Materials Transactions, Vol. 49, No. 4 (2008) pp. 787 to 791 #2008 The Japan Institute of Metals Analysis of Uniaxial Alignment Behavior of Nonmagnetic Materials under Static Magnetic Field with Sample Rotation* Jun Akiyama1 , Hidefumi Asano2 , Kazuhiko Iwai3 and Shigeo Asai4 1 Institute for Molecular Science, Okazaki 444-8585, Japan Department of Crystalline Materials Science, Nagoya University, Nagoya 464-8603, Japan 3 Department of Materials, Physics and Energy Engineering, Nagoya University, Nagoya 464-8603, Japan 4 Innovation Plaza Tokai Japan Science and Technology Agency, Nagoya 457-0063, Japan 2 A high magnetic field is a useful tool to control the crystal alignment of nonmagnetic materials such as metals, ceramics and polymers. However, the uniaxial alignment of hexagonal crystals with a magnetic susceptibility of c < a cannot be achieved under a static magnetic field, because the c-axis could lie along any arbitrary direction in the plane perpendicular to the direction of the magnetic field. For the uniaxial alignment of these materials, the imposition of a rotating magnetic field during a slip casting process has been proposed. In this study, both theoretical analysis and model experiment have been conducted for the elucidation of the crystal alignment phenomena under a rotating magnetic field and for the quantitative clarification of the optimum operating parameters such as magnetic field strength and viscosity of the medium surrounding the crystals. It has been found that the alignment time decreased with the magnetic field strength and/or with an increase in the viscosity of the surrounding medium. This relation is in contrary to the case of the crystal alignment under the static magnetic field. The result of the model experiment agrees well with that obtained by the theoretical analysis. [doi:10.2320/matertrans.MRA2007326] (Received December 21, 2007; Accepted February 13, 2008; Published March 19, 2008) Keywords: electromagnetic processing of materials, crystal alignment, rotating magnetic field, magnetic anisotropy 1. Introduction Crystal alignment control is one of the most important factors for improving the physical and chemical properties of polycrystalline materials with an anisotropic lattice structure.1–4) The alignment control has been investigated using various methods such as tape casting, pulse current pressure sintering and self-organization. However, it is difficult to obtain highly and uniformly aligned large-sized materials using such processes. On the other hand, the crystal alignment of a diamagnetic material has been obtained experimentally under a static high magnetic field5–10) or the simultaneous imposition of a static high magnetic field and sample rotation11–19) (rotating magnetic field) during a slip casting process. In this study, a theoretical analysis has been conducted and a model experiment has been performed for the elucidation of the crystal alignment phenomena under a rotating magnetic field and for the quantitative clarification of the optimum operating parameters such as the magnetic field strength and the viscosity of the medium surrounding the crystals. 2. Crystal Alignment under Magnetic Field The principle of crystal alignment of a nonmagnetic material under a magnetic field is explained from the perspective of magnetization energy. When a nonmagnetic material is submerged in a magnetic field, the magnetization energy U is approximately given by 1 1 U ¼ 0 V? H 2 0 VH 2 cos2 2 2 *This ð1Þ Paper was Originally Published in Japanese in J. Japan Inst. Metals 71 (2007) 108–112. where 0 is the magnetic permeability in vacuum, V is the volume of the material, ? is the magnetic susceptibility of the magnetic hard axis, is the difference in the magnetic susceptibility between the magnetic easy axis and the magnetic hard axis, H is the intensity of the imposed magnetic field and is the angle between the magnetic field direction and the magnetic easy axis. The magnetic torque acting on the substance is given by the differentiation of U with respect to as follows: T¼ @U 1 ¼ 0 VH 2 sin 2 @ 2 ð2Þ Thus, the