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Enhancing nanoparticle diffusion on a
unidirectional domain wall magnetic ratchet
Ralph L. Stoop,†Arthur V. Straube,‡,†and Pietro Tierno∗,†,¶,§
†Departament de Física de la Matèria Condensada, Universitat de Barcelona, Av.
Diagonal 647, 08028 Barcelona, Spain
‡Freie Universität Berlin, Department of Mathematics and Computer Science, Germany
¶Institut de Nanociència i Nanotecnologia, Universitat de Barcelona, Barcelona, Spain
§Universitat de Barcelona Institute of Complex Systems (UBICS), Barcelona, Spain
E-mail: ptierno@ub.edu
Keywords: Diffusion, Magnetic fields, Domain walls, Microfluidics, Ratchet effect.
Abstract
The performance of nanoscale magnetic devices is often limited by the presence
of thermal fluctuations, while in micro-nanofluidic applications the same fluctuations
may be used to spread reactants or drugs. Here we demonstrate the controlled motion
and the enhancement of diffusion of magnetic nanoparticles that are manipulated and
driven across a series of Bloch walls within an epitaxially grown ferrite garnet film.
We use a rotating magnetic field to generate a traveling wave potential that unidirec-
tionally transports the nanoparticles at a frequency tunable speed. Strikingly, we find
an enhancement of diffusion along the propulsion direction and a frequency dependent
diffusion coefficient that can be precisely controlled by varying the system parameters.
To explain the reported phenomena, we develop a theoretical approach that shows a
fair agreement with the experimental data enabling an exact analytical expression for
1
arXiv:1811.05762v1 [cond-mat.soft] 14 Nov 2018
the enhanced diffusivity above the magnetically modulated periodic landscape. Our
technique to control thermal fluctuations of driven magnetic nanoparticles represents a
versatile and powerful way to programmably transport magnetic colloidal matter in a
fluid, opening the doors to different fluidic applications based on exploiting magnetic
domain wall ratchets.
The ratchet effect emerged in the past as a powerful way to transport matter at the micro
and nanoscale, taking advantage of Brownian motion.1,2 The success of such concept comes
from the possibility of using thermal fluctuations to obtain useful work out of a thermody-
namic system, although such fluctuations produce noise, heat and randomize the motion of
nanoparticles which limits the efficiency of any device operating at such scale.3–5 Reducing
or controlling thermal fluctuations in nanoparticle systems may present different technolog-
ical advantages apart from providing important fundamental insight into the dynamics and
interactions of matter at such scale. In the first case, diffusion can be used as a mean for
mixing streams of fluids or for spreading reactants, drugs and biological species6,7 in micro-
and nano-fluidic applications.8–10 On a more fundamental level, the search for strategies
that enable controlling diffusion and noise has fascinated scientists for long time, since the
pioneering seminar of Richard Feynman on Brownian ratchet.11
In a typical ratchet system the random fluctuations of nanoparticles can be rectified
into a directed motion by an external potential. In overdamped systems, different ratchet
mechanisms can be sorted in two general classes, depending on the nature of the external
potential.2One class refers to the tilting ratchets that are typically specified by a station-
ary, spatially asymmetric potential landscape often accompanied by an external force. The
asymmetry of the landscape is able to rectify thermal fluctuations into net motion, and the
force, which determines the tilt of the total potential, can be used to further enhance the
particle current. Such a ratchet scheme can be realized by using, for example, the walls or
barriers in a microfluidic device.12 Another broad class are the pulsating ratchets that are
characterized by a time-dependent landscape, in which the net particle flux arises from the
2
periodic or random evolution of the landscape. Their particular subclasses are the flashing
ratchet with the landscape switching between two states, and the traveling wave ratchet,
where the landscape translates at a given speed. The latter allows dragging the particles
that are trapped in the energy minima of the moving landscape.
The generation of translating potential landscapes usually requires an external field,
which makes it appealing due to its programmable nature, that enables to remotely control,
direct and reverse the flux of particles at will.13 Recent realizations in this direction include
the use of electric,14–18 optical19–22 or magnetic23–25 fields to transport microscopic particles.
