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Brownian motion of ellipsoidal particles on a granular magnetic bath

C. Tapia-Ignacio1, R. E. Moctezuma2, F. Donado1, and Eric R. Weeks3∗1Instituto de Ciencias Basicas e Ingenierıa, Universidad

Autonoma del Estado de Hidalgo, Mineral de la Reforma 42184, Hidalgo, Mexico,2CONACYT-Instituto de Fısica “Manuel Sandoval Vallarta”,

Universidad Autonoma de San Luis Potosı, Alvaro Obregon 64, 78000 San Luis Potosı, S.L.P., Mexico3Physics Department, Emory University, Atlanta, Georgia 30322 USA

We study the Brownian motion of ellipsoidal particles lying on an agitated granular bath com-posed of magnetic particles. As we decrease the magnetic forcing, the bath takes characteristics ofa molecular system that is cooled towards the glass transition. We quantify the mobility of differ-ent floating ellipsoidal particles using the mean square displacement and the mean square angulardisplacement, and relate the diffusion coefficients to the bath particle motion. The ratio of transla-tional and rotational diffusion constants for the floating particles is forcing-independent, althoughthe particle shape matters (with longer floating particles rotating slower than shorter). Unusualaspects of the floating particle motion include non-Gaussian statistics for their displacements, anda shape-dependent and forcing-dependent anisotropy of translational diffusion coefficients paralleland perpendicular to the ellipsoid long axis.

I. INTRODUCTION

Brownian motion (diffusion) occurs in a wide varietyof processes in nature. While initially observed in regularliquids, its theory has been applied not only in physics,but in biology, chemistry, and even economics. In macro-scopic granular systems, external forcing can effectively“thermalize” the system and result in diffusive motion [1–5]. In super-cooled liquids, diffusion slows greatly, butwith rotational diffusion of non-spherical probes slow-ing even more than translational diffusion [12–16]. Somegranular experiments [5–8] and simulations [9, 10] showedthat granular systems can exhibit glassy behavior undersome conditions, with diffusive motion greatly slowed.However, rotational diffusion has not been studied inthese granular experiments.

In this manuscript, we study the translational and ro-tational behavior of floating ellipsoidal particles on amagnetic granular bath, and examine the influence ofthe bath particle intensity on the motion of the floatingparticles. By varying the forcing applied to the mag-netic bath particles, we can vary the diffusivity of thefloating particles by nearly two orders of magnitude. Forlow magnetic forcing, the bath particles behave like asupercooled fluid. Across the range of bath particle con-ditions, we find that the floating particles always haverotational diffusion and translational diffusion coupled,that is, their ratio is constant, unlike what has been seenin prior glassy experiments mentioned above. While thecoupling of the diffusion constants is to be expected forparticles in a “normal” fluid, we find other unusual as-pects of the floating particle motion such as non-Gaussianstatistics and a forcing-dependent anisotropy of transla-tional diffusion coefficients parallel and perpendicular tothe ellipsoid long axis.

∗Electronic address: erweeks@emory.edu

FIG. 1: The experimental setup consists of magnetic particlesin a horizontal container (diameter 132 mm) in the middle ofHelmholtz coils (inner diameter 152 mm, outer diameter 179mm). A sinusoidal signal fed into a power amplifier producesan alternating magnetic field. The spherical bath particleshave permanent magnetic dipoles so the changing field causesthe particles to roll on the surface in random directions. A flatparticle “floats” on the bath particles and moves due to therandom kicks it receives from the underlying bath particles.

