Langmuir 2004, 20, 5669-5672
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Uniaxial Alignment of the Smallest Diamagnetic Susceptibility Axis Using Time-Dependent Magnetic Fields Tsunehisa Kimura,*,†,‡ Masashi Yoshino,† Tsutomu Yamane,† Masafumi Yamato,† and Masayuki Tobita§ Department of Applied Chemistry, Tokyo Metropolitan University, 1-1 Minami-ohsawa, Hachioji, Tokyo 192-0397, Japan, Tsukuba Magnet Laboratory, National Institute for Materials Science, 3-13 Sakura, Tsukuba, Ibaraki 305-0003, Japan, and Research and Development Center, Polymatech Company, Limited, 5-10-5 Tabata, Kita, Tokyo 114-0014, Japan Received March 12, 2004. In Final Form: May 6, 2004 A diamagnetic particle with magnetic susceptibilities χ3 < χ2 ) χ1 < 0 was subjected to a rotating magnetic field to obtain an alignment of the χ3 axis (the smallest susceptibility axis) in the direction perpendicular to the plane of the rotating magnetic field. A polymer short fiber, whose fiber axis coincides with the χ3 axis, was suspended in a fluid with the same density, and then a rotating magnetic field generated by a rotation of a pair of permanent magnets was applied. The fiber axis, rotating following the applied field, finally ended up with an alignment perpendicular to the plane of the rotating magnetic field. The experimental data on the time course of the alignment was in good agreement with the numerical calculation based on the equation of rotation.
Introduction Materials having diamagnetic anisotropy such as fibers can undergo magnetic alignment. Polymer fibers,1,2 carbon fibers,3,4 cellulose fibers,5,6 carbon nanotubes,7-9 and crystallites10 suspended in a liquid medium can align under a magnetic field. Under a static magnetic field, the alignment occurs so that the axis corresponding to the largest diamagnetic susceptibility becomes parallel to the applied field. The alignment manner of fibers depends on the combination of the magnetic axis and the shape axis. The fiber axis of a short carbon fiber coincides with the axis of the largest diamagnetic susceptibility, resulting in the alignment with its fiber axis parallel to the applied magnetic field. In this case, χ⊥ < χ| < 0, where χ| and χ⊥ are the diamagnetic susceptibilities in the directions parallel and perpendicular to the fiber axis, respectively, with the suffixes | and ⊥ hereafter indicating the directions parallel and perpendicular to the fiber axis, respectively. The quantity defined by χa ≡ χ| - χ⊥ is referred to as the anisotropic diamagnetic susceptibility, and χa > 0 for carbon fiber. On the other hand, a short polyethylene fiber aligns perpendicular to the field because the direction * To whom correspondence should be addressed. E-mail:
[email protected]. † Tokyo Metropolitan University. ‡ National Institute for Materials Science. § Polymatech Company, Limited. (1) Kimura, T.; Yamato, M.; Koshimizu, W.; Koike, M.; Kawai, T. Langmuir 2000, 16, 858. (2) Yamato, M.; Aoki, H.; Kimura, T.; Yamamoto, I.; Ishikawa, F.; Yamaguchi, M.; Tobita, M. Jpn. J. Appl. Phys. 2001, 40, 2237. (3) Timbrell, V. J. Appl. Phys. 1972, 43, 4839. (4) Schmitt, Y.; Paulick, C.; Royer, F. X.; Gasser, J. G. Non-Cryst. Solids 1996, 205-207, 139. (5) Sugiyama, J.; Chanzy, H.; Maret, G. Macromolecules 1992, 25, 4232. (6) Revol, J.-F.; Godbout, L.; Dong, X.-M.; Gray, D. G.; Chanzy, H.; Maret, G. Liq. Cryst. 1994, 16, 127. (7) Fujiwara, M.; Oki, E.; Hamada, M.; Tanimoto, Y.; Mukouda, I.; Shimomura, Y. J. Phys. Chem. A 2001, 105, 4383. (8) Kimura, T.; Ago, H.; Tobita, M.; Ohshima, S.; Kyotani, M.; Yumura, M. Adv. Mater. 2002, 14, 1380. (9) Casavant, M. J.; Walters, D. A.; Schmidt, J. J.; Smalley, R. E. J. Appl. Phys. 2003, 93, 2153. (10) Kawai, T.; Iijima, R.; Yamamoto, Y.; Kimura, T. J. Phys. Chem. B 2001, 105, 8077.