crystal under a magnetic field rotates in order to reduce the magnetization energy so as to be in a stable conformation. In the case of imposing a magnetic field on a spherical crystal, which is surrounded by a viscous liquid medium, the alignment behaviour of the crystal is given by the following differential equation: 2 5 d2 d r ¼T þ 8r 3 2 5 dt dt ð3Þ where is the crystal density, r is the crystal radius and is the viscosity of the surrounding medium. In the case that the particle size is small, the inertial term can be neglected. Then, the temporal variation of is derived from eq. (3) as follows: t tan ¼ tan 0 exp ð4Þ s S ¼ 6 0 H 2 ð5Þ where 0 is the initial angle between the direction of the magnetic field and the magnetic easy axis of the crystal and S is the relaxation time for the crystal alignment 788 J. Akiyama, H. Asano, K. Iwai and S. Asai z a χa < χc c-axis Crystal with magnetic anisotropy of χ c < χa B ϕ (t ) θ (t ) c(t ) O y b χc < χa c-axis x B c-axis Fig. 1 Alignment of hexagonal crystal under static high magnetic field. (a) a < c (b) c < a under the static magnetic field20) (hereafter, denotes the ‘alignment time’). Let us consider the magnetic alignment of a crystal with a hexagonal lattice structure. The relationship between the imposed magnetic field and the magnetically stable direction of the hexagonal crystal is summarized in Fig. 1. For a magnetically anisotropic crystal with the relation a < c , the magnetization energy is the lowest when the c-axis of the crystal is parallel to the direction of the magnetic field (Uc < Ua ). Therefore, the c-axis of the crystal aligns parallel to the direction of the magnetic field (Fig. 1(a)). On the other hand, the a-axis of the crystal aligns parallel to the direction of the magnetic field for crystals with the relation c < a (Ua < Uc ). Therefore, with regard to the magnetic susceptibility of such a crystal, it is difficult to control the c-axis alignment under a static field (Fig. 1(b)). 3. Analysis of Crystal Alignment Behaviour under Rotating Field It has been experimentally proven that the uniaxial alignment of a crystal with the magnetic susceptibility of c < a can be obtained under a rotating magnetic field or by rotating the sample under a static magnetic field. We now define a system, as shown in Fig. 2, in order to analyze this phenomenon. A spherical crystal, whose magnetic anisotropy is c < a , is surrounded by a medium under a rotating magnetic field Hð!Þ. The equation of the change is described by the following two simultaneous differential equations: d 1 ¼ cos2 ð’ !tÞ sin 2 ð6Þ dt 2s d’ 1 ¼ sin 2ð’ !tÞ ð7Þ dt 2s where ! is the angular velocity of the rotating magnetic field and ’ is the angle between the c-axis and the x component. B (t ) ωt Fig. 2 Analytical system. From eqs. (6) and (7), the alignment behaviours can be classified based on s as follows. 3.1 S ! 1 (HðtÞ ! 0 and/or ! 1) For S ! 1, the right-hand term of eqs. (6) and (7) could be zero. d ¼0 dt d’ ¼0 dt ð8Þ ð9Þ Then, ðtÞ and ’ðtÞ are given by the constant values for this limit. The boundary conditions are ¼ 0 and ’ ¼ ’0 (initial angles); the resulting solutions are then given as ¼ 0 ’ ¼ ’0 ð10Þ ð11Þ It is understood that when the magnetic field is very small and/or the viscosity of the medium is very large, the magnetic field cannot supply sufficient torque to the crystal; subsequently, the angles and ’ do not vary from their initial states. 3.2 S ! 0 (HðtÞ ! 1 and/or ! 0) In this case, the left-hand term of eqs. (6) and (7) becomes zero. cos2 ð’ !tÞ sin 2 ¼ 0 ð12Þ sin 2ð’ !tÞ ¼ 0 ð13Þ Equation (13) gives the following solution: ’ !t ¼ ð2n þ 1Þ 2 ðn ¼ 0; 1; 2; Þ ð14Þ Hence, the angle ‘’ !t’ is always maintained at =2 rad. eq. (12) then becomes 0 sin 2 ¼ 0 ð15Þ Therefore, Equation (12) holds for all values of . In such a case, it is impossible to achieve crystal alignment because there is no -component magnetic torque acting on the substance. Analysis of Uniaxial Alignment Behavior of Nonmagnetic Materials under Static Magnetic Field with Sample Rotation 3.3 0 < S < 1 In the case that the crystal synchronously