Fluid to solid transition in magnetofluidized beds of fine powders

J. Appl. Phys. 108, 054903 (2010); https://doi.org/10.1063/1.3480989 108, 054903 © 2010 American Institute of Physics. Fluid to solid transition in magnetofluidized beds of fine powders Cite as: J. Appl. Phys. 108, 054903 (2010); https://doi.org/10.1063/1.3480989 Submitted: 16 December 2009 . Accepted: 21 July 2010 . Published Online: 07 September 2010 J. M. Valverde, M. J. Espin, M. A. S. Quintanilla, and A. Castellanos ARTICLES YOU MAY BE INTERESTED IN Rheology of magnetofluidized fine powders: The role of interparticle contact forces Journal of Rheology 54, 719 (2010); https://doi.org/10.1122/1.3380851 Fluid to solid transition in magnetofluidized beds of fine powders J. M. Valverde,1,a兲M. J. Espin,2M. A. S. Quintanilla,1and A. Castellanos1 1Department of Electronics and Electromagnetism, University of Seville, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain 2Department of Applied Physics II, University of Seville, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain 共Received 16 December 2009; accepted 21 July 2010; published online 7 September 2010兲 Experimental observations on the fluid to solid transition in beds of magnetized fine particles fluidized by gas are reported for different particle sizes 共dp兲. Contrarily to stability analysis prediction, the fluidized bed is stabilized by a sufficiently strong magnetic field in the cross-flow configuration. As the strength Hof the horizontally applied magnetic field is increased, particle chaining in the bubbling bed becomes apparent due to the induced attractive magnetostatic forces between the particles. In close analogy with magnetorheological fluids chain stability is determined by the balance between gas flow shear and the interparticle magnetostatic force. The jamming transition occurs at a gas velocity scaling proportionally to dp 2H2when the length of the stable chains reaches a critical size which is independent of particle size. © 2010 American Institute of Physics. 关doi:10.1063/1.3480989兴 I. INTRODUCTION Magnetofluidized beds 共MFBs兲basically consist of magnetic particles suspended in a vertical gas flow and subjected to a magnetic field. Relevant applications of MFBs can be found in numerous examples such as chemical reactors taking advantage of their high gas-solid contact efficiency or magnetically controlled valves.1In general, the flow properties of magnetized particles is a matter of interdisciplinary interest in a wide range of processes and scales besides of MFBs. For example, the flow alignment of magnetic particles is used to identify palaeoflow directions in volcanic rocks and sediments usually in the presence of external fields.2Recently, the control of the structuring of magnetizable nanoparticles on a mesoscopic scale has opened a new area of research to achieve structures for a wide variety of applications.3 Granular materials are generally characterized by a nonequilibrium kinetic transition from a fluidlike to a solidlike jammed regime.4,5At jamming the constituent particles are suddenly arrested in a metastable static state forming a solid disordered network that spans the system. The jamming transition has been described by a phase diagram parametrized by interparticle attractive force, temperature, particle volume fraction, and applied stress.4In the case of MFBs, jamming of athermal magnetic particles is driven by the magnetic stress induced by the externally applied magnetic field. Particle size in these systems is usually of the order of tens to hundreds of microns, thus van der Waals forces and Brownian motion are fully negligible. In MFBs particle sedimentation is avoided by subjecting the bed to a countergravity directed gas flow in such a way that the gas pressure drop across the powder balances the material weight per unit area. Fluidized beds are commonly unstable, being characterized by the rapid developments of gas bubbles and large fluctuations