Dynamics of a paramagnetic colloidal particle driven on a magnetic-bubble lattice

We present a theoretical study of the recently observed dynamical regimes of paramagnetic colloidal particles externally driven above a regular lattice of magnetic bubbles P. Tierno, T. H. Johansen, and T. M. Fischer, Phys. Rev. Lett. 99, 038303 2007 . An external precessing magnetic field alters the potential generated by the surface of the film in such a way to either drive the particle circularly around one bubble, ballistically through the array, or in triangular orbits on the interstitial regions between the bubbles. In the ballistic regime, we observe different trajectories performed by the particles phase locked with the external driving. Superdiffusive motion, which was experimentally found bridging the localized and delocalized dynamics, emerge only by introducing a certain degree of randomness into the bubbles size distribution.

Transport of particles in periodic pinning potentials is a general phenomenon common to numerous systems in condensed matter.Examples are various and include cold atoms in an optical lattice ͓1,2͔, flux flow in superconductors ͓3,4͔, biological motors ͓5,6͔, macromolecules ͓7,8͔, and colloidal particles in optical tweezers ͓9,10͔ or above magnetic structured substrates ͓11,12͔.Besides the scientific interest, there are also technological possibilities which derive from studying these systems, such as sorting colloidal particles ͓13͔, separating DNA chains ͓14͔, or designing novel electronic and/or semiconductor devices ͓15͔.To study such a phenomenon, colloidal particles represent a suitable choice due to the availability of methods to generate periodic potentials by using, e.g., holographic optical tweezers or patterned magnetic substrates, and the fact that particle motion can be easily followed in real time.In this context, some of us recently observed different dynamical regimes of paramagnetic colloidal particles externally driven above a hexagonal lattice of magnetic bubbles ͓16͔.The bubbles are cylindrical ferromagnetic domains of uniform magnetization located in a continuous film with reverse magnetization.The size of these domains can increase ͑or decrease͒ by application of a magnetic field parallel ͑opposite͒ to the bubble magnetization direction.The paramagnetic particles, dispersed in water, were deposited above the bubble array and subjected to a magnetic field precessing around the axis normal to the film.Such field controls the size of the bubbles and, at the same time, drives the particles through the lattice in different dynamical modes depending on its frequency and strength.In particular, for low field strengths and at a fixed frequency, the particles were localized around the bubbles describing circular orbits.Increasing the field reduces the size of the bubbles and the particles were able to escape through the array following a ballistic trajectory.Higher field strengths localized again the particle motion in triangular orbits above larger interstitial regions between the bubbles.Remarkably, superdiffusive trajectories were observed when passing from the localized circular orbits to the ballistic and, again, when crossing back from the ballistic regime towards the triangular orbits.
Here we present a theoretical study of such system which reproduces the particle dynamics.First we parametrize the potential generated by a hexagonal array of magnetic bubbles.We solve the overdamped equation of motion for a particle externally driven in such potential when superimposing a precessing magnetic field.Similarly to the experiment, we observe localized ͑circular or triangular͒ and delocalized ͑ballistic͒ trajectories.In contrast, we obtain superdiffusive motion only when we consider a magnetic disorder in the system in the form of a random distribution of the radius of the bubbles.
We consider a paramagnetic particle of volume V placed above a hexagonal array of magnetic bubbles with radius R and lattice vectors a 1 ϵ͑a ,0͒, a 2 ϵ͑ 1 2 a , ͱ 3 2 a͒ ͓see Fig. 1͑a͔͒.When subjected to a magnetic field H, the particle acquires a magnetic moment m = VH, proportional to the field, where is the effective particle magnetic susceptibility ͑ = 0.17͒.The total magnetic field acting over the particle is H tot = H l + H ex , where H l is the field due to the bubble lattice and H ex the external magnetic field which precesses with an angle and a frequency ⍀ around the z axis, H ex ϵ͑H 0 sin ⍀t , H 0 cos ⍀t , H z ͒.To parametrize H l we consider first the magnetic potential above the surface of an isolated cylindrical bubble of thickness t and uniform magnetization −M s ͑M s ͒ at z =0 ͑z =−t͒ ͓17͔.The magnetic field generated by this bubble is h =−ٌ, and the field due an array of such bubbles is H b = ͚ n,m h n,m where n , m cross the entire hexagonal lattice.To obtain H l we use the superposition principle and subtract to H b the contribution due to the opposite magnetized film H f ͓18͔.The knowledge of H tot allows us to calculate the energy of the paramagnetic particle as U͑r , t͒ , where is the magnetic susceptibility of the medium ͑water͒.Figure 1͑b͒ shows the energy contour plot calculated at an elevation z = 1.5 m ͑from the particle center͒ with the experimental parameters: a =11 m, R = 3.4 m, M s = 1.7ϫ 10 5 A / m, H z = 0.29M s and H 0 = 0.1M s .The main features to notice are the cylindrical hills centered at the position of the bubbles ͑black-red regions͒ and the energy minima around them.Application of a magnetic field normal to the film H z but antiparallel to the magnetization of the bubbles, enlarges the interstitial regions making the energy minima closer and the hills higher.This is illustrated in Fig. 1͑c͒, where we show a cut along the energy landscape for two fields: H z =0 A / m ͑continuous͒ and H z = 0.29M s ͑dashed line͒.Application of H z also alters the lattice constant a in the film due to magnetic repulsive interactions between the bubbles ͓19͔.To consider this effect we assume a linear dependence of a on H z ͓20͔.
