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Studies of mesoscopic lattices forming with magnetic fluid
Physica A 281 (2000) 87–92
www.elsevier.com/locate/physa
Studies of mesoscopic lattices forming with magnetic uid I.M. Jiang a; ∗ , M.S. Wang a , H.E. Horng b , C.Y. Hong c a Department
of Physics, National Sun Yat-sen University, Kaohsiung, Taiwan of Physics, National Taiwan Normal University, Taipei, Taiwan c Department of Mechanical Engineering, Da-Yeh University, Chang-Hwa, Taiwan b Department
Abstract The high-quality magnetic uid composition is capable of forming two-dimensional lattices of concentrated magnetic particle columns in a thin lm in magnetic elds. The extraordinary lattices, a kind of novel mesoscopic system, have been explored with optical microscopy, digital imaging, and computer–video technique in this study. The two-dimensional lattices present hexagonal phase with exotic topological defects due to the structure distortion under excitations. The ordering of the 2D lattices are classi ed with positional and bond-orientational correlation functions. Some dynamical observations of topological defects are also made in this study. c 2000 Elsevier Science B.V. All rights reserved.
Keywords: Magnetic uid; KT transition; Bond-orientational order; Hexatic phase
1. Introduction Inspired by the work of Kosterlitz and Thouless (KT) [1], there has been considerable interest in two-dimensional (2D) melting both in theoretical and experimental condensed matter physics [2,3]. They proposed that the 2D XY model can have topological defects which mediate a new type of phase transition. In contrast to the conventional order–disorder phase transition, this novel transition involves a di erent mechanism, the unbinding of vortex–antivortex pairs. Such a phase transition can be continuous. Subsequently, Halperin, Nelson (HN) [4,5] and Young [6] have extended the ideas of the KT mechanism to the 2D melting problem. They predicted the presence of a new 2D phase called the hexatic phase for systems with sixfold symmetry. The bond orientational order (BOO) parameter of the 2D crystal is a long-range order ∗
Corresponding author.
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 0 3 8 - 8
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(LRO), but the translational order parameter is only a quasi-long-range order (QLRO). The melting of a 2D crystal transits into the hexatic phase, which is driven by the breakup of thermally generated bound dislocation pairs at temperature Tm . Then the increase of the density of free dislocations above Tm will result in an exponential decay of the translational order parameter. However, the orientational order persists, in the sense that the BOO decays only algebraically and displays QLRO. Such a transition is a continuous phase transition. Subsequently, the spontaneous breakup of free dislocations into their constituent disclinations will drive a second continuous transition to an isotropic uid from the intermediate hexatic phase. Inspired by the theoretical ndings, some researchers use computer simulations to probe the exotic ordering and phase transition in many systems [7]. And some experimental systems for studying the novel KTHNY theory have also been explored by some researchers. They include electrons on helium [8], noble gases physisorbed on substrates [9], liquid crystals [10], polystyrene colloids [11], magnetic bubble arrays [12], dusty plasma [13], and vortex arrays in high-temperature superconductors [14]. These experimental systems can be divided into two classes. In some of them, the microscopic structural details are not likely to be observed directly with the existing technique, and only macroscopic properties are measured. The others are model systems which mimic the complex many-body properties on the atomic scale. The topologic defects can be visualized, and the microscopic structural details are directly accessible, allowing theoretical assumptions to be tested and veri ed. Although these model systems shed new light on the fundamental properties of KT theory, yet they also demonstrate anomalies due to the nature of interactions [15], substrate roughness and so on.
