Acoustically propelled nano- and microcones: fast forward and backward motion
AAcoustically propelled nano- and microcones: fast forward and backward motion
Johannes Voß and Raphael Wittkowski ∗ Institut f¨ur Theoretische Physik, Center for Soft Nanoscience,Westf¨alische Wilhelms-Universit¨at M¨unster, D-48149 M¨unster, Germany
We focus on cone-shaped nano- and microparticles, which have recently been found to showparticularly strong propulsion when they are exposed to a traveling ultrasound wave, and studybased on direct acoustofluidic computer simulations how their propulsion depends on the cones’aspect ratio. The simulations reveal that the propulsion velocity and even its sign are very sensitiveto the aspect ratio, where short particles move forward whereas elongated particles move backward.Furthermore, we identify a cone shape that allows for a particularly large propulsion speed. Ourresults contribute to the understanding of the propulsion of ultrasound-propelled colloidal particles,suggest a method for separation and sorting of nano- and microcones concerning their aspect ratio,and provide useful guidance for future experiments and applications.
I. INTRODUCTION
Ultrasound-propelled colloidal particles, havingbeen discovered in experiments in 2012, consti-tute a particularly advantageous type of activenano- and microparticles. The most importantadvantages of motile particles that are propelled byultrasound compared to particles with other propul-sion mechanisms are the fact that the former parti-cles can move in various types of fluids and soft materials,the bio-compatibility of their propulsion mechanism,and the easy way of supplying the particles permanentlywith energy. As a consequence, ultrasound-propellednano- and microparticles have a number of importantpotential applications. An example is their usage asself-propelled nano- or microdevices in medicine, e.g., for targeted drug delivery.
There exist twodifferent types of acoustically propelled particles: rigidparticles andparticles with movable components.
The rigid particles are easier to produce in large num-bers and thus of special relevance with respect to futureapplications, where usually a large number of particlesis required. There exist also some hybrid particlesthat combine acoustic propulsion with other propulsionmechanisms. ∗ Corresponding author: [email protected]
In recent years, ultrasound-propelled nano-and microparticles have been intensivelyinvestigated.
Besides only two articles that are based on analyticalapproaches and an article that relies on directcomputational fluid dynamics simulations, a largenumber of experimental studies have been published sofar. In the previ-ous work, mostly cylindrical particles with a concave endand a convex end were studied.
As a limiting case, which corresponds to a very shortcylindrical particle with concave and convex ends, alsohalf-sphere cups (nanoshells) were considered.
Recently, half-sphere-shaped particles, cone-shaped par-ticles, and spherical- as well as conical-cup-like particleswere compared with respect to their propulsion. In twoother studies, gear-shaped microspinners were ad-dressed, and there are a few additional publications thatfocus on particles with movable components.
Among the rigid particles with mainly translationalmotion that have been addressed so far, cone-shaped par-ticles and conical-cup-shaped particles showed the fastestpropulsion, where the speeds of cone-shaped and conical-cup-shaped particles differed only slightly. Since cone-shaped particles have a simpler shape, which facilitatestheir fabrication, and a larger volume, which is advan-tageous for delivery of drugs or other substances, thanconical-cup-shaped particles, the former particles havebeen identified as particularly suitable candidates for effi-cient ultrasound-propelled particles that could be used in a r X i v : . [ c ond - m a t . s o f t ] F e b future experiments and applications. Cone-shaped par-ticles can be produced, e.g., by electrodeposition, ordirectly be found, e.g., in the form of carbon nanocones, in large numbers. Up to now, however, only ultrasound-propelled cone-shaped particles with a particular aspectratio have been studied, although the aspect ratio canhave a strong influence on the efficiency of the particles’propulsion. Given that previous studies found for theshort spherical-cup-shaped particles motion towards theparticles’ convex end but for the longer cylindricalparticles with concave and convex ends motion towardsthe concave end, also the direction of propulsion candepend on the aspect ratio.Therefore, in this article we investigate the acousticpropulsion of cone-shaped nano- and microparticles inmore detail. Using direct acoustofluidic simulations, westudy how the propulsion of acoustically propelled nano-and microcones depends on their aspect ratio and wedetermine an aspect ratio that is associated with partic-ularly fast and thus efficient propulsion. II. METHODS
