- Email: [email protected]
deterministic dynamics
Physica B 294}295 (2001) 721}728
Diamagnetism and magnetic force: a new laboratory for granular materials and chaotic/deterministic dynamics J.S. Brooks*, J.A. Cothern Physics Department and NHMFL, Florida State University, 1800 E. Paul Dirac dr., Tallahassee, FL 32310, USA
Abstract We report studies on the dynamics of macroscopic particles in a low gravity `magnetic levitationa environment. Particle ensembles are held in a weak con"ning potential due to diamagnetic forces in a high-"eld resistive magnet. In such a case, the kinetic energy is not zero, and assemblies of particles undergo ergodic processes to "nd the lowest con"gurational ground state. Examples of such processes, as well as consideration of chaotic versus deterministic motion for single-particle orbits, will be presented. 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetic levitation; Graphite}epoxy composites; Chaotic dynamics; Granular materials
1. Introduction The use of high-"eld resistive and superconducting magnets to study various diamagnetic systems in a `levitateda condition is recently a subject of considerable interest. Following the report of Beaugnon and Tournier [1] of the levitation of high diamagnetic susceptibility materials, a body of literature in this area has developed which both explain the underlying mechanisms and report on applications of the e!ects of magnetic levitation [2}9]. We consider the application of the `low gravitya environment to an emerging area of science, namely granular matter [10}12]. Granular matter is represented by any collection of particles which typically range in size from 1 mm to 1 m, and where the interactions also scale down from gravi-
* Corresponding author. Fax: #1-850-644-5038. E-mail address: [email protected] (J.S. Brooks).
tational/frictional to electrostatic/magnetic to molecular/Brownian. Traditionally, these materials are studied in the presence of a gravitational "eld where the conditions are static (sand piles, silos, etc.), in #ows, or driven with oscillatory forces. Here, fundamental issues including force propagation [13], continuum hydrodynamic-treatments [14], and electrostatic interactions [15], respectively, are areas of current interest. In this paper we will "rst discuss the magnetic levitation environment, then examine a simple granular assembly, and "nally look at an example of single-particle dynamics.
2. The magnetic levitation environment For su$ciently high values B dB /dz, diamagX X netic materials will experience a repulsive force upwards along the axis of a vertically situated solenoid magnet (here cylindrical coordinates (z, r) are used where azimuthal symmetry is assumed). In
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 7 5 1 - 1
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Fig. 1. A phosphor}bronze cantilever is used to measure the vertical diamagnetic force on a water sample in a 19.5 T NMR magnet. (Schematic not to scale.) The maximum in BdB/dz appears at about 0.22 m above magnet center.
particular, for any material which is predominantly comprised of water, a "eld-gradient product of B dB /dz"1400 T/m will produce a diamagnetic X X force which equals the force of gravity. A simple expression for the axial balance condition is g" (/ ) BdB/dz where is the mass density, is the diamagnetic susceptibility, and g is the acceleration of gravity. In Fig. 1 we show the results of a simple measurement with a strain gauge/cantilever device which measures the diamagnetic force on a small water sample which can be moved along the axis of a solenoid magnet (in this case a 19.5 T NMR superconducting magnet). The results show that the maximum e!ective upwards acceleration is slightly less than that needed to levitate water. In Fig. 2 the maximum values of B dB /dz are X X shown for a selection of magnets at the National High Magnetic Field Laboratory (NHMFL). In addition to the axial force, radial forces that arise from the radial gradients are also necessary for equilibrium in magnetic levitation. We may visualize this condition by considering the speci"c total energy [4] E"B/ #gz (J/kg) of a speci"c object in a particular magnet bore. We have determined the local minima in E(z, r) for an amorphous graphite object where /"4.2;10\ kg\ in of the NHMFL 195 mm bore, 20 T resistive magnet. Values of the "eld B(z, r) in cylindrical coordinates from "nite element computations [16] were used.
Fig. 2. Maximum "eld-gradient products for di!erent magnets at the NHMFL.
