Orientation Behavior of Carbon Fiber Axes in Polymer Solutions under

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J. Phys. Chem. C 2008, 112, 15611–15622

15611

ARTICLES Orientation Behavior of Carbon Fiber Axes in Polymer Solutions under Magnetic Field Estimated in Terms of Orientation Distribution Function Yumiko Nakano and Masaru Matsuo* Graduate School of Humanities and Sciences, Nara Women’s UniVersity, Nara 630-8263, Japan ReceiVed: March 3, 2008; ReVised Manuscript ReceiVed: July 28, 2008

Orientation behavior of carbon fibers (CFs) in polymer solution/gel under magnetic field was estimated in terms of the orientation distribution function. The estimation was done by a series of spherical harmonics of the orientation distribution function of the reciprocal lattice vector of the (002) plane measured for the composite films. Despite a number of papers concerning the orientation of CFs and carbon nanotubes under magnetic field, all estimations have been done in terms of the second-order orientation factor of the (002) plane. The work described in this paper was first successful for the direct estimation of orientation of CF axes within the dried film in terms of orientation function by considering a random orientation of the CF axes around the magnetic field direction. In this experiment, poly(vinyl alcohol) (PVA) was adopted as the solution to disperse CFs and to prepare the composite films by evaporating solvent. The normalized function obtained experimentally was analyzed in comparison with the magnetic orientation of CFs in solution or gel at the equilibrium state calculated by the magnetic energy of CF. Furthermore, the orientation of the CFs up to the equilibrium state in the given systems was estimated as a function of time by a common diffusion equation. The calculated results indicated that the preferential orientation of CFs with respect to the magnetic field direction becomes promoted with elapsing time. This tendency became significant, when the contents of CFs and/or PVA were less than the critical concentration within the dispersion solution. The experimental and theoretical results, however, provided that the preferential orientation degree became lower drastically, when the PVA content and/or the CF content were beyond their critical concentration. This was due to a drastic decrease in viscosity of the dispersion solution before gelation/crystallization. 1. Introduction A number of established papers have reported that the magnetic effect is proportional to the square of the magnetic flux density B and the origin of the diamagnetism is the induced magnetization caused by the induced motion of electrons under the applied magnetic field.1-20 That is, the magnitude of the magnetization M induced on a material is proportional to the applied strength H; i.e., M ) χH, where χ is the diamagnetic susceptibility. The interaction of M with the applied field causes a repulsive force, repelling the particle away to the direction of a decreasing field strength. If the force is equivalent to the gravitational force, the particles levitate in the air. Accordingly, when a material has an anisotropy in diamagnetism, a magnetic torque acts on it without the rotation. The origin of the magnetic anisotropy is traced back to chemical bonds. The magnetic anisotropy due to chemical bonds provides that diamagnetic susceptibilities of the C-C bond along the bond (|) are smaller than those perpendicular to the bond (⊥). Accordingly, the anisotropic diamagnetic susceptibility defined by χ| - χ⊥ (χ| < χ⊥ < 0) is negative, and then the C-C bond tends to align in the direction perpendicular to the applied field.21 Here it may be noted that the orientation of the particles under magnetic * To whom correspondence should be addressed. Telephone and Fax: +81-742-20-3462. E-mail: [email protected].

field needs their volume to ensure thermal energy higher than thermal disturbance. The orientation of crystallites and particles under electric and magnetic field have been evaluated in terms of the second-order orientation factor. The orientation under magnetic fields is related to the apparent permanent dipole moment, while the orientation under electric field is due to the excess electrical polarizabilities along symmetry and transverse axes in addition to the apparent permanent dipole moment. The comparison between theoretical and experimental orientations was done by birefringence.22-24 However, the orientation factor obtained by birefringence is sort of the second moment of the orientation function. The preferential orientation of carbon fibers (CFs) within a polymer matrix has been realized by elongation of the composite films.25,26 Actually, carbon nanotube (CNT) and ultrahigh molecular weight polyethylene (UHMWPE) composites could be elongated more than 100 times, when the composites were prepared by gelation/crystallization and the drawn composites provided high electric conductivity and high modulus. However, the ultradrawn method is not effective in obtaining the composites where the CFs and CNTs are oriented predominantly with respect to the film normal (thickness) direction. To realize the preferential orientation, magnetic and electric fields must be applied to the direction parallel to the film thickness direction.

