ARTICLE pubs.acs.org/Langmuir
Ordered Complex Structures Formed by Paramagnetic Particles via Self-Assembly Under an ac/dc Combined Magnetic Field Yutaka Nagaoka, Hisao Morimoto, and Toru Maekawa* Bio-Nano Electronics Research Center, Toyo University, 2100, Kujirai, Kawagoe, Saitama 350-8585, Japan
bS Supporting Information ABSTRACT: We apply ac and dc magnetic fields simultaneously in orthogonal directions to each other to a solution, in which paramagnetic microparticles are dispersed, and show that complex secondary structures composed of oscillating chain clusters, that is, long linear clusters interconnected by T-, L-, and criss-crossjunctions, are self-assembled. Disklike clusters are formed at some junctions and the number of disklike clusters increases as the frequency of the ac magnetic field increases. We finally show that the angle between long linear clusters can be altered by changing the ratio of the intensities of the ac and dc magnetic fields.
’ INTRODUCTION Self-organization induced in polarizable nano/microparticle dispersed systems has attracted many researchers’ attention in recent years, since the interaction between polarizable particles can easily be tuned by external fields, and therefore, those systems are considered to be ideal models for studying phase transitions and conformational transformations occurring in atomic and molecular systems.1,2 When oscillating fields are applied to the systems, the particles or clusters formed by the particles will move following the fields’ oscillations.311 The secondary structures formed by such moving particles or clusters may be more complicated than those formed in dc fields. It has recently been revealed that active organisms such as swimming micro-organisms and living cells form self-organized patterns under certain conditions,1215 and thus, understanding self-organization induced by moving particles or clusters under oscillating fields may well give further insight into pattern formations observed in biological systems as well. Furthermore, nano/ microdevices such as micro total analysis systems (μ-TAS) and nano/micro-electromechanical systems (NEMS/MEMS), in which particles and clusters are actively controlled by external oscillating fields, have been developed for biochemical, biomedical, and electromechanical purposes. Rotating clusters, which are composed of magnetic particles, have been used as stirring components to enhance the mixing of fluids11,1619 and biochemical reactions.20 Micropumps2123 and artificial swimmers7,2426 have also been developed controlling the motion of clusters by external fields. Several methods of manipulating magnetic and nonmagnetic particles on magnetic or nonmagnetic substrates using rotational magnetic fields have been developed.2729 An innovative idea of nanosurgery of brain tumors was materialized utilizing the interactions between magnetic particles and the membranes of tumor cells in a rotational magnetic field.30 It is very important to investigate the cluster structures and dynamics induced by oscillating external fields since they provide us with not only new insight into physical, chemical, and biological r 2011 American Chemical Society
complex phenomena but also a wide range of applications in nano/ microtechnology, biotechnology, and bionano fusion technology. In this study, we show new structures formed by paramagnetic particles in an in-plane ac/dc combined magnetic field. In such an oscillating field, the particles form chain clusters, which move following the field’s oscillation and complex secondary structures such as long linear clusters, which are interconnected by T-, L-, and crisscross-junctions, are formed under certain conditions. We clarify the dependence of the cluster formation process and the length and orientations of those linear clusters on the oscillating magnetic field.