II. EXPERIMENTAL SETUP

The experimental configuration consists of a flat par-ticle floating on a forced granular medium; see Fig. 1.This granular medium is composed of a 2D ensemble ofstainless steel spherical particles of diameter 1 mm, mass4.2 mg, and area fraction 0.23. The system is placedin a time-dependent magnetic field. The steel particleshave been previously magnetized so that they possess apermanent magnetic dipole. The magnetic field is gener-ated by a pair of Helmholtz coils fed by a power amplifierwhich receives a sinusoidal signal. This alternating fieldcauses the particles to move randomly: each steel par-ticle tries to align its magnetic dipole moment with the

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FIG. 2: Images showing floating particles with aspect ratio (a)α = 1.00, (b) α = 1.44, (c) α = 1.48, and (d) α = 2.20. Thebath particles are the smaller spheres. Note that the image in(a) is viewed with reflected light, while the other are viewedwith transmitted light, leading to the different appearance of(a) from the other panels. The circular particle in (a) has awhite ellipse drawn on it to facilitate rotational tracking ofthis otherwise circularly symmetric particle.

instantaneous field (pointing up or down), causing theparticle to rotate in a random direction to implement thereorientation. When the field reverses, the particle triesto reorient, subject to the rotational inertia it alreadypossesses. The result is that the steel particle motionis quite random and plays the role of a thermal bath.Decreasing the magnitude of the magnetic field resultsin an effectively “cooler” granular bath. Although thissystem is highly dissipative, it reaches a stationary statewhich fits the necessary conditions to be considered asan Ornstein-Uhlenbeck stochastic process [19]. The dy-namics of a system similar to these bath particles hasalready been studied, and it has been demonstrated thatfor certain conditions the magnetic field is proportionalto an effective temperature [20]. The intensity of thebath particle motion is varied with the magnitude of theoscillating magnetic field: Bo = 66, 60, 54, 48, 42, 36and 30 G, all of them with a frequency of 10 Hz. In thissystem, for high intensities of the field, the bath particlesshow the dynamics similar to that of a Brownian gas, andthe velocity distributions are Maxwell-Boltzmann distri-butions [20]. For low intensities, the motion of the par-ticles decreases, showing freezing-like effects resemblingthe dynamics near the glass transition [5].

We use floating particles with aspect ratios α =1.00, 1.44, 1.48, and2.20, as shown in Fig. 2. Further de-tails of these particles are listed in Table I. The floatingparticles are non-magnetic, and do not interact with themagnetic field. Rather, they sit atop the magnetic bathparticles, and move due to the random kicks of the bath

aspect ratio 2rmajor 2rminor mass1.00 11.1 mm 11.1 mm 2.2 mg1.44 13.0 mm 9.0 mm 7.1 mg1.48 17.0 mm 11.5 mm 10.4 mg2.20 24.4 mm 11.1 mm 17.0 mg

TABLE I: Details of the floating particles, including semi-major diameter and semi-minor diameter.

FIG. 3: Mean square displacement 〈∆r2rmbath〉 for the bathparticles. The vertical dotted line indicates the time scale (0.1s) corresponding to the period of the magnetic forcing. Theintersection of the data with this vertical line sets the lengthscale L through L2 = 〈∆r2〉(∆t = 0.1s). The dashed lineindicates power law growth 〈∆r2〉 ∼ ∆t1.7 for the early timescale data. The lowest forcing (30 G) of the bath particlescorresponds to the purple curve at the bottom and increasesup to 66 G (red curve at the top).

particles.We obtain three series of three-minute videos for each

floating particle and magnetic forcing amplitude. Fromeach video we track the translational and rotational mo-tion of the floating particle, along with the bath parti-cle motion [21]. Our temporal resolution is 1/60 s, ourspatial resolution is better than 0.07 mm (x and y posi-tion), and our angular resolution is better than 0.006 rad(≈ 0.4◦).

