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Notes Magnetic Orientation of Polymer Fibers in Suspension Tsunehisa Kimura,* Masafumi Yamato, Wataru Koshimizu, Minako Koike, and Takahiko Kawai Department of Applied Chemistry, Faculty of Engineering, Tokyo Metropolitan University, Minami-ohsawa, Hachioji, Tokyo 192-0397, Japan Received June 14, 1999. In Final Form: September 2, 1999
Introduction Diamagnetic materials with magnetic anisotropy have a potential ability to align in magnetic fields.1 Magnetic orientation of liquid crystals and liquid crystalline polymers is well-known. Also, the magnetic orientation of a polyethylene single crystal,2 carbon fibers,3,4 and biological materials1,2,5 has been reported. In addition, recent studies have revealed that the magnetic orientation occurs during the crystallization of organic materials6 and proteins7,8 from solutions, during the induction period of the melt crystallization of crystalline polymers,9-12 and during the gelation process of an aggarose gel.13 Magnetic orientation of inorganic paramagnetic materials in molten states with residual magnetic anisotropy has been also reported.14 Macroscopic orientation occurs due to the rotation of ordered domains under the resistance of the hydrodynamic torque exerted by the surrounding viscous medium. To model this orientation kinetics, an equation of motion derived through the balance between the magnetic torque and the hydrodynamic torque has been employed,15,16 where the aligning domain is usually regarded as a sphere implicitly. However, in some cases, such as fibers, the shape of the aligning domain is far from a sphere, and hence the equation fails to describe the phenomenon. As a matter of fact, the fiber-length dependence of the * To whom correspondence should be addressed. (1) Maret, G.; Dransfeld, K. In Topics in Applied Physics; Herlach, F., Ed.; Springer-Verlag: Berlin, 1985; Vol. 57, Chapter 4. (2) Kawamura, Y.; Sakurai, I.; Ikegami, A.; Iwayanagi, S. Mol. Cryst. Liq. Cryst. 1981, 67, 77. (3) Timbrell, V. J. Appl. Phys. 1972, 43, 4839. (4) Schmitt, Y.; Paulick, C.; Royer, F. X.; Gasser, J. G. J. Non-Cryst. Solids 1996, 205-207, 139. (5) Yamagishi, A.; Takeuchi, T.; Higashi, T.; Date, M. J. Phys. Soc. Jpn. 1989, 58, 2280. (6) Fujiwara, M.; Chidiwa, T.; Tokunaga, R.; Tanimoto, Y. J. Phys. Chem. B 1998, 102, 3417. (7) Sazaki, G.; Yoshida, E.; Komatsu, H.; Nakada, T.; Miyashita, S.; Watanabe, K. J. Cryst. Growth 1997, 173, 231. (8) Wakayama, N. I.; Ataka, M.; Abe, H. J. Cryst. Growth 1997, 178, 653. (9) Sata, H.; Kimura, T.; Ogawa, S.; Yamato, M.; Ito, E. Polymer 1996, 37, 1879. (10) Sata, H.; Kimura, T.; Ogawa, S.; Ito, E. Polymer 1998, 39, 6325. (11) Ezure, H.; Kimura, T.; Ogawa, S.; Ito, E. Macromolecules 1997, 30, 3600. (12) Kimura, T.; Ezure, H.; Sata, H.; Kimura, F.; Tanaka, S.; Ito, E. Mol. Cryst. Liq. Cryst. 1998, 318, 141. (13) Yamamoto, I.; Matsumoto, Y.; Yamaguchi, M.; Shimazu, Y.; Ishikawa, F. Physica B 1998, 246-247, 408. (14) de Rango, P.; Lees, M.; Lejay, P.; Sulpice, A.; Tournier, R.; Ingold, M.; Germi, P.; Pernet, M. Nature 1991, 349, 770. (15) Moore, J. S.; Stupp, S. I. Macromolecules 1987, 20, 282. (16) Kimura, T.; Sata, H.; Ito, E. Polym. J. 1998, 30, 455.
Figure 1. Experimental setup used to monitor fiber orientation in magnetic field: S, suspension; P, polarizer; A, analyzer; CC, CCD camera; EM, electromagnet; LS, light source; VM, video monitor; PC, personal computer.
