Letter pubs.acs.org/NanoLett
Optimal Length of Low Reynolds Number Nanopropellers D. Walker,†,‡ M. Kübler,‡ K. I. Morozov,§ P. Fischer,*,†,‡ and A. M. Leshansky*,§ †
Max Planck Institute for Intelligent Systems, Heisenbergstraße 3, 70569 Stuttgart, Germany Institute for Physical Chemistry, University of Stuttgart, Pfaffenwaldring 55, 70569 Stuttgart, Germany § Department of Chemical Engineering and Russel Berrie Nanotechnology Institute, TechnionIsrael Institute of Technology, Haifa 32000, Israel ‡
S Supporting Information *
ABSTRACT: Locomotion in fluids at the nanoscale is dominated by viscous drag. One efficient propulsion scheme is to use a weak rotating magnetic field that drives a chiral object. From bacterial flagella to artificial drills, the corkscrew is a universally useful chiral shape for propulsion in viscous environments. Externally powered magnetic micro- and nanomotors have been recently developed that allow for precise fuel-free propulsion in complex media. Here, we combine analytical and numerical theory with experiments on nanostructured screw-propellers to show that the optimal length is surprisingly short−only about one helical turn, which is shorter than most of the structures in use to date. The results have important implications for the design of artificial actuated nano- and micropropellers and can dramatically reduce fabrication times, while ensuring optimal performance. KEYWORDS: Nanopropellers, magnetic nanomotors, microswimmers, viscous hydrodynamics, rotating magnetic field, glancing angle deposition (GLAD) Once suspended in a fluid medium, a relatively weak (on the order of a few milli Tesla) rotating uniform magnetic field acts on their magnetic moment, which is orthogonal to the direction of the screw-axis. This results in a rotation about the long-axis and consequently in corkscrew-like propulsion. However, the motion is in general more complex, as there is an additional longitudinal component of the magnetic moment. The above fabrication techniques usually allow for the prescribed amount of magnetic material to be deposited at the propellers of preprogrammed length. Therefore, given the magnetic torque, the fundamental question for practical applications is what it the optimal length/size of the helical propeller? The concept of propulsion16,17 or towing efficiency18,19 and optimal design based on power, which is often applied to gauge autonomous (internally actuated) biological and artificial swimmers, is less relevant for fuel-free micro- and nanopropellers powered by external fields. We here associate “optimal” propeller with the fastest one and provide simple scaling arguments that identify the optimal lengths of helical nanoscrews. Both ferromagnetic and superparamagnetic propellers exhibit the following generic dynamics when driven by a rotating magnetic field: at low frequency of the actuating field, they tumble in the plane of the field rotation without appreciable propulsion, while upon increasing the field frequency, the tumbling switches to wobbling, whereas the precession angle
T
he development of active artificial micro- and nanostructures that can be controllably steered through liquids promises a range of applications. Emergent approaches include catalytically driven (chemical-fuel-driven) nanowires and nanoparticles, tubular microjets, as well as thermally, light, and ultrasound-driven colloids (see ref 1 for a review). Magnetic actuation shows considerable potential for the navigation through complex biological media. Since the early work of using a millimeter-sized magnetic screw, which was drilled wirelessly through a bovine tissue sample,2 it is of interest to fabricate magnetic propellers that hopefully can achieve a similar feat at smaller scales. Various methods have been developed toward the fabrication of microsized ferromagnetic helices, including a physical vapor deposition method, known as glancing angle deposition (GLAD),3 or the fabrication of strained semiconductor layers,4 the delamination of magnetic strips,5 direct laser writing combined with vapor deposition,6 and the metallization of the xylem spiral vessels extracted from plants.7 In addition, superparamagnetic helices have been fabricated by two-photon polymerization of a curable polymer composite.8 The smallest nanopropellers to date have a diameter of about 70 nm and a length of under 400 nm and are small enough to penetrate the molecular network of biomimetic gels;9 they were fabricated by an improved GLAD process.10 GLAD permits excellent control over the shape and structure11−15 and in conjunction with seeding allows very large numbers of nanopropellers (tens to hundreds of billions) to be grown in just a few hours.10 © XXXX American Chemical Society
Received: February 23, 2015 Revised: June 1, 2015
A
DOI: 10.1021/acs.nanolett.5b01925 Nano Lett. XXXX, XXX, XXX−XXX