’ EXPERIMENTAL SECTION Sample Preparation. Spherical paramagnetic particles, the surface of which had been coated with carboxylic acid (M-PVA C22, Chemagen Co.), were diluted in deionized water of pH 7.0 and the volume fraction of the particles was 1.2 103. Each particle was composed of polyvinylalcohol, in which magnetite grains were suspended. The zeta potential of a particle was 31.9 ( 3.5 mV (Zetasizer Nano ZS, Malvern Instruments Ltd.). The average diameter of a particle, which was measured using digital image analysis software (WINROOF, MITANI Corp.), was 2.20 ( 0.54 μm and the density of the particle was 2.0 103 kg/m3. The magnetic susceptibility of the particle was 7.46 107 H/m, which was measured by a vibrating sample magnetometer (7407A, LakeShore Cryotronics Inc.). Test Cell and Experimental Setup. A schematic diagram of the test cell is shown in Figure 1a. The waterparticles solution was confined between two quartz substrates. The bottom substrate had a rectangular hollow of 23 mm (l) 10 mm (w) 100 μm (h), in which the paramagnetic particle solution was confined (see Figure 1a). The top surface of the solution was covered with another quartz plate. The particles dispersed stably without any coagulation in a two-dimensional Received: March 29, 2011 Revised: June 15, 2011 Published: June 27, 2011 9160
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Figure 1. Schematic diagrams of a test cell (a) and experimental setup (b). (a) A waterparamagnetic particles solution is confined in a cell fabricated on a quartz substrate. The top surface of the solution is covered with another quartz plate. Dc and ac magnetic fields are applied, respectively, in the OC and AB directions. (b) The test cell is placed at the center of two orthogonal pairs of coils. The magnetic field is generated by the two pairs of coils, a function generator and two amplifiers. horizontal plane in the solution due to electrostatic repulsion. The area fraction of the magnetic particles in the test cell, which was measured from the particles’ projection to the two-dimensional plane, was 0.08. A schematic diagram of the whole experimental setup is shown in Figure 1b. Ac and dc magnetic fields were applied simultaneously in the orthogonal directions to each other after equilibration in the absence of a magnetic field. A dc electric current was supplied to one pair of coils and an ac electric current to the other pair, using a function generator (HP8904A, Agilent Technologies Inc.) and two amplifiers (BOP 2010M, KEPCO Inc.). A dc magnetic field Hdc was applied in the OC direction and an ac magnetic field Hac sin ωt, where t and ω are the time and the angular frequency, was applied in the AB direction (see Figure 1a). As a result, the combined magnetic field H oscillated between OA and OB in such a way as H = Hdc + Hac sin ωt. The frequency of the ac magnetic field was changed from 1 to 10 Hz and the intensities of the ac and dc magnetic fields were set at |Hdc| = |Hac| = 12.7 kA/m, which corresponds to a magnetic flux density of 16 mT. The control parameter, which represents the ratio of the dipoledipole potential energy between particles to thermal energy, is λ = |m|2/4πμsd3kT, where m, μs, d, k, and T are, respectively, the dipole moment vector of a particle, the magnetic permeability of a solvent, the diameter of a particle, the Boltzmann constant, and the temperature.5,6 In the present experiment, the value of λ varied between 4025 and 5692 due to the oscillating intensity of the combined magnetic field. The cluster structures and dynamics were observed by CCD cameras (CCD-F9000, Shimadzu Co., and VCC-8000C, Digimo Co., Ltd.), and an optical microscope (VHZ450, Keyence Corp.) and were recorded on the hard disk of a computer and videotape.
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’ RESULTS AND DISCUSSION When the frequency was lower than 3 Hz, short chains were gradually formed with time and oscillated following the oscillation of the magnetic field. However, once the frequency reached 3 Hz, the short chains started to coagulate to form disklike clusters. Snapshots of clusters formed 5 min after the application of the combined magnetic field are shown in Figure 2ad, where the frequencies of the ac magnetic field were, respectively, 1, 4, 7, and 10 Hz. The movie of the self-assembly process in the case of 4 Hz is given in the Supporting Information. Magnified images of those clusters are shown in Figure 2eh, where the frequency of the ac magnetic field was 4 Hz. After the combined magnetic field was applied, short chains were formed via magnetic dipole dipole interactions and oscillated following the oscillation of the magnetic field. Those oscillating chain clusters gradually aligned to form long linear clusters, which were interconnected by T-, L-, and crisscross-junctions. Under relatively low frequency conditions, the average length between two junctions in the secondary structures; i.e., long linear clusters, ÆLæ, increased with an increase in the frequency. When the frequency was higher than 3 Hz, oscillating chain clusters located near the junctions coagulated to form disklike clusters. Under such high frequency conditions, the oscillation of the magnetic dipole moment of a particle is much faster than the translational motion, and therefore, the