alignment rate reported for carbon fibers in suspension4 cannot be explained in terms of a simple sphere model. In this Note, a modified equation of motion is presented. The shape of a fiber is approximated with a prolate ellipsoid, and the aspect ratio dependence of the hydrodynamic torque is considered explicitly. The solution of the equation gives the alignment rate as a function of the aspect ratio. The experimental observation of the fiberlength dependence of the alignment rate is discussed on the basis of the modified equation of motion presented in this Note. Experimental Section Superdrawn polyethylene fiber (supplied by Mitsui Chemical) with the diameter of 30 µm was cut into 1, 2, 3, and 4 mm lengths and suspended in a water/ethanol mixture (1/1 in volume) having the same density as the fiber. The viscosity of the mixture was 2.57 × 10-3 Pa s. Carbon fiber (Torayca T300, supplied by Toray) with diameter a of 10 µm was cut into 1, 2, 3, and 4 mm lengths and suspended in a carbon tetrachloride/1,2-dibromoethane mixture (1/5 in volume) with the same density as the fiber. The viscosity of the mixture was 1.06 × 10-3 Pa s. The number of fibers was not very large, so that there were no aggregates. Individual fiber pieces were sufficiently apart from each other (ca. 30 fibers/cm3). A glass cell containing the suspension was immersed in a glass container filled with water. This reduced largely the convection flow in the suspension caused by the temperature gradient. The glass container was put in the center of a Tamagawa TMWTF6215C electromagnet which generates horizontal magnetic fields up to 1.2 T. The orientation process was monitored by a CCD camera and the data were stored in a computer. The number of fibers of each length examined to get the data was about three to five. Experimental setup is displayed in Figure 1. A single polyethylene (PE) fiber lying approximately parallel to the magnetic field was brought to the center of the view of the CCD camera by moving the cell. Then, the electromagnet was turned
10.1021/la990761j CCC: $19.00 © 2000 American Chemical Society Published on Web 10/22/1999
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Figure 2. Orientation of (a) a polyethylene (PE) fiber and (b) a carbon fiber (CF) taken at the later stage of the application of the magnetic field. A PE fiber aligns with the fiber axis perpendicular to the magnetic field, while a CF aligns parallel to the magnetic field. on and the orientation process was recorded by the CCD camera. In the case of the PE fiber, a crossed polar setting was used to obtain a high contrast. In the case of the carbon fiber (CF), a fiber lying approximately perpendicular to the magnetic field was chosen as a starting one. The angle θ between the magnetic field and the fiber axis was measured on the picture recorded at various periods of orientation time. The lengths of the fiber on the picture were measured in the beginning and the end of the orientation, and the run which exhibited the difference in length was discarded because the difference indicates that the fiber axis in the initial state was out of the plane normal to the monitoring direction: In such a case, an error in estimation of the angle θ is large.
Theory The equation of motion for the magnetic orientation of a fiber in a viscous medium is given by the balance of the magnetic torque and the hydrodynamic torque as follows:
1 dθ ) - V χa µ0H2 sin 2θ L dt 2
(1)
The right-hand side of the equation represents the magnetic torque, where H is the magnetic field, µ0 is the magnetic permeability of the vacuum, χa is the volume anisotropic diamagnetic susceptibility, and V is the volume of the fiber. The magnetic torque depends on the volume of the fiber but it does not depend on the shape of the fiber. The left-hand side of the equation represents the hydrodynamic torque acting on the fiber through the surrounding viscous medium when the fiber is rotating at the angular velocity of dθ/dt. Here, θ is the angle between H and the fiber axis. The value of L depends on the volume and the shape of the fiber. Equation 1 is easily solved to give
tan θ ) tan θ0 exp(-t/τ)
(2)
where the alignment rate |τ-1| is defined as
τ-1 ) (V/L)χa µ0 H2
(3)
In the case of a sphere of radius a, L ) 8π ηa3 17 and V (17) Jeffery, G. B. Proc. R. Soc. London 1922, A102, 161.
) (4/3)πa3 then eq 3 becomes
τ-1 ) χaµ0H2/6η
(4)
where η is the viscosity of the medium. The sign of τ-1 coincides with that of χa. If χa > 0, then the angle θ decreases in time (parallel alignment), while if χa < 0, θ increases in time until it reaches π/2 (perpendicular alignment). In the case of an ellipsoid, a general formula of the hydrodynamic torque acting on an ellipsoid rotating in a viscous medium was first derived by Jeffery.17-19 Following the formula, we obtain
L ) 8πηa3/F(D)
(5)
with
F(D) )
(
3D -2DxD2 - 1 + (1 - 2D2) ln
D - xD2 - 1 D + xD2 - 1
4(D2 - 1)(D2 + 1)xD2 - 1
)
(6)
where 2a is the length of the short axis and D is the aspect ratio. With V ) (4/3)πa3D for the ellipsoid, we obtain from eq 3
τ-1 ) F(D)
χaµ0H2 6η
(7)
This equation accounts for the fiber-length dependence of τ-1. In the limit of D ) 1, F(D) becomes unity, resulting in eq 4. In the limit of large D values, F(D) approaches to (18) Perrin, F. J. Phys. Radium. 1934, 5, 497. (19) Doi, M.; Edwards, S. F. Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986; Chapter 8 and references therein.
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Notes
Figure 3. A typical logarithmic plot of the temporal change of the angle θ between the fiber axis and the magnetic field (0.8 T) obtained for PE (O) and CF (b). The slope is equal to -τ-1. The length of the fiber is 4 mm both for PE and CF.
Figure 4. Relation between τ-1 and the square of the magnetic field B2 measured for the PE (open symbols) and CF (filled symbols) with different fiber lengths indicated in the figure.
an expression derived with a cylindrical approximation for a fiber.19 This case has been reported in the literature.20 The above equation is rewritten in terms of magnetic flux density B as follows by using the relation, B ) µ0H
τ-1 ) F(D)
χaB2 6µ0η
(8)
If the MKSA unit is employed, then B is expressed in Wb/m2 ()[T]), µ0 ) 4π × 10-7 Wb/(A m), η in Pa s, and τ-1 in s-1. Results and Discussion Orientation of a PE fiber after the later stage of the alignment is displayed in Figure 2a. Its orientation direction is consistent with the report 2 for the magnetic orientation of a PE single crystal suspended in xylene. It is reported that the PE single crystal suspended in xylene aligns in a magnetic field with the c-axis oriented perpendicular to the magnetic field. This indicates that the direction of the c-axis coincides with the direction of the largest diamagnetic susceptibility |χ||. In the case of fiber, there is a symmetry around the fiber axis (parallel to the c-axis) and the other two susceptibilities normal to the fiber axis are averaged to give χ⊥ (