In Fig. 3 we show E(z, r) for two cases, B(0, 0)"14 and 16 T. Fig. 4 shows the corresponding actual experimental cases for the object in the magnet. At the higher "eld both computation and experiment place the object at the bore wall, and at the lower "eld the object is freely suspended. The speci"c energy E for the 14 T con"guration is 3.505 J/kg near the center, and E rises to 3.520 J/kg near the edge. The inwards radial acceleration ranges from 0 to about 0.6 m/s over this same range. In
J.S. Brooks, J.A. Cothern / Physica B 294}295 (2001) 721}728
Fig. 3. Stability conditions for magnetic levitation of a diamagnetic object. Here a 20 g aquadag (disordered, solidi"ed graphite) block of volume 20 cm is used for example. Left and right panels: false grayscale contour plot of speci"c energy E versus z and r for 16 and 14 T (see text). Dark contours highlight regions of minima in E. As the "eld is decreased, the regions of minima move to lower z values, and eventually move from the bore tube wall to the radial center of the magnet.
Fig. 4. Position of aquadag block for 16 T (left panel } on the bore tube wall) and 14 T (right panel } in dynamic levitation). View is down the bore of the 195 mm magnet from above.
Fig. 5 the computed inwards acceleration is shown for the 14 T case. For small displacements from the center, there is an approximate Hooke's law relation with radius. At larger displacements near the bore wall the radial acceleration becomes more complicated. (The small oscillations in a(r) arise from the computation due to the "nite grain in B versus r.) There are additional unique advantages of this `magnetic bottlea con"guration. By tuning the
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Fig. 5. E!ective radial acceleration (inwards) for the aquadag block at 14 T.
magnetic "eld, the radial restoring forces may be made arbitrarily small, i.e. they can be made smaller than any other inter-particle forces which may be of interest. Since the condition of stability of an individual particle depends on the ratio /, for a given magnetic "eld intensity, the vertical position of a particle will also depend on its speci"c value of /. Hence an assembly of non-identical particles such as the graphite}epoxy composites described in this paper will have a vertical distribution, and the assembly will be three-dimensional. In dynamical studies, particles can be introduced into the center potential by reducing the "eld; hence, gaining kinetic energy. For particles with a wide distribution of / values, by starting at high "elds and then continuously lowering the "eld, particles with sequentially increasing values of / will "rst enter, then exit the granular assembly.
3. Composite materials In the study of granular materials, particle size, shape, density, and the nature of the surface are important. To study these materials under magnetic levitation, there are additional factors to consider. (1) The materials must be diamagnetic, but
724
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also, it is the ratio / (not the size) which determines how large BdB/dz must be to levitate a material. For water-based materials, seeds, and other organic objects, a 50 mm bore Bitter-type magnet is needed. Inorganic, denser substances like sand are too dense for even a 30 mm bore 33 T magnet (see Fig. 2.) Graphite will levitate in most high "eld resistive or superconducting magnets. We have found that composite materials based on graphite powders mixed with epoxy are the most useful, since they may be used in the 195 mm bore magnet due to the high diamagnetic susceptibility. (2) Single-crystalline materials present special issues. A single crystal which levitates in one orientation may fall in another. This is true for the graphite powders, which must be randomly oriented due to the anisotropic susceptibility of the graphite microcrystals. Bismuth and antimony can also be used to enhance the susceptibility of other materials, but due to crystalline anisotropy, powders or amorphous forms of the materials may be required. (3) Uniformity is a critical issue. For instance, the graphite}epoxy cylinders we have used in the `nucleusa study described below are not identical. This is due to small variations in the ratio / which inevitably occur when the composition is prepared, and the particles are shaped and cured. This leads to a three-dimensional array of particles. To make nearly identical particles on the scale where they will all be in equilibrium at the same point in the magnetic "eld gradient is a challenge. However, for a su$cient number of particles, one can use the magnetic levitation mechanism to carefully select and ensemble of particles that satisfy the required degree of uniformity. (4) Conducting materials, which are in a dynamic state, may have a "nite Lorentz force, which in#uences the motion.
4. Experimental The particles used in the work reported here were composed of equal parts of graphite powders and epoxy. The single particles were spherical (1.1 cm in diameter), with a mass of 1.2 g. The multiple particles were right cylinders of 2.6 mm diameter and 2.6 mm length, with a mass of 0.21 g.