10.1021/jp8018793 CCC: $40.75  2008 American Chemical Society Published on Web 09/13/2008

15612 J. Phys. Chem. C, Vol. 112, No. 40, 2008 The orientation of CFs5,6 and CNTs9-14 under the magnetic field has been evaluated from the two methods. One is the direct observation by scanning electron microscopy (SEM),9 and the other is due to the prediction from orientation of the reciprocal lattice vector of the (002) plane within the graphite crystal unit.13,14 The measurements were done for the composite film prepared by evaporating solvents from polymer gels. A completely mathematical description for the orientation distribution function of crystallites was proposed by Roe and Krigbaum27-29 using the orientation functions measured for many crystal planes. The description was applied to polyethylene crystallites with orthorhombic unit by Krigbaum et al.28 and Matsuo et al.,30-33 to cellulose,34 poly(vinyl alcohol) (PVA),35,36 and nylon 637 crystallites with monoclinic unit by Matsuo et al., and poly(butylene terephthalate) (PBT)38 and poly(ethylene terephthalate) (PET)39 crystallites with a triclinic unit by Matsuo et al. The evaluation methods have been developed to obtain the three principle crystallographic axes (a-, b-, and c-axes) from the predicted orientation distribution function of crystallites. In doing so, the orientation functions of the reciprocal lattice vectors must be measured by X-ray diffraction precisely. This paper, as a first trial, is concerned with the application of the above mathematical evaluations to the derivation of the orientation function of the CF axis from the function of the reciprocal lattice vector of the (002) plane, when the graphite crystallites with a hexagonal crystal unit are oriented randomly around their CF axis as well as around a magnetic field direction. The composites were prepared by gelation/crystallization of PVA solution and then by evaporation of dimethyl sulfoxide (DMSO) and water (H2O) mixed solvent. The orientation of CFs with respect to the magnetic field direction was estimated by the orientation distribution function expressed in a series of spherical harmonics. The predicted functions were discussed with the theoretical function at the equilibrium state estimated by the energy associated with the volume anisotropic diamagnetic susceptibility and magnetic field. Further prediction was obtained as a function of time by diffusion equation associated with the rotational fractional coefficient. The orientation function was measured for the PVA-CF composite films prepared by evaporating solvent from gel on the basis of the assumption that the orientation of CFs within gels is hardly affected by the evaporation process of the solvent under magnetic field. Incidentally, it is impossible to obtain the orientation function of the reciprocal lattice vector of the (002) plane for the gel state by X-ray diffraction technique. The measurement was done under weak magnetic field to avoid perfect orientation of CF axes because of the difficulty in expanding a very sharp function of the reciprocal lattice vector of the (002) plane of graphite crystallites to a series of spherical harmonic without termination error. 2. Experimental Section 2.1. Preparation of PVA-CF Composites. In this present work, PVA (polymerization of 2000 and 98% hydrolyzed) was used as matrix. The CFs (MGII from Toho Tenax Co. Ltd.) used in the previous paper40 were used again in this experiment, and the characteristics have been described in detail. Their average length and average diameter were determined to be 45 and 5.0 µm, respectively, from the observation by scanning electron microscopy (SEM). The average volume of a CF was 8.8 × 102 µm3. The density was 2.10 g/cm3. PVA-CF composites were prepared by gelation/crystallization from solutions in dimethyl sulfoxide and water mixtures. The DMSO/water composition was set to be 60/40, assuring

Nakano and Matsuo

Figure 1. Specific viscosity ηsp of PVA solutions with the indicated content.