However, most of the proposed approaches remain difficult to apply at the nanoscale, due to
the large field gradients required to overcome thermal forces. Plasmonic landscapes have been
proposed as a successful strategy,21,26,27 however the created patterns are often composed of
fixed lithographic gold structures that cannot be easily changed by an external control. An
alternative solution is the use of patterned magnetic substrates that present high contrast in
the magnetic susceptibility on the particle scale and thus generate strong and localized field
gradients.28–31 While the trapping of magnetic nanoscale matter was recently demonstrated
with lithographic32 and epithaxial33 films, the complete control of particle transport, speed
and diffusion within the same functional platform still remains a challenging issue.
In this article we demonstrate all these features by using a periodic patterned substrate,
i.e. a uniaxial ferrite garnet film (FGF), composed of a series of mobile magnetic Bloch
walls. Controlling the dynamics of magnetic domain walls in thin films has recently led to
novel applications in disparate fields, including spintronics,34 logic devices, 35,36 nanowires37,38
and ultracold atoms.39 Here we use such magnetic walls to achieve the controlled transport
and, in particular, the diffusion enhancement of magnetic nanoparticles. The application of
an external rotating magnetic field induces the unidirectional translation of the otherwise
stationary magnetic landscape generated by the FGF. Due to the magnetic interaction with
the nanoparticles, the translation of the landscape is converted into directed particle motion.
This technique allows to rapidly traps and steers magnetic nanoparticles deposited on top of
3
it thus, with in situ real time control. Moreover, our magnetic ratchet allows us to precisely
control also the diffusivity of the nanoparticles, to enhance, to suppress it, by simply changing
the driving frequency of the applied field.
Our magnetic ratchet scheme is shown in Figure 1(a) and it is based on a series of
Bloch walls (BWs), narrow (∼20 nm wide) transition regions where the film magnetization
changes orientation by performing a 180 degrees rotation in the particle plane (x, y). Such
BWs emerge spontaneously in a single crystal FGF film after an epitaxial growth process
and they generate strong local stray fields on the surface. In this study we use an FGF of
composition Y2.5Bi0.5Fe5−qGaqO12 (q= 0.5−1), that was grown by dipping liquid phase
epitaxy on a h111ioriented single crystal gadolinium gallium garnet (Gd3Ga5O12) substrate.
In absence of external field, the BWs in the FGF are equally spaced at a distance λ/2being
λ= 2.6µmthe spatial periodicity, and separates domains of opposite magnetization, with
saturation magnetization Ms= 1.3×104A m−1.
Above this film we demonstrate the controlled transport of three three types of param-
agnetic polystyrene nanoparticles with diameters d= 540 nm,360 nm and 270 nm, charac-
terized by a ∼40% wt.iron oxide content (Microparticles GmbH). When placed above the
FGF, the nanoparticles show simple diffusive dynamics and are able to easily pass the BWs
due to the presence of a 1µmthick polymer coating that prevents sticking to the FGF sur-
face, see the Materials and Methods section. We apply an external magnetic field rotating in
the (x, z)plane with frequency fand amplitudes (Hx, Hz) = H0(√1 + β, √1−β), such that
Hac(t)=(Hxcos (2πf t),0,−Hzsin (2πf t)), Figure 1(a). The polarization of the applied
field is slightly elliptical, Hx6=Hz, as characterized by the amplitude H0=p(H2
x+H2
z)/2
and an ellipticity parameter β= (H2
x−H2
z)/(H2
x+H2
z). The value of β=−1/3is chosen to
minimize dipolar interactions between nearest particles,40 as they can promote the undesired
chaining at high density.41 Although for self-consistency we account for this small ellipticity
in a full numerical model, a more tractable semi-analytical model for the special case of
β= 0 is shown to work quantitatively well.
4
The applied field modulates the stray magnetic field generated by the FGF, leading to a
spatially periodic magnetic energy landscape Um(x, t) = −U0cos(k(x−v0t)) that translates
at a constant speed, v0=λf , Figure 1(a), see Materials and Methods section for further
details and definition of U0and k. Its evolution at the elevation corresponding to the 270 nm
particles is shown in Figure 1(b). One may intuitively expect that the particles would follow
the energy minima whose locations are given by the blue regions, and travel the distance of
one wavelength during one time period. However, the nature of motion and the velocity of
mean drift across the BWs,
hvi= lim
t→∞ hX(t)i
t, X(t) = x(t)−x(0),(1)
depend on the external frequency, as becomes evident for the simple case of no thermal
fluctuations, see Equation 4. The variation of hviwith the frequency fis also shown in Fig-
ure 1(c), where the experimental data (open symbols) are plotted along with the theoretical
predictions (dashed and solid lines) for a 270 nm particle.