The particle trajectory results from the interplay between the externally modulated energy landscape and the viscous friction of the medium.Thus we use an overdamped equation of motion, where f is the hydrodynamic drag coefficient ͑f =27 ͓21͔͒, b the particle radius ͑b = 1.4 m͒ and the water viscosity.We initially ͑t =0͒ place a particle at the center of a hexagonal array of typically 64ϫ 64 bubbles, and at elevation z = 1.1b, and solve this equation with a time interval dt = 1.4ϫ 10 −4 s and a total of ϳ10 5 time steps to ensure a steady condition ͓22͔.The obtained trajectories are analyzed in terms of the correlation function C͑͒ = ͓͗r͑t͒ − r͑t + ͔͒ 2 ͘ t ϳ ␣ with r͑t͒ the two-dimensional particle position at time t and where ͗•͘ t denotes a time average.Depending on the exponent ␣, we classify the particle motion as localized ͑␣ =0͒, ballistic ͑␣ =2͒, or superdiffusive ͑␣ Ͼ 1͒.
In Fig. 2 we show the phase diagram of the exponent ␣ computed by varying H z and ⍀, while keeping the plane component of the applied field H 0 fixed.We note that, for the field strengths considered here ͑H z Ͻ M s ͒, only H z controls the size of the bubbles.Representative images of the particle trajectories encountered are depicted at the bottom of Fig. 2. For small field intensities, 0 Ͻ H z Ͻ 0.25M s , the dynamics are localized and the particle orbits around the bubbles ͓Fig.2͑a͒, inset, bottom͔, as observed in the experiments.The particle follows the planar component H 0 which results in a synchronous rotation with the field frequency.Further increasing H z decreases the size of the bubbles and widens the energy minima around them which now become closer.For H z Ն 0.25M s the dynamics becomes delocalized and the particle settles into a stable ballistic trajectory ͓Fig.2͑b͒, inset, bottom͔.In this case, H 0 breaks the symmetry of the potential and the particle instead of circulating around the entire bubble passes to a consecutive one where it finds the closer energy minimum.The particle trajectory could be directed along any of the six crystallographic axis of the array, depending on the initial position of the particle relative to the orientation of H 0 .We also observe similarly to ͓16͔ that the ballistic region in the phase diagram closes at frequencies higher than ϳ200 s −1 , when the damped particle cannot follow the fast field dynamics and it localizes again.This feature together, with the reproduced phase diagram confirm that our model describes the experimental results.In our simulations we also find that the ballistic regime closes be- low ⍀Ͻ10 −3 s −1 .It is hence clear that the lagging of the particle behind the magnetic energy minima is an essential feature for the occurrence of ballistic motion.The experiments of Ref. ͓16͔ were all carried out well above this threshold.
Two striking results are worth mentioning at this point.First, we distinguish up to three ballistic trajectories in the delocalized region, 0.32M s Ͻ H z Ͻ 0.44M s , which were not previously observed in the experiments.For H z Ͼ 0.32M s , the ballistic path changes and the particle after jumping to a consecutive bubble moves first forward then backward along the circular domains and further passes to the next bubble ͓see Fig. 2͑c͒, bottom inset͔.Further increasing H z enlarges the part of the path which passes through the interstitial regions and correspondingly shortens the transit time of the particle around the bubbles ͓Figs.2͑d͒ and 2͑e͒ bottom in-sets͔.The numerical method employed allows us to resolve such different paths something which was not possible with the experimental resolution.When H z increases above 0.44M s the trajectory becomes again localized and the particle performs triangular orbits on the interstitial regions between the bubbles.Here the strength of H z is such that the energy minima between three nearest bubbles are interconnected and trap the particle in a closed orbit.
The second remarkable result is the absence of a superdiffusive regime that bridges the localized and ballistic motions.Indeed in the phase diagram of Ref. ͓16͔ we observed two superdiffusive regions, respectively, on each side of the ballistic region, spanning an area of 3 ϫ 10 −2 M s .We confirm the lacking of superdiffusion in our model by deeply exploring the transition lines between the two dynamical phases up to a precision of 10 −4 M s .The fact that these regimes were present in the experiments at the transitions between stable dynamical phases suggests the presence of some degree of noise in the system which could destabilize the dynamics during transition.We first conjectured that this effect could arise from thermal fluctuations that were not included initially in our model.However, by adding a thermal noise to Eq. ͑1͒ and solving this equation at different temperatures, we did not observe any change in the particle trajectories at the transition lines.Actually the large magnetic strengths by which the magnetic potential pins the particles ͑ϳ100 pN͒ compared to the Brownian forces ͑ϳfN͒ explains the negligible role played by the thermal noise.