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attain the optimal value, Umax*, while for L>L*, it will diminish again due to lowering of ωs−o, while Ch will remain nearly constant. It should be mentioned that ref 23 examined numerically the optimal shape of chiral micropropellers with a rigidly attached payload3,4 under the assumption of negligible wobbling, θmin ≈ 0, and propulsion along the helical axis, and found that geometry optimization of the helical propeller may result in considerably improved translational speeds. In this Letter we report the first experimental evidence of an optimal chiral structure in support of the theoretical arguments presented herein. The propulsion experiments were performed for different propeller lengths while keeping all other parameters (i.e., helical radius, pitch, and filament diameter) fixed. Nanopropellers with 0.7, 1.1, 1.6, and 2.2 helical turns, respectively, were grown on-wafer onto 500 nm nonmagnetic silica beads using physical vapor GLAD10 process (see Figure 1
gradually diminishes resulting in a corkscrew-like directed motion. In refs 20 and 21, the orientation and dynamics (e.g., tumbling/wobbling, synchronous/asynchronous transitions) of ferromagnetic and superparamagnetic helical nanomotors, respectively, were studied theoretically, offering some simple design rules for the choice of magnetic material, preferred magnetization (or easy-axis orientation) as well as the propeller geometry. In particular, the propulsion velocity of a ferromagnetic propeller in the wobbling regime reads20 UZ = −ω cos2 θ()∥/ξ∥), where θ is the precession angle between the propeller’s easy axis of rotation (the principal rotation axis corresponding to the lowest eigenvalue of the rotation matrix22) and the axis of the field rotation, and ξ∥ and )∥ are, respectively, the translational and the coupling viscous resistances with respect to the propeller’s easy rotation axis. The wobbling angle is given by20 sin θ = A/ω, where A = m∥H/ κ⊥ is the tumbling/wobbling transition frequency, H is the amplitude of the driving magnetic field, and κ⊥ is the transverse rotational resistance, i.e., in the plane perpendicular to the easy axis of rotation. The longitudinal (i.e., along the easy axis of rotation) magnetization m∥ is typically much smaller than the transverse magnetization m⊥, and for slender propellers κ∥ < κ⊥, so that A ≪ ωs−o ≈ m⊥H/κ∥. Thus, neglecting wobbling, θ ≈ 0, the propulsion velocity at frequencies ω ≫ A can be written as UZ ≈ Ch ωR, where R is the helical radius and Ch = )∥/(ξ R ) is the chirality coefficient of the propeller assembly. At the maximum available (step-out) frequency in the synchronous regime, ωs−o, one finds Umax ≈
m⊥H ⎛ )̃ ⎜⎜ r 2η ⎝ ξ ̃ κ ̃
⎞ ⎟⎟ ⎠
(1)
where the resistance coefficients (marked by a tilde) are scaled with the characteristic spatial dimension r, e.g., radius of the chiral filament. If wobbling is not entirely negligible, the righthand side of eq 1 should be multiplied by cos2 θmin, where the minimal precession angle is found from20 sin(θmin) = A/ωs−o = [1 + (κ⊥m⊥/κ∥m∥)2]−1/2. It follows that to maximize the speed one has to maximize the chirality coefficient Ch while minimizing the rotational friction κ∥. Let us consider the propeller design of refs 9 and 10 where the magnetic material is deposited onto the spherical bead/ head as a nonchiral segment followed by the helical nonmagnetic structure. For a long propeller, the net chirality Ch will be at a maximum (i.e., approaching a constant value); however, the rotational friction will also increase as κ∥ ∼ L, where L is the propeller’s length (along the helical axis). Therefore, Umax will decrease with L. On the other hand, the rotational friction for a short propeller will be determined by the bead (i.e., nearly constant), while the net chirality will be small, as the head is achiral. So, the net chirality Ch will diminish upon decreasing the propeller’s length L. Therefore, there must be an optimal propeller/head size ratio that gives a maximal speed at the step-out frequency Umax*. In other words, when the velocity is measured as a function of the actuation frequency, the following scenario takes place: when the propeller is short, then the slope of U vs ω will be small, but the synchronous regime will persist to higher frequencies. Gradually increasing the overall length of the chiral propeller will increase the slope of the velocity curve in the synchronous regime and ωs−o will decrease, so that overall Umax shall increase. At some optimal length L* the maximum speed will
Figure 1. (A) SEM image of the 1.6 turn-helices grown on wafer. The image was taken with an ESB detector, rendering the magnetic Nisection clearly visible (lighter). The scale bar is 1 μm. (B−E) SEM images of individual nanopropellers with 2.2 (B), 1.6 (C), 1.1 (D), and 0.7 (E) full helical turns. The scale bar in (E) is 500 nm and is the same for images B−E.