of the gas velocity in close analogy with the behavior of sedimenting particle suspensions.4,6Gas bubbles can be suppressed in MFBs by the application of a sufficiently strong magnetic field that causes the jamming of the particles. Usually, the field strength needed for jamming is presented as a function of the nondimensional number Nm, defined as the ratio of the solids kinetic energy per unit volume to the magnetic energy. According to Rosensweig stability analysis, based on spatially averaged mass and momentum conservation equations,1magnetic stabilization would take place under the condition NmNv⬍1. For magnetically linear solids, Nv=4 ␲ 关3−2共1− ␾ 兲兴2共1+ ␾ ␹ p兲/关共1− ␾ 兲2 ␾ cos2 ␸ 兴, being ␾ the particle volume fraction, ␸ the angle between the magnetic field and the direction of the gas flow, and ␹ p the particles magnetic susceptibility. As Nvis increased the growth factor of local voidage perturbations is increased. Stability in this continuum model is possible because the magnetic body force arising from gradients of the void fraction smoothes these perturbations. In the case of a cross-flow field 共 ␸ = ␲ /2兲the value of Nvbecomes infinite and magnetic stabilization would not be possible. Yet, experimental observations have evidenced that cross-flow fields are also capable of stabilizing MFBs.7Thus, the physical mechanism responsible for the jamming transition in MFBs is far from being fully understood and a widely accepted equation to predict the strength of the field necessary to stabilize MFBs is lacking. MFBs operated in the cross-flow field configuration bear a close resemblance to magnetorheological fluids 共MRFs兲,in which micron-sized magnetic particles are suspended in nonmagnetic liquids.8As in MFBs, particles in MRFs are magnetically multidomain and a magnetic field may induce intense interparticle interactions leading to a transition from a fluidlike to a solidlike regime. Essentially, it can be said that the jamming transition in both MRFs and MFBs is characterized by a common mechanism consisting of the formation of chains of magnetized particles due to the strong magnetostatic interaction induced between them. According to a basic chain model originally developed for MRFs, chain stability a兲Electronic mail: [email protected]. JOURNAL OF APPLIED PHYSICS 108, 054903 共2010兲 0021-8979/2010/108共5兲/054903/6/$30.00 © 2010 American Institute of Physics108, 054903-1 in MRFs is ruled by the balance between shear and interparticle magnetic forces.9We present here an experimental work aimed to look at the jamming transition of MFBs operated in the cross-flow configuration. In close analogy with the behavior of MRFs it will be shown here that a modified chain model is also able to capture the fundamentals of the physical mechanism behind jamming in MFBs. II. EXPERIMENTAL SETUP Figure 1schematizes the experimental setup used in our work. The magnetic powder sample is held in a vertically oriented cylindrical vessel made of polycarbonate 共D =2.54 cm internal diameter兲and rests on a porous plate that acts as gas distributor 共5 ␮ m pore size兲. By means of a mass flow controller 共MKS model 1179A, 2000 cm3/min full scale兲, a controlled flow of filtered and dried air is pumped through the powder bed while the gas pressure drop ⌬p across it is read from a differential pressure transducer 共MKS model 220CD, 10 Torr full scale兲. The height of the bed, which gives an average value of the particle volume fraction ␾ , is measured by means of an ultrasonic sensor 共Senix model Ultra-S兲placed on top of the vessel. This device can determine distance, with an accuracy smaller than local fluctuations in bed height, by sending an ultrasonic wave and measuring the time of reflection from the target. Our magnetic powders have been tested as affected by a horizontal uniform magnetic field externally imposed 共cross-flow configuration兲. This is accomplished by placing the bed in the center of a pair of square Helmholtz coils 