2. Theory and experiments Our unique high-quality magnetic uid composition is capable of forming twodimensional lattices of concentrated magnetic particle columns in a thin lm in magnetic elds [16]. The extraordinary lattices, which can be explored with microscope, are novel mesoscopic system suitable for the 2D melting studies. Magnetic uids typically consist of colloidal magnetic particles such as magnetite or manganese–zinc ferrites, dispersed in a continuous carrier phase [17,18]. The co-precipation technique makes the magnetic particles coated with a surfactant layer, allowing us to neglect the van der Waals force interaction between particles. As the particles interact with one another through the steric repulsion of surfactant coats and the dipole–dipole interaction of particle magnetic moments, one gets the stabilization of suspensions and homogeneity of magnetic uids. The average diameter of the dispersed particles ranges between 5 and 10 nm. To form a magnetic uid thin lm, the sample is sealed in a 8 mm ×4 mm rectangular glass cell of 6 m thickness. The perpendicular magnetic elds applied to the lm are generated by Helmholtz coils and by a uniform solenoid, respectively. The resulting magnetic elds are uniform in the exploration region. The dispersed magnetic particles subjected to external magnetic elds can form droplet aggregates. This may
I.M. Jiang et al. / Physica A 281 (2000) 87–92
89
be considered as the violation of thermodynamical stability and the occurrence of condensation. With further increase of the magnetic eld, the droplet aggregates lled in the thin lm will form columns of similar size with similar magnitude of magnetic moment. Through the dipole–dipole interaction between columns, they will form diverse patterns and ordered structures. The two-dimensional well-ordered structures of concentrated magnetic particle columns in the thin lm under the in uence of magnetic elds, which can be explored with optical microscope, are the novel mesoscopic system to be investigated in this study. Patterns are visualized with the transmission Olympus microscope. All of the patterns are recorded by CCD to produce digitized images with a threshold chosen to render the pattern faithfully, resulting in highly contrastive black and white images. The eld of view is typically ∼100 m in width, and is always much smaller than the active area of the magnetic uid lm. Digital images can locate the center of each concentrated magnetic particle columns, the position of the lattice point of 2D mesoscopic lattice. Both positional and orientational correlation functions are calculated to characterize the structure of the 2D system. The positional correlation function, G(r), describes the possibility of nding another point at relative distance r apart and displays the degree of the translational symmetry. G(r) can be de ned as follows: G(r) = h(0)(r)i : The local orientational order parameter ’6 (r) is de ned as follows: 1 X i6Â(r) e : ’6 (r) = 6 n; n Here, Â(r) is the angle between the bond and the reference axis, and the summation is taken over all the complex numbers of the nearest neighbors. Subsequently, the local bond orientational order related to that of r distance apart is de ned as the orientational correlational function as follows: G6 (r) = h’6 (0)’∗6 (r)i : In this study, our magnetic uid displays soft magnetic behavior and the saturated magnetization of the sample is 8:7 emu=g. Subjected to various external magnetic elds, the magnetic moment of the concentrated magnetic particle columns varies in magnitude and the interaction strength between columns will also undergo a variation. In this way, we simulate the situation corresponding to the temperature variation of the system. Subjected to the perpendicular magnetic eld of the range between 85 and 115 Oe, we obtain the nearly hexagonal lattices. For a well-ordered hexagonal lattice, the majority of the lattice points should be six-bond in bond number which is the number of the nearest-neighbor connecting lines. In this case 88% of lattice points is six-bond, 8% is ve-bond, 4% is seven-bond, and a low percentage of four- and eight-bond presents. The distribution of the bond number of the lattice points does not vary too much in the above-mentioned range of external magnetic elds. Fig. 1(a) displays a typical nearly hexagonal structure of magnetic uid thin lm subjected 110 Oe perpendicular
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Fig. 1. (a) Typical nearly hexagonal structure of magnetic uid thin lm subjected to a perpendicular magnetic eld. (b) FFT image. (c) Positional correlation function G(r). (d) Orientational correlation function G6 (r).
magnetic eld and Fig. 1(b) displays its fast Fourier transformation (FFT). The FFT image displays clear six spots and indicates fairly ordered hexagonal structure. The distance between columns can be calculated directly from the location of the columns or with d = 2=k from the k-space distance. The average distance between columns in this case is about 4:5 m. We increase the magnetic eld stepwise and accumulate the data from 20 pictures after equilibrium is reached at each magnetic eld. We calculate both the translational and orientational correlation functions. They display oscillatory behavior and ascertain ordered structure. Typical translation correlation function and orientational correlation function are plotted in Figs. 1(c) and (d), respectively. The oscillatory peak-height decreases with the distance between lattice points. The decay behavior may be distinguished with algebraic or exponential way. In the range of 85 –115 Oe magnetic elds, the algebraic tting, ∼ r −c , of the oscillatory peaks are proper both for the translational and orientational correlation functions. Both positional and orientational order parameters decay slowly with increasing distance. This means that the mesoscopic lattices of magnetic uid thin lms subjected to the perpendicular magnetic eld of 85 –110 Oe are nearly solid.