Electrical measurements showed that the composites were above percolation threshold, with a conductivity of 9.25;10\ S/cm. The experiments were carried out in the 195 mm bore, 20 T magnet. The video data (30 fps) were captured to JPEG frames and the particle positions were digitized with intensity analysis software. Each JPEG was loaded into the MATLAB environment as 640;480;3 matrix and reduced to a 640;480 intensity plot. For each image, a mask was created to remove extraneous information from the image. For the one body problem, a center of light calculation was then performed on the masked image to determine the X}Y position of the particle for each frame of the movie. The resulting trajectory data were then compared with the original video for accuracy. For the `nucleusa, the center of light technique was used to determine the center. The boundary of the `nucleusa was then determined using edge detection software in 12 segments (i.e. a dodecagon con"guration). To date all studies have been done with the bore open to the air. Complementary studies in vacuum are planned in the near future.
5. Granular matter in levitation } an example In Fig. 6, we present a study of the behavior of an assembly of 20 particles, on average, where the "eld is slowly lowered and particles continually enter and leave the approximately constant mass of the `nucleusa. This example represents a granular material assembly under the conditions where: (1) there is a solid, self organized state at the center and, (2) a kinetic state at the surface where particles are in motion and are sampling possible stable con"gurations at the surface. A schematic of the assembly is shown in Fig. 7. To explore methods to characterize this unique state of granular material, we have made a 12 section edge determination of the circumference of the nucleus and have computed the area A, the perimeter P, and the ratio P/A to that of a perfect dodecagon versus time. For a dodecagon and a circle, P/A"12.8615 and 4, respectively. The time dependence of this ratio (Fig. 8) has no discernible time correlation, and hence the #uctuations in P/A are random. The fact
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Fig. 6. Time sequence of cylindrical graphite}epoxy particles in magnetic levitation. The assembly is rotating ccw at 1 Hz, and particles are entering from the left.
6. Single-particle dynamics } an example
Fig. 7. Schematic representation of granular assembly. The core is solid, but the surface particles #uctuate in position. As the "eld is reduced, particles with lower / values enter the assembly "rst.
that P/A #uctuates temporally is a measure of the #uctuations in the geometry of the structure, a concept somewhat akin to #uctuations in the fractal dimension of a perimeter.
Since we have previously reported dynamical studies of single-particle trajectories [5,17], we will here only consider one example to review the most important concepts and "ndings. Here a single particle of 1.1 cm diameter is introduced into the stable levitation zone of the magnet with a "nite velocity. Of the many trajectory histories we have studied [5,17], this speci"c example shows the most complex behavior. The magnetic "eld is held "xed in this case. We compute the position, velocity, acceleration, and Fourier transforms of the oscillatory motion. The coordinate positions and total mechanical energy (in the X}Y plane) are shown in Fig. 9. We may consider such trajectories from the point of view of deterministic versus chaotic motion. To explore this, we have employed elementary tests for
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Fig. 8. Geometrical #uctuations in particle nucleus assembly versus time. Letters correspond to frames in Fig. 6.
chaotic behavior [18] to the data. Our analysis is given in Fig. 10. We "nd the PoincareH section Y"X, and then plot the successive radii R "(X#>) L L L in terms of the "rst return plot R versus R . The L> L Fourier transform of the coordinates is also computed. The "rst return plot gives an indication of the degree of deterministic motion, which for this particular trajectory, is relatively little. The "rst return plot is nearly random. There are several Fourier components in the oscillation spectrum, which appear over the lifetime of the trajectory. Returning to Fig. 9, the total mechanical energy of the single-particle system was computed by assuming the simple two-dimensional harmonic oscillator form 2E/m"v#cr. Here c is a constant which involves the ratio of the e!ective spring constant to the mass, under the assumption of a Hooke's law restoring force in the regime shown in Fig. 5. In Fig. 9, the X and Y coordinate oscillations are those which appear in the Fourier transform (0.3, 0.42, and 0.6 Hz). For a damped harmonic oscillator, the total energy should decrease monotonically with time, with no oscillatory behavior at the coordinate frequency. However, a Fourier component in the total energy does appear at 0.2 Hz, which is distinct from the coordinate spectrum. This implies that the oscillatory motion is amplitude modulated (AM) with a frequency of 0.2 Hz in the "rst half of its history.