perpendicular to the fiber axis coincides with the largest diamagnetic susceptibility χ⊥. Namely, χ| < χ⊥ < 0 or χa < 0. This difference between a carbon fiber and a polyethylene fiber becomes significant when a number of these fibers, starting with an initial random orientation, are subjected to a static magnetic field. Carbon fibers align uniaxially with respect to the field. In contrast, polyethylene fibers do not reach uniaxial alignment, but they exhibit random distribution within the plane perpendicular to the field (planar alignment). In many application areas, uniaxial alignment is more desirable than planar alignment because physical properties in the direction of the fiber axis are superior to those in the perpendicular direction. To reach a uniaxial alignment of polyethylene type fibers (χa < 0), we need to devise a means to uniaxially align the smallest diamagnetic axis. The use of a time-dependent magnetic field is not quite new. An alternating field is used as a cold crucible induction melting, where the Lorentz force is utilized. Also, it is reported that the rotating magnetic field induces a uniaxial c-axis alignment of a droplet of carbon pitch mesophase;11,12 the c axis corresponds to the smallest susceptibility axis. In this paper, we apply a rotating magnetic field on fibers suspended in a liquid to achieve the uniaxial alignment. The equation of rotation is derived and solved numerically to compare with the experimental results. It is shown that the alternating application of the field from two different directions brings about the same effect. Experimental Section A home-built apparatus generating a rotating magnetic field was used. A pair of permanent magnets (3.5 × 2.5 × 3.0 cm Nd-Fe-B permanent magnets), with the north and south sides put face to face, separated by 1.9 cm, were mounted on a rotating disk which allows rotation ranging 1-30 rpm. The field strength at the center was about 0.55 T. A sample cell (1.2 × 1.2 × 4.5 (11) Singer, L. S.; Lewis, R. T. 11th Biennial Conference On Carbon, Extended Abstracts; National Technical Information Service, U.S. Department of Commerce: Springfield, VA, 1973; p 207. (12) Imamura, T.; Honda, H.; Kakiyama, H. Tanso 1974, No. 76, 20 (in Japanese).
10.1021/la049347w CCC: $27.50 © 2004 American Chemical Society Published on Web 06/04/2004
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Figure 1. Definition of the principal axes χ1, χ2, χ3 of the diamagnetic tensor and the principal axes L1, L2, L3 of the hydrodynamic tensor L with respect to the symmetry axis n of the particle. cm) was hung from the top in the middle of two magnets. The motion of the fiber samples suspended in a liquid was recorded by a Keyence charge-coupled device camera (magnification of ×20). The length and angle measured on recorded pictures were used to determine various angles required for the comparison with the theoretical calculation. A long nylon fiber (a Toray fishing gut, Ginrin, diameter of 0.370 mm) was cut into 1.30, 1.60, and 1.90 mm and suspended in a NaCl solution whose density is adjusted to the density of the fiber. The magnetic torque acting on the fiber was measured using a Tamagawa torque meter to obtain the value of χa.
Theory Let us consider a particle rotating under an externally applied torque N. In the case that the particle is immersed in a viscous medium and, hence, the inertia term is ignored, the equation of rotation is described as a balance of N and a hydrodynamic torque M:
M+N)0
(1)
We consider here that N is caused by a time-dependent magnetic field B(t) acting on a particle with diamagnetic susceptibilities χ1 ) χ2 ≡ χ⊥ and χ3 ≡ χ|. We assume that the χ| direction coincides with the uniaxial direction n of the particle, for example, the axis of a short fiber, a rod, a prolate ellipsoid, and so forth, and the χ⊥ directions are perpendicular to n (Figure 1). A magnetic torque N is then expressed as
Figure 2. Polar coordinate describing the director n and a unit vector b in the direction of the rotating magnetic field. ξ is the angle between n and b.