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Fig. 2. Evolution of typical dissociation of dislocation defect into two disclinations.
Because the two-dimensional crystals are essentially a defected crystal with high concentration of dislocations, the dynamics of defects is much more interesting. Fig. 2 displays the evolution of typical dissociation of dislocation defect into two disclinations. In order to observe the dislocation dissociation, we have put alternative magnetic eld to agitate the movement of the lattice points. Figs. 2(a) – (d) display the pictures of starting time, 1.2, 2.1, and 3.4 s, respectively. In these gures, solid circle denotes the lattice point with six-bond, which is of perfect hexagonal lattice. − sign denotes the lattice point with ve-bond, and + sign denotes the lattice point with seven-bond. A dislocation of two pairs of ve- and seven-bond defects appears at the upper right corner of the pictures. The ve- and seven-bond are connected with solid lines for distinction. As time goes by the dislocation is dissociated into two separated disclinations.
3. Discussion The 2D mesoscopic lattices of magnetic uid thin lm subjected to perpendicular magnetic eld demonstrate hexagonal phase with exotic topological defects. There are some lattice points with ve- and seven-bond instead of exact six-bond. The total topological charge may not be conserved in a nite viewing region due to nite
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boundary e ect and impurity. Also, some pinning defects arising from substrate roughness may introduce domains with di erent lattice orientations and a series of defects with non-zero net topological charge along the domain boundary as shown in Fig. 2. Due to unsatisfactory sample treatments and environment control, both positional and orientation correlation functions demonstrate algebraically decay with distance in preliminary experiments. This is a somewhat slow decay compared to the hexatic phase of KTNHY theory. So we just demonstrate the nearly solid 2D crystal of the mesoscopic lattice of magnetic uid thin lm in perpendicular magnetic eld region. Further study of 2D melting with the novel magnetic uid thin lms is in progress. Acknowledgements This work was partially supported by the National Science Council, Taiwan. I.M. Jiang would like to thank Prof. C.C. Huang for valuable discussions. References [1] J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181. [2] D.R. Nelson, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 7, Academic Press, London, 1983. [3] K.J. Strandburg, Rev. Mod. Phys. 60 (1988) 161. [4] D.R. Nelson, B.I. Halperin, Phys. Rev. Lett. 41 (1978) 121. [5] D.R. Nelson, B.I. Halperin, Phys. Rev. B 19 (1979) 2457. [6] A.P. Young, Phys. Rev. B 19 (1979) 1855. [7] K.J. Strandburg, in: K.J. Strandburg (Ed.), Bond-Orientational Order in Condensed Matter Systems, Springer, New York, 1992. [8] C.G. Grimes, G. Adams, Phys. Rev. Lett. 42 (1979) 795. [9] S.E. Nagler, P.M. Horn, T.F. Rosenbaum, R.J. Birgeneau, M. Sutton, S.G.J. Mochrie, D.E. Moncton, R. Clarke, Phys. Rev. B 32 (1985) 7373. [10] C.C. Huang, in: K.J. Strandburg (Ed.), Bond-Orientational Order in Condensed Matter Systems, Springer, New York, 1992. [11] C.A. Murray, in: K.J. Strandburg (Ed.), Bond-Orientational Order in Condensed Matter Systems, Springer, New York, 1992. [12] R. Seshadri, R.M. Westervelt, Phys. Rev. B 46 (1992) 5142 and 5150. [13] C.H. Chiang, I. Lin, Phys. Rev. Lett. 77 (1996) 647. [14] C.A. Murray, P.L. Gammel, D.J. Bishop, D.B. Mitzi, A. Kapitulnik, Phys. Rev. Lett. 64 (1990) 2312. [15] A.E. Larsen, D.G. Grier, Nature 385 (1997) 230. [16] C.Y. Hong, I.J. Jang, H.E. Horng, C.J. Hsu, Y.D. Yao, H.C. Yang, J. Appl. Phys. 81 (1997) 4275. [17] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Camdridge, 1985. [18] H.E. Horng, I.J. Jang, K.L. Kung, Y.D. Yao, H.C. Yang, C.Y. Hong, Czechosovak, J. Phys. 46 (1996) 2023.