共50⫻50 cm兲, with each coil consisting of 500 turns of 2 mm diameter copper wire. The magnetic field strength Hin the close vicinity of the bed is measured by a Hirst magnetics Gauss meter using an axial probe with an accuracy less than 0.1 mT. Experimental measurements show that, within this experimental accuracy, the external field strength is constant in a volume larger than the bed volume. III. MAGNETIC POWDERS The magnetic powders used in the experiments consist of magnetite particles of similar permeability but different particle size dp共35, 50, and 65 ␮ m兲artificially made by Xerox Co. and used in practice as carriers of toner particles in the xerographic process. Solid density ␳ p, as measured by means of a AccuPyc 1330 pycnometer, is 5060 kg/m3.As seen in scanning electron microscopy 共SEM兲photographs 共Fig. 2兲these particles are spherically shaped and rather monodisperse. In the range of field strengths applied 共H⬍5kA/m兲, the magnetic response of the powders used in our experiments is linear and reversible. This can be observed in Fig. 3, showing experimental data on magnetic characterization of the magnetite powders performed by means of a superconducting quantum interference device 共SQUID兲magnetometer, where around 1 mg samples were tested. As expected the samples susceptibility ␹ obtained were similar: ␹ =2.98共dp =35 ␮ m兲, 2.78共dp=50 ␮ m兲, and 2.60共dp=65 ␮ m兲. Accord- FIG. 1. 共Color online兲Sketch of the experimental setup used in the magnetofluidization experiments reported in this work. FIG. 2. SEM photograph of 35 ␮ m共top兲and 65 ␮ m共bottom兲sized magnetite beads used in the experiments. FIG. 3. 共Color online兲Magnetization of bulk samples of the magnetite powders used in the experiments vs applied field as measured by SQUID magnetometer. The main graph shows the magnetorheological behavior in the interval of field intensities used in our magnetofluidization experiments. The inset shows the magnetorheological behavior in an extended range of field strengths, showing that the magnetization saturates for H ⲏ100 kA/m. 054903-2 Valverde et al. J. Appl. Phys. 108, 054903 共2010兲 ing to effective medium theories and numerical analysis on random granular materials,10 the susceptibility ␹ of a bed of magnetically linear spherical particles of susceptibility ␹ plocated in a homogeneous environment of susceptibility ␹ 0can be calculated from the Bruggeman mixing rule 共1− ␾ 兲 ␹ 0− ␹ 3+ ␹ 0+2 ␹ + ␾ ␹ p− ␹ 3+ ␹ p+2 ␹ =0, 共1兲 where in our case ␹ 0=0. While the SQUID tests served to check the magnetically linear behavior of the powders tested it is difficult to derive ␹ pfrom these measurements since ␾ cannot be accurately known for the tiny amount of powder samples used. A complementary method employed to obtain ␹ pconsisted of measuring the change in self-inductance Lof a single layer solenoid 共number of turns: 300, radius: 41 mm, and length: 16 cm兲as the powder was poured into it. In order to measure Lwe used a precision LCR bridge 共model LCR400 Thurlby Thandar Instruments兲. An advantage of the L-method is that the average particle volume fraction ␾ of the tested samples 共of mass about 450 g兲filling the inside of the coil could be precisely obtained, which allowed us to estimate ␹ pfrom the averaging Bruggeman equation. The results were: ␹ =2.37, ␾ =0.425, ␹ p=9.52共dp=35 ␮ m兲; ␹ =2.33, ␾ =0.417, ␹ p=9.72共dp=50 ␮ m兲; and ␹ =2.42, ␾ =0.431, ␹ p=9.48共dp=65 ␮ m兲. Hunt et al.11 has reviewed data from the literature reported for magnetite particles of size between 0.01 and 100 ␮ m made from either crushing or crystal growing, which ranged between 2.5 and 10. Thus our results fit in the range of reported values. IV. EXPERIMENTAL RESULTS In the experimental procedure the bed is first initialized by subjecting it to a gas velocity large enough to drive it into a bubbling state in which the powder looses memory of previous processes.12 Once the bed reaches a stationary bubbling regime the horizontal magnetic field is