III. RESULTS

To characterize the motion of the bath particles, wecalculate the mean square displacement (MSD) for eachmagnetic field amplitude we consider. The results areshown in Fig. 3. It is interesting to note that for allof the data, at short ∆t, the MSD of the bath particlesis superdiffusive with 〈∆r2〉 ∼ ∆t1.7. This implies thatover a short time scale particles are more likely to move instraight lines, rather than being purely diffusive randomwalks [22]. At high magnetic field intensities, the motionof the bath particles is diffusive (top curves of Fig. 3).As the magnetic field intensity decreases, bath particle

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FIG. 4: Trajectories for floating particles with aspect ratioα = 2.20 for a high excitation case with L = 1.82 mm (a,b)and a low excitation case with L = 0.90 mm (c,d). Panels(a,c) show the x position as a function of time. The insetsshow the full trajectory in x and y with scale bars of length 5mm. Panels (b,d) show the corresponding angular orientationas a function of time. Note the vertical scales are significantlyreduced in panels (c,d) as compared to (a,b).

motion is slower. At the lowest forcing, the MSD has aplateau at long time scales (bottom curve of Fig. 3). Thisresembles a supercooled fluid close to the glass transition[23–26].

We likewise consider the floating particle motion,which has both translational and rotational components.Example translational and rotational trajectories of afloating particle with aspect ratio α = 2.20 for a high anda low excitation case are shown in Fig. 4. Both the ro-tational orientation and the position in x show a greatervariation for higher excitations than for the lower ones.The full trajectories for high excitations expand notice-ably over a larger area compared with the trajectories atlow excitations (insets of Fig. 4).

Figure 5 shows the MSD 〈∆r2〉 and the mean squareangular displacement (MSAD) 〈∆θ2〉 for the floating par-ticle with aspect ratio α = 2.20. The vertical dotted linecorresponds to the period of the magnetic forcing, and co-incides with a crossover region from short time scale be-havior to long time scale behavior. An interesting featurethat stands out in these graphs is the diffusive behaviorat large time scales for both MSD and MSAD, regardlessof the excitation of the bath. This result is interesting,since it indicates that, although the bath particles goesfrom diffusive to glassy states as we decrease the mag-netic forcing (Fig. 3), the floating particles maintain theirdiffusive behavior. Intriguingly, the bath particle MSDsall have a constant initial power law growth (Fig. 3, con-stant initial slope on the log-log plot) whereas the floatingparticle MSDs exhibit an initial slope dependent on theforcing. Thus, within the time scales ∆t < 0.1 s (theforcing period), the coupling between the bath particlesand floating particles is nontrivial. The ∆t → ∞ behav-ior is also interesting: for the bath particles, the slope

FIG. 5: (a) Mean square displacement 〈∆r2〉 for the longestfloating particle (α = 2.20). The color of each curve relates tothe intensity of the bath particle motion; see text for details.The vertical dotted line indicates the time scale (0.1s) corre-sponding to the period of the magnetic forcing. The dashedlines indicate linear dependence 〈∆r2〉 ∼ ∆t. (b) Mean squareangular displacement 〈∆θ2〉. The colors are the same as in(a).

of the MSD changes with the magnetic forcing intensity,whereas for the floating particles their behavior is alwaysdiffusive at long time scales (slope = 1 on the log-logplot).

The differing behavior at long time scales between thebath particles and the floating particles suggests thatthe direct influence of the bath is only at shorter timescales; that the diffusive behavior of the floating parti-cles is emergent from the random influence of the bathat shorter time scales. To investigate this, we fit the∆t → ∞ portion of the floating particle MSD and MSADcurves to determine the diffusion coefficients through〈∆r2〉 = 4DT∆t, 〈∆θ2〉 = 2DR∆t. We then examinethe dependence of the measured diffusion coefficients onvarious quantities related to the bath particles. In partic-ular we use the bath particle motions (quantified by theMSD) using a natural choice for the time scale, ∆t = 0.1 s(the period of the magnetic forcing). We define a length

scale L =√

〈∆r2bath〉 using displacements over this timescale. Figure 6 shows a linear relationship between L andthe logarithms of all of the diffusion coefficients. This re-lationship, ln(D) ∼ L/L1, holds for all our results with acommon length scale L1 = 0.22 mm; this is clear from theleast square fit lines in Fig. 6 where the fit is constrainedto have a common slope and differing intercepts. Defin-