www.elsevier.com/locate/physa
Studies of mesoscopic lattices forming with magnetic uid I.M. Jiang a; ∗ , M.S. Wang a , H.E. Horng b , C.Y. Hong c a Department
of Physics, National Sun Yat-sen University, Kaohsiung, Taiwan of Physics, National Taiwan Normal University, Taipei, Taiwan c Department of Mechanical Engineering, Da-Yeh University, Chang-Hwa, Taiwan b Department
Abstract The high-quality magnetic uid composition is capable of forming two-dimensional lattices of concentrated magnetic particle columns in a thin lm in magnetic elds. The extraordinary lattices, a kind of novel mesoscopic system, have been explored with optical microscopy, digital imaging, and computer–video technique in this study. The two-dimensional lattices present hexagonal phase with exotic topological defects due to the structure distortion under excitations. The ordering of the 2D lattices are classi ed with positional and bond-orientational correlation functions. Some dynamical observations of topological defects are also made in this study. c 2000 Elsevier Science B.V. All rights reserved.
Keywords: Magnetic uid; KT transition; Bond-orientational order; Hexatic phase
1. Introduction Inspired by the work of Kosterlitz and Thouless (KT) [1], there has been considerable interest in two-dimensional (2D) melting both in theoretical and experimental condensed matter physics [2,3]. They proposed that the 2D XY model can have topological defects which mediate a new type of phase transition. In contrast to the conventional order–disorder phase transition, this novel transition involves a di erent mechanism, the unbinding of vortex–antivortex pairs. Such a phase transition can be continuous. Subsequently, Halperin, Nelson (HN) [4,5] and Young [6] have extended the ideas of the KT mechanism to the 2D melting problem. They predicted the presence of a new 2D phase called the hexatic phase for systems with sixfold symmetry. The bond orientational order (BOO) parameter of the 2D crystal is a long-range order ∗
Corresponding author.
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 0 3 8 - 8
88
I.M. Jiang et al. / Physica A 281 (2000) 87–92
(LRO), but the translational order parameter is only a quasi-long-range order (QLRO). The melting of a 2D crystal transits into the hexatic phase, which is driven by the breakup of thermally generated bound dislocation pairs at temperature Tm . Then the increase of the density of free dislocations above Tm will result in an exponential decay of the translational order parameter. However, the orientational order persists, in the sense that the BOO decays only algebraically and displays QLRO. Such a transition is a continuous phase transition. Subsequently, the spontaneous breakup of free dislocations into their constituent disclinations will drive a second continuous transition to an isotropic uid from the intermediate hexatic phase. Inspired by the theoretical ndings, some researchers use computer simulations to probe the exotic ordering and phase transition in many systems [7]. And some experimental systems for studying the novel KTHNY theory have also been explored by some researchers. They include electrons on helium [8], noble gases physisorbed on substrates [9], liquid crystals [10], polystyrene colloids [11], magnetic bubble arrays [12], dusty plasma [13], and vortex arrays in high-temperature superconductors [14]. These experimental systems can be divided into two classes. In some of them, the microscopic structural details are not likely to be observed directly with the existing technique, and only macroscopic properties are measured. The others are model systems which mimic the complex many-body properties on the atomic scale. The topologic defects can be visualized, and the microscopic structural details are directly accessible, allowing theoretical assumptions to be tested and veri ed. Although these model systems shed new light on the fundamental properties of KT theory, yet they also demonstrate anomalies due to the nature of interactions [15], substrate roughness and so on.
