Direct Measurement of Lighthill's Energetic Efficiency of a Minimal Magnetic Microswimmer
Carles Calero, José García-Torres, Antonio Ortiz-Ambriz, Francesc Sagués, Ignacio Pagonabarraga, Pietro Tierno
JJournal Name
Direct Measurement of Lighthill’s Energetic Efficiencyof a Minimal Magnetic Microswimmer † Carles Calero, ‡ a , b José García-Torres, ‡ a , b , Antonio Ortiz-Ambriz, a , b Francesc Sagués, c , b ,Ignacio Pagonabarraga, a , d , e and Pietro Tierno a , b , e ∗ The realization of artificial microscopic swimmers able topropel in viscous fluids is an emergent research field of fun-damental interest and vast technological applications. Forcertain functionalities, the efficiency of the microswimmerin converting the input power provided through an externalactuation into propulsive power output can be critical. Herewe use a microswimmer composed by a self-assembledferromagnetic rod and a paramagnetic sphere and directlydetermine its swimming efficiency when it is actuated bya swinging magnetic field. Using fast video recording andnumerical simulations we fully characterize the dynamicsof the propeller and identify the two independent degreesof freedom which allow its propulsion. We then obtainexperimentally the Lighthill’s energetic efficiency of theswimmer by measuring the power consumed during propul-sion and the energy required to translate the propeller atthe same speed. Finally, we discuss how the efficiency ofour microswimmer could be increased upon suitable tuningof the different experimental parameters.
The realization of faster and smaller micro/nanopropellers isan active topic with direct applications in the emerging fieldsof drug delivery , microsurgery and lab-on-a-chip technol- a Departament de Física de la Matèria Condensada, Universitat de Barcelona, 08028Barcelona, Spain. b Institut de Nanociència i Nanotecnologia, Universitat de Barcelona, 08028 Barcelona, Spain. c Departament de Ciència de Materials i Química Física, Universitat de Barcelona,Barcelona, Spain. d CECAM, Centre Européen de Calcul Atomique et Moléculaire, École PolytechniqueFédérale de Lasuanne, Batochime, Avenue Forel 2, Lausanne, Switzerland. e Universitat de Barcelona, Institute of Complex Systems (UBICS), Universitat deBarcelona, 08028 Barcelona, Spain. * E-mail: [email protected] † Electronic Supplementary Information (ESI) available: Two experimental videos(.WMF) showing the dynamics of the nanorod-colloid micropropeller played at dif-ferent speed. See DOI: 00.0000/00000000. ‡ These authors contributed equally to this work. ogy . The main challenge arises from the low Reynolds num-ber regime in which these micropropellers operate, where viscousforces dominate over inertial ones and the Stokes equation be-comes time reversible. Thus, in order to attain propulsion, theseprototypes should avoid reciprocal motion, namely periodic back-ward and forward displacement. As described by Purcell, thiscondition can be satisfied by a minimum of two independent de-grees of freedom, when their change in time describes a finitearea in the parameter space .Recent years have witnessed a variety of theoretical proposi-tions , and a number of experimental advances which haveprovided multiple micropropeller prototypes based on differentpropulsion mechanisms, including chemical reactions , mag-netic or light fields . Most of these prototypes havebeen characterized and compared mainly in terms of their achiev-able speed, a simple observable that can be calculated directlyfrom particle tracking. However, choosing the correct propul-sion scheme for a specific task requires a suitable measure of thepropeller efficiency not limited to the sole speed. The control ofswimming efficiency is of paramount importance both in artificialmicroswimmers to, e.g. avoid excessive heating due to dissipa-tion, and in microorganisms to optimize energy release .In Lighthill quantified the swimmer efficiency in convert-ing input energy to thrust power through a single parameter e ,which compares the external power (cid:104) Φ (cid:105) , required to induce amean velocity V in a medium of viscosity η , to the power neededto rigidly drag the swimmer at the same speed with an externalforce F drag , e = F drag V (cid:104) Φ (cid:105) (1)Although other measures of the efficiency have been pro-posed , especially to account for collective ciliary motions,the Lighthill energetic efficiency remains the standard measureto