effective interaction between particles can be considered as the dipoledipole interaction averaged over one cycle of the oscillating field. The dipoledipole interaction between particles, the dipole moments of which are aligned in the direction of the field, is either attractive or repulsive depending on the angle between the dipole moment and the line connecting the centers of the particles. The average potential, on the other hand, is attractive irrespective of the difference in the relative positions of the dipole moments, which means that the anisotropy in the interaction between particles is remarkably reduced under highfrequency conditions. Therefore, once disklike structures are formed under high frequency conditions, they remain stable. It is known that in MR fluids which are subjected to rotational magnetic fields, disklike structures are formed under high concentration conditions.6 In the present case, the particle’s concentration was locally high near the junctions, and therefore, disklike clusters were formed at the junctions. Note that the relaxation time for the present system to reach its steady state may be extremely long, but it is supposed that the cluster structures formed 5 min after the application of the magnetic field are, qualitatively speaking, similar to those at the steady state. As the frequency increased, both the number and mean size of the disks increased, absorbing surrounding chains (see Figure 3 and the movie in the Supporting Information). As a result, the average length of the straight parts between two junctions in the long linear clusters, ÆLæ, decreased with an increase in the frequency (Figure 2ad). To analyze the above tendency more quantitatively, we evaluated the average length, ÆLæ, and the number density of disklike clusters as a function of the frequency of the ac magnetic field, which are shown in Figure 4. The average length, ÆLæ, increased as the frequency increased up to 3 Hz and then gradually decreased once the frequency exceeded 3 Hz, while the number density of disklike clusters increased with an increase in the frequency beyond 3 Hz. The secondary clusters; that is, long linear clusters, are composed of short chains, which are formed by paramagnetic particles. The length of those short chains decreases with an 9161
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Figure 2. Ordered complex structures formed by paramagnetic particles under an ac/dc combined magnetic field. (ad) Snapshots taken 5 min after the application of the magnetic field. The frequencies of the ac magnetic field are (a) 1 Hz, (b) 4 Hz, (c) 7 Hz, and (d) 10 Hz. The scale bar represents 1.0 mm. (eh) Motion of chain clusters forming long linear clusters (indicated by arrows): (e) 0 s, (f) 0.04 s, (g) 0.08 s, (h) 0.124 s. The frequency of the ac magnetic field is 4 Hz. The short chain clusters oscillate following the oscillation of the magnetic field and form long, linear clusters aligning in the directions of OA0 and OB0 , which are slightly inward with respect to OA and OB. The inset scale bar is 100 μm.
Figure 3. Growth of a disklike cluster at a T-junction. The frequency of the ac magnetic field is 4 Hz. (a) 0 s, (b) 7.2 s, (c) 14.7 s. The chain cluster indicated by an arrow is gradually absorbed into the disk, and as a result, the linear cluster composed of chains is shortened. The inset scale bar is 100 μm.
increase in the frequency of the ac magnetic field due to viscous drag. When those short chains are positioned in parallel, repulsive forces act on those chains, and therefore, translational motion is induced when the frequency of the ac magnetic field is low,31 whereas those short chains cannot move in any translational directions when the frequency is high. Therefore, short, oscillating chains tend to form long linear clusters when the frequency is high, which explains qualitatively why the length of those long linear chains increased with an increase in the frequency of the ac magnetic field up to 3 Hz. However, once the frequency exceeded 3 Hz, the short chains are absorbed by disklike clusters formed at junctions in the long linear clusters as mentioned above, and as a result, the length of the long linear clusters started to decrease. We will be investigating the dependence of the length of the long linear clusters and the number density of disklike clusters on the frequency of the ac magnetic field in more detail. It is clearly shown in Figure 2eh that long linear clusters composed of oscillating chain clusters were not aligned in either OA or OB direction but slightly tilted inward, that is, in OA0 and
Figure 4. Average length of the straight parts between two junctions in the secondary structures, ÆLæ, and the number density of disklike clusters as a function of the frequency of the ac magnetic field. Top panel: Definition of the length of the straight parts between two junctions, L. Disk clusters are formed at two junctions. Bottom panel: Dependence of the average length ÆLæ (circle) and the number density of the disk clusters (square) on the frequency of the ac magnetic field. ÆLæ and the number density of disks were measured 5 min after the application of the combined magnetic field. ÆLæ becomes maximum at a certain frequency, at which the number density of disks starts to increase.