Fig. 9. (a) X and (b) Y coordinates versus time. (c) Total mechanical energy versus time.
Fig. 10. First return plot (large panel) for the PoincareH section of the trajectory (open circles in lower right inset). Upper inset: Fourier transform of X coordinate.
J.S. Brooks, J.A. Cothern / Physica B 294}295 (2001) 721}728
The AM e!ect described above leads to the conclusion that additional energy is being transmitted to the oscillator with a characteristic period. The most likely cause is convective air currents in the magnet bore tube. They can arise from the temperature gradient (of order 103C over a meter) since the magnets are water-cooled by uniaxial #ow down from the top of the magnet. Estimates of the onset of convection [19] for the parameters of the NHMFL Bitter-type magnets show that for bore tubes greater than about 25 mm, convection will occur due to the temperature gradient. It is interesting to note that the convection e!ects can be separated in the analysis, since they enter only as the amplitude modulation of the otherwise harmonic motion of the orbits. Hence, the trajectory dynamics may be used as a probe of the aerodynamics in the magnet bore.
727
modulation of the magnetic "eld, by acoustic excitation, or by collisions. Finally, there are several sources of interactions between particles. At the macroscopic level there are `hard spherea interactions, induced magnetic moment interactions, and hydrodynamic e!ects in the presence of air. For su$ciently small kinetic energies and for su$ciently weak con"ning radial potentials, electrostatic e!ects, adhesion, and adsorbed water will become accessible to measurement.
Acknowledgements This work has been supported in part by NHMFL/NSF/IHRP 5033. The NHMFL is supported by a contractual agreement between the NSF and the State of Florida under contract NSFDMR-95-27035
7. Discussion and summary References We have presented several examples of the dynamics and con"gurations of granular materials, taken either as isolated particles, or in interacting assemblies, under unique conditions where the gravitational force is balanced by a diamagnetic force. The diamagnetic force and details of the con"ning potential can be tuned by either "eld intensity, and/or by the ratio / of composite materials. This unique condition of granular matter presents new challenges for both recording the physical state of such assemblies, and reducing the observed dynamics and con"gurations to an employable mathematical representation. Future work in this area will involve certain considerations. Clearly, performing the measurements in vacuum will be essential for certain experiments. Another consideration is the conducting nature of the graphite-epoxy composites. In all cases shown here they were conducting, and induced currents must be present as they move in the magnetic "eld gradient. We have observed highly unusual dynamics and rotational motion of such objects that cannot be explained by simple air convection models. Many granular materials studies involve shaking or vibration. This can be accomplished in the present case by one of several methods such as
[1] E. Beaugnon, R. Tournier, Nature 349 (1991) 470. [2] A.K. Geim, Physics Today 51 (1998) 36. [3] A.K. Geim, M.D. Simon, M.I. Boamfa, L.O. He#inger, Nature 200 (1999) 323. [4] M.D. Simon, A.K. Geim, J. Appl. Phys. 87 (2000) 6200. [5] J.S. Brooks, J.A. Reavis, R.A. Medwood, T.F. Stalcup, M.W. Meisel, E. Steinberg, L. Arnowitz, C. Stover, J.A.A.J. Perenboom, J. Magn. Magn. Mater. 87 (2000) 6194. [6] J. Nakagawa, S.-I. Horinouchi, N. Hirota, Y. Ikezoe, K. Kitazawa, Trans. Inst. Elect. Eng. Japan 119-A (1999) 1142. [7] M. Motokawa, I. Mogi, M. Tagami, M. Hamai, K. Watanabe, S. Awaji, Physica B 256}258 (1998) 618. [8] M. Tagami, M. Hamai, I. Mogi, K. Watanabe, M. Motokawa, J. Crystal Growth 203 (1999) 594. [9] N. Kitamura, M. Makihara, M. Hamai, T. Sato, I. Mogi, S. Awaji, K. Watanabe, M. Motokawa, Jpn. J. Appl. Phys. 39 (2000) L324. [10] P.G.d. Gennes, Rev. Mod. Phys. 71 (1999) S374. [11] L.P. Kadano!, Rev. Mod. Phys. 71 (1999) 435. [12] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68 (1996) 1259. [13] C.-h. Liu, S.R. Nagel, D.A. Schecter, S.N. Coppersmith, S. Majumdar, O. Narayan, T.A. Witten, Science 269 (1995) 513. [14] C. Bizon, M.D. Shattuck, J.B. Swift, H.L. Swinney, Phys. Rev. E 60 (1999) 4341.
