can be used to prevent the fouling of microfluidic devices. © 2014 American Chemical Society

Received: March 4, 2014 Accepted: April 24, 2014 Published: April 24, 2014 1691

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Figure 1. (a) 3D snapshot from simulation with a rigid particle. Black filaments denote cilia, and the red sphere is the particle. The inset shows superimposed 3D snapshots of a cilium (stiffness E = 3E0), taken at equal time intervals, during one cycle of forcing. Cilium is moving counterclockwise. It is colored black when the tip moves in the negative x direction (effective stroke) and red when the tip moves in the positive x direction (recovery stroke). (b) 3D snapshot from simulation with a soft particle. Black filaments denote cilia, and the magenta object is the soft particle. The adhesive cilia “heads” are shown as red spheres. The inset shows nodes (in yellow) on a tip, some of which can form tacky bonds with the particle.

the beads. With N = 9 and l = 1 LB (lattice Boltzmann) unit, the cilium length is L = 9 LB units; the intercilium spacing is δx = δz = 3 LB units. Because we explicitly account for the bending and stretching energies of the chain, we can assign a Young’s modulus E and cross-sectional moment of inertia I for each cilium. The hydrodynamic coupling of a cilium bead with the fluid is captured using a frictional force that is proportional to the slip velocity between the bead and fluid.16,25 The hydrodynamic interactions between the spherical particles and fluid are captured using bounce-back boundary conditions for the LB distribution functions at the particle−fluid surface,24 mimicking no slip boundary conditions at the particle surface. To model an adhesive interaction between the particles and cilia, the top three beads of each cilium experience an attractive Morse potential interaction with the particles, given as

microscale can be useful for designing antifouling strategies at the macroscale. Herein, we review recent theoretical and computational modeling studies focusing on the interaction of actuated, synthetic cilia with passive16−18 and active particles.19 In these studies, an external, periodic force is applied to tops of the cilia that drives them to mimic the motion of biological cilia.23 These artificial cilia also encompass adhesive or “sticky” tips. As we show below, in addition to generating net flows in microchannels, these artificial cilia can regulate the motion of microscopic particles in the system, separate particles based on their size, and expel both passive particles and actively swimming agents from the layer and thus can be used as an efficient antifouling mechanism. In the ensuing discussion, we first focus on systems encompassing uniformly sized, rigid particles16 and then examine systems containing two distinct particle sizes.17 Next, we investigate the interaction of cilia with compliant, soft particles18 and finally simulate systems involving “active” particles, which undergo self-directed motion.19 By relating our simulation parameters to relevant experimental values, we can provide guidelines to facilitate the experimental realization of these different findings and thereby aid in the fabrication of ciliated surfaces that play a vital role in a range of applications. Transport of Rigid Particles by Actuated Cilia. Figure 1a shows a snapshot from our simulations for the interactions between a ciliary array and a hard particle.16 The equally spaced cilia are tethered to the bottom wall (y = 0) in a three-dimensional simulation box, which is filled with a fluid of shear viscosity η. No slip boundary conditions are imposed at the top and bottom walls, and periodic boundaries are applied in the lateral (x and z) directions. The fluid velocity is evolved using the lattice Boltzmann Method (LBM),24 which is an efficient solver for the Navier−Stokes equation. Each cilium is modeled as a chain of N beads (of radius a), connected by linear springs.16 The elastic potential energy of the chain is given as N

U=

⎛1⎞ i i−1 ⎜ ⎟[k (|r − r | − l)2 + k b(θi − π )2 ] s 2⎠

∑⎝ i=1

V = D[1 − exp{−λ(Δr − re)}]2

(2)

when Δr ≥ re. Here, D is the adhesion strength, Δr is the distance between the surface of the particle and cilium, re is the equilibrium bond length, and λ characterizes the range of interaction. The repulsive exclusion interactions among the beads, particles, and walls are modeled via a similar Morse potential, when Δr < re. The cilia are actuated by a periodic force applied at the cilial tip given by Fext (t ) = Fx0 cos(ωt ) cos(ψ )i + F y0 sin(ωt )j + Fx0 cos(ωt ) sin(ψ )k

(3)

where ω is the angular frequency of actuation and F0x and F0y are the respective forcing amplitudes. The force oscillates harmonically in a plane that is tilted by an angle ψ with respect to the x−y plane. The cilial tips undergo an elliptic trajectory (inset in Figure 1a), so that they are located farther away from the wall during the effective stroke (in the negative x direction) and closer to the wall during the recovery stroke. This asymmetric beating pattern induces a net fluid motion in the negative x direction that, over several cycles, causes net advection of the rigid particle in the same direction, even when the particle is not in direct contact with the cilia. When the particles do come sufficiently close to the cilia, they are caught by the adhesive cilia tips during the effective stroke. Thus, we are able to mimic the salient behavior exhibited by cilia on