parallel to the direction of L⊥ in the present case, eq 4 is rewritten as
Ω ) (V/L⊥)µ0-1χaη-1(n‚B)(n × B)
(5)
Using the relation dn/dt ) Ω × n, we obtain
dn/dt ) τ-1(n‚b)[b - n(n‚b)]
(6)
where b ) B/B and τ-1 is defined as
τ-1 ) (V/L⊥)µ0-1χaη-1B2
(7)
In the case of a sphere of radius a, we have V/L⊥ ) (4π/3)a3/8πa3 ) 1/6. In the case of a prolate ellipsoid with the aspect ratio of D, we obtain V/L⊥ ) F(D)/6, where F(D) is the same function as in the previous paper (eq 6 in ref 1). Using polar coordinates to describe n and b (see Figure 2), we rewrite eq 6 in terms of the angles θ and φ as follows:
dθ/dt ) (2τ)-1 cos2(φ - ωt) sin 2θ
(8a)
dφ/dt ) -(2τ)-1 sin 2(φ - ωt)
(8b)
Here, it is assumed that the field B is rotating on the xy plane at the angular velocity of ω (rad/s). Results and Discussion
-1
-1
2
N ) Vµ0 χa(n‚B)(n × B) ) Vµ0 χaB cos ξ sin ξ e (2) Here, µ0 is the magnetic permeability of a vacuum, V is the volume of the particle, χa is the anisotropic diamagnetic susceptibility, ξ is the angle between n and B, e is a unit vector parallel to n × B, and B is the norm of B. The hydrodynamic torque M acting on the particle, when it is rotating at an angular velocity Ω, is written as
M ) -ηLΩ
(3)
Here, η is the viscosity of the medium surrounding the particle and L is a tensor that is dependent on the particle shape. In the coordinate system imbedded on a particle, L is diagonal, with its components being L1 ) L2 ≡ L⊥ and L3 ≡ L|, as shown in Figure 1. Combining eqs 1-3, we obtain
Ω ) Vµ0-1χaη-1(n‚B)L-1(n × B)
(4)
where L-1 is the inverse tensor of L. Because n × B is
In the stationary state, where dθ/dt ) 0, we have two solutions for eq 8a: θ ) 0 and π/2. The first solution indicates that n aligns in the direction of the z axis in a long run, irrespective of the initial condition or the rotation speed. The director n remains on the xy plane only when starting with the initial value of θ ) π/2. This happens because there is no torque component pushing n toward the z axis if n is on the xy plane (θ ) π/2). The second solution, θ ) π/2, is realized only when the initial condition is θ ) π/2. For any other initial conditions ranging 0 < θ < π/2, the value of θ decreases in time because dθ/dt < 0 in the case of τ < 0 (i.e., χa < 0), which finally leads to the stationary state of θ ) 0. In experiments, the initial value of θ cannot be controlled to be exactly π/2 but may fluctuate because of an inevitable convection of the medium and the thermal fluctuation of the particle. Therefore, the final stationary state is θ ) 0 for any actual experimental cases. From eq 8b we obtain sin 2δ ) 2τω for a stationary state of dδ/dt ) 0, where we define δ as δ ≡ φ - ωt. This stationary state is possible only if |τω| e 1/2. The physical meaning of this stationary state is that n rotates synchronously around the z axis at the same angular velocity ω of the applied rotating field. If this inequality does not
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Figure 5. Uniaxial alignment of χa < 0 fibers. Fibers initially oriented randomly (a) turn out to align uniaxially (b) by applying a static field (4 min) followed by exposure to a rotating field for 1 min.
Figure 3. Experimental results for the temporal change of θ(t), δ(t), and ξ(t) obtained for a 1.9-mm fiber at 3 and 5 rpm, compared with the numerical calculation based on eq 8.
corresponds to µ0-1χaη-1B2/6. From separate evaluation13 of the viscosity η () 1.5 × 10-3 Pa s) and the anisotropic diamagnetic susceptibility χa () -1.1 × 10-7), we obtained µ0-1χaη-1B2/6 ) -2.9 s-1 which should be compared to K ) -4.5 s-1. The agreement is good despite the experimental errors involved in the present experiment. Figure 5 demonstrates a uniaxial alignment attained by a rotating magnetic field. We apply a static field (4 min) to a suspension of the fibers (diameter of 0.104 mm, length of 1 mm) that are randomly distributed (Figure 5a), followed by the application of a rotating field (ω ) 5 rpm). We finally obtain a uniaxial alignment (Figure 5b). Rotating field is not the only solution to the achievement of the alignment of the smallest diamagnetic susceptibility. A time-dependent field, a set of static fields (Bx, 0, 0) and (0, By, 0) in the x and y directions, respectively, applied alternately in these two directions for a given period of time, can meet the purpose. The response of n upon a static field on the x axis and the y axis is given by solving eq 8a,b to give
tan φ ) tan φ0 exp(-t/τ) Value of |τ-1
Figure 4. | plotted against the aspect ratio D. The solid line shows F(D) multiplied by 4.5 s-1.