rapid gelation and stiff gel reported elsewhere.41 The contents of CFs against the mixed solvent were 0.3, 0.5, 1.0, and 2.5%, in which the concentration of PVA against the mixed solvent was fixed to be 10 g/(100 mL). Beyond 10 g/(100 mL), the uniform dispersion of CFs could not be assured macroscopically. To obtain further information for the viscosity dependence of PVA, 13 g/(100 mL) solution was prepared, in which the CF content against the mixed solvent was 1.0%. For the sample preparation, CFs were added into a mixed solvent of DMSO/water and treated for 1 h with ultrasonic treatment at room temperature. CFs were well-dispersed in solvent. After then, the PVA powders were put into the dispersion solution, and the mixture was stirred and heated to 105 °C under nitrogen and maintained for 40 min at 105 °C. The hot homogenized solution was poured into a Petri dish set under the magnetic field. The magnetic field (0.46 T) was produced by the magnetic-wave device (Niroku Seisakusho Co. Ltd.) in which the Petri dish was set between two permanent magnetic plates, the magnetic field direction being vertical. The gelation occurred within 2 h after cooling the solution to room temperature. The gels were kept at room temperature to evaporate the mixed solvent for 3 days, and then they were immersed in a water bath for 1 day. After then, the gels were dried at room temperature. The orientation behavior of CFs within the resultant composite was estimated by X-ray diffraction technique. Here we must emphasize that the gelation is very important to maintain the original orientation of the CFs in the solution. 2.2. Measurements and Results. SEM. Dispersion of CFs in the composite films, which were prepared by evaporating solvent under the magnetic field, was observed with field emission scanning electron microscopes (FE SEM JSM-6700F of JEOL) at accelerating voltage of 5 kV. Samples were spit under liquid nitrogen to get a natural cross-section. Viscosity of the CF Dispersed Solutions. Viscosity measurements were carried out with an Ubbelohde type capillary viscometer, put into silicon oil baths controlled at 105 °C. The specific viscosity ηsp was determined as the average value of three measurements, in which flow time with a variation not exceeding (0.1 s was taken for each solution. Incidentally, it was impossible to measure the viscosity of the CF dispersion solutions, precisely. Figure 1 shows the specific viscosity against PVA concentration. The viscosity increases drastically with increasing PVA content. These data are important for pursuing theoretical

Carbon Fiber Axes in Polymer Solutions

J. Phys. Chem. C, Vol. 112, No. 40, 2008 15613

calculation of the orientation distribution function of the CFs. Unfortunately, the measured values of ηsp were too scattered to obtain the creditable values, when the CFs were admixed into PVA solution. Even so, it was confirmed that the 10% PVA solution (against the solvent) containing CFs in the range of 0.3-2.5% contents did not provide a drastic increase in the ηsp values. Orientation Distribution Function of Oriented CFs under Magnetic Field. X-ray measurements were carried out using a 12 kW rotation-anode X-ray generator (Rigaku RAD-rA) with monochromatic Cu KR radiation (wavelength of 0.154 nm). As discussed elsewhere,32,35 the orientation of crystallites around the film normal direction cannot be obtained by using the usual X-ray diffraction technique. We developed a small but refined instrument to stack a number of thin films as shown in Figure 2. The wide-angle X-ray diffraction (WAXD) patterns could be observed easily when the X-ray beam was directed parallel to the film surface. As for the diffraction intensity in such a stacked condition, measurements of the X-ray diffraction intensity could be performed using a horizontal scanning type goniometer, operating at a fixed time step scan of 0.1°/(40 s) over a range twice the Bragg angle 2θB from 10 to 35°. The intensity distribution was measured as a function of a given rotational angle θj by rotating around the film thickness direction at 1° interval from 0 to 90°. The sample arrangements at θj ) 0 and 90° are shown in Figure 2a,b, respectively. Under the evaporation process of the solvent, the magnetic field direction corresponded to the film thickness direction. Corrections of X-ray diffraction intensity were made for air scattering, polarization, absorption, amorphous contribution, and background noise. The intensity curve thus obtained was assumed to be due to the overlapped contribution from PVA crystallites and graphite crystallites. The intensity curveIcryst.(2θB) was separated into the contribution from the (110), (11j0), and (200) planes of PVA crystallites and the (002) plane of graphite crystallites within CFs. The separation was done by assuming that each peak had a symmetric form given by a Lorenzian function of eq 1, where Ioj is the maximum intensity of the jth peak.

