938-2947(1994) R. Y Chiao and A M. Steinberg, Tunneling Times and Superluminal- bM. W. Mitchell andR.Y negative group delays in ity, in Progress in Optics, edited by E. Wolf(Elsevier, Amsterdam, simple bandpass amplifie 997),Vol.37,pp.347-405 17A. M. Steinberg and R. Y. Chiao, "Dispersionless, highly superluminal >See p 943 of R. P. Feynman, "A Relativistic Cut-Off for Classical Elec propagation in a medium with a gain doublet, "Phys. Rev. A 49, 2071 urodynamics, Phys. Rev. 74, 939-946(1948) 075(1994) P. A M. Dirac, The Principles of quantum Mechanics(Clarendon, Ox R. Y. Chiao, ""Population Inversion and Superluminality, " in Amazing ford, 1958), 4th ed, Sec. 4 Forces in complex fluids Bruce J. Ackerson) and Anitra N. Now Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078-3072 (Received 25 August 2000; accepted 18 December 2000) [DO:10.1119/11351151 . PROBLEM IL SOLUTION Ferrofluids are stable suspensions of magnetic particle Magnetic body forces, surface tension, viscous drag, and gravity combine to produce the peculiar behavior observed having linear dimension on the order of 10 nm. Due to vig- here. We focus on the magnetic body forces as the primary orous Brownian motion the magnetic particles assume ran- explanation for the question posed above. Figure 3 shows a dom orientations rendering the suspension as a whole para- side view of one of the"cones"of ferrofluid which form the magnetic. These complex fluids show a variety of phe- two-dimensional crystalline array beneath the magnet. These teachers alike. Because these fluids are used in a variety of face with the symmetry axis parallel to the applied field hey are commercially available immiscible fluid deform into ellipsoids and align parallel to Figure I shows the experimental apparatus for viewing the direction of a uniform external magnetic field 6 solutions and recording the response of a ferrofluid film trapped at an have long existed for the magnetic (electric)field of an el air-water interface. Figure 2 shows recorded images for a lipsoid of permeability u(dielectric constant e) subjected to drop(-40 ul)of mineral-oil-based ferrofluid introduced to a uniform external field in a surrounding medium of perme- the surface of clean, filtered, de-ionized (18 MOD)water. The ability uo(dielectric constant Eo).When the symmetry axis hydrophobic ferrofluid spreads uniformly over the surface of of the ellipse is aligned parallel to the external field direction water contained in a Petri dish. We gently stir the surface to the field inside the ellipse is uniform and parallel to the ap- emulsify the film, creating a collection of dark flat circular plied field(at large distances). Thus, to first order the"ellip- drops of ferrofluid as recorded in Fig. 2(a). Figure 2(b) soidal cones"behave like little magnets oriented parallel to shows the film 1 min after a cylindrical magnet having a the external field and so move along the water surface to the radius of I cm is introduced with the axis of symmetry ver- strongest field regions located directly beneath the cylindri tical and the lower end 3.3 cm above the ferrofluid film. The cal magnet. However, the magnetic field induced in each of ferrofluid film clears from directly beneath the magnet but these cones is aligned parallel with the neighboring cone moves radially inward at large distances, forming tear- fields; therefore, the cones repel one another in the plane of shaped drops with the clearer regions streaming outward. the interface just like parallel oriented permanent magnets. A The ferrofluid collects in a ring structure at a finite radius crystal lattice results. These results are qualitative, but in- (which is most dense at radius -1.0 cm) from the center of tuitive, given our experience playing with permanent mag he magnetic field symmetry axis. As the ferrofluid builds up, nets clumps or cone-shaped structures develop. As the cones How do we understand the quite different behavior of th grow, they become unstable and migrate one at a time into film? Rosensweig gives a general derivation of the body the central region. Figure 2(c)taken at 3 min shows the force f or force per unit volume, which reduces for ferrofluid clumping in the ring-shaped structure with one cone at two suspensions to o'clock escaping to the central region. Finally in Fig. 2(d) taken 21 min after introducing the magnet, a regular"crys f=Ho(M-V)H-AoMVH, talline"array of well-separated ferrofluid cones has formed. where uo is the vacuum permeability, H is the magnetic field Yet there remains a ferrofluid film ring surrounding this crys- strength, and M is the magnetization in the film volume el talline structure ement. This functional form suggests the Kelvin force den- How is it possible that the ferrofluid is both attracted to sity on an isolated body, except that the local field H re- (cones)and repelled from(film) the region directly below the places the applied field Ho. Intuitively, we understand this body force to be like the force acting on a magnetic dipole Am J. Phys. 69(5), May 2001 http://ojps.aiporg/ajp/ c 2001 American Association of Physics Teachers15E. L. Bolda, J. C. Garrison, and R. Y. Chiao, ‘‘Optical pulse propagation at negative group velocities due to a nearby gain line,’’ Phys. Rev. A 49, 2938–2947 ~1994!. 16M. W. Mitchell and R. Y. Chiao, ‘‘Causality and negative group delays in a simple bandpass amplifier,’’ Am. J. Phys. 68, 14–19 ~1998!. 