applied. When the magnetic field is applied, the initial gas velocity must be sufficiently large to keep the bed in the bubbling regime. The formation of macroscopic chains at this high gas velocity is thus still offset by the intense hydrodynamic shear. Then the gas velocity vgis slowly decreased while the gas pressure drop across the bed ⌬pis measured. Figure 4illustrates data on ⌬pversus vg, showing the effect of the applied field on the behavior of the system. As vgis slowly decreased and the bed is still in the fluidlike bubbling regime, ⌬pbalances the powder weight per unit area W⯝510 Pa. However, chaining becomes progressively noticeable in the magnetized bed as evidenced by Fig. 5, which illustrates particle chains elutriated from the MFB. Image samples were obtained by approaching a card of adhesive tape to the free surface. Remarkably, the layer of powder adhered to the tape revealed the presence of chains oriented at a preferential angle ␪ ⯝60° with respect to the direction of the applied field. As a first relevant result these images show that particle chaining does not occur along field lines in the cross-flow MFB. As vg is further decreased the hydrodynamic shear becomes less intense and at a critical gas velocity vg=vcthe system is jammed because of the prevailing magnetostatic attractive force between the particles, thus transiting to a solidlike regime. Further decrease in vgbelow vccauses ⌬pto fall below Wsince part of the weight is sustained by the enduring network of interparticle contacts in the jammed bed. This allows us to identify vcfrom the gas pressure drop measurements. The effect of the field on the jamming transition velocity vcis seen in Fig. 4, showing that the presence of the field induces jamming at higher gas velocities as the strength of the field is increased. Figure 6illustrates ⌬pversus vg data for fixed field strength and different particle sizes. As can be seen the effect of the field becomes more relevant as particle size is increased. Data of vcare plotted in Fig. 7as a function of the magnetic field strength Hfor the three particle sizes tested 共each data point corresponds to an average over around 15 measurements兲. FIG. 4. Gas pressure drop across the powder bed 共made nondimensional with the powder weight per unit area W兲as a function of gas velocity during the fluidization-defluidization cycles in the absence of magnetic field 共top兲 and presence of a magnetic field 共bottom兲of strength H⯝2.4 kA/m. The bed is jammed at vg=vc, which is indicated. FIG. 5. Optical microscope images of the magnetite particles taken from the bubbling fluidized bed in the presence of a magnetic field. Top: dp=35 ␮ m, H⯝3.2 kA/m, and vg=2 cm/s. Bottom: dp=65 ␮ m, H ⯝5.2 kA/m, and vg=6.6 cm/s. The angle ␪ depicted is 60°. 054903-3 Valverde et al. J. Appl. Phys. 108, 054903 共2010兲 The effect of the magnetic field on bed expansion is illustrated by Fig. 8, where the particle volume fraction ␾ measured as the gas velocity is decreased from the bubbling regime is plotted against the gas velocity. In the bubbling regime 共vg⬎vc兲the bursting of bubbles at the free surface cause strong fluctuations of bed height, which give rise to strong fluctuations of ␾ . Note however that when the bed is jammed 共vgⱕvc兲the intensity of fluctuations is suddenly dropped since bed height remains stable. A remarkable feature revealed by Fig. 8is that, in the absence of field, ␾ increases as the gas velocity is increased in the bubbling regime 共vg⬎vc兲, indicating an increase in the size and frequency of bubbles, which is a well known property of bubbling fluidized beds.13 In contrast, we observe that, in the presence of a magnetic field, ␾ remains approximately constant in the bubbling regime, which suggests that the particle chains formed are able to limit the growth of gas bubbles. As seen in Figs. 4, Fig. 8shows that chaining causes the bed to be jammed at higher gas velocities. Moreover, it demonstrates that jamming