As it follows from our model (see Equation 4 in Materials and Methods), and as confirmed
by the experimental data, there exist two different dynamic states separated by the critical
frequency fc. At low frequencies, here f < fc≈13 Hz, the nanoparticles indeed translate
consistently with the landscape with the maximum possible speed, ˙x(t) = hvi=v0=λf; see
the range of perfectly linear increase of hviwith fin Figure 1(c). We also note that at any
0< f < fcin this “locked” regime, the particle lags behind the minimum of the potential
energy landscape Um, whose relative position is determined by minimizing the potential given
by Equation 5. For f > fc, the particle is unable to move together with the landscape. It
starts to slip and cannot remain localized within the same minimum. In this “sliding” regime,
every time period the particle covers a distance smaller than λ, resulting in a reduction of the
mean speed. At high frequencies f, the speed of mean drift decays as hvi ' λf 2
c/(2f). Note
that at low and high frequencies, thus far away from fc, the predictions of the model with
5
thermal fluctuations and the experimental data are highly consistent with the deterministic
(zero temperature) results given by Equation (4). However, the deterministic model overshot
the experimental data close to fc. Therefore, we conclude that thermal fluctuations start
to significantly affect the particle speed close to the critical frequency, blurring the sharp
transition from the locked to the sliding dynamics and effectively decreasing the value of fc
as compared to the relative to the deterministic model.
While our domain wall magnetic ratchet enables full control over the mean speed of
particle moving across BWs, the instant position of the particles is affected by unavoidable
thermal fluctuations, as evidenced by the particle trajectory in Figure 1(d). The role of such
fluctuations is typically quantified by measuring the mean squared displacement (MSD),
which can be calculated from the positions of the nanoparticles. Since we are interested in
the general effect of how the diffusive properties of a nanoparticle are influenced by the ratchet
mechanism, we focus on investigating these quantities along the propulsion direction, namely
the xaxis. Because of the non vanishing mean drift, we define the MSD as the variance
of the corresponding particle position, σ2
x(t) = hδx2(t)i ∼ tα, where δx(t) = X(t)− hX(t)i.
For our statistical analysis we averaged over more than ∼50 independent experimentally
measured trajectories, for each of which we subtracted its mean drift, hX(t)i=hvit. The
exponent of the power law αcan be used to distinguish the diffusive α= 1 from anomalous
(sub-diffusive α < 1or super-diffusive α > 1) dynamics. Our case corresponds to normal
diffusion, with an effective diffusion coefficient across the BWs evaluated as
Deff = lim
t→∞
σ2
x(t)
2t.(2)
In Figure 2 we show that effectively, the magnetic ratchet provides a strong enhancement
of the diffusion coefficient at an optimal frequency. At low frequencies, the diffusive motion
of the nanoparticles with d= 270 nm corresponds to a Deff much lower than the diffusivity
in absence of the external field, D0= 1.0µm2s−1. At high frequencies, the diffusivity
6
becomes close to D0. At intermediate frequencies, the effective diffusivity increases (note
that Deff ≈0.6D0for f= 11.5 Hz and f= 70 Hz) with an enhancement of almost one order
of magnitude relative to free diffusivity, Deff ≈8.5D0, at a frequency of f= 15.5 Hz. In
particular, as shown in the inset of Figure 2, after subtracting the mean drift we calculate
the effective diffusion coefficient using Eq. 2 and performing a linear fit only for the region
of data where the mean square displacement displays a power law behavior with α= 1. This
regime is always found in the long-time limit. Overall, the dependence of Deff on ffollows a
well defined trend, characterized by an initial sharp raise above a value of f≈8 Hz, and an
exponential-like decay above a peak at 15.5 Hz, which is close to the critical frequency value
measured for this size of nanoparticle, fc= 13.4 Hz.