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[15] I.S. Aranson, D. Blair, V.A. Kalatsky, G.W. Crabtree, W.-K. Kwok, V.M. Vinokur, U. Welp, Phys. Rev. Lett. 84 (2000) 3306. [16] Y.M. Eyssa (http:www.magnet.fsu.edu, 2000). [17] J.S. Brooks, J.A. Cothern, J. Magn. Magn. Mater., in press.
[18] G.P. Williams, Chaos Theory Tamed, National Academy Press, Washington DC, 1997. [19] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon Press, London, 1959.
Diamagnetism and magnetic force: a new laboratory for granular materials and chaotic/deterministic dynamics J.S. Brooks*, J.A. Cothern Physics Department and NHMFL, Florida State University, 1800 E. Paul Dirac dr., Tallahassee, FL 32310, USA
Abstract We report studies on the dynamics of macroscopic particles in a low gravity `magnetic levitationa environment. Particle ensembles are held in a weak con"ning potential due to diamagnetic forces in a high-"eld resistive magnet. In such a case, the kinetic energy is not zero, and assemblies of particles undergo ergodic processes to "nd the lowest con"gurational ground state. Examples of such processes, as well as consideration of chaotic versus deterministic motion for single-particle orbits, will be presented. 2001 Elsevier Science B.V. All rights reserved. Keywords: Magnetic levitation; Graphite}epoxy composites; Chaotic dynamics; Granular materials
1. Introduction The use of high-"eld resistive and superconducting magnets to study various diamagnetic systems in a `levitateda condition is recently a subject of considerable interest. Following the report of Beaugnon and Tournier [1] of the levitation of high diamagnetic susceptibility materials, a body of literature in this area has developed which both explain the underlying mechanisms and report on applications of the e!ects of magnetic levitation [2}9]. We consider the application of the `low gravitya environment to an emerging area of science, namely granular matter [10}12]. Granular matter is represented by any collection of particles which typically range in size from 1 mm to 1 m, and where the interactions also scale down from gravi-
* Corresponding author. Fax: #1-850-644-5038. E-mail address: [email protected] (J.S. Brooks).
tational/frictional to electrostatic/magnetic to molecular/Brownian. Traditionally, these materials are studied in the presence of a gravitational "eld where the conditions are static (sand piles, silos, etc.), in #ows, or driven with oscillatory forces. Here, fundamental issues including force propagation [13], continuum hydrodynamic-treatments [14], and electrostatic interactions [15], respectively, are areas of current interest. In this paper we will "rst discuss the magnetic levitation environment, then examine a simple granular assembly, and "nally look at an example of single-particle dynamics.
2. The magnetic levitation environment For su$ciently high values B dB /dz, diamagX X netic materials will experience a repulsive force upwards along the axis of a vertically situated solenoid magnet (here cylindrical coordinates (z, r) are used where azimuthal symmetry is assumed). In
0921-4526/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 7 5 1 - 1
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Fig. 1. A phosphor}bronze cantilever is used to measure the vertical diamagnetic force on a water sample in a 19.5 T NMR magnet. (Schematic not to scale.) The maximum in BdB/dz appears at about 0.22 m above magnet center.