Another possibility is to consider a magnetic disorder in the film which arises during application of the field.Indeed the planar component of the rotating field H 0 , while unable to change the volume of the bubbles, could induce very small radial deformations of the magnetic domains ͓23͔, unobservable with the microscope system used in ͓16͔.One computationally efficient way which we found was to add some degree of randomness to the distribution of the radii of the bubbles and periodically refresh it with a time constant ref .
Figure 3 shows a superdiffusive trajectory at the transition line from a circularly localized to a ballistic trajectory ͑H z = 0.25M s , ⍀ =20 s −1 ͒ by using a uniform random distribution of the radii of the bubbles, between R + R and R − R with = 0.1 and ref = 0.02 s.Computing C͑͒ we determined a superdiffusive exponent ␣ = 1.4. in the long-time limit of our simulation.It should be emphasized that this behavior occurs only close to the transition point.Numerical simulation with the same value of inside either of the two localized or the ballistic regions do not show any feature of superdiffusive behavior.
The superdiffusive dynamics was observed by varying periodically in time the radii of the bubbles in the film.This means that this particular regime can be characterized in terms of two parameters, the radius dispersion and the refreshing time ref .We performed a series of numerical integrations close to the two transition points at fixed frequency, ⍀ =20 s −1 .The results are shown in Fig. 4  situation happens when the polydispersity of the bubble is frozen ͑ ref → ϱ ͒ since the motion in this case is permanently localized without the freedom to explore the changing landscape.
Transition from localized to delocalized dynamics above periodic pinning potentials has been observed ͓24͔ and theoretical described ͓25͔ in the magnetotransport of electrons through an array of scatterers, in the so-called "electron pinball."Also theoretical work has been done in the context of vortex motion in a pinning potential ͓26͔ and particle transport trough periodic substrates ͓27͔.Of the cited manuscripts, our simulation shows similar features with Ref. ͓26͔ under the condition of zero dc drive, where transport into a defined direction is obtained via symmetry breaking, since the system has no preferred direction of motion.Also ballistic particle motion phase locked with the external field for an overdamped driven system was shown in Ref. ͓27͔.Where our work differs, apart from the physical system to which it refers, is in the implementation of some degree of disorder in the pinning potential which allows one to obtain the experimentally observed superdiffusive particle trajectories.
In summary, we theoretically studied the motion of paramagnetic colloidal particles externally driven above a magnetic bubble lattice.We explored the particles dynamics in a wide range of fields and frequencies and obtain, in most of the cases, satisfactory agreement with the experimental data.
Where our finding differs is in the various ballistic trajectories of the particles and in the absence of a superdiffusive regime.We attributed the latter to dynamical disorder in the magnetic potential.Implementing this hypothesis in our model in terms of a time changing random distribution in the effective radii of the bubbles allows one to capture the superdiffusive trajectories.The simple model described in this Rapid Communication can be extended in various ways.For example one can study the motion of particles with different size and/or magnetic content and observe different dynamics ͑coexistence of localized and delocalized states͒ useful in sorting and fractionating colloidal particles.Also plans to investigate motion of the particles above magnetic lattices with frozen defects, different geometry of the magnetic domains, or lattice morphologies ͑such as square, oblique, etc.͒ are on the way.

FIG. 1 .
FIG.1.͑Color online͒ ͑a͒ Schematic of a lattice of magnetic bubbles ͑radius R͒ with a paramagnetic particle subjected to a magnetic field H ex precessing with frequency ⍀ and angle around the z axis.͑b͒ Contour plot of the magnetic potential generated at height z = 1.5 m above the surface ͑z =0͒, of the bubble array ͑lattice vectors a 1 , a 2 ͒ with H z = 0.29M s , H 0 = 0.1M s .Energy minima correspond to gray ͑yellow͒ regions and maxima to black ͑red͒ regions.͑c͒ Magnetic potential along the ͑dashed line͒ cut in the phase diagram for H z =0 ͑black͒ and H z = 0.29M s ͑blue͒.
FIG.3.͑Color online͒ Superdiffusive trajectory of a paramagnetic particle subjected to a field with H z = 0.25M s , H 0 = 0.1M s , and ⍀ =20 s −1 .The radii of the bubbles were periodically varied every 0.02 s ͑see text͒.Lower inset shows the correlation function C͑͒ with a superdiffusive exponent ␣ = 1.4.