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FIG. 6: (a) Translational diffusion coefficients DT as a func-tion of the forcing L. The different symbols are for differentaspect ratio floating particles as indicated. (b) Rotationaldiffusion coefficients DR as a function of the forcing L. Thesymbols are the same as (a). For both panels, all fit lineshave a constant slope: ln(D) ∼ L/L1 with L1 = 0.22 mm.The uncertainty in D is the symbol size.

ing L with a different ∆t gives results that have a largerleast square fit error.

We note that fitting the diffusion coefficients toln(D) ∼ L2/L2

2 works nearly as well, with L2 = 0.78 mm.The least square error is slightly larger, but the overallappearance is similar to the results shown in Fig. 6. Thetwo fitting constants L1 and L2 are both similar in orderof magnitude to the bath particle radius 0.5 mm. Thedependence of ln(D) on L or L2 suggests something likean activated process, although it is not apparent why thiswould be the case.

An alternate way of thinking of an activated process isinspired by an analysis technique introduced by C. A. An-gell for analyzing glassy data [27, 28]. He argued thatviscosity η (at high temperature) is an Arrhenius processwith some activation energy Ea, so η ∼ exp(Ea/kBT ),with kB being Boltzmann’s constant and T being theabsolute temperature. By plotting ln(η) as a function of1/T , a straight line will be found (with slope Ea/kB).We do not measure viscosity in our experiment – in fact,it may not be a relevant concept for our particles floatingon the granular bath. Nonetheless, in analogy with dif-fusion coefficients for fluids, we note for fluids η ∼ T/D(for both translational and rotational diffusion). If ourL acts like a temperature, then plotting ln(L/D) as afunction of 1/L should give a straight line. We show

FIG. 7: Angell plot of the floating particles with different as-pect ratios for (a) translational diffusion and (b) rotationaldiffusion. Straight lines suggest an activated process, withthe forcing L playing the role of a temperature; see text fordetails. The symbols correspond to particles with aspect ra-tios as labeled in (a). The dashed lines are the same fittingshown in Fig. 6. Note that in the original paper by Angell(Ref. [27]) the horizontal axis is non-dimensionalized by theglass transition temperature. Here we do not identify a glasstransition for the floating particles, so we do not attempt tonon-dimensionalize the horizontal axis. The uncertainty isthe symbol size.

this in Fig. 7; here the linear dependence works plausi-bly well for the “hot’ data (left symbols). The linear fitdoes not pass through the points corresponding to thelowest forcing (right-most points) and in fact we do notinclude those points in the least square fit. The lines havea variety of slopes, which is plausible as the “activationenergy” would be expected to depend on the floating par-ticle shape, size, and motion (translational or rotational).That being said, the fitting shown in Fig. 6 has fewer freeparameters and seems more compelling, albeit less phys-ically motivated. The fitting of Fig. 6 is shown as thedashed lines in Fig. 7.We next wish to compare DT to DR. In a 3D New-

tonian fluid, the rotational motion of ellipsoids followsDT ∼ 1/rmajor, DR ∼ 1/r3major, in part because the twodiffusion coefficients have different units. Accordingly, weconsider the non-dimensional ratio r2majorDR/DT , plot-ted as a function of L in Fig. 8. Given that DT and DR

have the same functional dependence on L (Fig. 6), itmust be true that their ratio is a constant, as Fig. 8shows (within the uncertainty). The largest value of

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FIG. 8: The ratio of diffusion coefficients for floating particleswith aspect ratios as indicated, as a function of the forcingintensity L. The ratio is nondimensionalized by multiplyingby r2major, the square of the semi-major axis. Representativeerror bars are shown for two data sets; the uncertainties ofthe other two data sets are identical. The horizontal linesindicate the mean value of the ratio for each particle type:from top to bottom, these values are 3.2, 2.0, 1.5, and 1.0 rad2,corresponding to α = 1.00, 1.48, 1.44, and 2.20.