and Supporting Information for details). The wafers were magnetized in-plane (i.e., orthogonal to the propellers’ long axis) in a 1.8 T electromagnet, and their ferromagnetic response was observed by SQUID magnetometry. Propulsion experiments were performed using a custom-built three-axis Helmholtz coil array providing a magnetic field strength up to 10 mT. For each velocity measurement only the motion parallel to the axis of rotation of the magnetic field was taken into account. For a given set of conditions (frequency/number of helical turns), the velocity of at least 20 nanopropellers was measured over a minimum distance of 40 μm (at high propulsion speeds) or for a minimum time of 5 s (at low propulsion speeds) and ensemble averaged. To fabricate B
DOI: 10.1021/acs.nanolett.5b01925 Nano Lett. XXXX, XXX, XXX−XXX
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increases the propulsion speed, here by at least ∼25% when shorter (about one-turn) propellers are used instead of 2.2 turn propellers. In the literature24,25 even longer up to 4 turn helical micropropellers are reported, which are even further away from the optimal length. When the propulsion speed of 2.2 turn helices (without a payload, empty symbols) is compared to their towing speeds (with the payload, full symbols) in Figure 3A, it can be readily seen that the cargo has very little effect on the speed, indicating that the propellers are too long and mostly pulling themselves, rather than towing a cargo, in agreement with the arguments presented herein. For a short headless 0.7 turn propellers, the slope of the velocity curve is close to that of the propeller with a cargo (see Figure 3B); however, the absence of a payload shifted the step-out to higher frequency yielding a slightly higher maximal propulsion speed. Cargo that has a larger cross sectional area than the propeller will affect the propulsion speeds differently. To provide further theoretical support for the experimentally determined optimal length, we computed the dimensional slope of the curves in Figure 2, 2πr()̃ ∥/ξ∥̃ ), applying the particlebased algorithm9,19−21 (see Supporting Information for details) to determine the resistance coefficients of the propeller/ payload structure. The cargo is modeled as a sphere with radius rc positioned at the helical centerline and rigidly attached to the propeller of radius R made of spheres with radii r = 1 (see Figure 4). The translational, ξ∥̃ , and the coupling, )̃ ∥, viscous resistance coefficients are found as the corresponding force and torque exerted on the propeller/payload structure in the frame of the principal rotation axes22 when dragging it along the easy axis of rotation (which, in general, does not coincide with the propeller’s helical axis). The rotational friction κ̃∥ is determined by computing the torque exerted on the propeller carrying cargo when it is rotated about the easy axis of rotation. We use the experimental geometry, i.e., helical radius R/r = 2.0, cargo radius rc/r = 2.2, and the helical pitch angle θ = 53°, upon varying the propeller’s length L/rc (i.e., varying the number of spheres composing the propeller). The computed values of the slope U/ν are 0.073, 0.128, 0.125, and 0.139 μm for helices with 0.7, 1.1, 1.6, and 2.2 helical turns, respectively. The latter are in very good agreement with the corresponding
propellers without a head, a thin layer of sodium chloride was deposited on top of the silica particle monolayer before growing the magnetic Ni section and the helical silica tail. Dissolution in water detaches the head from the magnetic helix. For all micropropellers the helical radius was R ≈ 210 nm, filament radius r ≈ 110 nm, pitch angle θ ≈ 53°, and the head radius rc ≈ 240 nm. The measured speed of the propellers of different lengths is depicted in Figure 2 as a function of the actuation frequency ν
Figure 2. Experimental results for the propulsion speed vs rotation frequency of the external magnetic field with amplitude 40 G. The chiral propellers of different lengths have, respectively, 0.7, 1.1, 1.6, and 2.2 helical turns. The continuous lines are the analytical predictions from ref 20. The best-fit of the step-out frequency are 100, 80, 75, and 65 Hz, respectively. The values of the tumbling/wobbling transition frequency are approximately 8, 4, 3, and 1 Hz. The dimensional (μm) slopes of the curves are 0.072, 0.133, 0.118, and 0.128, respectively. The error bars represent the variation between different propellers.