rotates with the magnetic field, the following relation holds true: ð16Þ The integration of eq. (16) with respect to t gives ’ðtÞ ¼ !t þ ð17Þ Here, is the integration constant. Equation (7) can be rewritten in terms of as d d 1 ¼ ð’ !tÞ ¼ sin 2 ! ¼ 0 dt dt 2s 104 τR τ S (η =100Pa • s) 1 τ S (η =1Pa • s) 10-1 ð19Þ Fig. 3 4 τ S (η =10Pa • s) 6 8 10 12 Relation between alignment time and magnetic flux density, 0 H. ∞ (1) τ s 0.5 t tanθ = tanθ 0 exp − τR Boundary curve τ s ω = 0.5 (2) τ s ↑ The alignment times S and R , which are calculated from eqs. (5) and (22) under various experimental conditions ( ¼ 1, 10 and 100 Pas, ¼ 1:0 105 and ! ¼ 0:3 rad/ s), are plotted in Fig. 3. It is found that R decreases with H and/or an increase in ! and . This relation is in contrary to the case of the crystal alignment under the static magnetic field S . Step-out 0 θ = arbitrary value Synchronous O Angular velocity, ω / rad • s-1 ∞ Fig. 4 Relation between alignment time and angular velocity !. Figure 4 schematically shows the analyzed region in this time. The alignment behaviour of the crystal under the rotating field is clarified from this figure, except for that under the step-out mode. 4. ð25Þ (3) τ s ω Alignment time, ð22Þ ∞ θ = θ0 τS/s ð20Þ Here, the boundary condition is ¼ 0 at t ¼ 0. The angle decreases monotonically with time except under the initial condition 0 ¼ =2. R denotes the alignment time for the crystal alignment under a rotating magnetic field in the synchronous mode. The optimum operating condition is S ! ¼ 0:5, and in this case, the alignment time is twice of that under the static field (R ¼ 2S ). The differentiation of R with respect to H, and ! gives the following equations: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @R Hð1 þ 1 4ðS !Þ2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > 0 ¼ ð23Þ @H S !2 1 4ðS !Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @R 1 þ 1 4ðS !Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < 0 ¼ ð24Þ @ 2S !2 1 4ðS !Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @R 2ðS !Þ2 ð1 þ 1 4ðS !Þ2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ @! S !3 1 4ðS !Þ2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2S 1 þ 1 4ðS !Þ2 <0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S ! 3 ! 1 4ðS !Þ2 2 ↑ tan 2S pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ! 1 1 4ðS !Þ2 (η =100Pa • s) Magnetic flux density, µ 0 H / T It can be integrated with respect to t and this gives the temporal variation of the angle : cos2 !t ð21Þ t ¼ tan 0 exp tan ¼ tan 0 exp s tan R ¼ (η =10Pa • s) 10 ð18Þ Hence, the synchronous condition is given as S ! < 0:5; otherwise, it can be step-out. Substituting eq. (17) into eq. (6) gives the differential equation under this condition. d 1 ¼ cos2 sin 2 dt 2s (η =1Pa • s) τR 0 1 1 12! arcsinð2s !Þ ¼ arcsin 2 2 0 H 2 τR 102 is then given by ¼ ∆χ = 1.0 × 10 -5 ω = 0.3 rad • s -1 103 Alignment time, τ / s d’ ¼! dt 789 Experimental We have performed the following model experiment to verify the analytically derived relation among R , H and of the medium. A schematic view of the experimental apparatus is shown in Fig. 5. An acrylic vessel (inside diameter: 20 mm, height: 40 mm) was filled with 4 mL of glycerine solution and placed on a rotating platform set at the centre of a bore in a superconducting magnet generating a horizontal static magnetic field. The viscosity of the glycerine solution was 5 Pas. The magnetic field and mould rotation were simultaneously imposed on a polymeric fibre 790 J. Akiyama, H. Asano, K. Iwai and S. Asai PC (a) µ 0 H =4T Light Source Recorder Reflecting Prism Vessel Video Camera 0s (b) µ0 H =5T 5s 10s 0s 10s 20s 10s 20s B Turn Table Superconducting Magnet Fig. 5 Schematic view of the experimental apparatus. (c) µ0 H =6T Magnetic Susceptibility. Direction Susceptibility [] c-axis 1:08 105 a-axis 8:11 106 Table 2 Anisotropy [] 2:7 106 0s Experimental condition. Upper view Sample Magnetic Flux Density B/T Rotating velocity !