Analysis of Uniaxial Alignment Behavior of Nonmagnetic Materials

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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’’.
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Appendix
B: Magnetic flux density (T)
d: Diameter (m)
H: Magnetic field strength (Am1 )
l: Length of fiber (m)
r: Radius (m)
t: Time (s)
T: Magnetic torque (Nm)
U: Magnetization energy (J)
V: Volume (m3 )
x; y; z: Cartesian coordinate (-)
: Angle (rad)
: Magnetic anisotropy ja c j (-)
: Viscosity (kgm1 s1 )
: Angle (rad)
0 : Magnetic permeability in vacuum (Hm1 )
: Density (kgm3 )
R ; S : Alignment time (s)
’: Angle (rad)
: Magnetic susceptibility (-)
a : Magnetic susceptibility of a axis (-)
c : Magnetic susceptibility of c axis (-)
? : Magnetic susceptibility of hard axis (-)
!: angular velocity (rads1 )
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