= ω/2π and clearly shows that there is an optimal propeller with about one helical turn. A longer propeller with ∼2.2 turns yields a similar dependence on the actuation frequency (i.e., about the same net chirality) but has a lower step-out frequency. A shorter propeller with ∼0.7 helical turns is characterized by a higher step-out frequency, but a lower net chirality resulting in a lower maximum speed Umax at νs−o. Thus, the experiments clearly show that an optimal geometry
Figure 3. Effect of the payload on the propulsion upon varying the rotation frequency of the external magnetic field; full symbols stand for experimental results for chiral motors with the attached payload and empty symbols correspond to the headless motors (the head detached by dissolution of a sacrificial NaCl layer in water); the continuous lines are the analytical predictions from ref 20. (A) Propulsion speed for the external magnetic field strength of 40 and 75 G for chiral motors with 2.2 helical turns; the inset shows SEM images of 2.2 turn helix with an intact head (bottom), and the headless propeller (top). The scale bar is 500 nm. (B) Propulsion speed of the 0.7 turn chiral motors with/without a payload in comparison to longer 2.2 turn helices for the external magnetic field strength of 40 G. C
DOI: 10.1021/acs.nanolett.5b01925 Nano Lett. XXXX, XXX, XXX−XXX
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Figure 4. Illustration of the particle-based propellers with payload used in numerical computations. For all propellers the helical radius (normalized by the filament radius) is R/r = 2, the cargo radius rc/r = 2.2 and the pitch angle Θ = 53° and (A) 2.2; (B) 1.6; (C) 1.1 and (D) 0.7 helical turns.
Figure 5. Normalized maximal propulsion speed, Ũ max = Umax/(m⊥H/ r2η), vs the propeller’s length L/rc for the nanomotors with R/r = 2, rc/ r = 2.2, and θ = 53°. The solid line stands for the results ()̃ ∥/ξ∥̃ κ∥̃ ) of numerical particle-based computations, while the symbols for the experimental results for 0.7, 1.1, 1.6 and 2.2 turn helices; error bars correspond to an experimental deviation between different samples.