The observed enhancement of diffusion can be explained by formulating a reduced theo-
retical model that explicitly takes into account the interaction of the nanoparticle with the
magnetic landscape generated by the FGF surface and the thermal noise. Details of the
derivation are given in the Materials and Methods section. In the reference frame moving
with the magnetic potential, the behavior of the nanoparticle is equivalent to the motion
in an effective tilted periodic potential V(u)as given by Equation 5. For such a potential,
the effective diffusion coefficient, defined via Equation 2, can be expressed as Equation 7,
which admits an accurate numeric evaluation. We used Equation 7 in Figures 3(a)-3(c) to
fit the experimental data for the three types of nanoparticles, and find that our theoretical
approach captures very well the observed diffusion enhancement for all cases.
The explicit form of the effective potential, Equation 5, with the tilt ∝fand the am-
plitude of the landscape ∝fcadmits an intuitively clear interpretation of the observed
frequency-dependent diffusive regimes, as also illustrated in Figures 3(d)-3(f). Indeed, at
subcritical frequencies, f≤fc, the potential barrier the particle needs to overcome to es-
cape from a minimum drops with the frequency as ∆V/kBT≈∆υm(1 −f /fc)3/2, where
∆υm=λ2fc/(πD0)is double the amplitude of the landscape relative to thermal energy, cf.
Equation 5. At low frequencies (ffc), the tilt is negligible and the particle is strongly
7
trapped in a minimum of the magnetic landscape, as shown in Figure 3(d). Since in this
case the potential barrier is maximum, ∆V/kBT≈∆υm, and is too high for the particle to
escape, the effective diffusivity is nearly vanishing, Deff D0. At intermediate frequencies,
as ftends to fcbut remains smaller than fc, the potential barrier decreases with fand dis-
appears at f=fc. In this frequency range, the barriers become progressively more accessible
for the Brownian particle, explaining the observed significant enhancement of the effective
diffusivity at frequencies f≈fc, see Figure 3(e). Beyond f=fc, there exits no minima in
the potential, and at high frequencies ffc, the particle does not feel the landscape, as
shown in Figure 3(f). As a result, the diffusive motion occurs effectively at a constant force
∝fand therefore corresponds to free diffusion, when ideally Deff =D0.
Further, since the reduced model was developed for the zero field ellipticity, β= 0, we
have performed control Brownian dynamic simulations of the full system with β=−1/3to
confirm the negligible effect of βon the basic physics. The simulation (blue lines) agree very
well with the model and the experimental data. The only difference is that the system with
β=−1/3is characterized by slightly higher values of the critical frequency compared to the
case β= 0, which is in agreement with our earlier observation that the value of fcat any
β6= 0 is generally smaller than that for β= 0 at otherwise identical conditions.40 Taken
together, our findings show that each type of nanoparticle investigated exhibit an enhanced
diffusive behavior, with the highest diffusion coefficient measured for the intermediate size,
d= 360 nm. This outcome can be understood by considering the balance between two
opposite effects. First, reducing the nanoparticle diameter dincreases thermal fluctuations
and its free diffusion coefficient D0=kBT/ζ, where ζis the friction coefficient of a spherical
particle immersed in water. Note that if in the bulk ζ= 3πηd with the dynamic viscosity
η= 10−3Pa s, the presence of the FGF surface leads to effectively larger values of ζ. Second,
for our system, smaller particles come closer to the FGF surface, and thus they are strongly
attracted by its stray field, which results in a reduction of their thermal fluctuations. For
example, for a 270 nm particles an effective enhancement of the friction coefficient due to
8
the presence of the FGF surface is estimated to be 25 −30%, as follows from the reduction
of the diffusion coefficient (cf. the inset of Figure 2) relative to the bulk diffusivity.
While we have analyzed the transport of single particles, our magnetic ratchet enables
also the collective motion for an ensemble of nanoparticles. We demonstrate this feature
in Figure 4(a), where a dense suspension of 270 nm particles are trapped and transported
across the FGF surface, see also corresponding VideoS2 in the Supporting Information. In the
absence of applied field (ratchet off) the colloidal suspension shows simple diffusive dynamics
as illustrated by the Gaussian distribution of the displacements perpendicular (along the x
axis) and parallel (along the yaxis) to the magnetic stripes, Figures 4(b,c). Here we used
P(δx) = (2πσ2
x)−1exp(−δx2/2σ2
x)with the variance σ2
x(t) = hδx2(t)i,δx(t) = X(t)− hX(t)i
and X(t) = x(t)−x(0) defined as earlier and considered at a sufficiently large time. The
distribution P(δy)is defined in a similar way, for which hYi ≡ 0, implying no mean drift and
free diffusion in the ydirection. We note that when the ratchet is off, H0= 0, the magnetic
landscape Um∝H0becomes effectively flat and does not affect the dynamics of the particle.