account for swimming efficiency of single propellers at the mi-croscale. This parameter has been employed in different theoret-ical works to analyze the performance of simple artificial designs Journal Name, [year], [vol.] , a r X i v : . [ c ond - m a t . s o f t ] O c t ike three-link flagella , three-sphere swimmers , squirm-ers , necklace-like propellers and undulating magnetic sys-tems . However, to date Lighthill’s efficiency has been only ex-perimentally estimated in few biological systems , where thecomplex flagellar dynamics precludes the possibility to directlydetermine the relevant degrees of freedom of the microswimmer,and indirect methods such as optical tweezers, are used.In this article we directly measure the energetic Lighthill effi-ciency of a minimal magnetic microswimmer cyclically actuatedby an external magnetic field. The microswimmer is composed bya ferromagnetic nanorod and a paramagnetic microsphere whichself-assemble due to magnetic dipolar interactions. The pair issubjected to a swinging magnetic field that forces the rod to tiltand to slide at the surface of the microsphere. Such an actuatingfield would not induce propulsion for a single body swimmer in aviscous fluid, unless composed by a flexible tail . The simplicityof the microswimmer allows us to show that its locomotion is en-abled by only two independent degrees of freedom, which are ex-perimentally identified and characterized. The precise measure-ment of the microswimmer dynamics allows for a direct, accurateand systematic measurement of its energetic Lighthill efficiency.By combining real space/time experiments with theory and nu-merical simulations, we determine speed and efficiency over alarge range of frequencies and amplitudes. We will first describethe microswimmer and indicate how we exploit its simplicity todescribe its behavior both experimentally and theoretically. Weexploit subsequently this detailed control to quantify its efficiencyand characterize its regime of motion.Our hybrid microswimmer is composed by a spherical param-agnetic particle of radius R = . µ m and a ferromagnetic Nickelnanorod that is synthesized via template-assisted electrodepo-sition. The latter has a length of L = µ m , a diameter of D = and displays a permanent magnetic moment alongits long axis, m n = . · − Am − . In contrast, the spheri-cal particle is characterized by a magnetic volume susceptibility χ = . , and under an external field BBB acquires an induced mo-ment mmm p = π R χ BBB / ( µ ) which points along the field direction,being µ = π · − Hm − . Both elements are dispersed in highlydeionized water and allowed to sediment above a glass substrate,which is placed at the center of a set of orthogonal coils arrangedon the stage of a brightfield optical microscope, more details arein the Material and Methods section. To further understand theexperimental data, we perform numerical simulations of a simpletheoretical model of the microswimmer which captures the ge-ometry of the particles and considers their mutual magnetic andhydrodynamic interactions, see Materials and Methods section formore details.We actuate our prototype by using a swinging magnetic fieldcomposed by an oscillating component B y with frequency ν and aperpendicular, static field B x , BBB ≡ [ B x , B y sin ( πν t )] . The effect ofthe applied field on the relative displacement of the two elementsduring a half-cycle, t ∈ [ / ( ν ) , / ( ν )] , is shown in Fig.1(a). Ini-tially, both the permanent, m n , and the induced, m p , moments arealigned with the field direction, B x , B y > . As B y oscillates, m p fol-lows instantaneously the external field. The nanorod rotates anangle θ about the contact point with the paramagnetic particle Fig. 1 (a) Sequence of schematics showing the colloid-nanorod pairsubjected to the swinging magnetic field