experimental best-fitted values 0.072, 0.133, 0.118, and 0.128 μm (see Figure 2). Focusing on the optima in Figure 2 at the step-out frequencies, we also compute the scaled maximal propulsion speed, Ũ max = Umax/(m⊥H/r2η). Since the value of the magnetization m⊥ is not readily available from the experiment, we estimate its value from the experimentally observed step-out frequency, 2πνs−o ≈ m⊥ H/κ∥, where κ∥ = κ̃∥ηr3 is the longitudinal rotation friction resistance with η = 1 cP being the dynamic viscosity of the liquid and r ≈ 110 nm the chiral filament radius. The dimensionless rotational resistance coefficient is computed for the experimental geometries using the particle-based algorithm resulting in κ̃∥ ≈ 660, 840, 970, and 1420 for propellers with 0.7, 1.1, 1.6, and 2.2 turns, respectively. This yields the following estimates of m⊥ ≈ 1.4 × 10−13, 1.4 × 10−13, 1.6 × 10−13, and 1.9 × 10−13 emu. The discrepancy in magnetic moment is due to uncertainty in the experimental measurement of the step-out frequency as a result of ensembleaveraging over many propellers with sightly different magnetic properties. Indeed, in the synchronous regime for ω < ωs−o the propulsion is of purely geometric nature and independent of the magnetic properties of the propeller (relatively small error bars in Figure 2), while ωs−o itself depends on m∥. The experimental values of Ũ max are shown (symbols) in Figure 5 for four different propeller lengths, L/rc together with the corresponding theoretical predictions )̃ ∥/(ξ∥̃ κ∥̃ ) from eq 1 (solid line). It can readily be seen that the agreement between the experiment and the particle-based computations is excellent and the best propeller is the one with ≈1.1−1.2 helical turns with L/rc ≃ 5−5.5. This is also in agreement with the optimal propeller shape in ref 23 where for r/rc = 0.4 the optimal propeller half-length (along the helical centerline) was 3 ≈ 4rc . In our case r/rc = 0.45 and the optimal shape corresponds to 3 ≈ 3.6rc . The headless propellers are modeled in a similar way with a smaller spherical cargo with rc/r = 1.5 replacing the large spherical payload; this smaller cargo is mimicking the residual magnetic part (see the inset in Figure 3). In agreement with experiment, our theory found no difference between a headless motor and one with cargo for a propeller with 2.2 turns. The slope and the maximal propulsion velocity are very similar in both cases as the payload has a little effect on a long propeller.
For the shorter motors with a 0.7 turn helix, we find that the slope is U/ν as 2πr()̃ ∥/ξ∥̃ ) = 0.08 μm compared with the experimental value of 0.078 μm. The same value of the slope of 0.078 μm found for propeller towing a payload (see Figure 3B) implies that some reduction in the coupling coefficient )̃ ∥ is compensated for by the corresponding reduction in ξ̃∥. However, for the short 0.7 turn propeller the reduction in the rotational friction is significant, κ̃∥ ≈ 490 (vs κ̃∥ ≈ 660 for a propeller with the payload) so that the step-out occurs at the higher frequency of ∼120 Hz (see Figure 3B) yielding a slightly higher maximum propulsion speed. The computed value of Umax/(m⊥H/r2η) ≈ 0.23 × 10−3 is in excellent agreement with the experimental result (0.23 ± 0.07) × 10−3. The axial/longitudinal magnetic moment m∥ is estimated from the frequency of the tumbling/wobbling transition, 2πνc = m∥H/κ⊥ (see Figure 2), where the transverse (in plane of the helical axis) rotational resistance of the propeller/payload structure κ⊥ = κ̃⊥ηr3. The values from the best-fit of the transition frequency for propellers with 0.7, 1.1, 1.6, and 2.2 turns are, respectively, 8, 4, 3, and 1 Hz. The numerically computed values of the scaled rotational resistance are κ̃⊥ ≈ 1620, 2430, 4370, and 10120. This yields the longitudinal magnetic moment m∥ ≈ 2.7 × 10−14, 2.0 × 10−14, 2.7 × 10−14, and 2.2 × 10−14 emu, respectively. To justify the assumption of negligible wobbling, we determine the minimal precession angle θmin as20 sin−1[1 + (κ⊥m⊥/κ∥ m∥)2]−1/2, where we used the numerically determined values of κ⊥, κ∥, and the estimated values of m⊥ and m∥. The resulting values of θmin are less than 1° for all propellers, and hence wobbling is indeed negligible. Using