particular, for any material which is predominantly comprised of water, a "eld-gradient product of B dB /dz"1400 T/m will produce a diamagnetic X X force which equals the force of gravity. A simple expression for the axial balance condition is g" (/ ) BdB/dz where is the mass density, is the diamagnetic susceptibility, and g is the acceleration of gravity. In Fig. 1 we show the results of a simple measurement with a strain gauge/cantilever device which measures the diamagnetic force on a small water sample which can be moved along the axis of a solenoid magnet (in this case a 19.5 T NMR superconducting magnet). The results show that the maximum e!ective upwards acceleration is slightly less than that needed to levitate water. In Fig. 2 the maximum values of B dB /dz are X X shown for a selection of magnets at the National High Magnetic Field Laboratory (NHMFL). In addition to the axial force, radial forces that arise from the radial gradients are also necessary for equilibrium in magnetic levitation. We may visualize this condition by considering the speci"c total energy [4] E"B/ #gz (J/kg) of a speci"c object in a particular magnet bore. We have determined the local minima in E(z, r) for an amorphous graphite object where /"4.2;10\ kg\ in of the NHMFL 195 mm bore, 20 T resistive magnet. Values of the "eld B(z, r) in cylindrical coordinates from "nite element computations [16] were used.
Fig. 2. Maximum "eld-gradient products for di!erent magnets at the NHMFL.
In Fig. 3 we show E(z, r) for two cases, B(0, 0)"14 and 16 T. Fig. 4 shows the corresponding actual experimental cases for the object in the magnet. At the higher "eld both computation and experiment place the object at the bore wall, and at the lower "eld the object is freely suspended. The speci"c energy E for the 14 T con"guration is 3.505 J/kg near the center, and E rises to 3.520 J/kg near the edge. The inwards radial acceleration ranges from 0 to about 0.6 m/s over this same range. In
J.S. Brooks, J.A. Cothern / Physica B 294}295 (2001) 721}728
Fig. 3. Stability conditions for magnetic levitation of a diamagnetic object. Here a 20 g aquadag (disordered, solidi"ed graphite) block of volume 20 cm is used for example. Left and right panels: false grayscale contour plot of speci"c energy E versus z and r for 16 and 14 T (see text). Dark contours highlight regions of minima in E. As the "eld is decreased, the regions of minima move to lower z values, and eventually move from the bore tube wall to the radial center of the magnet.
Fig. 4. Position of aquadag block for 16 T (left panel } on the bore tube wall) and 14 T (right panel } in dynamic levitation). View is down the bore of the 195 mm magnet from above.
Fig. 5 the computed inwards acceleration is shown for the 14 T case. For small displacements from the center, there is an approximate Hooke's law relation with radius. At larger displacements near the bore wall the radial acceleration becomes more complicated. (The small oscillations in a(r) arise from the computation due to the "nite grain in B versus r.) There are additional unique advantages of this `magnetic bottlea con"guration. By tuning the
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Fig. 5. E!ective radial acceleration (inwards) for the aquadag block at 14 T.
magnetic "eld, the radial restoring forces may be made arbitrarily small, i.e. they can be made smaller than any other inter-particle forces which may be of interest. Since the condition of stability of an individual particle depends on the ratio /, for a given magnetic "eld intensity, the vertical position of a particle will also depend on its speci"c value of /. Hence an assembly of non-identical particles such as the graphite}epoxy composites described in this paper will have a vertical distribution, and the assembly will be three-dimensional. In dynamical studies, particles can be introduced into the center potential by reducing the "eld; hence, gaining kinetic energy. For particles with a wide distribution of / values, by starting at high "elds and then continuously lowering the "eld, particles with sequentially increasing values of / will "rst enter, then exit the granular assembly.