BBB ( t ) during half-cycle, m n and m p denote, respectively, the permanent and the induced moments of thenanorod and the colloid. (b) Corresponding microscope images showingrelative movement of the pair during half cycle. The pair is propelled by afield with frequency ν = and amplitudes B x = . , B y = . and recorded at fps, see corresponding VideoS1 and VideoS2 inthe Supporting Information. The scale bar is µ m . (c,d) Cycles in the( φ , θ ) plane at different frequencies of the applied field for experiments(c) and numerical simulation (d). The small schematic on the right pro-vides the definition of the rotational angle of the nanorod ( θ ) , and theangle between the x -axis and the contact point between the nanorod andthe particle, ( φ ) .due to the torque exerted by the field. Simultaneously, the attrac-tive dipolar interaction between the pair forces the rod to slideover the particle surface to align with m p , minimizing the mag-netic energy of the pair. Such sliding motion can be characterizedby a second degree of freedom φ (see Fig 1), which representsthe angle between the x -axis and the contact point between thetip of the rod and the particle. The uncoupled rotation and slid-ing motions are able to break the time-reversal symmetry of theflow, leading to the microswimmer net translation. This mecha-nism is observed in our numerical simulations and demonstratedexperimentally in Fig.1(b), where the pair moves with the micro-sphere in front of it for these geometric parameters. Therefore,the propulsion of the microswimmer is allowed by the two decou-pled rotational modes and it does not require the presence of abounding wall, as opposed to the mechanism reported in Ref. .We record high frame-rate videos to directly measure the twoangular degrees of freedom ( φ , θ ) . The non-reciprocity of thecyclical motion is evident from the trajectories in the parameterspace ( φ , θ ) shown in Fig.1(c) for experiments, and in Fig.1(d) forsimulations. Indeed, the trajectories follow different paths in thefirst and second half-cycles of the actuating field B ( t ) , describinga closed asymmetric region. As the frequency of the field is re-duced, the rotation and sliding motions of the rod become more Journal Name, [year], [vol.][vol.]
BBB ( t ) during half-cycle, m n and m p denote, respectively, the permanent and the induced moments of thenanorod and the colloid. (b) Corresponding microscope images showingrelative movement of the pair during half cycle. The pair is propelled by afield with frequency ν = and amplitudes B x = . , B y = . and recorded at fps, see corresponding VideoS1 and VideoS2 inthe Supporting Information. The scale bar is µ m . (c,d) Cycles in the( φ , θ ) plane at different frequencies of the applied field for experiments(c) and numerical simulation (d). The small schematic on the right pro-vides the definition of the rotational angle of the nanorod ( θ ) , and theangle between the x -axis and the contact point between the nanorod andthe particle, ( φ ) .due to the torque exerted by the field. Simultaneously, the attrac-tive dipolar interaction between the pair forces the rod to slideover the particle surface to align with m p , minimizing the mag-netic energy of the pair. Such sliding motion can be characterizedby a second degree of freedom φ (see Fig 1), which representsthe angle between the x -axis and the contact point between thetip of the rod and the particle. The uncoupled rotation and slid-ing motions are able to break the time-reversal symmetry of theflow, leading to the microswimmer net translation. This mecha-nism is observed in our numerical simulations and demonstratedexperimentally in Fig.1(b), where the pair moves with the micro-sphere in front of it for these geometric parameters. Therefore,the propulsion of the microswimmer is allowed by the two decou-pled rotational modes and it does not require the presence of abounding wall, as opposed to the mechanism reported in Ref. .We record high frame-rate videos to directly measure the twoangular degrees of freedom ( φ , θ ) . The non-reciprocity of thecyclical motion is evident from the trajectories in the parameterspace ( φ , θ ) shown in Fig.1(c) for experiments, and in Fig.1(d) forsimulations. Indeed, the trajectories follow different paths in thefirst and second half-cycles of the actuating field B ( t ) , describinga closed asymmetric region. As the frequency of the field is re-duced, the rotation and sliding motions of the rod become more Journal Name, [year], [vol.][vol.] , ynchronized and the deformation approaches reciprocity. As aresult, the first and second half-cycles of the trajectories shown inFigs.1(c,d) approach one another, coming close to a straight linefor the lowest frequencies. Fig. 2 (a,b) Velocity of the magnetic propeller as a function of the fre-quency ν for different amplitudes of the applied field from experiments(a) and simulations (b). In both cases the static field component is keptconstant to B x = . . Fig.2(a) displays