(1)

where θi is the angle between neighboring links in the chain, ri is the location of the ith node in the chain, r0 is the tether point of the cilia, while ks, kb, and l are the respective stretching modulus, bending modulus, and equilibrium distance between 1692

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suspension feeders.23 Within the simulations, the positions of the cilia bead and the particles are evolved via a velocity-Verlet algorithm; a separate Quaternion algorithm was used to evolve the particle orientation.16 Using this model, we focused on a 15 × 12 cilia array and introduced a rigid particle within reach of this array to investigate how the particle motion is controlled by the cilia stiffness (E), actuation frequency (ω), and adhesion strength (D). We fixed F0x/F0y = 4 and kept the external forcing proportional to the cilia stiffness, that is, F0x ∝ E. Notably, the properties of the cilia are characterized by the dimensionless Sperm number, Sp, which represents the ratio of viscous to elastic forces26 and is given by Sp = L[4πηω/EI]1/4. By setting F0x ∝ E, we ensure that for low Sp, the trajectory of the cilium tip does not change with E and depends only on Sp. Therefore, varying cilium stiffness E (for a given frequency ω) is equivalent to varying frequency ω (for a given cilium stiffness E). Here, we fixed ω and varied only E and D within the following ranges: E0 ≤ E ≤ 6E0 and D0 ≤ D ≤ 10D0, where E0 was chosen such that Sp(E0) = L[4πηω/E0I]1/4 = 4.3 and the adhesion scale D0 = F0x/ 2λ ∝ E. The dimensionless values for the relevant parameters used in the simulations are a/L = 0.016, Re = ωL2/ν = 0.1, F0xL2/EI = 64, R/L = 0.58, λL = 189, rc/L = 0.065, and re/L = 0.055. Our simulations can be experimentally realized by using the following sets of values for the relevant parameters: cilium length L = 20 μm, actuation frequency ω/2π = 40 Hz, cilium cross-sectional diameter a = 333 nm, kinematic fluid viscosity ν = 10−6 m2 s−1, fluid density ρ = 103 kg/m3, particle radius R = 12 μm, the parameter characterizing the range of the Morse potential λ = 9.6 × 106 m−1, and the scale for the Young’s modulus E0 = 0.132 MPa. The results of our simulations show that the particle trajectories converge to one of the following three states: “released”, “propelled”, or “trapped”. Moreover, the state of particle motion depends greatly on both the adhesion strength and cilia stiffness. Figure 2a displays a trajectory for low adhesion, where the particle is pushed away from the cilia layer and, eventually, attains the released state at a height R + L with respect to the lower wall. At this height, the particle is transported almost completely by fluid advection. Importantly,

this result indicates that over this range of parameters, the actuated cilia can expel particles from a substrate and thus be used for antifouling applications. An increase in the adhesion strength, D, leads to the propelled state, which is characterized by the particle trajectory in Figure 2b (for the same number of actuation cycles as in Figure 2a). Interestingly, the particle travels faster in this state than in the released state. By analyzing the forces acting on the particle,16 we found that in the propelled state, the particle is effectively pushed and pulled forward by the cilia during the effective stroke. For a given E, the net particle speed reaches a maximum at a particular value of the adhesion strength. At this optimal adhesion, the cilia release the particle exactly at the end of the effective stroke, thus maximizing the propulsion of the particle with each ciliary cycle. Below this optimal adhesion, the cilia tips release the particle before the end of the effective stroke; above the optimal adhesion, the cilia remain attached to the particle even during the recovery stroke, canceling the extra push/pull from the effective stroke. Notably, the particle does not reach the substrate even at these values of D, and thus, even the more adhesive cilia are effective at preventing fouling. Finally, Figure 2c shows a trajectory at an even higher adhesion level. At this high level of adhesion, the attached cilia do not release the trapped particle. Consequently, the timeaveraged velocity of the particle is small. To further differentiate the particle trajectories in the different states, we determined the variance h′ = [⟨(rsy − h)2⟩]1/2 of the particle height, rsy(t), with respect to its moving s average, h(t) = (1/T)∫ t+T/2 t−T/2 ry(t′) dt′, where T = 2π/ω is the time period of cilia actuation. By measuring the height variance h′(D,E) of the particle and the mean stream-wise velocity U = |⟨ṙsx⟩|, we could construct the phase map on the (D,E) plane (see Figure 3), which is divided into regions corresponding to the released (I), propelled (II), and trapped (III) states. The I/ II boundary marks a critical adhesion level Dc(E), above which the particle trajectory makes a sharp transition in its height variance, going from a released to a propelled state,

Figure 2. Particle trajectories (solid black curves), starting at green squares and ending at red squares, projected on the x−y plane, for E = 3E0. Plots have been made for the number of actuation cycles. Filtered particle trajectories, with height h(t), shown by dash−dot red curves. The trajectory at (a) low adhesion, D = 0.66D0, shows the released state, (b) medium adhesion, D = 1.33D0, shows the propelled state, and (c) high adhesion, D = 5D0, shows particle trapping.