occurs in states of larger expansion 共smaller ␾ 兲. Figure 10 shows data of the particle volume fraction at jamming as a function of the applied field strength. It is observed that, in the absence of field, the bed jams in more open structures as particle size is decreased: ␾ c⯝0.48共dp=65 ␮ m兲, 0.42共dp=50 ␮ m兲, and 0.39共dp=35 ␮ m兲. This result has been usually reported in the literature.14 The decrease in particle size implies an increase in natural cohesiveness thus producing relatively looser packed structures. We observe however that as the strength of the field is increased, particle chaining blurs the effect of natural cohesiveness associated to particle size, indicating the progressive prevalence of magnetostatic forces growing with particle size. V. DISCUSSION MFBs operated in the cross-flow configuration are similar systems to MRFs. Typically, the suspension of magnetic particles in MRFs is placed between two horizontal plates and a magnetic field His applied in the vertical direction solidifying the structure. The plates are sheared in the horizontal direction at constant shear rate and the force per unit surface needed to initiate flow is defined as the yield stress. Basically, the dynamics of both systems is ruled by the competition between particle chaining induced by the magnetic field and the shear exerted by the fluid essentially in the direction perpendicular to the field. In this section we will show that a chain model originally developed for MRFs can be extended to rationalize the jamming transition in MFBs. If the ratio of the magnetic permeability of the particles to that of the surrounding fluid 共 ␣ =1+ ␹ p= ␮ p/ ␮ f兲is large, multipolar interactions between the magnetized particles play a relevant role. According to the multipolar expansion, the force between two spheres of diameter dplinearly magnetized by a field of strength Hcan be calculated by15 fm=fm 0 冉 dp r 冊 4 关共2f 储 cos2 ␪ −f⬜sin2 ␪ 兲u ˆr+f⌫sin 2 ␪ u ˆ ␪ 兴. 共2兲 Here ris the distance between the centers of the two spheres, fm 0=共3/16兲 ␲ ␮ fdp 2 ␤ 2H2, where ␤ =共 ␮ p− ␮ f兲/共 ␮ p+2 ␮ f兲and ␪ is the angle between the chain and the field direction. The force coefficients fican be calculated in terms of the multipole moments15,16 and depend on the magnetic permeability and the distance between the particles. In the dipolar approximation 共r/dp→⬁兲,fi=1. In the case of two spheres at FIG. 6. Gas pressure drop across the powder bed 共made nondimensional with the powder weight per unit area W兲as a function of gas velocity during the fluidization-defluidization cycles in the presence of a magnetic field of strength H=2.8 kA/m. The bed is jammed at vg=vc, which is indicated. FIG. 7. Gas velocity at jamming transition vs the strength of the magnetic field applied. Solid lines are obtained from Eq. 共6兲using Lc⯝0.35 cm as the best fitting parameter for the three particle sizes. FIG. 8. Particle volume fraction ␾ of the fluidized bed 共dp=35 ␮ m兲as the gas velocity vgis decreased from the bubbling regime in the absence of applied magnetic field and in the presence of applied magnetic field of strength H=3.7 kA/m. The inset shows snapshots of the jammed bed in the absence 共left兲and presence 共right兲of magnetic field. 054903-4 Valverde et al. J. Appl. Phys. 108, 054903 共2010兲 contact 共r/dp=1兲and ␣ =10, which approximately corresponds to our case, it is f 储 =7.287, f⬜=0.6192, and f⌫ =1.5035.15 The main effect of increasing the magnetic permeability of the particles is the increase in f 储 , which raises up to f 储 =182.21 for ␣ =100, thus further enhancing the attractive interaction. The mechanical stability of particle chains in MRFs has been analyzed from a chain model based on the balance between the drag force exerted by the shearing fluid on a particle chain and the interparticle magnetostastic force.9In our case, particle chains in the bubbling heterogeneous regime are strained by gas velocity fluctuations. A strong fluctuation of the gas velocity will produce on the particle chain a shear force fsthat opposes the magnetostatic attractive force fm between the particles thus limiting the growth of the chain. Let us consider a linear chain of Nparticles oriented at an angle ␪ with H. Following the chain model, the angle that the chain makes with the field can be computed by balancing the tangential component of these forces, fmu ˆ ␪ =fsu ˆ ␪ . Using the multipolar expansion for the force between spheres at contact 关Eq. 