the resulting propulsion speed as a functionof frequency and amplitude B y of the driving magnetic field, withfixed longitudinal component B x = . . The experimentalcurves follow the same trends obtained in numerical simulationsof the system shown in Fig. 2(b). Both set of results demonstratethat the propeller velocity increases with the frequency of thetransverse oscillating field, consistent with the observed increasein the area enclosed by the ( φ , θ ) trajectories.The microswimmer displacement is determined by the motionof the contact point of the rod and the sphere. This motion canbe resolved using resistive force theory and Stokes’s law to ac-count for the overdamped interaction of the rod and spherical col-loid with the fluid, respectively. In this formulation, the propul-sion speed is controlled by three dimensionless parameters, theratio between the particles sizes δ = R / L , α ind = µ m n / ( R B x ) thatcompares the magnetic field induced by the ferromagnet on the paramagnetic particle with the external field, and the susceptibil-ity χ of the paramagnetic particle. The propeller dynamics canbe solved perturbatively for a weak oscillating transversefield, B y / B x ≡ ε (cid:28) .The averaged propeller velocity V , to leading order in ε reads V / V = ε b ( ν / ν ) a ( ν / ν ) + a ( ν / ν ) + a + O ( ε ) , (2)where V = R ν , and ν = B x m n πη L is a characteristic frequency of theproblem. b , a , a , a are coefficients which depend on δ , α ind and χ (see Supporting Information). Eq. 2 shows a non-monotonicdependence on frequency, leading to an optimal propulsion at ν maxV = ν ( a / a ) / . For the experimentally relevant frequencyrange, one can approximate V / V = ε ( b / a )( ν / ν ) , exhibitinga quadratic dependence on the frequency of the transverse field,consistent with the results obtained both experimentally and com-putationally, see Fig. 2. Although the experiments need to beperformed in conditions where B y / B x ≈ to be able to track indetail the motions of the magnetic particles, the leading order so-lutions of the theoretical model in ε , Eq. 2 (and Eq. 3 below),provide insights on the functional form and relevant parametersof the problem. The perturbative analytical solutions, Eq. 2, arevalidated by comparing with numerical simulations of the model,exhibiting good agreement even for B y / B x ≈ .In order to extract the Lighthill efficiency, e , from the experi-mental data, we measured independently the external power P B exerted on the system, and the equivalent power P required totranslate the propeller. Since the only work exerted by the exter-nal field is due to the torque on the ferromagnetic rod (whichis then partly transferred to the sliding motion), we can ex-press the instantaneous external power as P B ( t ) = τττ B ( t ) · ˙ ααα , be-ing τττ B ( t ) = m n ( t ) × B ( t ) the instantaneous torque applied on therod, and ˙ α the rod angular velocity. The average power in oneperiod of the magnetic field is given by ¯ P B = ν (cid:82) / ν P B ( t ) dt . Thisquantity is determined from the orientation and angular veloc-ity of the rod at each instant of time, see Materials and Meth-ods for more details. We calculate the phase between m n and B by coupling to a light emitting diode the signal of the ac cur-rent flowing through the Helmholtz coils. The power needed torigidly drag the swimmer at a given speed v is obtained by us-ing controlled magnetic gradients, P = v m n · ∇ B . The resultingexperimental data are shown in Fig.3(a), where e is measured inthe frequency range ν ∈ [ , ] Hz and at different amplitudes B y with fixed B x = . . In the efficiency calculation we neglectthe contribution due to the rotational motion of the paramagneticsphere. From the analysis of the experimental videos we find thatthat such rotation was ∼ ◦ and its contribution negligible respectto the rotation of the rod.The perturbative solution of the dynamical model provides ananalytical expression for the Lighthill efficiency at small drivingamplitudes ε , e = ε n ( ν / ν ) d ( ν / ν ) + d ( ν / ν ) + d ( ν / ν ) + d + O ( ε ) , (3)which reduces to e = ε ( n / d )( ν / ν ) in the range of experimen- Journal Name, [year], [vol.] , ally accessible frequencies. The coefficients n , d , d and d are given by the parameters of the problem δ , α ind , χ . Figs. 3show a qualitative agreement between the measured values ofthe Lighthill efficiency and the