OB0 directions. In other words, the angle between two linear clusters was smaller than 90. We calculated the potential energy between two oscillating chain clusters in order to understand 9162
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Figure 6. Effect of the ratio |Hac|/|Hdc| on the cluster structures. |Hdc| and the frequency of the ac magnetic field were, respectively, 12.7 kA/m and 4 Hz. (a) Combined magnetic field. The angle of the fields’ oscillation, R, was altered by changing |Hac|. (b) |Hac|/|Hdc| = 0.5, R = 0.23 π, (c) |Hac|/|Hdc| = 1.0, R = 0.5 π, (d) |Hac|/|Hdc| = 1.5, R = 0.63 π. The scale bar is 1.0 mm. Figure 5. Average dipoledipole interaction energy between two chain clusters. (a) Calculation system. Each chain cluster is assumed to be an ellipsoid of aspect ratio A. Two ellipsoids oscillate between OA and OB following the external ac/dc combined magnetic field. The distance between the centers of the ellipsoids is set equal to the length of the long axis of an ellipsoid. (b) Dependence of the nondimensional potential energy between two ellipsoids averaged over one cycle of the oscillating field, Æεddæ, on the azimuthal angle j and the aspect ratio of an ellipsoid A. Two ellipsoids are likely to align in the direction, at which the energy becomes minimum, and therefore, the angle difference between the local minima of the energy, Δj, gives the angle between two linear clusters. (c) Dependence of Δj on the aspect ratio of an ellipsoid. Δj increases and approaches 90 with an increase in the aspect ratio.
qualitatively the orientations of those linear clusters. The schematic diagram of the calculation model is shown in Figure 5a. Since the shape of each oscillating chain is considered to be ellipsoidal (see Figure 2eh), we estimated the average dipole dipole interaction energy between two magnetized ellipsoids over one cycle of their movement following the oscillation of the external magnetic field, changing the aspect ratio of the ellipsoid, A, and the azimuthal angle of the line connecting the centers of two ellipsoids, j. Note that the distance between two ellipsoids, R, was set equal to the length of the long axis of an ellipsoid. We assumed that the ellipsoids oscillate without any delay following the field oscillation, in other words, ellipsoids are always aligned in the field direction. When both the ellipsoid and the magnetic field are directed to the x-axis, a magnetic field produced by a magnetized ellipsoid under an external magnetic field, Hell, can be expressed as follows, using the ellipsoidal coordinate system (ξ, η, ζ)32,33 Hell ¼ grad ϕ ϕ¼
3 jmjϕ0 1 2 jHj 4πμs
Z
∞ ξ
ds ðs þ a2 ÞRs
ð1Þ
where m, H, ϕ0, a, and μs are, respectively, the magnetic moment of an ellipsoid, the external magnetic field, the potential of the external magnetic field, the length of the long axis of the ellipsoid, and the magnetic permeability of the solvent. Rs is defined as
Rs = [(s + a2)(s + b2)2]1/2, where b is the length of the short axis of the ellipsoid. Note that, in the present experiment, the intensity of the external field oscillated as |H| = H*(1 + sin2 ωt)1/2, where ω is the angular frequency of the ac field and H* = |Hdc| = |Hac|. As was done by Singh et al.,33 we approximately estimated the dipoledipole interaction energy between two ellipsoids, Edd, using the following equation Edd ¼ m 3 Hell
ð2Þ
We nondimensionalized the potential energy as εdd Edd/(m*2/ 4πμsb3), where m* is the magnetic moment of an ellipsoid induced in a field intensity of H* (= |Hdc| = |Hac|). Finally, we calculated the average nondimensional energy between the ellipsoids over one cycle of the external field as follows Z 1 2π εdd dτ ð3Þ Æεdd æ ¼ 2π 0 where τ is the nondimensional time defined by τ ωt. Note that Æεddæ is a function of the aspect ratio of the ellipsoid, A a/b, the distance between the ellipsoids, R, and the azimuthal angle of the line connecting the centers of the ellipsoids, j (see Figure 5a). The dependence of the normalized mean potential energy between two ellipsoids, Æεddæ, on the azimuthal angle j and the aspect ratio of the ellipsoid is shown in Figure 5b. If the rotational motion of