In this situation, the mean drift disappears, hvi= 0, and therefore hXi=hvit= 0. As a
result, when the ratchet is off, both distributions (greens lines in Figures 4(b,c)) are identical,
with σ2
x=σ2
yand thus featuring an isotropic free diffusion dynamics.
Upon application of the rotating field, H0>0, the particles are immediately localized
along the BWs, forming parallel chains and being consecutively transported across the mag-
netic platform. The ratchet transport features a similar dispersion along the perpendicular
direction as in the free case, as shown by the blue line in Figure 4(b). Along the transport
directions we observe a stronger confinement with a narrower distribution of displacement
P(δx), where the mean drift is subtracted by putting hX(t)i=hvit. VideoS2 in the sup-
porting information shows the versatility of our magnetic ratchet approach, as now the
nanoparticles can be easily trapped or released and transported to the right or left by just
inverting the chirality of the rotating field, Hx→ −Hx, which keeps H0and βunchanged.
While these features have been previously reported for micro-scale systems,23,24,29,30 our
9
experimental realization proves its potential for further miniaturization of the transported
elements and opens the doors towards applications in magnetic drug delivery systems using
the higher surface to volume ratio of nanoparticles.
To conclude, we have reported the controlled transport and diffusion enhancement of
nanoparticles in a magnetic ratchet generated at the surface of a ferrite garnet film. In con-
trast to previous experimental works on microscopic particles confined in a rotating optical
ring,42,43 above patterned plasmonic21 or lithographic44 landscapes, our nanoparticles are
trapped and controlled on an extended surface in absence of any topographic relief that can
perturb the transport via steric interactions. The advantage of using the garnet film as a
source of magnetic background potential is that the domain wall motion is essentially free
from intermittent behavior and hysteresis. This makes the cyclic displacements and the de-
vice performance fully reversible. We have also reported in the past the giant diffusion in a
magnetic ratchet of microscopic colloids.45 However, the significant advantage of the present
implementation is that the particle fluctuations can also be controlled along the direction
of motion and remain completely independent of the deformation of the BWs in the lateral
directions. This crucial difference has important implications for the design of channel-free
nanofluidic devices where the colloidal motion can be confined to a straight line. Further,
our work goes beyond a previous one,33 centered on trapping fluctuating nanoparticles along
the BWs in an FGF. Here, we not only demonstrate the possibility to transport nanoscale
objects at a well defined speed, but also that our magnetic ratchet system can be used to
tune the diffusive properties of the particles, increasing their effective diffusion constant by
almost an order of magnitude with respect to the previous case. Another future avenue of
this work is to investigate the dynamics of dense suspensions of interacting nanoparticles,
and how collective effects alters the average particle flow. Increasing their density, however,
requires a visualization procedure different from fluorescent labeling, to avoid artifacts dur-
ing the tracking mechanism. While with relatively larger particles, the average speed of a
colloidal monolayer was found to decrease with respect to the individual case, 46 the presence
10
of stronger thermal fluctuations for smaller particles may cause different non-trivial effects
on the overall system dynamics. Finally, our controlled enhanced diffusion in a transported
ratchet may be used as a pilot system for fundamental studies related to transport, diffusion
and their complex relationship at the nanoscale.
Materials and Methods
Experimental System and Methods
We give further details on the sample preparation and experimental setup. In order to
decrease the strong magnetic attraction, we coat the FGF film with a h= 1 µmthick layer
of a photoresist (AZ-1512 Microchem, Newton, MA), i.e. a light curable polymer matrix,
using spin coating at 3000 rpm for 30 s (Spinner Ws-650Sz, Laurell). Before the experiments,
each type of particle is diluted in highly deionized water and deposited above the FGF, where
they sediment due to the magnetic attraction to the BWs.