predictions from simulations andthe dynamical model. Fig. 3 (a) Propeller efficiency e versus frequency ν of the swinging fieldfor different amplitudes B y as measured in the experiments. Inset showsresults for the efficiency obtained from simulations. (b) Normalized effi-ciency e / ε versus frequency ν as obtained from the theoretical model(solid line) and simulations (dots). Inset shows e / ε for a wide range offrequencies to visualize a peak at ν maxe = . We note that the quantitative discrepancies between the ex-periments and the computational predictions can be attributed todifferent factors. Specifically, paramagnetic colloids, composedof magnetic domains in a non-magnetic matrix, can present aresidual magnetic anisotropy . As a result, the rotation of thespherical colloid due to the interaction with the external field andthe field induced by the ferromagnetic rod can contribute to thedissipation, without affecting the overall swimmer propulsion. Inaddition, as a consequence of such interaction, the ferromagneticrod could acquire a higher dephasing with the external field, re-sulting in a larger input power. Other factors such as the interac-tion with the surface and the action of gravity on the couple couldalso be a source of discrepancy.Eq. 3 predicts the existence of an optimal efficiency at a fre-quency ν maxe ( ∼ Hz for the parameters of the experimen-tal system), where the efficiency is over three orders of magni-tude higher than the one measured at the experimentally acces-sible frequencies (see inset of Fig. 3). Additionally, the analyti- cal model allows for an optimal design of the microswimmer thatmaximizes its efficiency for a given field B ( t ) . While the efficiencyis not very sensitive to δ and α ind , it exhibits a strong dependenceon the susceptibility of the paramagnetic particle (see SupportingInformation). Indeed, Eq. 3 predicts that the efficiency of the mi-croswimmer has a maximum for a susceptibility χ max ≈ . if onekeeps the other parameters of the microswimmer constant, forwhich the efficiency is more than an order of magnitude higher.Note that for χ (cid:29) , the phase difference between the rotationand sliding motions of the ferromagnetic rod vanishes generatinga reciprocal motion. On the other hand, for χ (cid:28) the slidingmotion does not occur, leaving only one independent degree offreedom and thus no propulsion. Note also that the frequencyof maximum efficiency is inversely proportional to the viscosityof the solvent, which provides a possible route to optimize theefficiency of the swimmer.In conclusion, we have determined the Lighthill energetic effi-ciency of a minimal self-assembled microswimmer composed bya paramagnetic microsphere and a ferromagnetic nanorod, andhave developed a theoretical scheme that captures the essen-tial dynamics of the system. The simplicity of the synthetic mi-croswimmer allows a quantitative and precise measurement of itsefficiency, showing a value smaller than flexible magnetic swim-mers ? but larger than other artificial non-propelling engines .We show that the efficiency of the microswimmer is much moresensitive to the internal degrees of freedom, which are not rele-vant to determine its propulsion speed. The theoretical study hasshown the existence of an optimal efficiency, which is three or-ders of magnitude larger than the one obtained experimentally.While the peak efficiency could be observed only in the modelgiven the experimental limitations, we show that it could be con-trolled by varying the field parameters or the magnetic suscep-tibility of the paramagnetic colloid. The realization of minimal,artificial prototypes that avoid the complexity of biological sys-tems and that can be controlled through their independent de-grees of freedom, is still an open challenge in the field, in spiteof the amount of theoretical propositions. More in general, whencompared to other prototypes driven by external fields, magneticmicroswimmers have the advantage of being easily controlled andsteered trough the fluid without altering the composition of thedispersing medium, all features that make them rather appeal-ing for practical applications. Finally, the possibility to character-ize the swimming efficiency of nano/micro propellers in terms ofthe constituent degrees of freedom represent a key issue in manytechnological contexts ? . Conflicts of interest
There are no conflicts to declare’.