the estimated values of the magnetic moments m⊥ and m∥, we further obtain the average ratio m⊥/m∥ ∼ 6.7 ± 1.5; it is in agreement with the SQUID measurements (on wafer) showing m⊥/m∥ ≈ 2.4/0.3 = 8. This ratio controls the misalignment angle φ0 between the direction of the net magnetic moment with its transverse component that can be computed as20 φ0 = cos−1[1 + (m∥/ m⊥)2]−1/2 ≈ 8.9° ± 2°; it is in agreement with the measurements in a weak static magnetic field giving φ0 ≈ 9 ± 4°. We note that a bacterial flagellum has many turns, and its length is clearly much longer. This illustrates the difference between a flexible, internally driven (torque-balanced) flagellum D
DOI: 10.1021/acs.nanolett.5b01925 Nano Lett. XXXX, XXX, XXX−XXX
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(10) Schamel, D.; Pfeifer, M.; Gibbs, J. G.; Miksch, B.; Mark, A. G.; Fischer, P. J. Am. Chem. Soc. 2013, 135, 12353−12359. (11) Malac, M.; Egerton, R. F.; Brett, M. J.; Dick, B. J. Vac. Sci. Technol. B 1999, 17, 2671−2674. (12) Zhao, Y.-P.; Ye, D.-X.; Wang, G.-C.; Lu, T.-M. Nano Lett. 2002, 2, 351−354. (13) Hawkeye, M.; Brett, M. J. J. Vac. Sci. Technol. A 2007, 25, 1317− 1335. (14) He, Y.; Fu, J.; Zhang, Y.; Zhao, Y.; Zhang, L.; Xia, A.; Cai, J. Small 2007, 3, 153−160. (15) Huang, Z.; Bai, F. Nanoscale 2014, 6, 9401−9409. (16) Lighthill, M. J. J. Fluid Mech. 1960, 9, 305−317. (17) Childress, S. J. Fluid Mech. 2012, 705, 77−97. (18) Purcell, E. M. Proc. Natl. Acad. Sci. U.S.A. 1997, 94, 11307− 11311. (19) Raz, O.; Leshansky, A. M. Phys. Rev. E 2008, 77, 055305(R). (20) Morozov, K. I.; Leshansky, A. M. Nanoscale 2014, 6, 1580− 1588. (21) Morozov, K. I.; Leshansky, A. M. Nanoscale 2014, 6, 12142− 12150. (22) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Prentice-Hall: Englewood Cliffs, NJ, 1965. (23) Keaveny, E. E.; Walker, S. W.; Shelley, M. J. Nano Lett. 2013, 13, 531−537. (24) Ghosh, A.; Paria, D.; Singh, H. J.; Venugopalan, P. L.; Ghosh, A. Phys. Rev. E 2012, 86, 031401. (25) Ghosh, A.; Mandal, P.; Karmakar, S.; Ghosh, A. Phys. Chem. Chem. Phys. 2013, 15, 10817−10823.
rotating in the proximity of counter-rotating bacterial body and a rigid magnetic propeller corotating with the rigidly attached cargo. Moreover, it is anticipated that the swimming gait of a microorganism optimizes power rather than speed (see, e.g., ref 18). To conclude, we experimentally demonstrate that an optimal helix length exists for ferromagnetic nanopropellers. Our simple physical argument can readily be applied to other experimental systems. The optimum length results from a trade-off between maximizing the net chirality and minimizing the rotational viscous friction of the propeller. While common intuition suggests that an efficient nanoscrew should possess multiple helical turns, we however find in agreement with our qualitative arguments and rigorous particle-based computations that the optimal helical propeller has about one full turn.
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ASSOCIATED CONTENT
S Supporting Information *
Detailed experimental procedures, additional theoretical considerations, description of the numerical algorithm. This material is available free of charge via the Internet at http:// pubs.acs.org/.The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.nanolett.5b01925.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail: fi
[email protected]. *E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors (D.W., M.K., and P.F.) thank the Department Schütz, Max-Planck-Institute for Intelligent Systems, for access to the SQUID magnetometer, and the Department Spatz, MaxPlanck-Institute for Intelligent Systems, for SEM access. A.M.L. thanks Victor Kaganov for useful discussions on drills, screws, and bolts. This work was supported in part by the German− Israeli Foundation (GIF) via the grant no. I-1255-303.10/2014, “Dynamics of Artificial Magnetic Nanopropellers” (P.F. and A.M.L.), Rubin Scientific and Medical Research Fund (A.M.L.), and the Israel Ministry for Immigrant Absorption (K.I.M.).
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REFERENCES
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DOI: 10.1021/acs.nanolett.5b01925 Nano Lett. XXXX, XXX, XXX−XXX