共2兲兴, this gives fs fm 0=f⌫ sin 2 ␪ cos ␪ .共3兲 Besides, mechanical stability requires that the radial component of fmprevails against the radial component of the shear force, fmu ˆrⱖfsu ˆr,or fs fm 0ⱕ2f 储 cos2 ␪ −f⬜sin2 ␪ sin ␪ .共4兲 By combining Eqs. 共3兲and 共4兲, the critical stability angle ␪ c is obtained as tan2 ␪ c=2f 储 2f⌫+f⬜ .共5兲 As can be seen in Fig. 9, where 兩 ␪ c兩is plotted as a function of ␣ using reported values of the force coefficients,15,16 兩 ␪ c兩 shows a logarithmic increase with ␣ 共兩 ␪ c兩共°兲⯝ ␪ 0+9.6 ln ␣ 兲, where ␪ 0⯝39.2° corresponds to the dipolar approximation兲. In our case 共 ␣ ⯝10兲it is 兩 ␪ c兩⯝60°, and it reaches values close to 90° as ␣ approaches ␣ =100. Further elaboration of the model should take into account the hydrodynamic interactions between particles in the chain, local field corrections, long range interactions and polydispersity. The effects of these corrections are considered in Ref. 9for MRFs. Nevertheless it is worth noting that the chain angle observed by us from a bubbling MFB 共Fig. 5兲is close to the value predicted by the simplest approach. On the other hand, Eq. 共5兲predicts that the critical stability angle must be independent of gas velocity, particle size, and magnetic field strength. Accordingly, we observe that the orientation of the chains is about the same in pictures obtained for magnetite particles of different particle size, fluidized at different gas velocities and for fields of different strength. A thorough experimental work remains to be performed to confirm these observations. This will be the scope of an independent work in which a noninvasive visualization technique should be also devised. Moreover, the effects of cooperative behavior of the chains or boundary conditions remain also to be addressed. Let us now calculate the dependence of chain length on field strength, which will serve us to have a deeper insight into the jamming transition. According to experimental observations gas velocity fluctuations in fluidized beds are typically of the order of gas velocity 共 ␦ vg⯝vg兲and may extend to macroscopic sizes.17,18 In our MFB chain model the strain on the chain will be thus assumed to be caused by a fluctuation of the gas velocity ␦ vgextending over the whole length of the chain. We label particles in the chain from 1 to N. Following the chain model, the shear force acting on the particle at the end of the chain will be a sum over contributions from all particles further out in the chain fs ⯝兺i=1 i=N3 ␲ ␩ dpi⌬vg, where ⌬vgis the velocity gradient between two contacting particles in the chain 共N⌬vg= ␦ vg =vg兲, thus fs⯝3 ␲ ␩ dpvgN/2. Using Eqs. 共3兲and 共4兲, the chain length Lc=Ndpcan be thus related to the magnetic field strength Hand the gas velocity vg, Lc⬇dp 2 4 ␩ vg ␤ 2 ␮ 0H2f⌫冑2f 储 2f 储 +2f⌫+f⬜ .共6兲 It must be taken into account that near jamming, chains are crowded and there will be a strong interaction between the chains which is not accounted for by this simple model of an isolated chain. Furthermore the magnetic field strength within the bulk of the bed Hbwill be decreased by demagnetization. Thus, it will be Hb=H− ␬ M, where ␬ is the demagnetization factor 共 ␬ =0.5 for an infinite cylinder magnetized transversely to its axis兲and Mis the bulk magnetization of the bed. In the case of a linearly magnetizable material Hb=H/共1+ ␬ ␹ 兲, where ␹ =M/Hb. Using data on the particle volume fraction at jamming ␾ = ␾ c共Fig. 10兲 and the Bruggeman equation 关Eq. 