This work is based on a similar setup and procedure asRef. 30. We consider a particle that is surrounded by wa-ter and exposed to ultrasound. Using direct acoustoflu-idic simulations, where the compressible Navier-Stokesequations are solved numerically, the propagation of ul-trasound through the water and the interaction with theparticle are calculated. These calculations allow to deter-mine the sound-induced forces and torques acting on theparticle, from which in turn we calculate the particle’stranslational and angular propulsion velocities.Figure 1 shows the setup in detail. We consider a par-ticle with a conical shape in two spatial dimensions. Theparticle is oriented perpendicular to the direction of wavepropagation, has a fixed cross-section area A , and is de-scribed by a particle domain Ω p . Its aspect ratio χ = h/σ with the particle’s height h and diameter σ is varied. Theposition of the particle is fixed. This means that the re-sults of the simulations are valid for a particle which isheld in place. Such a particle can be seen as a free movingparticle in the limiting case of an infinite mass density.This limiting case can, in turn, be considered as an up-per bound for a free moving particle made of a materialwith a high mass density like gold, which is a widely usedmaterial for such particles. The particle is positioned in the middle of a water-filled rectangular domain so that the center of mass ofthe rectangle and the center of mass S of the particle co-incide. One edge of the rectangular domain has length l = 200 µ m and is perpendicular to the direction of ul-trasound propagation. The other edge is parallel to thedirection of ultrasound propagation and has length 2 l .We choose a Cartesian coordinate system such that the x axis is parallel to the direction of ultrasound prop-agation and the x axis is perpendicular to that direc- v in ( t ) p in ( t ) i n l e t directionof wavepropagation slipslip l l l F k F ⊥ n o s li p particle o u t l e t fluid Ω p x x σ h A = σ h /2 TS l /2 FIG. 1. The considered setup. A rigid cone-shaped particleis in the middle of a fluid-filled rectangular domain. The par-ticle has width σ , height h , and a fixed cross-section area A ,and is described by a particle domain Ω p . Furthermore, therectangular domain has width 2 l and height l and the centerof mass S of the particle is in the middle of the rectangular do-main. At the inlet, a traveling ultrasound wave is entering thefluid-filled domain. For this purpose, an inflow velocity v in ( t )and pressure p in ( t ) are prescribed at the inlet. The ultrasoundwave propagates through the fluid, where slip boundary con-ditions are used for the lateral boundaries of the rectangulardomain. At the particle, for which no-slip boundary condi-tions are used, the ultrasound exerts a propulsion force withtime-averaged components F (cid:107) and F ⊥ parallel and perpen-dicular to the particle’s orientation, respectively, as well as atime-averaged torque T . When the ultrasound wave reachesthe end of the domain, it leaves the domain through the out-let. tion, i.e., the coordinate axes are parallel to the edges ofthe rectangular domain. The ultrasound wave has fre-quency f = 1 MHz and enters the rectangular domainat the edge perpendicular to the sound-propagation di-rection. We prescribe the incoming ultrasound wave bya time-dependent inflow pressure p in ( t ) = ∆ p sin(2 πf t )and velocity v in ( t ) = (∆ p/ ( ρ c f )) sin(2 πf t ) perpendicu-lar to the inlet with the pressure amplitude ∆ p = 10 kPa,density of the initially quiescent fluid ρ = 998 kg m − ,and sound velocity in the fluid c f = 1484 m s − .The acoustic energy density from this wave is E =∆ p / (2 ρ c ) = 22 . − . After a distance l = λ/ λ is the wavelength of the ultrasound wave λ =1 .