Icryst.(2θB) )

Ijo

∑ 1 + (2θj - 2θ )2 ⁄ β 2 j

o

B

(1)

j

Here βj is the half-width of the jth peak at half of the peak

intensity, and θjo is the Bragg angle at which the maximum peak of the jth peak appears. Using the same process at a given θj in the range from 0 to 90°, the intensity distribution Icryst.(2θB) can be determined for the respective jth plane after integrating Icryst.(2θB) by 2θB at each θj, and consequently the orientation distribution function qj(cos θj) of the jth reciprocal lattice vector may be given by

2πqj(cos θj) )

Ij(θj)

∫0π Ij(θj) sin θj dθj

(2)

The orientation distribution function of crystallites can be calculated on the basis of the orientation functions of the reciprocal lattice vectors which were measured by X-ray diffraction according to the simple application of the method proposed by Roe and Krigbaum.27,28 In this paper, the orientation distribution function of the reciprocal lattice vector of the (002) plane for the CFs is adopted to obtain the orientation function of the CF axis for the composites prepared by different conditions. 3. Results and Discussion Figure 3 shows wide-angle X-ray diffraction patterns (WAXD) of the dried composites with 0.3, 0.5, 1.0, and 2.5% CF contents, when an X-ray beam was directed parallel to the film surface (end view), and the technique is shown in Figure 2. The WAXD patterns on the left and right sides were taken for the composites prepared under no applied and applied magnetic fields, respectively. The WAXD patterns reveal that PVA chain axes and CF axes are oriented predominantly parallel to the film surface, which is independent of magnetic field. Namely, the diffraction arcs from the (110) and (11j0) planes were detected in the horizontal and vertical directions, respectively, indicating that the (110) and (11j0) planes are predominantly oriented parallel and perpendicular to the film surface, respectively. This indicates the preferential orientation of the crystal chain axes of PVA crystallites parallel to the film surface. In contrast, the diffraction from the (002) plane of graphite crystallites within CFs shows the arcs in the vertical direction under no magnetic field indicating preferential orientation of CF axes parallel to the film surface, while the arcs appeared in the horizontal direction under the magnetic field indicating preferential orientation of CF axes perpendicular to the film surface. In the evaporation process of

Figure 2. Number of the stacked films to measure X-ray diffraction intensity distribution as a function of the twice Bragg angle at (a) θj ) 0° and (b) θj ) 90°.

15614 J. Phys. Chem. C, Vol. 112, No. 40, 2008

Figure 3. WAXD patterns (end view) for the PVA-CF composites, in which the films were prepared from the 10% PVA solution with 0.3, 0.5, 1.0, and 2.5% CF contents under (a) no magnetic field and (b) magnetic field.

solvent, the parallel orientation of the CF axes under no magnetic field is due to the cylindrical structure with large aspect ratio, while the characteristic orientation perpendicular to the CF axes under the magnetic field is due to the anisotropic diamagnetic susceptibility of CFs. The characteristic properties of CFs shall be discussed by using eq 2 in terms of the orientation distribution function in comparison with the experimental results measured on the basis of the procedures described in the Experimental Section. Figure 4 shows SEM images of the composites containing 0.3, 1.0, and 2.5%, in which the concentration of CFs against the solvent was 10% PVA. It is seen that the preferential orientation of CFs oriented with respect to the magnetic field direction were maintained under the gelation/crystallization process of PVA by the evaporation of mixed solvent. The preferential orientation of CFs is significant at 0.3-1.0% CF contents, but it is obviously poor at the 2.5% CF content. This is thought to be probably due to collision between CFs under the preferential orientation process, which indicates the existence of an optimum maximum content of CFs in the dispersed solution with a fixed PVA concentration. To make clear conclusive evidence, the quantitative estimation of the CF orientation under the gelation/crystallization and solvent evaporation processes are studied in terms of the orientation function of CF axes. 3.1. Orientation Distribution Function of CF Axes Estimated from the Orientation Function of the Reciprocal Lattice Vector of the (002) Plane. Figure 5 shows the geometrical interrelations of Cartesian coordinates O-X1X2X3 and O-U1U2U3 fixed within the bulk specimen and crystal structural unit, respectively. The orientation of the structural unit within the space of the specimen may be specified by using three Euler angles, θ, φ, and η, as shown in diagram a. The