Figure 3. Phase map of particle motion as a function of (D,E). Simulations were run at each point marked by the green symbol. Crosses mark phase I, squares mark phase II, and circles mark phase III. Solid magenta lines divide the phases. The yellow dashed line marks the I/II boundary predicted by the low-order model. A contour plot of the height variance of the particle from LBM simulations, h′(D,E), is shown in the background. 1693

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adhesion strengths, these cilia can be effective at repelling microparticles and cells away from the substrate and thus can be used to prevent the fouling of microfluidic devices. Size Separation of Microparticles Using Artif icial Cilia. Marine suspension feeders use arrays of adhesive, beating cilia to discriminate between different sized particles,2−8 feeding on the smaller particles, and expelling the larger ones. In the following studies,17 we examined if the cilia in Figure 1 could mimic this ability to differentiate between particles of different sizes. The artificial cilia interact with the particles solely through their “sticky” tips and the flow fields generated by their actuation. Thus, these findings could reveal the role of fundamental physicochemical processes in this vital biological function. Using the model described above, we simulated the interaction of adhesive, actuated cilia with a mixture of rigid particles that differed in size. The larger particles had a radius of RL = 6 LB units, and the smaller particle radius was RS = 3 LB units. Initially, the particles were distributed in the simulation box so that they all interacted with the adhesive cilia tips (see the inset in Figure 4). The periodic external force given in eq 3

independent of the initial particle depth in the array. Because Dc(E)/D0 shows little variation with E, especially for high cilia stiffness, and because D0 itself is proportional to E (as noted above), the phase map in Figure 3 implies that Dc(E) increases almost linearly with E. In other words, softer cilia can “capture” particles at lower adhesion. Alternatively, stiff cilia will inhibit sticky particles from settling near the surface and hence can be used to repel particulates away from the layer. To gain additional insight into the phase map in Figure 3, we formulated a lower-order analytical model,16 where we assumed that the depth of the sphere in the ciliary layer is small and, thus, ignored the momentum equation for the sphere in the direction parallel to the wall (the x direction). The elastic behavior of the average cilia “rod” is lumped into the bending and stretching moduli, which are calibrated using the LBM simulations described above. Resistive force theory27 is used to model the viscous drag on these rods. The average attached cilia interact with the particle via a Morse potential identical to the one used in the simulation; the force due to the Morse potential per cilium is then simply given by FMorse (r ys , ry̅ ) y

−∂V (|r ys − R − ry̅ |) = ∂ ry̅

(4)

The number of cilia−particle bonds at any instant in time is taken as n = ρcπd2, where ρc is the grafting density of the cilia on the lower wall. The motion of the spherical particle due to the Morse force from n bonds in the overdamped regime can be written as ryṡ =

−nFyMorse(r ys , ry̅ ) β 6πηR

(5)

where the factor β accounts for the decrease in mobility of the sphere due to the presence of confining walls. (As we show further below, β in fact depends on the particle radius R; this in turn can lead to the dependence of Dc on R.) We evolve eq 5 for several actuation cycles, with the same frequency as that in the LBM simulation. As in the simulation results, we see that above a critical adhesion level, the variation in particle height suddenly increases as the particle attains a propelled state. Figure 3 shows a comparison of the I/II phase boundary calculated from the LBM-based simulations and the lower-order model. We find reasonable agreement between the two predictions, implying that the critical adhesion level primarily depends on cilia−particle interaction forces in the wall-normal direction. Using the physical values mentioned above, we can specify the I/II boundary in terms of the surface energy γ = D/πa2 and interfacial tensile strength σ = Dλ/2πa2 of the cilia tips per unit area. Our results indicate that for E = E0 = 0.132 MPa, the I/II boundary will occur at σ = 293 Pa and γ ≈ 61 μJ/m2, while for E = 6E0 = 0.792 MPa, the boundary will occur at σ = 967 Pa and γ ≈ 200 μJ/m2. We note that cilia stiffness corresponding to the values used in our simulations is comparable to the synthetic cilia stiffness (∼2 MPa) used in experiments.16,17 The predicted values of the adhesion energies also correspond well to typical adhesion energies for microcapsules and biological cells (∼20−100 μJ/m2).16,17 These results indicate that by appropriately tuning the cilia stiffness and the cilia−particle adhesion, the artificial cilia can be used to transport microscopic particles and biological cells within microchannels. The findings also suggest that for a range of cilia stiffness and cilia−particle

Figure 4. (Top inset) Snapshot of the simulated system. Black filaments represent the cilia, and red and blue spheres are the particles of two different sizes. (Bottom) Sample trajectory of a small and a large particle, beginning at filled squares, projected on the x−y plane.