the ellipsoids is much faster than the translational one, the ellipsoids are likely to align in the direction at which the average interaction energy becomes a minimum. Therefore, the angle difference between the two potential energy minima, Δj (see Figure 5b for the definition), gives the angle between two linear clusters composed of oscillating chain clusters. Figure 5c shows the dependence of the angle, Δj, on the aspect ratio of an ellipsoid. Δj increases and approaches 90 with an increase in the aspect ratio of the ellipsoid. According to our experimental observation, the actual aspect ratio of the oscillating chain clusters decreased from 6.8 to 3.2 as the frequency was increased from 1 to 10 Hz, and in such a range of the aspect ratio, the angle Δj is smaller than 90, which agrees with the experimental result. The change in the angle Δj caused by the difference in the frequency of the magnetic field is as small as 2.7. In other words, 9163
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Langmuir the angle Δj does not change dramatically in the frequency range 110 Hz, which also coincides with our experimental observations (see Figure 2ad). Although the angle between long linear clusters is not dramatically altered in such a frequency range, it can be actively controlled by changing the ratio of the intensities of the dc and ac magnetic fields. We carried out experiments changing the ratio |Hac|/|Hdc|, the result of which is shown in Figure 6. Note that |Hdc| was fixed at 12.7 kA/m and the ratio |Hac|/|Hdc| was altered by changing |Hac|. The angle between long linear clusters composed of oscillating chain clusters changed depending on the angle of the fields’ oscillation, R = 2 tan1(|Hac|/|Hdc|) (see Figure 6a).
’ CONCLUSIONS We investigated the cluster structures formed by paramagnetic particles subjected to an ac/dc combined magnetic field. We found that unique complex secondary structures composed of oscillating chain clusters, that is, long linear clusters interconnected by T-, L-, and crisscross-junctions, are self-assembled. As the frequency of the oscillating magnetic field increased, disk clusters were formed and the number of disks gradually increased. We clarified the dependence of the average length between two junctions in the long linear clusters and the number density of disklike clusters on the frequency of the ac magnetic field. We also found that the angle between long linear clusters can be altered by changing the ratio of the intensities of the dc and ac magnetic fields. We will be investigating the effect of the area or volume fractions of paramagnetic particles, the value of the control parameter λ, the frequency of the ac magnetic field, and the ratio of the intensities of the dc and ac magnetic fields on the cluster formation process and the structures of clusters created in the present system both experimentally and numerically. There are varieties of possible combinations of dc and ac magnetic fields, which can be applied to magnetic particle dispersed systems, so that different patterns may be created. Ferromagnetic nanoparticle dispersed systems, in which each particle possesses a permanent magnetic dipole moment unlike paramagnetic microparticles, should also be examined. ’ ASSOCIATED CONTENT
bS
Supporting Information. Movie SI1, showing the time evolution of the cluster structures formed by paramagnetic particles in an ac/dc combined magnetic field. The frequency and intensity of the magnetic field are 4 Hz and 12.7 kA/m. The play speed is 5 times faster than the real speed. The scale bar is 1.0 mm. Movie SI2, showing the linear long chains absorbed into disk clusters. The frequency of the ac magnetic field is 4 Hz. The play speed is real. The scale bar is 100 μm. This material is available free of charge via the Internet at http://pubs.acs.org.
’ AUTHOR INFORMATION
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’ ACKNOWLEDGMENT Part of this study has been supported by a Grant for the HighTech Research Centers, which is organized by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan, since 2006. 9164
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