Nakano and Matsuo

Figure 4. SEM images of the PVA-CF composites, in which the films were prepared from the 10% PVA solution with 0.3, 1.0, and 2.5% CF contents under magnetic field.

angles θ and φ, which define the orientation of the U3-axis of the unit within the space, are the polar and azimuthal angles, respectively, and η specifies the rotation of the unit around its own U3-axis. The orientation of the jth axis with respect to O-X1X2X3 is given by the polar angle θj and the azimuthal angle φj, as shown in panel b. Moreover, the jth axis within the structural unit is specified by the polar angle Θj and the azimuthal angle Φj shown in panel c, in which Θj and Φj are given defined values. Based on the three Cartesian coordinates, a model for the orientation of a CF within the bulk specimen can be proposed as panel d. Here the proposed model can be used in the case where the reciprocal lattice vector of the (002) plane is perpendicular to the CF axis perfectly. Without this assumption, the following quantitative evaluation cannot be discussed. A complete mathematical description for the orientation of crystallites can be treated by using the orientation functions of the reciprocal lattice vectors of crystal planes. In accordance with the general evaluation by Roe and Krigbaum,27-29 Matsuo et al. proposed the concrete general description for the relationship between the orientation distribution function of crystallites and the orientation factors of the crystal planes measured by X-ray diffraction, when crystallites are oriented randomly around the reference axis (X3-axis) of the speciemn.38 That is, l

j Fl0 ) Fl00Pl(cos Θj) + 2

n)! (Fl0n cos(nΦj) ∑ (l(l +- n)!

n)1

Gl0n sin(nΦj))Pln(cos Θj) (3)

Carbon Fiber Axes in Polymer Solutions ∞

[ {

2l + 1 1 4π2ω(cos(θ,η)) ) + 2 Fl00Pl(ξ) + 2 2 l)2

∑

l

2

J. Phys. Chem. C, Vol. 112, No. 40, 2008 15615 ∞

∑

}]

n)! (Fl0n cos(nη) + Gl0n sin(nη))Pln(ξ) ∑ (l(l +- n)!

n)1

l

(4)

∑

(5)

where ω(cos(θ,η)) and qj(cos θj) are the orientation distribution function of crystallites and the orientation function of the reciprocal vector of the jth crystal plane measured by X-ray diffraction technique, respectively. WithFjl0, Θj, and Φj thus determined for the jth crystal plane, the coefficients Fl0n and Gl0n can be calculated by solving simultaneous equations of eq 3, changing j up to at least 2l + 1. In turn, the orientation distribution function of crystallites, ω(cos(θ,η)) can be determined from eq 4 with the approximation of finite series of expansion up to l ) (j - 1)/2 (j, odd; l, even), instead of the infinite series of expansion. As an example, in the case of the orthorhombic unit such as polyethylene, a given vector rj at Θj and Φj is equivalent to following seven vectors: rj (1) at Θj and -Φj, (2) at Θj and π - Φj, (3) at Θj and π + Φj, (4) at π - Θj and Φj, (5) at π Θj and -Φj, (6) at π - Θj and π + Φj, and (7) at π - Θj and π - Φj. Accordingly, n must be even, and consequently Gl0n becomes zero. Thus, eqs 3 and 4 reduce to30-33 ∞