hold, the rotation of the field passes n. Once this stationary state is reached, eq 8 becomes independent of φ, being reduced to an equation describing an alignment under a static field (see eq 1 in ref 1, where the definition of θ differs by π/2). Experiments are carried out for fibers with the lengths 1.3, 1.6, and 1.9 mm at rotation speeds of ω ) 3 and 5 rpm. In Figure 3 are shown the results for a 1.9-mm fiber at rotation speeds of 3 and 5 rpm. The angles of θ(t), δ(t), and ξ(t) (the angle between the director n and b, Figure 2) are shown. The value of τ () -1.3 s) is determined by a stationary value of φ for the 3-rpm experiment and used to simulate the experimental result. In the case of the 5-rpm experiment, this same τ value is used because the φ value does not reach the stationary value. At this rotation speed, the rotating field passes the director n. In both cases, the calculated results are in good agreement with the experimental results. No passing occurs for the other experiments for 1.3- and 1.6-mm fibers at 3 and 5 rpm. The value of τ is determined using the stationary value of φ to simulate the experimental results. Fairly good agreements are observed. In many cases, θ ends up being 0. In Figure 4, |τ-1| values thus determined are plotted against the aspect ratio D, along with the theoretical curve for F(D) multiplied by a factor of |K| ) 4.5 s-1 to obtain the best fit to the experimental data. From eq 7, this factor
(9a-1)
tan θ ) tan θ0 x{1 + exp(2t/τ) + [-1 + exp(2t/τ)] cos 2φ0}/2 (9a-2) for the case of a static field applied in the x direction and
tan φ ) tan φ0 exp(t/τ)
(9b-1)
tan θ ) tan θ0x{1 + exp(2t/τ) + [1 - exp(2t/τ)] cos 2φ0}/2 (9b-2) for the case of a static field applied in the y direction, where τ has the same meaning as in eq 7 and (θ0, φ0) is the initial condition. Using these equations, we find that the alternation of static magnetic fields in the x and y directions can push n to the z axis when τ < 0. This allows us to reach a uniaxial alignment of fibers with χa < 0 initially oriented randomly. This situation is intuitively understood by considering a simple two-step case. Namely, we first apply the field on the x axis until a planar alignment on the yz plane is reached. Then, we turn off the field on the x axis and turn on the field on the y axis. Because there is not a torque acting to make fibers deviate out from the yz plane, all the fibers align perpendicular to the y axis, staying on the yz plane. This means that all the fibers align in the z direction. The effect of the Brownian motion becomes larger if the particle size becomes smaller and then disturbs the motion due to the applied field. Therefore, the scheme described (13) Kagaku Binran; The Chemical Society of Japan: Tokyo, 1958 (in Japanese).
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here is only effective for particles with sizes larger than some critical value. Comparison of the thermal energy and the anisotropic magnetic energy leads us to a criterion for these phenomena to be effective, that is, VχaB2/(2µ0) > kBT. A typical size in the linear dimension is about several tens of nanometers in the case of B ≈ 10 T at room temperature. The scheme described here is also applied to whatever systems contain or form anisotropic structures if their size satisfies the above criterion. Examples are carbon pitch mesophase,11,12 liquid crystalline polymers, crystalline polymers, suspensions of crystallites, and so forth. Conclusions It is demonstrated experimentally and theoretically that the time-dependent magnetic fields, including a rotating magnetic field and an alternating exposure of static magnetic fields from different directions, provide an effective means to align the smallest diamagnetic susceptibility axis χ3 (< χ2 ) χ1 < 0) in a specific direction.
Letters
In the case that χ3 coincides with the fiber axis, a uniaxial alignment of fibers becomes possible, which is never possible under a static magnetic field. For example, polyethylene fibers, whose smallest diamagnetic axis (χ| ) χ3) and the largest one (χ⊥ ) χ1, χ2) are parallel and perpendicular to the fiber axis, respectively, that is, χa ) χ| - χ⊥ < 0, align uniaxially under a rotating magnetic field. This method is applied not only to fiber systems but also in many other fields where “precise” alignment is necessary. Acknowledgment. This work was partially supported by Japan Society for the Promotion of Science through the Research for the Future Program. This work was partially supported by Grant-in-Aid for Scientific Research on Priority Area “Innovative utilization of strong magnetic fields” (Area 767, no. 15085207) from MEXT of Japan. The authors thank Tamakawa Co., Ltd., for torque measurement. LA049347W