17A. M. Steinberg and R. Y. Chiao, ‘‘Dispersionless, highly superluminal propagation in a medium with a gain doublet,’’ Phys. Rev. A 49, 2071– 2075 ~1994!. 18R. Y. Chiao, ‘‘Population Inversion and Superluminality,’’ in Amazing Light, edited by R. Y. Chiao ~Springer-Verlag, New York, 1996!, pp. 91–108. 19R. Y. Chiao and A. M. Steinberg, ‘‘Tunneling Times and Superluminal￾ity,’’ in Progress in Optics, edited by E. Wolf ~Elsevier, Amsterdam, 1997!, Vol. 37, pp. 347–405. 20See p. 943 of R. P. Feynman, ‘‘A Relativistic Cut-Off for Classical Elec￾trodynamics,’’ Phys. Rev. 74, 939–946 ~1948!. 21P. A. M. Dirac, The Principles of Quantum Mechanics ~Clarendon, Ox￾ford, 1958!, 4th ed., Sec. 4. Forces in complex fluids Bruce J. Ackersona) and Anitra N. Novyb) Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078-3072 ~Received 25 August 2000; accepted 18 December 2000! @DOI: 10.1119/1.1351151# I. PROBLEM Ferrofluids1 are stable suspensions of magnetic particles having linear dimension on the order of 10 nm. Due to vig￾orous Brownian motion the magnetic particles assume ran￾dom orientations rendering the suspension as a whole para￾magnetic. These complex fluids show a variety of phe￾nomena and instabilities that amuse and delight students and teachers alike.2 Because these fluids are used in a variety of applications including rotary seals, sensors, and actuators,3 they are commercially available.4 Figure 1 shows the experimental apparatus for viewing and recording the response of a ferrofluid film trapped at an air–water interface. Figure 2 shows recorded images for a drop ~;40 ml! of mineral-oil-based ferrofluid5 introduced to the surface of clean, filtered, de-ionized ~18 MV! water. The hydrophobic ferrofluid spreads uniformly over the surface of water contained in a Petri dish. We gently stir the surface to emulsify the film, creating a collection of dark flat circular drops of ferrofluid as recorded in Fig. 2~a!. Figure 2~b! shows the film 1 min after a cylindrical magnet having a radius of 1 cm is introduced with the axis of symmetry ver￾tical and the lower end 3.3 cm above the ferrofluid film. The ferrofluid film clears from directly beneath the magnet but moves radially inward at large distances, forming tear￾shaped drops with the clearer regions streaming outward. The ferrofluid collects in a ring structure at a finite radius ~which is most dense at radius ;1.0 cm! from the center of the magnetic field symmetry axis. As the ferrofluid builds up, clumps or cone-shaped structures develop. As the cones grow, they become unstable and migrate one at a time into the central region. Figure 2~c! taken at 31 4 min shows the clumping in the ring-shaped structure with one cone at two o’clock escaping to the central region. Finally in Fig. 2~d!, taken 21 min after introducing the magnet, a regular ‘‘crys￾talline’’ array of well-separated ferrofluid cones has formed. Yet there remains a ferrofluid film ring surrounding this crys￾talline structure. How is it possible that the ferrofluid is both attracted to ~cones! and repelled from ~film! the region directly below the cylindrical magnet? II. SOLUTION Magnetic body forces, surface tension, viscous drag, and gravity combine to produce the peculiar behavior observed here. We focus on the magnetic body forces as the primary explanation for the question posed above. Figure 3 shows a side view of one of the ‘‘cones’’ of ferrofluid which form the two-dimensional crystalline array beneath the magnet. These clumps have a nearly ellipsoidal shape above the water sur￾face with the symmetry axis parallel to the applied field. Other studies show that ferrofluid droplets submerged in an immiscible fluid deform into ellipsoids and align parallel to the direction of a uniform external magnetic field.6 Solutions have long existed for the magnetic ~electric! field of an el￾lipsoid of permeability m ~dielectric constant e! subjected to a uniform external field in a surrounding medium of perme￾ability m0 ~dielectric constant e 0!. 7 When the symmetry axis of the ellipse is aligned parallel to the external field direction, the field inside the ellipse is uniform and parallel to the ap￾plied field ~at large distances!. Thus, to first order the ‘‘ellip￾soidal cones’’ behave like little magnets oriented parallel to the external field and so move along the water surface to the strongest field regions located directly beneath the cylindri￾cal magnet. However, the magnetic field induced in each of these cones is aligned parallel with the neighboring cone fields; therefore, the cones repel one another in the plane of the interface just like parallel oriented permanent magnets. A crystal lattice results.8 These results are qualitative, but in￾tuitive, given our experience playing with permanent mag￾nets. How do we understand the quite different behavior of the film? Rosensweig1 gives a general derivation of the body force f or force per unit volume, which reduces for ferrofluid suspensions to f5m0~M"¹!H5m0M“H, ~1! where m0 is the vacuum permeability, H is the magnetic field strength, and M is the magnetization in the film volume el￾ement. This functional form suggests the Kelvin force den￾sity on an isolated body, except that the local field H re￾places the applied field H0 . Intuitively, we understand this body force to be like the force acting on a magnetic dipole. 614 Am. J. Phys. 69 ~5!, May 2001 http://ojps.aip.org/ajp/ © 2001 American Association of Physics Teachers 614