was applied to the cilial tips, and we set F0x/F0y = 2. We also fixed E = 3E0 and varied the adhesion strength in the range 0.5D0 ≤ D ≤ 3D0, with D0 = F0x/2λ ∝ E. Four independent simulations were carried out for each value of D for a total of 60 actuation cycles. In the following discussion, we use ϕ to denote the area fraction of the particles (i.e., ratio of the total projected area of the large and small particles to the area of the simulation box) and f to denote the ratio of the projected area of the large particles to the total projected area of all particles. For a mixture of particles with NL large particles and NS small particles, ϕ = π(NLRL2 + NSRS2)/LxLz and f = NLπRL2/(NLπRL2 + NSπRS2), where Lx and Lz are the length and the width of the simulation box. Thus, ϕ is a measure of total concentration of the particles in the fluid, and f is a measure of the composition of the mixture. Figure 4 shows sample trajectories of a large and a small particle from a mixture of particles (ϕ = 0.33, f = 0.53) for D/ D0 = 1.5. Both particles undergo net motion in the x direction; however, the particle motion in the y direction differs significantly based on particle size. To characterize this difference, we referred to the particles as “captured” or “noncaptured”. A particle is captured if it is pulled below the adhesive part of the cilia layer, that is, ⟨r̃sy⟩ < 0.66L + R. (With 1694

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of adhesion strengths yielding size selectivity is clearly related to this difference in the critical adhesion strengths required to capture the single large and small particles. The cilia stiffness for these simulations was fixed at E = 3E0; however, Figure 6 shows

respect to the terminology in the previous section, the captured particles are in the propelled state, while the noncaptured particles are in the released state.) Notably, it is the small particles that are captured by the cilia; they display periodic variation in their y positions while remaining at a low average height in the cilia layer. In contrast, the large particles show little fluctuations in their y positions and are pushed away from the layer. Thus, the adhesive, artificial cilia selectively capture small particles from the mixture and act as a size-based particle sorter. Figure 5a shows the fraction of small and large particles captured by the cilia for mixtures having similar compositions f

Figure 6. Critical adhesion strength (D) with cilia stiffness (E) for two different sizes of the particles. Symbols are the simulation results, and lines are predictions from the analytical model. For a given cilia stiffness, the minimum adhesion to capture the large particle is greater than that for the small particle. (Inset) Variation of factor β that accounts for the increase in the Stokes drag due to the presence of confining walls (obtained from separate simulations) with particle radius R. The symbols are simulation results, and the line is a fit to the data.

that the critical adhesion strength required to capture the isolated large particle is higher than that to capture the isolated small particle for any given stiffness of the cilia (for the range considered here). The higher value of critical adhesion strength needed to capture the large particle can be explained by the particle’s motion (eq 5) due to the adhesive bonds formed with the cilia layer. As mentioned above, the factor β in eq 5 accounts for the increased drag due the presence of confining walls. We estimate this factor β for different particle sizes using separate LBM simulations;17 these computed values of β match well with the theoretical values for the correction to the Stokes drag obtained by linearly superimposing the contribution from each flat wall.28,29 As shown in the inset in Figure 6, β shows a quadratic increase with particle size, indicating a higher drag coefficient for the larger particles. We use the fitted values of β from Figure 6 in the lower-order analytical model and obtain the critical adhesion above which the particles are captured by the cilia. The critical adhesions for different cilia stiffnesses obtained from the LBM simulations (using the height variance criterion mentioned in the previous section) are shown as symbols in Figure 6 for the two particle sizes. The analytical model predictions, shown as solid lines in Figure 6, compare well with the simulation results, especially for the higher stiffness of the cilia. The deviation from the simulation results for low cilia stiffness is expected because the assumption that the cilia remains straight during the effective stroke is valid only for the highly stiff cilia. Both the analytical model and simulations reveal that the critical adhesion above which the large particles are captured is higher than that for the smaller particle. The behavior can be understood by considering the adhesive and hydrodynamic forces acting on the particles. Equation 5 states that the

Figure 5. Fraction of small (filled symbols) and large (open symbols) particles captured for different adhesion strengths for (a) different total concentrations of the particles with f ≈ 0.54 and (b) different compositions of mixture for ϕ = 0.33. N is the number of small (large) particles captured, and N0 is the total number of the small (large) particles in the mixture. Broken vertical lines indicate the critical adhesion values for isolated particles for small and large particles.