External magnetic fields were applied via custom-made Helmholtz coils connected to
two independent power amplifiers (AMP-1800, Akiyama), which are controlled by a wave
generator (TGA1244, TTi). Particle positions and dynamics are recorded using an upright
optical microscope (Eclipse Ni, Nikon) equipped with a 100 ×1.3 NA oil immersion objective
and a CCD camera (Basler Scout scA640-74fc, Basler) working at 75 frames per second. The
resulting field of view is 65 ×48 µm2.
Videomicroscopy and particle tracking routines47 are used to extract the particle positions
{xi(t), yi(t)}, with i= 1, . . . , N , from which the mean speed hviis obtained performing
both time and ensemble averages. To calculate the mean square displacement and diffusion
coefficient, we use Ntrajectories with length lthreshold = 200 frames, corresponding to a
measurement time of ∆t= 200/75 = 2.6s.
11
Theoretical model
The motion of paramagnetic colloidal particles above the FGF can be well interpreted within
a simple model. In an external magnetic field H, a spherical magnetically polarizable particle
of volume Vbehaves as an induced magnetic dipole with the moment m=V χHand
the energy of interaction Um=−µ0m·H/2,48 where χis the effective susceptibility and
µ0= 4π×10−7H m−1is the magnetic permeability of free space. The total magnetic field
above the FGF His given by the superposition Hac +Hsub of the external modulation
of elliptic polarization, Hac =H0(√1 + βcos ωt, 0,−√1−βsin ωt), and the stray field of
the garnet film Hsub ≈(4Ms/π) exp(−kz)(cos kx, 0,−sin kx),49 where Msis the saturation
magnetization, ω= 2πf is the angular frequency, k= 2π/λ is the wave number, H0is the
amplitude and βis the ellipticity of the modulation.
Being interested in the transport properties across the magnetic stripes and evaluating
the magnetic force exerted on the particle, F(x, t) = −∂xUm(x, t)with Um=−µ0V χ(Hac +
Hsub)2/2, in the overdamped limit we obtain an equation of motion,
˙x(t) = ζ−1F(x, t) + ξ(t),(3)
with F(x, t) = −ζλfc(√1 + βcos ωt cos kx−√1−βsin ωt sin kx)and a random variable ξ(t)
taking account of thermal fluctuations. Here, fc= 8µ0χV MsH0exp(−kz)/(ζ λ2), and ξ(t)
obeys the properties hξ(t)i= 0,hξ(t)ξ(t0)i= 2D0δ(t−t0), where D0is the free diffusion
coefficient, kBTis the thermal energy and ζis the friction coefficient. By using fcand D0as
fitting parameters, we numerically integrate Equation 3 and evaluate the velocity of mean
drift hviand effective diffusion Deff, which capture well the experimental data.
To gain further insights, we approximate the general model for an arbitrary β, Equation
3, by a more tractable special case β= 0, which corresponds to a traveling wave potential,
Um(x, t) = Um(x−v0t)∝cos(k(x−v0t)). In the reference frame co-moving with the speed
v0=λf, in terms of a new variable u(t) = −x(t)+v0twe obtain, ˙u(t) = λf −λfcsin ku+ξ(t).
12
The deterministic velocity of mean drift is known to be:49
hviT=0 =
λf, f ≤fc,
λf −λpf2−f2
c, f > fc.
(4)
Here, fcplays the role of the critical frequency that separates the low frequency domain
with the maximum possible speed of mean drift, hvi=v0∝f(f < fc), from the high
frequency domain where its efficiency progressively drops down, hvi ∝ f−1< v0(f > fc). For
nanoparticles, thermal fluctuations are, however, inevitable, and we consider their thermal
motion in the associated potential,
V(u)
kBT=−λf
D0
u−λfc
D0kcos ku, (5)
which admits evaluation of the velocity of mean drift and effective diffusion50,51
hvi=λf −D0
λ
1−exp(−λ2f/D0)
hI±(u)iu
,(6)
Deff =D0hI2
±(u)I∓(u)iu
hI±(u)i3
u
.(7)
Here, I±(u) = hexp[±(V(u)−V(u∓u0))/kBT]iu0are evaluated for potential (Equation 5)
and h·iu=λ−1Rλ
0·dudenotes the average over one wavelength of the landscape.