j Fl0 ) Fl00Pl(cos Θj) + 2

2

n)! Fl0nPln(cos Θj) cos(nΦj) ∑ (l(l +- n)! l)2

(6) and

Figure 5. Cartesian coordinates illustrating the geometrical relation. (a) Euler angles θ and η which specify the orientation of coordinate 0-U1U2U3 of the structural unit with respect to coordinate 0-X1X2X3 of the specimen. (b) Angles θj and φjwhich specify the orientation of given jth axis of the structural unit with respect to the coordinate 0-X1X2X3. (c) Angles Θj and Φj which specify the orientation of the given jth axis of the structural unit with respect to the coordinate 0-U1U2U3.. (d) Geometrical arrangement of the reciprocal lattice vector of the (002) plane and fiber axis within the film coordinate 0-X1X2X3.

]

n)! Fl0nPln(cos θ) cos(nη) ∑ (l(l +- n)!

n)2

∞

2l + 1 j 1 2πqj(cos θj) ) + 2 Fl0Pl(cos θj) 2 2 l)2

[

2l + 1 1 4π2ω(cos(θ, η)) ) + 2 Fl00Pl(cos θ) + 2 2 l)2

(7)

in which l becomes even. As for a CF, the reciprocal lattice vector of the (002) plane takes a random orientation around the magnetic field direction, in which the rotational angle η specifies a random rotation around the U3-axis in Figure 5a and Θj takes 90° in Figure 5c. Then, a schematic diagram of a CF can be represented as Figure 5d. Equations 6 and 7 reduce to j Fl0 Fl00 ) Pl(1)

(8)

∞

2l + 1 1 2πω(cos θ) ) + 2 Fl00Pl(cos θ) 2 2 l)2

∑

(9)

The orientation factorF[002] of the reciprocal lattice vector of l0 the (002) plane in eq 8 can be obtained by X-ray diffraction directly, and the predicted orientation function ω(cos θ) of a CF is estimated by eq 9. Here, we must emphasize that the orientation function of CF axes can be obtained by eqs 8 and 9 in a very special case that most of the c-axes of graphite crystallites are aligned perfectly along a CF axis. However, actual orientation of the c-axes has the slight fluctuation,42 and further detailed theory must be taken into consideration as future work. Panels a and b in Figure 6 show the orientation distribution function 2πqj(cos θj) of the reciprocal lattice vector of the (002) plane estimated for the PVA-CF films by using eq 2. The functions in panel a were measured for the composite films containing 0.3, 0.5, 1.0, and 2.5% CF contents prepared from 10% PVA solution, while the two functions in panel b for 10 and 13% PVA solutions containing 1.0% CF are represented as the comparison with panel a. On the other hand, panels c and d show the orientation distribution function 2πω(cos θ) of the CF axes predicted by eqs 8 and 9. The functions of the CF axes in panels c and d were calculated from the functions for the (002) plane in columns a and b, respectively, by using eqs 8 and 9. The estimation of CF orientation in terms of the distribution function has never been reported, and the treatment must be done by eqs 8 and 9 on the basis of the established general concept for the orientation distribution function of crystallites.29,38 Incidentally, the very small waves of the function 2πω(cos θ) in the range of θj ) 15-50° that are attributed to the vibration of the Legendre polynomial Pn(u) appeared at higher order n. As shown in panel c for 10% PVA solution, the orientation functions of the CFs are almost independent of CF contents up to 1.0% but the function becomes much broader at 2.5%. This indicates that the collision between adjacent CFs becomes considerable at 2.5%, but the magnetic field is too weak to overcome the collision and to achieve high orientation of CF axes. As shown in panel d, the orientation of CF axes becomes less pronounced with increasing PVA content at 1.0% CF content. Namely, the CF orientation at 13% PVA solution is much lower than that at 10% PVA solution. This is obviously due to a drastic increase in viscosity of the dispersed solution shown in Figure 1.