of large and small particles but different total concentrations ϕ of the particles, ϕ = 0.33 (NS = 28, NL = 8, f = 0.53) and ϕ = 0.24 (NS = 20, NL = 6, f = 0.55). Figure 5b shows the fraction captured for three different compositions f of the mixture for the same total concentration of particles (ϕ = 0.33). While none of the particles are captured by the cilia for very low adhesion strengths, an increase in the adhesion leads to the capture of small particles and a nearly complete separation of the particles. Figure 5 suggests that there exists a range of adhesion strengths for which only small particles are selectively captured by the cilia. This size separation effect is quite robust and is observed for different compositions of the mixture and for different total concentrations of the particles, including a mixture with an equal number of large and small particles (Figure 5b, f = 0.8; NS = NL = 12). Notably, further increases in the adhesion strength cause the large particles to also be captured, indicating that the cilia become less selective at higher values of D. To obtain a better understanding of the observed behavior, we calculated the critical adhesion strength required to capture a single, isolated particle for the two different particle sizes. The dashed vertical lines in Figure 5 indicate the critical adhesion values obtained from these isolated particle studies. The range 1695

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adhesion force on the particle due to the n bonds formed with the cilia layer is balanced by the hydrodynamic drag force on the particle. Due to the presence of channel walls, the viscous drag for the larger particle increases substantially compared to that of the small particles (i.e., β increases with R). Hence, a higher adhesion between the cilia and the particle is needed to compensate for the increased drag force on the particle and, hence, capture the larger particles. Thus, the size-selective capture of particles can be achieved simply due to the competition of adhesive and hydrodynamic forces. Moreover, for a given cilia adhesive strength, there exists a particle size beyond which the adhesion force is insufficient to capture the particles.

In these simulations, the soft particles are introduced above a 10 × 10 array of cilia. The Young’s modulus of the cilia is fixed at E = 6E0, while the stiffness of the particle, EPart, as well as the adhesion strength of the bond, DBond, is varied. We rescale (EPart,DBond) in terms of F0x, Δ (the shell thickness), and λ to obtain the nondimensional quantities EPart* = EPart/EΔ and DBond* = DBond/D0, where EΔ = F0x/Δ2 and D0 = F0x/2λ. The stiffness of the particle is varied over a factor of 20, ranging from floppy, soft particles (EPart* = 0.26) to nearly rigid particles (EPart* = 5.2). The adhesion strength is varied over 0 < DBond* < 7.2 for the rigid particle and over 0 < DBond* < 6.0 for the soft particle. Most of the important nondimensional parameters for the cilium are the same as those in the previous section, except for the tip forcing, F0xL2/EI = 126.5, and length scale for the Morse potential, λL = 72. Experimentally, these studies can be realized using the parameters listed in the previous section. Additionally, EΔ and D0 in these simulations correspond to the following experimental values: EΔ = 0.6 KPa and D0 = 1.62 × 10−15 J; the range of the particle’s Young’s modulus therefore extends from EPart = 0.16 to 3.12 KPa.18 We find that for a given DBond*, the nature of particle transport depends strongly on EPart*. In particular, at intermediate adhesion levels (DBond* = 1.8), the rigid particle stays in the released state, while the floppy particle undergoes a propelled trajectory.18 Hence, the adhesive cilia can also be used to separate the rigid and floppy particles by tuning the adhesive interaction between the cilia and the particle. At higher adhesion levels (DBond* = 4.8), both floppy and rigid particles undergo propelled trajectories. At this high adhesion, however, the floppy particle travels significantly slower (by roughly 40%) than the rigid particle. The modulation of the particle velocity by its stiffness (EPart) is illustrated in Figure 7, which shows plots of the average particle velocity in the x direction, U = |⟨ṙsx⟩|, with respect to adhesion strength, DBond*, for floppy and rigid particles. Clearly, below a threshold adhesion strength, DBond* ≈ 3, the floppy particle is transported with a higher net velocity than the stiff particle.

These findings reveal that actuated adhesive ciliary arrays can be used as an efficient means of sorting microscopic particles by size. These findings reveal that actuated adhesive ciliary arrays can be used as an efficient means of sorting microscopic particles by size. In contrast to size separation mechanisms that utilize inertial migration of the particles30−32 and, hence, are of limited use at low Reynolds numbers, adhesive ciliated surfaces can be used for sorting even at low Re. Because the physical values of D used here are comparable to estimated values of adhesion energies for biological cells,17 these studies indicate that artificial cilia could be useful in separating cells of different sizes and thus would be valuable for various pharmaceutical applications.33 Transport of Sof t Particles by Active Cilia. Up to this point, we have only considered the interaction of actuated cilia with rigid particles. To model the interaction of soft particles with the active, adhesive cilia (Figure 1b),18 we modified our LBM-based approach so that both the compliant particle and cilia are discretized in terms of nodes. Moreover, each cilium consists of a “head”, which itself consists of a cluster of nodes (Figure 1b, inset). The cilia heads were introduced to evenly distribute contact forces between the soft particle and cilia tips. We assumed zero twist along the length of the cilia to obtain the orientation of the head. The soft particle is modeled as a hollow sphere of outer radius R with a shell thickness of Δ = L/9. The other new features of the model can be summarized as follows. The particle surface is tessellated with roughly equilateral triangles, with the nodes forming the vertices of the triangles. All 1284 nodes in the soft particle are discretized into nonintersecting tetrahedral volumes. To account for large deformation in the soft particle, a neo-Hookean elastic model is assumed for the strain energy function.34,35 (Constants in the neo-Hookean model can be related to the Young’s modulus EPart and Poisson’s ratio νPart of the soft particle for small deformation.) Given that UPart is the strain energy of the whole particle (summed over all tetrahedra), the elastic force at a node i in the particle is now given by FEl,i = (∂UPart/∂rij).18 j Nodes on the particle’s outer surface can form breakable bonds with nodes on the cilium head (Figure 1b, inset). While the bond potential V(r) is still given by eq 2, a single cilium can now form multiple bonds with the particle. Forces acting on the particle surface are linearly interpolated onto the surface nodes.