3. Composite materials In the study of granular materials, particle size, shape, density, and the nature of the surface are important. To study these materials under magnetic levitation, there are additional factors to consider. (1) The materials must be diamagnetic, but
724
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also, it is the ratio / (not the size) which determines how large BdB/dz must be to levitate a material. For water-based materials, seeds, and other organic objects, a 50 mm bore Bitter-type magnet is needed. Inorganic, denser substances like sand are too dense for even a 30 mm bore 33 T magnet (see Fig. 2.) Graphite will levitate in most high "eld resistive or superconducting magnets. We have found that composite materials based on graphite powders mixed with epoxy are the most useful, since they may be used in the 195 mm bore magnet due to the high diamagnetic susceptibility. (2) Single-crystalline materials present special issues. A single crystal which levitates in one orientation may fall in another. This is true for the graphite powders, which must be randomly oriented due to the anisotropic susceptibility of the graphite microcrystals. Bismuth and antimony can also be used to enhance the susceptibility of other materials, but due to crystalline anisotropy, powders or amorphous forms of the materials may be required. (3) Uniformity is a critical issue. For instance, the graphite}epoxy cylinders we have used in the `nucleusa study described below are not identical. This is due to small variations in the ratio / which inevitably occur when the composition is prepared, and the particles are shaped and cured. This leads to a three-dimensional array of particles. To make nearly identical particles on the scale where they will all be in equilibrium at the same point in the magnetic "eld gradient is a challenge. However, for a su$cient number of particles, one can use the magnetic levitation mechanism to carefully select and ensemble of particles that satisfy the required degree of uniformity. (4) Conducting materials, which are in a dynamic state, may have a "nite Lorentz force, which in#uences the motion.
4. Experimental The particles used in the work reported here were composed of equal parts of graphite powders and epoxy. The single particles were spherical (1.1 cm in diameter), with a mass of 1.2 g. The multiple particles were right cylinders of 2.6 mm diameter and 2.6 mm length, with a mass of 0.21 g.
Electrical measurements showed that the composites were above percolation threshold, with a conductivity of 9.25;10\ S/cm. The experiments were carried out in the 195 mm bore, 20 T magnet. The video data (30 fps) were captured to JPEG frames and the particle positions were digitized with intensity analysis software. Each JPEG was loaded into the MATLAB environment as 640;480;3 matrix and reduced to a 640;480 intensity plot. For each image, a mask was created to remove extraneous information from the image. For the one body problem, a center of light calculation was then performed on the masked image to determine the X}Y position of the particle for each frame of the movie. The resulting trajectory data were then compared with the original video for accuracy. For the `nucleusa, the center of light technique was used to determine the center. The boundary of the `nucleusa was then determined using edge detection software in 12 segments (i.e. a dodecagon con"guration). To date all studies have been done with the bore open to the air. Complementary studies in vacuum are planned in the near future.
5. Granular matter in levitation } an example In Fig. 6, we present a study of the behavior of an assembly of 20 particles, on average, where the "eld is slowly lowered and particles continually enter and leave the approximately constant mass of the `nucleusa. This example represents a granular material assembly under the conditions where: (1) there is a solid, self organized state at the center and, (2) a kinetic state at the surface where particles are in motion and are sampling possible stable con"gurations at the surface. A schematic of the assembly is shown in Fig. 7. To explore methods to characterize this unique state of granular material, we have made a 12 section edge determination of the circumference of the nucleus and have computed the area A, the perimeter P, and the ratio P/A to that of a perfect dodecagon versus time. For a dodecagon and a circle, P/A"12.8615 and 4, respectively. The time dependence of this ratio (Fig. 8) has no discernible time correlation, and hence the #uctuations in P/A are random. The fact
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Fig. 6. Time sequence of cylindrical graphite}epoxy particles in magnetic levitation. The assembly is rotating ccw at 1 Hz, and particles are entering from the left.
6. Single-particle dynamics } an example
Fig. 7. Schematic representation of granular assembly. The core is solid, but the surface particles #uctuate in position. As the "eld is reduced, particles with lower / values enter the assembly "rst.
that P/A #uctuates temporally is a measure of the #uctuations in the geometry of the structure, a concept somewhat akin to #uctuations in the fractal dimension of a perimeter.