2. Theory and experiments Our unique high-quality magnetic uid composition is capable of forming twodimensional lattices of concentrated magnetic particle columns in a thin lm in magnetic elds [16]. The extraordinary lattices, which can be explored with microscope, are novel mesoscopic system suitable for the 2D melting studies. Magnetic uids typically consist of colloidal magnetic particles such as magnetite or manganese–zinc ferrites, dispersed in a continuous carrier phase [17,18]. The co-precipation technique makes the magnetic particles coated with a surfactant layer, allowing us to neglect the van der Waals force interaction between particles. As the particles interact with one another through the steric repulsion of surfactant coats and the dipole–dipole interaction of particle magnetic moments, one gets the stabilization of suspensions and homogeneity of magnetic uids. The average diameter of the dispersed particles ranges between 5 and 10 nm. To form a magnetic uid thin lm, the sample is sealed in a 8 mm ×4 mm rectangular glass cell of 6 m thickness. The perpendicular magnetic elds applied to the lm are generated by Helmholtz coils and by a uniform solenoid, respectively. The resulting magnetic elds are uniform in the exploration region. The dispersed magnetic particles subjected to external magnetic elds can form droplet aggregates. This may
I.M. Jiang et al. / Physica A 281 (2000) 87–92
89
be considered as the violation of thermodynamical stability and the occurrence of condensation. With further increase of the magnetic eld, the droplet aggregates lled in the thin lm will form columns of similar size with similar magnitude of magnetic moment. Through the dipole–dipole interaction between columns, they will form diverse patterns and ordered structures. The two-dimensional well-ordered structures of concentrated magnetic particle columns in the thin lm under the in uence of magnetic elds, which can be explored with optical microscope, are the novel mesoscopic system to be investigated in this study. Patterns are visualized with the transmission Olympus microscope. All of the patterns are recorded by CCD to produce digitized images with a threshold chosen to render the pattern faithfully, resulting in highly contrastive black and white images. The eld of view is typically ∼100 m in width, and is always much smaller than the active area of the magnetic uid lm. Digital images can locate the center of each concentrated magnetic particle columns, the position of the lattice point of 2D mesoscopic lattice. Both positional and orientational correlation functions are calculated to characterize the structure of the 2D system. The positional correlation function, G(r), describes the possibility of nding another point at relative distance r apart and displays the degree of the translational symmetry. G(r) can be de ned as follows: G(r) = h(0)(r)i : The local orientational order parameter ’6 (r) is de ned as follows: 1 X i6Â(r) e : ’6 (r) = 6 n; n Here, Â(r) is the angle between the bond and the reference axis, and the summation is taken over all the complex numbers of the nearest neighbors. Subsequently, the local bond orientational order related to that of r distance apart is de ned as the orientational correlational function as follows: G6 (r) = h’6 (0)’∗6 (r)i : In this study, our magnetic uid displays soft magnetic behavior and the saturated magnetization of the sample is 8:7 emu=g. Subjected to various external magnetic elds, the magnetic moment of the concentrated magnetic particle columns varies in magnitude and the interaction strength between columns will also undergo a variation. In this way, we simulate the situation corresponding to the temperature variation of the system. Subjected to the perpendicular magnetic eld of the range between 85 and 115 Oe, we obtain the nearly hexagonal lattices. For a well-ordered hexagonal lattice, the majority of the lattice points should be six-bond in bond number which is the number of the nearest-neighbor connecting lines. In this case 88% of lattice points is six-bond, 8% is ve-bond, 4% is seven-bond, and a low percentage of four- and eight-bond presents. The distribution of the bond number of the lattice points does not vary too much in the above-mentioned range of external magnetic elds. Fig. 1(a) displays a typical nearly hexagonal structure of magnetic uid thin lm subjected 110 Oe perpendicular
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I.M. Jiang et al. / Physica A 281 (2000) 87–92
Fig. 1. (a) Typical nearly hexagonal structure of magnetic uid thin lm subjected to a perpendicular magnetic eld. (b) FFT image. (c) Positional correlation function G(r). (d) Orientational correlation function G6 (r).
magnetic eld and Fig. 1(b) displays its fast Fourier transformation (FFT). The FFT image displays clear six spots and indicates fairly ordered hexagonal structure. The distance between columns can be calculated directly from the location of the columns or with d = 2=k from the k-space distance. The average distance between columns in this case is about 4:5 m. We increase the magnetic eld stepwise and accumulate the data from 20 pictures after equilibrium is reached at each magnetic eld. We calculate both the translational and orientational correlation functions. They display oscillatory behavior and ascertain ordered structure. Typical translation correlation function and orientational correlation function are plotted in Figs. 1(c) and (d), respectively. The oscillatory peak-height decreases with the distance between lattice points. The decay behavior may be distinguished with algebraic or exponential way. In the range of 85 –115 Oe magnetic elds, the algebraic tting, ∼ r −c , of the oscillatory peaks are proper both for the translational and orientational correlation functions. Both positional and orientational order parameters decay slowly with increasing distance. This means that the mesoscopic lattices of magnetic uid thin lms subjected to the perpendicular magnetic eld of 85 –110 Oe are nearly solid.
I.M. Jiang et al. / Physica A 281 (2000) 87–92
91
Fig. 2. Evolution of typical dissociation of dislocation defect into two disclinations.