共1兲兴, it is obtained that ␹ shows little variation around ␹ ⯝2. Thus, the field strength is decreased within the bulk of the bed by a factor of 0.5 approximately. In Fig. 7we plot vc=vc共0兲+vc共H兲, where vc共0兲is the measured gas velocity at jamming in the absence of field and vc共H兲is calculated from Eq. 共6兲, where Lchas been taken as a fitting parameter. The values of the force coefficients used are the values reported for ␣ =10 in the literature16 and demagnetization has been considered by using Hbinstead of H in Eq. 共6兲. The same value of Lc共Lc⯝0.35 cm兲gives a good fit to the data for the three particle sizes tested. It must be FIG. 9. Critical stability angle of a chain from Eq. 共5兲, and using reported values in the literature for the force coefficients, as a function of the relative permeability of the particles ␣ = ␮ p/ ␮ f. The solid line is a logarithmic fit to the data. 054903-5 Valverde et al. J. Appl. Phys. 108, 054903 共2010兲 recognized that a single-chain model is too simple to yield a fully predictive equation even though it is remarkable that a single Lcvalue serves to fit experimental data independently of particle size. In particular, the interaction between chains near the jamming transition should be considered in a more elaborated analysis. Albeit, the single-chain model proposed in this work is a first step enabling us to rationalize the scaling of the gas velocity at jamming with particle size and field strength observed experimentally. VI. CONCLUSIONS In conclusion, we have studied the jamming transition of fluidized beds of magnetic particles as affected by an externally applied cross-flow magnetic field. In close analogy with MRFs, the dynamics of MFBs in the cross-flow configuration is seen to be mainly ruled by the balance between the attractive magnetostatic force between the magnetized particles and the shear force due to the fluidizing gas. As the strength of the magnetic field His increased, or the gas velocity is decreased, particles tend to chain due to the progressively growing prevalence of the magnetostatic attractive force on the drag force produced on the chains by the gas flow. It has been observed that particle chains are oriented with respect to the applied field at a preferential angle, which according to a chain model based on the balance between magnetostatic and shear forces only depends on the relative magnetic permeability of the particles. Contrarily to the predicted result from stability analysis, it is observed that the MFB may be stabilized by a cross-flow magnetic field. The growth of particle chains as the gas velocity is decreased drives the system to a jamming transition at a critical gas velocity. Experimental data on the gas velocity at jamming are shown to be in accordance with the prediction by a single-chain model when the length of particle chains reaches a critical size which is independent of particle size dp and showing that the gas velocity at jamming scales proportionally to dp 2H2. ACKNOWLEDGMENTS This work was supported by Spanish Government Agency Ministerio de Ciencia y Tecnologia 共Contract No. FIS2006-03645兲and Junta de Andalucia 共Contract No. FQM 421兲. 1R. E. Rosensweig, Ferrohydrodynamics 共Dover, New York, 1997兲. 2D. K. Potter and A. Stephenson, Geophys. Res. Lett. 15, 1097 共1988兲. 3Y. Lalatonne, J. Richardi, and M. P. Pileni, Nature Mater. 3,121共2004兲. 4P. N. Segrè, E. Herbolzheimer, and P. M. Chaikin, Phys. Rev. Lett. 79, 2574 共1997兲. 5J. M. Valverde, M. A. S. Quintanilla, and A. Castellanos, Phys. Rev. Lett. 92, 258303 共2004兲. 6B. Herzhaft and E. Guazelli, J. Fluid Mech. 384, 133 共1999兲. 7J. Y. Hristov, Powder Technol. 87,59共1996兲. 8T. B. Jones, Electromechanics of Particles 共Cambridge University Press, Cambridge, 1995兲. 9J. E. Martin and R. A. Anderson, J. Chem. Phys. 104, 4814 共1996兲. 10K. Karkkainen, A. Sihvola, and K. Nikoskinen, IEEE Trans. Geosci. Remote Sens. 39, 1013 共2001兲. 11C. P. Hunt, B. M. Moskowitz, and S. K. 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