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The magnetic ratchet system. (A) Schematic of the magnetic traveling wave: A sinusoidal potential (wavelength = 2.6 m) is generated above the surface of an FGF. The potential translates at a speed v m = f under the action of an elliptically polarized rotating field H with frequency f and ellipticity . (B) Calculated energy landscape of one driven particle showing the time evolution of low (high) energy corridors in blue (white). (C) Normalized single-particle speed v x versus frequency f from experiments (open symbols) and numerical simulation (filled symbols). Continuous lines are fit to the synchronous (blue) and sliding (red) regimes. (D) Normalized position (x − x 0 )/ versus time t of a single particle (red line, bottom image) and a particle in a rhombic cluster (blue line, top image). In both cases, f = 8 Hz and = −0.4, which corresponds to asynchronous regime for the individual particle, and x 0 is the position at time t = 0 s. The movement of pair of particles in the asynchronous regime is shown in movie S1.
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Ratchet transport systems are widespread in physics and biology; however, the effect of the dispersing medium in the collective dynamics of these out-of-equilibrium systems has been often overlooked. We show that, in a traveling wave magnetic ratchet, long-range hydrodynamic interactions (HIs) produce a series of remarkable phenomena on the transpo...
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... driven colloidal system is based on a ferrite garnet film (FGF), which displays at zero applied field a pattern of parallel ferromag netic domains with alternating up and down magnetization, and a spatial periodicity of = 2.6 m (Fig. 1A). On the surface of the FGF, the stray field generates a sinusoidallike magnetic potential composed of a series of equispaced minima located at a distance . Above this platform, we deposit paramagnetic polystyrene micro spheres with diameter d = 2.8 m and magnetic volume susceptibility ∼ 0.4 (Dynabeads M270, Invitrogen). These ...
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... , which will be used to tune the dipolar interactions. Here, > 0 ( < 0) corresponds to H x > H z (H x < H z ), i.e., a higher in plane (outofplane) field component. This timedependent field modulates the stray magnetic field on the FGF surface and leads to a translating spatially periodic magnetic energy landscape, Fig. 1B. Here, U 0 is the potential amplitude (see later) and v m = f is the speed of the traveling wave. As a consequence of this modulation, the magnetic landscape transports colloidal particles that are trapped in its energy minima. Figure 1C illustrates the main feature of the singleparticle trans port and combines experiments and ...
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... a consequence of this modulation, the magnetic landscape transports colloidal particles that are trapped in its energy minima. Figure 1C illustrates the main feature of the singleparticle trans port and combines experiments and simulation data (see later), demonstrating the quantitative agreement between both. By raising the driving frequency, we find two dynamic regimes, separated by a critical frequency f c = 6.7 Hz. ...
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... driving frequency, we find two dynamic regimes, separated by a critical frequency f c = 6.7 Hz. The first regime is a phaselocked mo tion (f < f c ) where the particle moves with the speed of the travel ing wave, v x = v m . For f > f c , the particle desynchronizes with the traveling wave (sliding regime), and its average speed decreases as (Fig. 1C). In the latter regime, the traveling wave becomes too fast and the loss of synchronization re sults from the viscous drag that overcomes the magnetic driving. As we are interested in the collective resynchronization effect due to HIs, we drive our particles above f c , fix for all experiments the total amplitude H 0 = 850 A m −1 , ...
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... due to HIs, we drive our particles above f c , fix for all experiments the total amplitude H 0 = 850 A m −1 , and vary mainly and the normalized surface density ~ = N (d / 2) 2 / A , where N is the number of parti cles located in area A. An illustrative example of the difference be tween synchronous and asynchronous regimes is shown in Fig. 1D (see also movie S1), which shows the evolution of the position along the driving direction for a single particle and a particle in a rhombic like cluster. In both cases, the driving frequency is f = 8 Hz ( = − 0.4) so that the position of the individual particle (image at the bottom) displays a series of characteristic oscillations ...
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... emerging dynamics observed in our driven colloidal system result from the combined action of different interactions, including magnetic dipolar and hydrodynamics ones. To understand their relative role in the system, we perform Brownian dynamic simula tions (see Fig. 1C). For each particle i at position r i , we integrate the overdamped equation of ...
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... the friction coefficient, F i ext is the external driving force re sulting from the traveling wave, F i dip is the total force due to mag netic dipolar interactions, F i int accounts for the steric force with the rest of the particles, and is a Gaussian white noise. These forces reproduce the isolated particle experimental speed, as shown in Fig. 1C. More details on F i ext , F i dip , and F i int and the parameters used are given in Materials and Methods. To model HIs, we assume that the particles are embedded in a solvent and dragged by the fluid flow of velocity v i H , generated by the net force acting on the rest of the suspended ...
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... In Eq. 6, the contribution of the HIs appears from the second term in the right hand side, while in the absence of HIs, we obtain the singleparticle behavior characterized in Fig. 1C. In this case, the solution q ̇ = 0 is only possible when ~ f < ~ f c , where the particle is synchronized with the traveling wave. To analyze the im pact of HIs, for simplicity, we assume that the particles are equidis tantly distributed above the traveling wave with periodicity , which allows factorizing Eq. 6 ...
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... typical experimental values, we use = 2.6 × 10 −8 m N −1 s −1 and F M = 0.1 pN. The simulation parameters are estimated to be h sub = 15.3, F d = 56.1, Yukawa force strength U 0 /F M = 300, Pe = 150, and Y = 1. Further, comparing the simulations for a single particle and the experiments (Fig. 1C) as a function of the frequency, we can estimate the characteristic time as D = 0.075 ...
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... A trajectory of the particle moving in V(x) and being acted upon by f ⊥ > 0 is sketched in Fig. 1c. The asymmetric sawtooth-like potential V(x) is qualitatively similar to potentials used in various theoretical and experimental studies of ratchets [59][60][61][62][63][64][65][66][67][68][69][70][71] . A distinctive feature of the ratchet mechanism discussed below is that the emergent current of particles in the x-direction is induced by the transverse force f ⊥ applied perpendicular to it in the y-direction. ...
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... is qualitatively similar to potentials used in various theoretical and experimental studies of ratchets. [59][60][61][62][63][64][65][66][67][68][69][70][71] A distinctive feature of the ratchet mechanism discussed below is that the emergent current of particles in the x-direction is induced by the force f ⊥ applied perpendicular to it in the y-direction. Although here we focus on the two-dimensional case, in an actual experiment, the force f ⊥ can point in any direction in the yz-plane. ...
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