Since we have previously reported dynamical studies of single-particle trajectories [5,17], we will here only consider one example to review the most important concepts and "ndings. Here a single particle of 1.1 cm diameter is introduced into the stable levitation zone of the magnet with a "nite velocity. Of the many trajectory histories we have studied [5,17], this speci"c example shows the most complex behavior. The magnetic "eld is held "xed in this case. We compute the position, velocity, acceleration, and Fourier transforms of the oscillatory motion. The coordinate positions and total mechanical energy (in the X}Y plane) are shown in Fig. 9. We may consider such trajectories from the point of view of deterministic versus chaotic motion. To explore this, we have employed elementary tests for
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Fig. 8. Geometrical #uctuations in particle nucleus assembly versus time. Letters correspond to frames in Fig. 6.
chaotic behavior [18] to the data. Our analysis is given in Fig. 10. We "nd the PoincareH section Y"X, and then plot the successive radii R "(X#>) L L L in terms of the "rst return plot R versus R . The L> L Fourier transform of the coordinates is also computed. The "rst return plot gives an indication of the degree of deterministic motion, which for this particular trajectory, is relatively little. The "rst return plot is nearly random. There are several Fourier components in the oscillation spectrum, which appear over the lifetime of the trajectory. Returning to Fig. 9, the total mechanical energy of the single-particle system was computed by assuming the simple two-dimensional harmonic oscillator form 2E/m"v#cr. Here c is a constant which involves the ratio of the e!ective spring constant to the mass, under the assumption of a Hooke's law restoring force in the regime shown in Fig. 5. In Fig. 9, the X and Y coordinate oscillations are those which appear in the Fourier transform (0.3, 0.42, and 0.6 Hz). For a damped harmonic oscillator, the total energy should decrease monotonically with time, with no oscillatory behavior at the coordinate frequency. However, a Fourier component in the total energy does appear at 0.2 Hz, which is distinct from the coordinate spectrum. This implies that the oscillatory motion is amplitude modulated (AM) with a frequency of 0.2 Hz in the "rst half of its history.
Fig. 9. (a) X and (b) Y coordinates versus time. (c) Total mechanical energy versus time.
Fig. 10. First return plot (large panel) for the PoincareH section of the trajectory (open circles in lower right inset). Upper inset: Fourier transform of X coordinate.
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The AM e!ect described above leads to the conclusion that additional energy is being transmitted to the oscillator with a characteristic period. The most likely cause is convective air currents in the magnet bore tube. They can arise from the temperature gradient (of order 103C over a meter) since the magnets are water-cooled by uniaxial #ow down from the top of the magnet. Estimates of the onset of convection [19] for the parameters of the NHMFL Bitter-type magnets show that for bore tubes greater than about 25 mm, convection will occur due to the temperature gradient. It is interesting to note that the convection e!ects can be separated in the analysis, since they enter only as the amplitude modulation of the otherwise harmonic motion of the orbits. Hence, the trajectory dynamics may be used as a probe of the aerodynamics in the magnet bore.
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modulation of the magnetic "eld, by acoustic excitation, or by collisions. Finally, there are several sources of interactions between particles. At the macroscopic level there are `hard spherea interactions, induced magnetic moment interactions, and hydrodynamic e!ects in the presence of air. For su$ciently small kinetic energies and for su$ciently weak con"ning radial potentials, electrostatic e!ects, adhesion, and adsorbed water will become accessible to measurement.
Acknowledgements This work has been supported in part by NHMFL/NSF/IHRP 5033. The NHMFL is supported by a contractual agreement between the NSF and the State of Florida under contract NSFDMR-95-27035
7. Discussion and summary References We have presented several examples of the dynamics and con"gurations of granular materials, taken either as isolated particles, or in interacting assemblies, under unique conditions where the gravitational force is balanced by a diamagnetic force. The diamagnetic force and details of the con"ning potential can be tuned by either "eld intensity, and/or by the ratio / of composite materials. This unique condition of granular matter presents new challenges for both recording the physical state of such assemblies, and reducing the observed dynamics and con"gurations to an employable mathematical representation. Future work in this area will involve certain considerations. Clearly, performing the measurements in vacuum will be essential for certain experiments. Another consideration is the conducting nature of the graphite-epoxy composites. In all cases shown here they were conducting, and induced currents must be present as they move in the magnetic "eld gradient. We have observed highly unusual dynamics and rotational motion of such objects that cannot be explained by simple air convection models. Many granular materials studies involve shaking or vibration. This can be accomplished in the present case by one of several methods such as
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