Acknowledgements
This work has received funding from the Horizon 2020 researchand innovation programme, Grant Agreement No. 665440. F.S.acknowledges support from MINECO under project FIS2016-C2-1-P AEI/FEDER-EU. I.P. acknowledges support from MINECO un-der project FIS2015-67837-P and Generalitat de Catalunya underproject 2017SGR-884 and SNF Project No. 200021-175719. P.T.
Journal Name, [year], [vol.][vol.]
Journal Name, [year], [vol.][vol.] , cknowledges the European Research Council (ENFORCE, No.811234), MINECO (FIS2016-78507-C2-2-P, ERC2018-092827)and Generalitat de Catalunya under program "Icrea Academia". Materials and Methods
Experimental System and Methods
The Ni nanorods are synthesized by template-assisted electrode-position from a single electrolyte, . · dm − NiCl solution(Sigma Aldrich), prepared with distilled water treated witha Millipore (Milli Q system). The electrosynthesis was con-ducted potentiostatically using a microcomputer-controlled po-tentiostat/galvanostat Autolab with PGSTAT30 equipment, GPESsoftware and a three electrode system. A polycarbonate (PC)membrane with pore diameter ∼ (Merck-MilliPore) andsputter-coated with a gold layer on one side to make it conductiveis used as the working electrode. The reference and the counterelectrodes are a Ag/AgCl/KCl ( · dm − ) electrode and a plat-inum sheet respectively. After synthesis, the Ni nanorods are re-leased from the membrane by first removing the gold layer with aI /I − aqueous solution, and then by wet etching of the PC mem-brane in CHCl . Nanorods are then subsequently washed withchloroform ( times), chloroform-ethanol mixtures ( times),ethanol ( times) and deionised water ( times). Finally, sodiumdodecyl sulphate (Sigma Aldrich) is added to disperse nanorods.The typical length of the fabricated Ni nanorods used in thisstudy is around L = µ m . Structural and morphological anal-ysis were carried out with scanning and high-resolution trans-mission electron microscopes. The permanent moment of thenanorod is measured by following its orientation under a staticmagnetic field, as described in previous works . The valueobtained for the magnetic moment of the ferromagnetic rod is m n = . × − Am − .The spherical colloids used are paramagnetic microsphereswith radius R = . µ m , ∼ iron oxide content and surfacecarboxylic groups (ProMag PMC3N, Bang Laboratories). The par-ticles are characterized by a magnetic volume susceptibility equalto χ = . , as measured in separate experiments . The particlesand the nanorods are dispersed in highly deionized water (MilliQ,Millipore) and allowed to sediment above a glass substrate. Thesubstrate is placed in the center of five orthogonal coils arrangedon the stage of a light microscope (Eclipse Ni, Nikon), equippedwith a Nikon × objective with . NA. The coils are connectedto a wave generator (TGA1244, TTi) feeding a power amplifier(IMG AMP-1800). The particle dynamics are recorded with aCCD camera (scA640-74fc, Basler) working at around framesper second (fps), with a CMOS camera (MQ003MG-CM, Ximea)working at fps, or in color at fps (acA640-750uc, Basler). Measurement of the phase of the field.