484 mm, the wave reaches the fixed rigid particle. Theinteraction of the ultrasound with the particle leads totime-averaged forces F (cid:107) parallel and F ⊥ perpendicularto the particle orientation as well as to a time-averagedtorque T relative to the reference point S acting on theparticle. After a further distance l the wave leaves thedomain through an outlet at the edge of the water domainopposing the inlet. The particle boundaries are describedby a no-slip condition and at the edges of the water do-main parallel to the direction of sound propagation weassume a slip condition.In the simulations, we numerically solve the continu-ity equation for the mass-density field of the fluid, thecompressible Navier-Stokes equation, and a linear con-stitutive equation for the fluid’s pressure field. Thuswe are avoiding approximations like perturbation ex-pansions that are used in most previous studies usinganalytical or numerical methods. For solv-ing these equations, we used the finite volume methodimplemented in the software package OpenFOAM. Weapplied a structured mixed rectangular-triangular meshwith about 300,000 cells, where the cell size ∆ x is verysmall close to the particle, and larger far away from it.Concerning the time integration, an adaptive time-stepmethod is used with a time-step size ∆ t such that theCourant-Friedrichs-Lewy number C = c f ∆ t ∆ x (1)is smaller than one. To get sufficiently close to the sta-tionary state, we simulated a time interval with a dura-tion of t max = 500 τ or more, where τ is the period ofthe ultrasound wave. An individual simulation requireda computational expense of typically 36 ,
000 CPU corehours. The reason for this expense is the necessary finediscretization in space and time relative to the large spa-tial and temporal domains.Through the simulations, we calculated the time-dependent force and torque acting on the particle in thelaboratory frame. Since the particle has no-slip bound-ary conditions and is fixed in space, the fluid velocityis zero at the fluid-particle interface. So the force andtorque can be calculated by the integral of the stress ten-sor Σ over the particle surface. The force (cid:126)F ( p ) + (cid:126)F ( v ) andtorque T ( p ) + T ( v ) consist of two components, namely apressure component (superscript “( p )”) and a viscositycomponent (superscript “( v )”) with F ( α ) i = (cid:88) j =1 (cid:90) ∂ Ω p Σ ( α ) ij d A j , (2) T ( α ) = (cid:88) j,k,l =1 (cid:90) ∂ Ω p (cid:15) jk ( x j − x p ,j )Σ ( α ) kl d A l (3)for α ∈ { p, v } . Here, Σ ( p ) and Σ ( v ) are the pressure com-ponent and the viscous component of the stress tensor,respectively. d (cid:126)A ( (cid:126)x ) = (d A ( (cid:126)x ) , d A ( (cid:126)x )) T is the normaland outwards oriented surface element of ∂ Ω p at position (cid:126)x when (cid:126)x ∈ ∂ Ω p , (cid:15) ijk the Levi-Civita symbol, and (cid:126)x p theposition of S. To obtain the time-averaged stationaryvalues, we locally averaged over one period and extrapo-lated towards t → ∞ using the extrapolation proceduredescribed in Ref. 30.With this procedure, we get the force (cid:126)F = (cid:126)F p + (cid:126)F v withpressure component (cid:126)F p = (cid:104) (cid:126)F ( p ) (cid:105) and viscous component (cid:126)F v = (cid:104) (cid:126)F ( v ) (cid:105) as well as the torque T = T p + T v withcomponents T p = (cid:104) T ( p ) (cid:105) and T v = (cid:104) T ( v ) (cid:105) acting on theparticle, where (cid:104)·(cid:105) denotes the time average. To calculatethe translational-angular velocity vector (cid:126) v = ( (cid:126)v, ω ) T withthe particle’s translational velocity (cid:126)v and angular velocity ω , we define the force-torque vector (cid:126) F = ( (cid:126)F , T ) T . Then the values of (cid:126) v can be calculated with the Stokes law as (cid:126) v = 1 ν s H − (cid:126) F (4)with the fluid’s shear viscosity ν s and the hydrodynamicresistance matrix H = (cid:18) K C TS C S Ω S (cid:19) . (5)Here, K S , C S , and Ω S are 3 × µ m to the particle, so that H canbe calculated. The general structure of H for a particlewith a shape as we study here is H = K C Ω , (6)where the values of the nonzero elements are given inTab. I for each aspect ratio considered in this work. Byneglecting the contributions K , C , Ω , and Ω thatcorrespond to the lower and upper surfaces of the par-ticle, we can then use the three-dimensional versions ofEqs. (2)-(4).We determine the components of (cid:126)F and (cid:126)v parallel andperpendicular to the particle’s orientation, i.e., paral-lel to the x and x axes, respectively. These compo-nents are the parallel force F (cid:107) = ( (cid:126)F ) = F (cid:107) ,p + F (cid:107) ,v ,with its pressure component F (cid:107) ,p = ( (cid:104) (cid:126)F ( p ) (cid:105) ) and vis-cous component F (cid:107) ,v = ( (cid:104) (cid:126)F ( v ) (cid:105) ) , perpendicular force F ⊥ = ( (cid:104) (cid:126)F ⊥ (cid:105) ) = F ⊥ ,p + F ⊥ ,v with the components F ⊥ ,p = ( (cid:104) (cid:126)F ( p ) (cid:105) ) and F ⊥ ,v = ( (cid:104) (cid:126)F ( v ) (cid:105) ) , parallel speed v (cid:107) = ( (cid:126)v ) , and perpendicular speed v ⊥ = ( (cid:126)v ) .Nondimensionalization of the governing equationsleads to four dimensionless numbers: the Helmholtz num-ber He, a Reynolds number corresponding to the shearviscosity Re s , another Reynolds number correspondingto the bulk viscosity Re b , and the number Ma Eu cor-responding to the pressure amplitude ∆ p of the ultra-sound wave entering the simulated system, where Ma isthe Mach number and Eu is the Euler number. Table IIshows the names and symbols of the parameters that arerelevant for our simulations and their values that we havechosen in analogy to the values used in Ref. 30. By us-ing the parameter values from Tab. II, the dimensionless χ K / µ m K / µ m K / µ m C / µ m C / µ m Ω / µ m Ω / µ m Ω / µ m .
25 8 .
58 11 .
29 8 . − .
38 0 .
61 3 .
44 4 .
76 4 . . .
49 9 .
72 8 . − .
14 0 . .
36 2 . .
75 8 .
76 9 .
07 7 . − .
03 0 .
11 3 .
06 3 .
03 2 .
221 9 .
03 8 .
77 7 . − . − .
14 3 .
18 2 .
96 2 . . .
69 8 .
57 8 .
13 0 . − .
54 3 .
68 2 .
97 2 .
682 10 .
15 8 .
55 8 .
39 0 . − . .
25 3 .
08 3 . . .
61 8 .
56 8 .
59 0 . − .
21 4 .
86 3 .
17 3 .
973 11 .
02 8 .
64 8 .
84 0 . − . .
53 3 .
29 4 . . .
41 8 .
75 9 .
08 0 . − .
79 6 .
23 3 .
41 5 .
474 11 .
67 8 .
87 9 .
33 1 . − .
07 6 .
94 3 .
51 6 . H for a particle with a triangular cross section as shown in Fig. 1,a thickness of 1 µ m in the third dimension, and the center of mass as the reference point for different aspect ratios χ = h/σ with the particle’s height h and diameter σ . Name Symbol Value
Particle cross-section area A . µ m Particle diameter-height ratio χ = h/σ . σ (cid:112) A/χ
Particle height h σχ
Sound frequency f c f − Time period of sound τ = 1 /f µ sWavelength of sound λ = c f /f .
484 mmTemperature of fluid T .
15 KMean mass density of fluid ρ
998 kg m − Mean pressure of fluid p
101 325 PaInitial velocity of fluid (cid:126)v (cid:126) − Sound pressure amplitude ∆ p
10 kPaAcoustic energy density E = ∆ p / (2 ρ c ) 22 . − Shear/dynamic viscosity of fluid ν s .
002 mPa sBulk/volume viscosity of fluid ν b .