15616 J. Phys. Chem. C, Vol. 112, No. 40, 2008

Nakano and Matsuo

Figure 6. Orientation distribution functions of 2πqj(cos θj) of the reciprocal vector of the (002) crystal plane, in which the functions in panel a were related to dispersion solutions containing CFs with the indicated contents and 10% PVA and those in panel b were related to dispersion solutions containing 1.0% CF admixed in 10 and 13% PVA. Orientation distribution function of 2πqj(cos θj) of CF axes calculated by using eqs 8 and 9, in which the curves in panels c and d were obtained from 2πω(cos θ) of the functions of the (002) plane in columns a and b, respectively.

3.2. Orientation Function of CFs at the Equilibrium State. It is well-known that the magnetic orientation of a single CF with susceptibility anisotropy at the equilibrium state may be given by

ω(θ) )

e-U⁄κT

∫02π ∫0π e-U⁄kT sin θ dθ dη

χaB2V cos2 θ χ⊥B2V 2µ0 2µ0

2πω(θ) )

(10)

eγ(cos

∫0π eγ(cos

2

2

θ)

θ)

(12) sin θ dθ

where

where k and T are the Bolzmann constant and the absolute temperature, respectively, and U is the magnetic energy under the applied magnetic flux density B ()(1 + χ)H), which is given by

U)-

side in eq 11 is independent of the profile of ω(θ) (ω(cos θ)). Then, eq 10 may be given by

(11)

where µ0 and V are the magnetic permeability of vacuum and the volume of the particle under consideration, and χa is associated with anisotropic diamagnetic susceptibility related to |χ| - χ⊥|, in which χ| and χ⊥ are the anisotropic susceptibilities parallel and perpendicular to the CF, respectively. θ is the angle between the CF and the applied magnetic field directions. Generally, it may be noted that a macroscopic orientation can occur when the first term exceeds the thermal energy kT represented as V > 2kTµ0/χaB2. This indicates that there exists the estimation of minimum critical volume V needed for the orientation given as a function of the applied magnetic flux density B and the anisotropic diamagnetic susceptibility χa. Substituting eq 11 into eq 10 to calculate the orientation function ω(θ) of the CFs, the second term on the right-hand

γ ) χa

V B2 2µ0kT

(13)

Figure 7 shows the curves of orientation functions of CFs calculated by eq 12 at the equilibrium state, and the functions shown in Figure 6c,d were replotted as open circles, in which the PVA concentration was 10%. The function ω(cos θ) provides a sharper profile with an increase in γ. The theoretical curves are in good agreement with the results predicted for four kinds of CF content by selecting the value of γ, but they are slightly broader than the predicted results. The difference between the profiles is due to two reasons. One is the experimental error of the orientation distribution function of the (002) crystal plane estimated by using eqs 1 and 2. Namely, the orientation function of the reciprocal lattice vector of the (002) plane contains experimental error, for example, on estimating the baseline of the scattering from the amorphous phase, air scattering, and the fluctuation of the X-ray intensity accumulation. Furthermore, the orientation function of the c-axes must be used by using eqs 8 and 9. The experimental function contains the termination error of the Legendre function. The other is due to the fact that the orientation functions by X-ray

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J. Phys. Chem. C, Vol. 112, No. 40, 2008 15617

Figure 7. 2πω(cos θ) of CF axes in the composites prepared from 10% PVA solution containing the indicated CF contents. Open circles are plotted values predicted by eq 9, and solid curves are calculated by eq 10.