r2majorDR/DT appears for the floating particle with as-pect ratio α = 1.00, with the non-dimensional ratio be-ing 3.2. For spheres diffusing in 3D, this ratio would be3/4 [29–31], so a somewhat similar order of magnitude.For our larger aspect ratio floating particles the ratior2majorDR/DT decreases. This decreasing trend with in-creasing aspect ratio disagrees with numerical results for3D rod diffusion [32, 33]. This disagreement suggests,not surprisingly, that the mechanism leading to diffusionin our 2D granular experiment are qualitatively differentfrom normal 3D diffusion in liquids.

According to the classic Stokes-Einstein-Sutherlandand Stokes-Einstein-Debye equations for DT and DR re-spectively, the ratio DT /DR should be a constant in-dependent of temperature and viscosity; the exact con-stant will depend on the floating particle shape. Inglass-forming systems, prior observations found “decou-pling” of translational and rotational diffusion, such thatDR/DT is not constant but rather decreases as the glasstransition is approached. This has been seen in experi-ments with small molecule glass-formers [34–36], colloidalglass experiments [15, 16, 37], and simulations [38–41].Our bath particles become somewhat glassy as the mag-netic forcing is decreased, as seen in Fig. 3. Despite ourbath particles becoming glassy, in our experiments we donot observe decoupling of the translational and rotationaldiffusion: there is no systematic dependence of the datain Fig. 8. This is further evidence that the long-time be-havior of the bath particles does not connect with thelong-time behavior of the floating particles.

To understand the influence of the anisotropic shapeof our floating particles, we examine motion along the

FIG. 9: Dpar/Dperp measured based on short lag time motion.As expected, motion along the axis (Dpar) is generally faster.The ratio decreases toward 1 for larger forcing (larger L).

instantaneous axes parallel and perpendicular to the el-lipsoid. We determined the displacements between agiven frame and the next, with the axis defined at thefirst frame [17]. We use this short time motion to de-fine Dpar in the direction parallel to the ellipsoid longaxis, and Dperp perpendicular to the long axis. (That is,Dpar = 〈∆r2par〉/∆t with ∆t = 1/60 s, and similarly forDperp. We use this short time interval so that the tracerbarely rotates, and thus the axes stay well defined.) Fig-ure 9 shows Dpar/Dperp for the four tracers as a functionof the forcing. As one would expect for ellipsoids in 2Dor 3D [17, 18], the parallel motion is faster than perpen-dicular motion and as aspect ratio increases, this ratio islarger.A surprising observation is that for the circular floating

particles (α = 1.00, violet diamonds in Fig. 9) the ratio isnot identically 1. For these tracers, the orientation is in-dicated by a mark on the circular particle [see Fig. 2(a)].It is likely that the particle is somehow asymmetric, forexample due to the ink printing the mark, or the particlehaving deformed slightly due to collisions with the bathparticles. Note that we initially define directions paralleland perpendicular to the mark on the circular particle,but then find the axes relative to that mark that max-imize the anisotropy; it is this coordinate system thatleads to the data shown in Fig. 9.Figure 9 shows that for all floating particles the ra-

tio depends on the forcing. The larger the force is, thelower the ratio Dpar/Dperp is, in contrast to ellipsoids inregular fluids where this ratio should be constant, inde-pendent of the fluid viscosity [17]. In all cases, we observeDpar/Dperp < 2; 2 is the limiting value for a long ellip-soid diffusing in 3D [18].To better understand particle motion (both bath and

floating particles), we show representative probabilitydistribution functions of displacements in Fig. 10. Ineach panel the blue data are for the lowest excitationand are the narrow distributions; the red data are for the