Figure 7. Average particle velocity U versus adhesion strength DBond for the different particle Young’s modulus (EPart). The black dashed curve is for the floppy particle, and the solid blue curve is for the stiff particle. The error bars denote minimum and maximum values over four random seeds. Insets (a) and (b) show snapshots of the simulation during the effective stroke for the floppy and rigid particle, respectively, at DBond/D0 = 1.2. 1696

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difference of π/2 between the two bonds, as illustrated in Figure 8a. This results in a small displacement after each period

Above this threshold, the trend reverses. Indeed, at DBond* = 6, the floppy particle almost comes to a halt, while the stiff particle continues moving. The modulation in particle velocity by EPart can be explained by the fact that during the first half of the effective stroke, when the cilia array pushes the particle upward, the softer particle tends to get deformed into an (oblate) ellipsoidal shape (Figure 7, inset). Upon measuring the number of bonds,18 we find that the average number of bonds is indeed much larger (by around a factor of 5−10) for the floppy particle, at both intermediate and higher adhesion levels. Thus, the softer particle deforms significantly (due to the lower elastic penalty associated with its deformation) and allows the formation of more cilia−particle bonds at the lower surface of the softer particle. As a result, the cilia are able to pull a softer particle to a lower height more easily and “propel” it faster at lower adhesion. Beyond a certain adhesion strength, however, the excess bonds begin to slow down the softer particle (in relation to the stiff particle) during the recovery stroke. Therefore, at higher adhesion, the rigid particle tends to travel faster.

Figure 8. Illustration of simulation components. (a) Sequence of configurations of the three-sphere model swimmer showing the beat cycle of over one period of motion. The darker sphere indicates the front of the swimmer. (b) Beat cycle of an externally forced cilium superimposing the cilium positions from different stages of one cycle. (c) Snapshot of a simulation showing the swimmer, cilia, and channel walls.

Our results show that actuated cilia arrays with adhesive tips can be used as a novel way to separate particles based on their stiffness.

of oscillation, as predicted by Najafi and Golestanian,36 who first proposed this minimal model for a swimmer at low Reynolds number. In our simulations, the mean swimmer length is Lswim = 25 μm = 10Δx, where we set the LB grid spacing Δx = 2.5 μm. Using the LB time step Δt = 1 μs and an oscillation period Tswim = 1000Δt, we find that far from the wall, the swimmer moves at an average speed of vswim = 250 μm/s. Crucially, we also observe that the swimmer is hydrodynamically attracted to walls and, in the absence of background flows and interactions with other objects, eventually swims parallel to a wall at a fixed distance irrespective of the starting conditions. This behavior has been noted for a variety of other model swimmers37−39 and is in agreement with experimental observations.40 We also emphasize that the length and the average speed of the simulated swimmers are comparable to those of biological swimmers.41,42 Cilia are modeled using an unconstrained Kirchhoff rod formulation.43 The cilium length is fixed as Lcil = 25 μm, and the following external force is applied to the free tips of the cilia: FExt(t) = −Fx0 sin(ωt) cos ψ i + Fy0 cos(ωt)j + Fx0 sin(ωt) sin ψ k. The resulting motion of the cilia is shown in Figure 8b. The actuation period is Tcil = 2π/ω = 3 × 104Δt, corresponding to the frequency f = 33 Hz. Here, the spacing between cilia is W = 2Lcil = 2Lswim, which is large enough to allow the swimmer to move within the ciliary layer without constant contact with cilia. The actuation of the cilia generates a flow field with a strong shear component in the region occupied by the cilia. The mean flow is necessarily zero at the wall and increases to a maximum value, ucil = 300 μm/s, at the top of the ciliary layer. Note that this value is comparable to the swimmer speed. In addition to hydrodynamic interactions, steric repulsion is introduced between cilia and swimmer nodes when the separation approaches the sum of their hydrodynamic radii. We performed 80 independent simulations up to a maximum time corresponding to 6 s with random initial positions and orientations of the swimmer with the constraints that the swimmer height is midway between the two walls and the tilt