To measure the phase between the instantaneous value of the ap-plied field and the orientation of the propeller, we modify theexperimental set-up by introducing two LEDs to the optical path,just above the observation objective. The two LEDs are connectedin an anti-parallel configuration to an alternate current (AC) volt-age source, which is produced by the same waveform generator that powers the magnetic coil system. We use a phase lock pro-gram to synchronize the oscillations coming from the two signals.In this configuration, the green LED emits light during the positivecycle of the applied field, while the red LED emits on the nega-tive one. The tube lens of the objective allows to distribute thecolored light over the whole sample view. Even if the transmittedintensity appears as relatively small, it can be distinguished fromthe experimental image. From the color video in RGB format, wecalculate the average value of all the pixels in the red and in thegreen channels as a function of time.We then perform a least squares fit using the function f ( t ) = A + B ( sin ( π f t + φ ) + | sin ( π f t + φ ) | ) from which we extract thephase φ , being A and B two amplitudes. The value of φ allowsus to calculate an instantaneous value of the field for each frame.Further, we track three points of the swimmer using the publicprogram ImageJ (National Institutes of Health). These points arethe outermost tip of the nanorod, the point of contact between thenanorod and the colloidal particle, and the center of the colloidalparticle. From these three points, we extract the relative angles,and using the instantaneous direction of the applied field B , wehave all the information over the different degrees of freedominvolved. Numerical simulations
The paramagnetic particle is modeled as a spherical bead of ra-dius R with an induced magnetic moment mmm p located in its cen-ter. The ferromagnetic rod is described as a group of N equallyspaced spherical beads of diameter D along a straight line. Everybead carries a fixed magnetic moment mmm n / N directed along theaxis of the rod. The external magnetic field exerts a torque onthe ferromagnet τττ B = mmm n × BBB , which is implemented as an artifi-cial force pair applied perpendicular to the axis of the rod. Theparamagnetic particle and the ferromagnetic rod interact throughmagnetic dipolar interactions, which are calculated as a sum ofdipolar interactions between the paramagnetic sphere and the N beads composing the ferromagnetic rod. In addition, the spheri-cal beads composing the swimmer interact with other beads andwith the bounding plane at z = through short ranged steric inter-actions, which are described using the Weeks-Chandler-Andersen(WCA) potential ? . Such interactions provide the beads with ex-tension and prevent overlaps.Due to the dimensions of the swimmer we assume that its dy-namics is overdamped and governed by the hydrodynamic dragwith the viscous fluid. The interaction of bead i of the microswim-mer with the viscous fluid is given by the hydrodynamic frictionforce F H , i = − γ i ( v i − u ( r i )) , (4)where γ i is the bead’s friction coefficient, v i its velocity, and u ( r i ) is the induced fluid flow at the bead’s position. The flow field u ( r ) is generated by the action of the net (non hydrodynamic) forceson the different elements of the microswimmer, which is treatedin the far field approximation. In fact, we approximate the hydro-dynamic behavior of the particles composing the microswimmer Journal Name, [year], [vol.] , o that of point particles, which gives u ( r i ) = πη ∑ j G ( r i ; r j ) · F j . (5)Here, F j is the non-hydrodynamic force acting on particle j , η is the viscosity of the fluid, and G ( r i ; r j ) is the Green’s functionof the Stokes equation. We have considered both the case of anunbound fluid, for which we use the Oseen tensor G Oseen ( r i ; r j ) ,and also the case with a no-slip, flat and infinite boundary forthe fluid flow, for which we use the Blake tensor G Blake ( r i ; r j ) .The actuation on our microswimmer by the applied external fieldsdominates over thermal effects ( m n B (cid:29) k B T ). For this reason, ournumerical simulations do not consider the effect of temperatureon the dynamics of the magnetic couple.The dynamics of the swimmer evolves following Newton’sequations of motion, which are solved using a Verlet algorithmadapted for cases with forces which depend on the velocity . Notes and references
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