87 mPa sInlet-particle or particle-outletdistance l λ/ l µ mMesh-cell size ∆ x
15 nm-1 µ mTime-step size ∆ t t max τ TABLE II. Parameters that are relevant for our simulationsand their values, which are chosen similar to those in Ref.30. The values of the speed of sound c f , mean mass density ρ , shear viscosity ν s , and bulk viscosity ν b are calculated forwater at rest at normal temperature T and normal pressure p . numbers for our simulations have the following values:He = 2 πf √ A/c f ≈ . · − , (7)Re s = ρ c f √ A/ν s ≈ , (8)Re b = ρ c f √ A/ν b ≈ , (9)Ma Eu = ∆ p/ ( ρ c ) ≈ . · − . (10)Note that the Reynolds number Re = ρ (cid:113) A ( v (cid:107) + v ⊥ ) /ν s < · − , which characterizesthe particle motion through the fluid, is close to zero. III. RESULTS AND DISCUSSION
Figure 2 shows our results for the propulsion-force components F (cid:107) and F ⊥ , propulsion torque T ,translational-propulsion-velocity components v (cid:107) and v ⊥ ,and angular propulsion velocity ω as functions of the as-pect ratio χ ∈ [0 . , F (cid:107) and velocity v (cid:107) have a strong depen-dence on the aspect ratio χ . This includes even a signchange. Both curves have qualitatively the same course.They start at χ = 0 .
25 with negative propulsion force F (cid:107) = − .
32 fN and velocity v (cid:107) = − . µ m s − . Increas-ing χ leads to a positive sign of F (cid:107) and v (cid:107) until about χ = 1 where the force is maximal with F (cid:107) = 0 .
76 fN andalso the speed reaches its maximum v (cid:107) = 0 . µ m s − .Afterwards, the propulsion force and velocity decrease toand remain at negative values. The globally maximal am-plitude is reached at χ = 2 .
5, where F (cid:107) = − .
84 fN and v (cid:107) = − . µ m s − . In the further course of the curves,the values rise until χ = 3 and then decrease again until χ = 4, where the values saturate at F (cid:107) = − .
73 fN and v (cid:107) = − . µ m s − .According to amount, the largest velocity v (cid:107) = − . µ m s − , found here for χ = 2 .
5, is about 80%larger than the velocity of cone-shaped particles withaspect ratio χ = 0 . Sincethe energy density E = 0 . − used in the presentwork is strongly smaller than the largest energy density E max = 4 . − allowed by the U.S. Food and DrugAdministration for diagnostic applications in the humanbody, the results can be extrapolated to higher ener-gies. To be able to do this, we need to make some assump-tions. In experiments, the dependence of the velocity wasmeasured to be proportional to the squared amplitude ofthe driving voltage, which is the same scaling as forthe energy density. Therefore, we can assume that thevelocity is proportional to the energy density. This as-sumption leads to a factor E max /E = 216. Rescaling thehighest magnitude of the velocity, which is attained at χ = 2 .
5, to the higher energy density E max results in v rescaled = 21 . µ m s − . For χ = 2 .