Our results show that actuated cilia arrays with adhesive tips can be used as a novel way to separate particles based on their stiffness. Our work also provides insight into how biological cilia capture other cells and organisms. For instance, in mammals, active cilia, located at the entrance of the oviduct, are responsible for transporting egg cells into the oviduct.10 Our results indicate that both the stiffness of the cilium and stiffness of the egg are crucial for determining whether or not the egg is captured by the oviduct cilia. Hence, diseases that affect the stiffness of the cilia can affect the transport of the egg cells. Expulsion of Swimming Micro-organisms by Actuated Cilia. In the preceding sections, we focused on passive particles that arrived at the surface by Brownian diffusion, fluid advection, or sedimentation. Biofouling, however, typically involves active migration of motile organisms that seek out suitable surfaces to colonize. These species are able to move in a directed fashion even in the absence of background flows. Recently, we investigated whether such synthetic cilia can be effective at expelling actively swimming particles. Our results demonstrate that externally actuated synthetic cilia can be used as an effective mechanism for expelling model swimmers and thus be used to prevent biofouling.19 As in the previous studies, the LBM is used to compute the fluid flows in a periodic domain bounded by upper and lower walls. The fluid is coupled to the dynamics of model cilia and swimmers using the immersed boundary method.19 This technique approximates a no-slip condition on a diffuse boundary and allows a single immersed boundary node to represent a bead with a hydrodynamic radius on the order of the fluid lattice spacing. The model swimmer consists of three such beads arranged along a line with adjacent beads joined together by stiff bonds. The equilibrium bond lengths oscillate in time with a phase 1697

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for a much longer time (Figure 9c). Inspection of these trajectories reveals that the swimmers adopt a stable configuration and swim parallel to the wall indefinitely, as in the case with no imposed flow.

direction is toward the lower wall. A snapshot of the system at early times is shown in Figure 8c. In nearly all cases, the swimmer enters the ciliary layer (y < Lcil), and we record the time spent by the swimmer in this layer before it leaves. Histograms of these residence times are given in Figure 9a.

We find that the proposed system of actuated cilia combines high local shear rates with the steric repulsion provided by a continually moving, dynamic boundary to drive away active swimmers that are attracted to bare walls. This antifouling method may be effective against a range of species as it utilizes physical mechanisms rather than specific biochemical interactions. Furthermore, additional simulations showed that none of the swimmers could be expelled from the surface by this pressuredriven flow when the swimming speed was increased by a factor of 5. These fast swimmers were, however, effectively expelled by cilia whether or not the ciliary motion generated a net flow. Thus, we conclude that the local interactions with the continually waving cilia may be sufficient to expel swimmers with a range of swimming speeds. If the collective motion of the cilia generates a net fluid flow comparable to the speed of the swimmer, then expulsion of the swimmer can be greatly enhanced. The same shear rate from a pressure-driven flow through the channel is able to prevent most swimmers from residing close to the channel walls. Depending on the application, however, it may be unfeasible to rely on high flow rates to prevent biofouling. Actuated cilia, however, are able to generate high fluid shear rates near a wall without necessarily causing a large flux through the channel. In summary, we find that the proposed system of actuated cilia combines high local shear rates with the steric repulsion provided by a continually moving, dynamic boundary to drive away active swimmers that are attracted to bare walls. This antifouling method may be effective against a range of species as it utilizes physical mechanisms rather than specific biochemical interactions. Biological cilia display a remarkable ability to control the motion of microscopic particles and thereby perform such vital functions as propelling food to the mouths of mollusks, transporting egg cells to the fallopian tube, expelling particulates out of the respiratory tract, and clearing the surface of marine organisms from fouling agents. Our findings reveal that actuated artificial cilia can perform analogous biomimetic functions; namely, these motile filaments can regulate the motion of microparticles in fluid-filled microchannels. By tuning their stiffness and “stickiness”, the cilia can be designed to completely repel particles away from the surface or propel the particle rapidly along the top of the layer. (Notably, the adhesion strength could be tailored by anchoring biomolecules onto the cilia tips.) The results also indicate that these adhesive cilia can effectively separate microparticles differing in size and hence could be used for sorting and separating microbeads and biological cells. Our studies dealing with the interaction of the

Figure 9. Distributions of residence times for swimmers within a distance Lcil from the lower wall in three scenarios: (a) actuated cilia driving a net flow, (b) actuated cilia producing no net flow, and (c) pressure-driven ambient flow. Each data set is for n = 80 independent swimmers. Some swimmers, which all start at the center of the channel and initially point downward with a random angle, turn away without coming close to the lower wall (“no entry”). In contrast to the other two cases, cilia generating a net flow do not allow swimmers to stay in the layer after 2.5 s.