/rads1 Viscosity /Pas Relaxation time S /s Relaxation time R /s a 4 0.3 5 0.85 11.7 b 5 0.3 5 0.56 19.3 c 6 0.3 5 0.39 28.3 (d ¼ 0:235 mm, l ¼ 8 mm) after it was placed in the vessel. In this case, the added mass effect due to the non-spherical object shape in eq. (3) is negligible. The alignment behaviour of the fibre was recorded by a high-speed video camera to evaluate the angles and . The magnetic susceptibility of the fibre was previously measured by a vibrating sample magnetometer (VSM). The resulting data are presented in Table 1. The magnetic easy axis is in the radial direction, and the value of the magnetic anisotropy is 2:7 106 [-]. The experimental conditions and the theoretical alignment time obtained using eqs. (5) and (22) are listed in Table 2. (Through the reflecting prism) Side view Fiber Fig. 6 Alignment of polymeric fibers under high magnetic field and mold rotation. (a) 0 H ¼ 4T, (b) 0 H ¼ 5T, (c) 0 H ¼ 6T 20 18 4T (experimental) 16 Angle, 90 − δ /degree Table 1 4T (theoretical) 14 5T (experimental) 12 10 5T (theoretical) 8 6 6T (theoretical) 6T (experimental) 4 5. Results and Discussions Figure 6 shows the time variation of the orientation of the fibre. The upper and lower halves of each image show the top view through the reflecting prism and the side view of the sample, respectively. In the case of imposing a magnetic field of 4 T (Fig. 6(a)), the long axis of the fibre becomes almost parallel to the axis of the rotating magnetic field within 10 s. The alignment time is prolonged when the strength of the magnetic field is increased (Fig. 6(b)), and in the case of a magnetic field of 6 T, the alignment is not achieved even after 20 s of the imposition of the magnetic field and rotation (Fig. 6(c)). The time variation of (¼ ’ !t), obtained from Fig. 6, is illustrated in Fig. 7. The alignment in the first four seconds 2 0 0 5 10 15 20 25 30 35 Time, t /s Fig. 7 Time dependence of the angle . of the imposition of the magnetic field is in the step-out mode because of the change in . After 4 s, becomes constant and the alignment behaviour changes from the step-out mode to the synchronous mode. Figure 8 illustrates the time dependence of . The solid lines denote the calculated values, and the dotted lines denote the actual measurement values. R decreases with the Analysis of Uniaxial Alignment Behavior of Nonmagnetic Materials under Static Magnetic Field with Sample Rotation 80 70 , , Angle, θ /degree 60 : Experimentally observed : Calculated 50 40 6T 30 5T 20 4T 10 0 -10 0 5 10 15 20 25 30 35 Time, t /s Fig. 8 Time dependence of the angle . magnetic flux density, and these experimental results quantitatively agree with the calculated ones. 6. Conclusion In this study, a theoretical analysis has been conducted and a model experiment has been performed for elucidating the crystal alignment phenomena under a rotating magnetic field. It has been analytically found that the alignment time decreases with the magnetic field strength and/or with an increase in the viscosity of the surrounding medium. This relation is in contrary to the case of the crystal alignment under a static magnetic field. The result of the model experiment agrees well with that obtained by the theoretical analysis. Acknowledgement This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, Grantin-Aid for Exploratory Research (No. 16656209), ‘‘Creation of Nature-Guided Materials Processing’’ of the 21st Century COE Program, Research support program of SEKISUI CHEMICAL Company and JSPS Asian Core Program ‘‘Construction of the World Center on Electromagnetic Processing of Materials’’. REFERENCES 1) S. Li, K. Sassa and S. Asai: J. Am. Ceram. Soc. 87 (2004) 1384–1387. 2) T. Kuribayashi, M. G. Sung, K. Sassa and S. Asai: CAMP-ISIJ 18 (2005) 801–802. 3) J. Akiyama, M. Hashimoto, H. Takadama, F. Nagata, Y. Yokogawa, K. Sassa, K. Iwai and S. Asai: Key. Eng. Mater. 309–311 (2005) 53–56. 791 4) J. Akiyama, M. Hashimoto, Hiroaki Takadama, Fukue Nagata, Yoshiyuki Yokogawa, Kensuke Sassa, Kazuhiko Iwai and Shigeo Asai: Mater. Trans. 46 (2005) 2514–2517. 5) H. Morikawa, K. Sassa and S. Asai: Mater. Trans. JIM 39 (1998) 814– 818. 6) T. Suzuki, Y. Sakka and K. 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