5, the values of the -20-15-10-505101520 F / f N F , p / f N F , v / f N F F F F F F F Fx x a F = 10 F , p + 10 F , v F , p F , v -0.2-0.15-0.1-0.0500.050.10.150.2 v / ( m s ) v = ( H / s ) Simulation data-0.2-0.100.10.20.30.40.50.6 F / f N F , p / f N F , v / f N F F F F F F F Fx x b F = F , p + F , v F , p F , v -0.02-0.0100.010.020.030.040.050.06 v / ( m s ) v = ( H / s ) Simulation data0 0.5 1 1.5 2 2.5 3 3.5 4= h /-0.5-0.4-0.3-0.2-0.100.10.20.3 T / (f N m ) T p / (f N m ) T v / (f N m ) T T T T T T T Tx x c T = T p + T v T p T v -0.1-0.08-0.06-0.04-0.0200.020.040.06 / s = ( H / s ) Simulation data
FIG. 2. (a) Simulation data for the forces F (cid:107) ,p and F (cid:107) ,v acting on a particle with triangular cross section and aspect ratio χ parallel to its orientation, their sum F (cid:107) = F (cid:107) ,p + F (cid:107) ,v , and the corresponding propulsion velocity v (cid:107) for various values of χ . (b)The corresponding forces F ⊥ ,p and F ⊥ ,v for the direction perpendicular to the particle orientation, their sum F ⊥ = F ⊥ ,p + F ⊥ ,v ,and the velocity v ⊥ for different values of χ . (c) Results of the simulation for the torque components T p and T v acting on theparticle, their sum T = T p + T v , and the corresponding angular velocity ω for different values of χ . parameters describing the particle size are σ = 0 . µ mand h = 1 . µ m. With the higher energy density theparticle would thus move with a speed of roughly 19body lengths per second. For the largest positive ve-locity, which corresponds to χ = 1, the rescaled speedhas the value v rescaled = 18 . µ m s − . Since the size pa-rameters are now σ = 0 . µ m and h = 0 . µ m, thisspeed equals 26 . χ = 2 . χ = 1 to reach a maximalabsolute speed or a maximal speed related to the particlelength, respectively. For medical applications, compactparticles in a certain size range are necessary suchthat they do not block the blood flow. Therefore, theaspect ratio χ = 1 could be preferable for these types ofapplications.The dependence of F (cid:107) ,p and F (cid:107) ,v on the aspect ratio issimpler. Both values keep their sign with F (cid:107) ,p as a nega-tive force and F (cid:107) ,v as a positive force. The magnitude ofboth values increases until χ = 1 fast to F (cid:107) ,p = − .
56 fNand F (cid:107) ,v = 11 .
32 fN. Afterwards, the magnitude oscil-lates a little bit but with the tendency to increase slowlytowards F (cid:107) ,p = − .
60 fN and F (cid:107) ,v = 12 .
87 fN at χ = 4.We now consider the propulsion parallel to the direc-tion of propagation of the ultrasound wave. The per-pendicular propulsion force F ⊥ increases for increasing χ slowly until χ = 1 . F ⊥ = 0 .
25 fN. Then it is con-stant until χ = 2 .
5. From χ = 2 . χ = 3 a strongincrease occurs to F ⊥ = 0 .
41 fN. Subsequently, a slowfurther increase follows. The perpendicular velocity v ⊥ can be seen as roughly constant with v ⊥ = 0 . µ m s − until χ = 2 . χ = 3with v ⊥ = 0 . µ m s − . Afterwards, it is roughly con-stant again. This behavior can be understood as follows:In the direction of ultrasound propagation, two opposingforces act on a particle. These are the acoustic radiationforce and the acoustic streaming force. Typically, for aparticle with a size of about a micrometer, the acous-tic radiation force is the dominant one. The scalingbehavior of the acoustic radiation force is nontrivial fornonspherical shapes, but for a sphere it scales linearlywith the particle volume. Since we kept the particlevolume constant, it is therefore reasonable that the ve-locity v ⊥ shows no strong change when the aspect ratioof the particle is varied.The value of the pressure component F ⊥ ,p of the force F ⊥ increases for increasing χ and the value of the viscouscomponent F ⊥ ,v decreases roughly until χ = 2 .
5, exceptfor a slight intermediate growth of F ⊥ ,v near χ = 1 .
5. At χ = 3, there is a downwards oriented peak in the am-plitude for both components. Afterwards, the amplitudeincreases again for both components, and from χ = 3 . χ from T = − .
005 fN µ m to T = − .
43 fN µ m, where the curve has a small local maximumat χ = 2. The corresponding angular velocity decreases from ω = − .
004 s − at χ = 0 .
25 to ω = − .
033 s − at χ = 1 .
5, increases afterwards to ω = − .
023 s − at χ = 2 .
5, decreases again to ω = − .
06 s − at χ = 3, andremains there for larger values of χ . The small valuesof the torques are negligible compared to the rotationalBrownian motion of the particles and indicate that thereis either no preferred orientation of the particles or a sta-ble orientation is close to the orientation considered inthe present work. Considering the aspect ratio χ = 4,where we found, according to amount, the largest torque T = − .