None of the swimmers stayed in the ciliary layer for more than 2.5 s. The cilia are therefore effective at expelling the swimmers, which would otherwise remain near the wall indefinitely. To determine the cause of swimmer expulsion, we constructed two test scenarios. In the first, we alternate the direction of the cilium forcing function after each cycle. The flow generated by the beating cilia in one cycle is canceled out by the flow from the next cycle, so that on average, the cilia do not generate any net fluid flow. The range of motion of each cilium is not substantially changed, however; therefore, there is approximately the same capability for the cilia to “knock” the swimmer away via steric repulsion. Indeed, we find that the swimmers are still expelled from the ciliary layer (Figure 9b), though the average residence time is higher than that in the case where the cilia generate a net flow. In the second test scenario, we remove the cilia and instead impose a pressure-driven background flow through the channel. The shear rate close to the wall is matched with that produced by the cilia, and in this manner, we can determine if it is the induced shear flow that causes swimmers to turn away and swim out of the ciliary layer. The histogram of residence times indicates that most swimmers are quickly turned away after approaching the wall, but a fraction (14%) remain at the surface 1698

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nent fluids in confined geometries. Web: http://www.engineering.pitt. edu/AnnaBalazs/; Web: http://grace.che.pitt.edu/

adhesive cilia with soft, deformable particles suggest a novel mechanism for particle separation based on elastic properties and can be a valuable component in microfluidic devices used for detecting and separating diseased cells with abnormal Young’s moduli. One of the most important insights gained from these studies is that the synthetic cilia can be used as an efficient antifouling strategy against a wide range of fouling agents. Our results indicate that the artificial cilia can prevent settlement of passive adhesive particles by actively repelling them away from the surface. In addition, the synthetic cilia can also expel active swimmers entering in the cilia layer and thus might be used as an antifouling coating to prevent the biofouling due to many biological organisms. We emphasize that an antifouling strategy like this might be applicable for a wide range of applications due to the nonchemical nature of the mechanism. The values of the cilia length, stiffness, and actuation frequency used in our studies compare well with magnetically actuated synthetic cilia.11,12 Using these experimentally realizable values, we relate our simulation parameters to predict the adhesion energy between the cilia and the particle. We find that the predicted adhesion energies correspond well with the adhesion energy of model capsules/biological cells reported in the literature,16−18 indicating that the above effects can be observed by conducting experiments44 with the already manufactured synthetic cilia and model capsules. Similarly, because the length and the swimming speed of the model swimmers used in our study correspond well to those of biological examples, it would be interesting to investigate the interaction biological swimmers with actuated synthetic cilia. It is noteworthy that we recently demonstrated that the nonactuated synthetic cilia can still be harnessed as antifouling surfaces by utilizing the oscillations in the ambient flow.45 The passive cilia in this case are driven to undulate by the oscillating ambient flow and are found to be effective in repelling sticky adhesive particles away from the ciliated surfaces. These results indicate that passive cilia can be used to create self-cleaning surfaces, utilizing oscillations in the flow to prevent the attachment of microparticles and biological cells. Finally, we emphasize that there have been few systematic experimental studies focused on varying the properties of the cilia, particles, and flow to determine the factors that control the fundamental interactions within the system. Furthermore, to the best of our knowledge, despite the significant utility and functionality of sticky cilia in directing particle motion in biological systems, there have been few experiments aimed at harnessing adhesive artificial cilia to direct the motion of particulates. Hence, the studies presented here could serve as a springboard to new experimental investigations.

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Amitabh Bhattacharya is an Assistant Professor in the Department of Mechanical Engineering at the Indian Institute of Technology, Bombay. He obtained his Ph.D. in Theoretical and Applied Mechanics from the University of Illinois, Urbana−Champaign. His research interests include modeling and simulation of microfluidic flows, soft materials, and turbulence. Web: http://www.me.iitb.ac.in/wiki/doku. php?id=bhattach Anurag Tripathi is an Assistant Professor of Chemical Engineering at the Indian Institute of Technology (I.I.T) Kanpur. He graduated from I.I.T Bombay and worked as a postdoctoral researcher in the Balazs Group at the University of Pittsburgh. His current research interests include the rheology of complex fluids and processing of granular mixtures and particulate solids. Web: http://www.iitk.ac.in/che/at.htm Henry Shum completed his D.Phil degree in Mathematical Sciences at the University of Oxford, U.K. and is currently a postdoctoral scholar at the University of Pittsburgh, U.S.A. His research interests involve modeling and simulation of microscale systems, including microorganism motility and collective dynamics, and active antifouling systems.

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ACKNOWLEDGMENTS A.C.B. gratefully acknowledges financial support from the ONR and helpful conversations with Profs. Julia Yeomans and Joanna Aizenberg.

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REFERENCES

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. Biographies Anna C. Balazs is the Distinguished Professor of Chemical Engineering and the Robert von der Luft Professor at the University of Pittsburgh. She received her Ph.D. in Materials Science at the Massachusetts Institute of Technology. Her research involves developing theoretical and computational models to capture the behavior of polymeric materials, nanocomposites, and multicompo1699

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