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FIG. 10: Probability distribution functions P (∆x) for (a)bath particles, (b) floating particles with aspect ratio α =1.00, and (c) floating particles with aspect ratio α = 2.20. Ineach case the blue data correspond to L = 0.90 mm (the low-est excitation) and the red data correspond to L = 1.82 mm(the highest excitation). ∆t = 0.1 s is used to define the dis-placements. The dashed and dotted lines are Gaussian fitswith the same standard deviation as the experimental data.The non-Gaussian parameter α2 has the values (a) 1.4, 0.1;(b) 6.2, 0.5; (c) 2.4, 0.6 for small and large L.

highest excitation and are the broad distributions. Thedotted and dashed black lines are Gaussian fits, and it isimmediately apparent that the distributions are broaderthan a Gaussian. That is, large displacements are rare,but more common than would be expected if the distri-bution was a Gaussian of the same standard deviation.Non-Gaussian distributions are often seen in supercooledsystems [11, 23, 24], so the result is plausible for the bathparticles. It is intriguing that the distribution functionsare non-Gaussian for the floating particles even thoughthese particles are diffusive at long time scales (Fig. 5).That is, often strong non-Gaussian behavior is associatedwith a more pronounced plateau in the mean square dis-placements; Fig. 5 shows a weak plateau, which is mostobvious for the lowest forcing data. In all cases, thedistributions are close to Gaussian for higher excitation.The same qualitative results are found for the rotational

FIG. 11: (a) Non-Gaussian parameter for translational mo-tion, evaluated at time scale ∆t = 0.1 s. The symbols havethe same meaning as in prior figures. The + symbols are forthe bath particles. (b) The same, for rotational motion.

displacements (data not shown).The relative width of the displacement probability dis-

tributions can be quantified with the non-Gaussian pa-rameter α2(∆t) [42],

α2(∆t) =〈∆r4〉(∆t)

2[〈∆r2〉(∆t)]2− 1. (1)

This quantity is zero for a Gaussian distribution, andpositive for distributions with broad tails such as thoseshown in Fig. 10; the specific values of α2 are given inthe caption to the figure.The non-Gaussian parameter is shown in Fig. 11(a) for

translational motion, and (b) for rotational motion, as afunction of the forcing strength L. For these results, thetime scale ∆t = 0.1 s is used to define displacements,although the qualitative behavior is not too sensitive tothis choice. For low forcing, the non-Gaussian parameteris largest, reflecting the extreme width of the blue datain Fig. 10 as compared to the Gaussian fit. α2 drops forlarger L in Fig. 11. In this respect the behavior is similarto glassy liquids, which are more non-Gaussian close tothe glass transition, and more Gaussian when well awayfrom the glass transition [23, 43, 44].

IV. CONCLUDING REMARKS

In our experiments the granular bath particles becomeglassy at the lowest forcing L, while the floating parti-cles are still diffusive. We conjecture this is because the

7

floating particle does not interact with the bath parti-cles directly. Typically one thinks of glassy behavior asarriving due to crowding of molecules or particles. Inour experiment, no matter how the bath particles arecrowded or interact with one another, the floating parti-cle sits atop them and does not experience the crowding.It is interesting that the floating particle’s long-time mo-tion continues being diffusive even though the particles inthe bath are not, and even though the floating particles’displacement probability distributions are non-Gaussian.As described above, the bath particles are forced peri-

odically with a period of 0.1 s, and this time scale appearsto play a key role in determining the floating particle mo-

tion. It suggests that in our experiment we have a novelsource of random energy acting like some sort of effectivetemperature. This may provide insight into other situa-tions with non-thermal “effective temperature” baths.

Acknowledgments

Partial financial support by CONACyT, Mexico,through grant 256176 (SEP-Ciencia Basica) is acknowl-edged. The work of E.R.W. was supported by the Na-tional Science Foundation (DMR-1609763).

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