43 fN µ m and angular velocity ω = − .
06 s − ,a change of the particle orientation by 90 degrees takesroughly 26 s. On the other hand, calculating the parti-cle’s diffusion tensor D = ( k B T /ν s ) H − , where k B is theBoltzmann constant, leads to a rotational diffusion coef-ficient D = 0 .
68 s − of the particle, which implies thatthe particle orientation changes significantly by Brownianrotation on the time scale 1 .
46 s. This estimate clearlyshows that the torques resulting from the ultrasound areso weak that they are dominated by Brownian rotation.However, if the angular velocity is rescaled in the sameway as the translation velocity to an energy density E max ,a change of the particle orientation by 90 degrees needsonly 0 .
12 s and is thus dominant compared to the Brow-nian rotation.Concerning the components of the torque, the pres-sure component T p decreases (with superimposed fluc-tuations) for increasing χ from T p = 0 .
02 fN µ m to T p = − .
36 fN µ m, whereas the viscous component T v fluctuates (with stronger amplitude than for T p ) aroundzero.In summary, the force and velocity perpendicular tothe direction of propagation of the ultrasound wave havea sign change at a particular value of the aspect ratio ofthe particle, the force and velocity parallel to the prop-agation direction do not change sign, and the torque isvery weak.Our results are in line with the available theoreticalresults on ultrasound-propelled particles from the litera-ture. In the theory of Collis et al. , the propulsion direc-tion depends strongly on the acoustic Reynolds number β = ρ σ πf / (2 ν s ), which ranges for our work between β = 0 . χ = 4 and β = 3 . χ = 0 .
25. They foundthat particles can change their propulsion direction up totwo times when increasing the acoustic Reynolds num-ber, which is exactly what happens here. According totheir theory, this should happen for β ∼ O (1), which isperfectly in line with the interval of values for β we inves-tigated. A similar result was found experimentally. There, long cylinders with spherical caps at the ends cor-responding to an aspect ratio χ = 4 . χ = 17 . whereas the shorthalf-sphere cups with aspect ratio χ = 0 . This is qualitatively thesame behavior as for our particles, where the long oneswith χ (cid:38) . . (cid:46) χ (cid:46) IV. CONCLUSIONS
We have studied the acoustic propulsion of nano-and microcones powered by a traveling ultrasound wavethrough direct numerical simulations. Our results showthat the propulsion of the particles depends sensitively ontheir aspect ratio and includes both fast forward and fastbackward motion. The strong dependence of the propul-sion on the aspect ratio could be used to separate andsort artificial and natural cone-shaped particles (such ascarbon nanocones) in an efficient and easy way with re-spect to their aspect ratio. For later applications of cone-shaped ultrasound-propelled particles, e.g., in medicine,an aspect ratio of χ = 1 was identified as a very suitablechoice, since it combines a compact particle shape with alarge body-lengths-per-time speed. This finding also sug-gests to use cone-shaped particles with this aspect ratioas a more efficient particle design for future experiments.The obtained results are in good agreement withthe literature and expand the understanding of acous-tically propelled colloidal particles, which is helpfulwith regard to future experiments and applications innanomedicine or materials science. Furthermore, theknowledge about the particle propulsion can be used tomodel this propulsion when describing the dynamics ofthe particles via Langevin equations or field theories based on symmetry-based modeling, the interaction-expansion method, classical dynamical density func-tional theory, or other analytical approaches on timescales that are much larger than the period of the ultra-sound.In the future, this study should be extended by con-sidering other particle orientations and studying how thepropulsion depends on the angle between the particle ori-entation and the direction of propagation of the ultra-sound. Furthermore, the dependence of the propulsionon parameters like the ultrasound frequency, fluid viscos-ity, and pressure amplitude still need to be investigated. CONFLICTS OF INTEREST
There are no conflicts of interest to declare.
ACKNOWLEDGMENTS
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