Acknowledgment
R. L. S. acknowledges support from the Swiss National Science Foundation grant No.
172065. A. V. S. acknowledges partial support by Deutsche Forschungsgemeinschaft (DFG)
through grant SFB 1114, Project C01. P. T. acknowledges support from the ERC starting
grant "DynaMO" (335040) and from MINECO (FIS2016-78507-C2) and DURSI (FIS2016-
78507-C2).
13
Supporting Information Available
The following files are available free of charge. Two experimental videos (.WMF) showing
the dynamics of magnetic colloidal particles above the FGF film.
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Figure 1: (a) Schematic illustrating a single magnetic nanoparticle above an FGF subjected
to a rotating magnetic field polarized in the (x, z) plane. The applied field helps to generate
a periodic potential (blue line) that translates at constant speed v0=λf and transports the
nanoparticle. Bottom illustrates the rotation of magnetization in a Bloch wall. (b) Color
coded energy landscape (Um/kBT) versus particle position and time calculated for a 270 nm
nanoparticle (effective magnetic volume susceptibility χ= 1, elevation from surface z=
1.34 µm,f= 2 Hz), low energies in blue and high energies in yellow. The particle transport
occurs consistently with the blue regions. (c) Particle mean speed hviacross the BWs versus
frequency f(top) and normalized with respect to λf (bottom), for a 270 nm nanoparticle
subjected to a rotating field with amplitude H0= 1200 A m−1. The experimental data
(empty circles β=−1/3) are plotted together with the theoretical lines: The dashed line
shows the deterministic limit, Equation 4, while the red solid line also accounts for the effect
of thermal fluctuations, Equation 6, see the Materials and Methods section. (d) Polarization
microscope image showing a 270 nm particle trajectory transported by the rotating field at
an average speed hvi= 5.2µm s−1. The field parameters are f= 2 Hz,H0= 1200 A m−1.
The trajectory (red line) is superimposed to the image, λ= 2.6µm, see VideoS1 in the
Supporting Information.
18
Figure 2: Effective diffusion coefficient Deff measured along the propulsion direction versus
driving frequency ffor particles with d= 270 nm. For all data, the particles were driven
above the ratchet by a rotating field with amplitude H0= 1200 A m−1and β=−1/3. The
green dashed line corresponds to the free diffusion coefficient measured above the FGF and
in absence of applied field. The small inset displays the mean square displacements (MSDs)
divided by the time, hδx2i/(2t)for three different frequencies with locations indicated by
open symbols in the main plot. The continuous red line through the f= 14.5Hz data shows
a linear fit to calculate Def f .
19
Figure 3: (a-c) Effective diffusion coefficient Deff versus frequency ffor three types of parti-
cles with diameters 270 nm (a), 3600 nm (b) and 540 nm (c). All panels show the experimen-
tal data (open circles), predictions of the reduced theoretical model with β= 0 according
to Equation 7 (red line) and estimates of the effective diffusion coefficient from numerical
simulations of the full model with β=−1/3, Equation 3 (blue lines). The experimental
parameters used are H0= 1200 A m−1and β=−1/3that reflect the values used for the
simulations. For the three types of particles we also indicate the critical frequencies fcused
for the model (red) and simulation (blue). (d-f) Magnetic potential V(u)in units of thermal
energy kBTin the moving reference frame (Equation 5) evaluated for a 270nm particle and
at a frequency of f= 1Hz (d), f= 11Hz (e) and f= 15Hz (f). Here the critical frequency
is fc= 13Hz.
20
Figure 4: (a) Sequence of snapshots showing the collective transport of 270 nm particles.
At time t= 7.2 s the magnetic ratchet is switched on and all the nanoparticles are trans-
ported to the right (arrow at bottom) at an average speed hvi= 2.6µm s−1(field parameters
f= 1 Hz,H0= 1200 A m−1,β=−1/3). The green (middle figure) and blue (bottom figure)
lines are trajectories of a single particle. Scale bar is 5µmfor all images, see also VideoS2
in the Supporting Information. (b) and (c) Probability distributions of the particle position
perpendicular P(δx)(b) and parallel P(δy)(c) to the BWs. The scattered points are exper-
imental data while the solid lines are Gaussian fits with σ2
xand σ2
yfor the variance in the
corresponding directions. In (b) the distribution was calculated by subtracting the drift as
described in the text.
21