Because the two-dimensional crystals are essentially a defected crystal with high concentration of dislocations, the dynamics of defects is much more interesting. Fig. 2 displays the evolution of typical dissociation of dislocation defect into two disclinations. In order to observe the dislocation dissociation, we have put alternative magnetic eld to agitate the movement of the lattice points. Figs. 2(a) – (d) display the pictures of starting time, 1.2, 2.1, and 3.4 s, respectively. In these gures, solid circle denotes the lattice point with six-bond, which is of perfect hexagonal lattice. − sign denotes the lattice point with ve-bond, and + sign denotes the lattice point with seven-bond. A dislocation of two pairs of ve- and seven-bond defects appears at the upper right corner of the pictures. The ve- and seven-bond are connected with solid lines for distinction. As time goes by the dislocation is dissociated into two separated disclinations.
3. Discussion The 2D mesoscopic lattices of magnetic uid thin lm subjected to perpendicular magnetic eld demonstrate hexagonal phase with exotic topological defects. There are some lattice points with ve- and seven-bond instead of exact six-bond. The total topological charge may not be conserved in a nite viewing region due to nite
92
I.M. Jiang et al. / Physica A 281 (2000) 87–92
boundary e ect and impurity. Also, some pinning defects arising from substrate roughness may introduce domains with di erent lattice orientations and a series of defects with non-zero net topological charge along the domain boundary as shown in Fig. 2. Due to unsatisfactory sample treatments and environment control, both positional and orientation correlation functions demonstrate algebraically decay with distance in preliminary experiments. This is a somewhat slow decay compared to the hexatic phase of KTNHY theory. So we just demonstrate the nearly solid 2D crystal of the mesoscopic lattice of magnetic uid thin lm in perpendicular magnetic eld region. Further study of 2D melting with the novel magnetic uid thin lms is in progress. Acknowledgements This work was partially supported by the National Science Council, Taiwan. I.M. Jiang would like to thank Prof. C.C. Huang for valuable discussions. References [1] J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181. [2] D.R. Nelson, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 7, Academic Press, London, 1983. [3] K.J. Strandburg, Rev. Mod. Phys. 60 (1988) 161. [4] D.R. Nelson, B.I. Halperin, Phys. Rev. Lett. 41 (1978) 121. [5] D.R. Nelson, B.I. Halperin, Phys. Rev. B 19 (1979) 2457. [6] A.P. Young, Phys. Rev. B 19 (1979) 1855. [7] K.J. Strandburg, in: K.J. Strandburg (Ed.), Bond-Orientational Order in Condensed Matter Systems, Springer, New York, 1992. [8] C.G. Grimes, G. Adams, Phys. Rev. Lett. 42 (1979) 795. [9] S.E. Nagler, P.M. Horn, T.F. Rosenbaum, R.J. Birgeneau, M. Sutton, S.G.J. Mochrie, D.E. Moncton, R. Clarke, Phys. Rev. B 32 (1985) 7373. [10] C.C. Huang, in: K.J. Strandburg (Ed.), Bond-Orientational Order in Condensed Matter Systems, Springer, New York, 1992. [11] C.A. Murray, in: K.J. Strandburg (Ed.), Bond-Orientational Order in Condensed Matter Systems, Springer, New York, 1992. [12] R. Seshadri, R.M. Westervelt, Phys. Rev. B 46 (1992) 5142 and 5150. [13] C.H. Chiang, I. Lin, Phys. Rev. Lett. 77 (1996) 647. [14] C.A. Murray, P.L. Gammel, D.J. Bishop, D.B. Mitzi, A. Kapitulnik, Phys. Rev. Lett. 64 (1990) 2312. [15] A.E. Larsen, D.G. Grier, Nature 385 (1997) 230. [16] C.Y. Hong, I.J. Jang, H.E. Horng, C.J. Hsu, Y.D. Yao, H.C. Yang, J. Appl. Phys. 81 (1997) 4275. [17] R.E. Rosensweig, Ferrohydrodynamics, Cambridge University Press, Camdridge, 1985. [18] H.E. Horng, I.J. Jang, K.L. Kung, Y.D. Yao, H.C. Yang, C.Y. Hong, Czechosovak, J. Phys. 46 (1996) 2023.