were measured after evaporating the solvent under magnetic field, since the X-ray measurement under magnetic field is impossible for the dispersion solution. In the equilibrium state concerning the magnetic orientation of CFs, however, all curves must be the same independent of the CF contents, as long as eq 12 for a single CF is adopted. However, the preferential orientation function of CFs in the dispersed solution containing 2.5% CF content shows an exceptional profile. The three functions at 0.3, 0.5, and 1.0% CF contents are almost the same within the experimental error, which are in good agreement with the SEM images in Figure 4. The differentγ values among the three specimens are within the experimental error of the measured plots (open circles). In contrast, the γ value at 2.5% is much lower in comparison with those at 0.3, 0.5, and 1.0% contents. The 2.5% CF content is too high to pursue smooth orientation of CFs. Obviously, the equilibrium theory supports that a number of CFs cause the collision between neighbor CFs under the magnetic orientation. Namely, as discussed before, the magnetic flux density B is too low to overcome the orientation disturbance by the collision and achieve the same saturated orientation distributions as in 0.3, 0.5, and 1.0%. Judging from the γ value, it is obvious that the CF orientation at 2.5% cannot reach the equilibrium state. Figure 8 shows the theoretical and experimental functions for 10 and 13% PVA solutions, in which CF content was fixed to be 1.0% against the mixed solvent. As can be seen in Figure 8, the γ value at 13% PVA concentration (against the mixed solvent) is lower than that at 10% concentration. This is obviously due to very high viscosity of 13% PVA content as shown in Figure 1. In this case, the present magnetic flux density is too low to pursue the perfect orientation of CFs with respect to the applied magnetic field direction. Hence, it is of interest to consider the magnetic orientation of CFs up to the equilibrium state as a function of time in terms of solution viscosity and CF content.

Figure 8. 2πω(cos θ) of CF axes in the composites prepared from 10 and 13% PVA solutions with 1.0% CF content. Open circles are plotted values predicted by eq 9, and solid curves are calculated by eq 10.

15618 J. Phys. Chem. C, Vol. 112, No. 40, 2008

Nakano and Matsuo

3.3. Time Dependence of the Orientation Function of CFs. To pursue further rigorous estimation for the magnetic orientation of CFs, it may be expected that the orientation of CF axes are attributed to the rotation of the CF axes in the magnetic field direction. In this case, the distribution function ω(cos θ) may satisfy the following diffusion equation.24,43

∂ω τ ) Θ∇2ω - div ω ∂t ζ

( )

(14)

where Θ is the rotational diffusion coefficient of CF axes in the solution, τ is torque, ζ is the rotational fractional coefficient, and the symbol ∇2 is the Laplacian operator. Θ is given bykT/ ζ. Since a CF is surely a rigid cylinder such as tobacco-mosaic virus, Θ may be given by44

Θ)

2 1 3kT kT ln2p - 1.57 + 7 - 0.28 ( 0.25 ) 3 ln2p ζ 8πη0a

[

(

)

]

{

}

d dh (1 - u2) exp(-γu2) + 2γ(3u2 - 1) exp(-γu2)h + du du λ exp(-γu2)h ) 0 (25) This equation is a kind of the Liouville equation with two singular points at u ) 1 and u ) -1.24 Denoting the solution by hn for λ ) λn and λ ) λm, we multiply the equation for hn by hm and the equation for hm by hn and subtract the one from the other. Integrating the resulting equation from -1 to +1, the following relation can be obtained.

(λn - λm)

τ)-

χaV 2 B sin θ cos θ µ0

(16)

To carry out the solution of eq 14, the following parameters γ and c are given by

χaV 2 γ) B ) cB2 2µ0kT

( )

{

}

∂ ∂ω 1 ∂ω (1 - u2) - 2(1 - u2)γωu ) ∂u ∂u Θ ∂t

(18)

u ) cos θ

(19)

where To solve this partial differential equation, we attempt to separate variables by the substitution

ω(u, t) ) g(t) h(u)

}

1∂ ∂h 1 ∂g (1 - u2) - 2(1 - u2)γhu ) h ∂u ∂u Θg ∂t

(21)

{

}

{

}

dyn d (1 - u2) + {c(3u2 - 1)B2 + c2(u4 - u2)B4}yn + du du λnyn ) 0 (28) In the limit as Bf0, eq 28 reduces to

{

}

dyn d (1 - u2) + λnyn ) 0 du du

(29)

For the solution of eq 29 to be finite in the range -1 e u e +1, λn ) n(n + 1) with n ) 0, 1, 2, 3,.... The function of yn corresponding to λn is the nth-order Legendre function Pn(u). An approximate solution of eq 28 can be obtained by using the perturbation method. By expanding λn and yn in terms of B, we have

λn ) n(n + 1) + AnB + BnB2 + CnB3 + DnB4 + EnB5 + FnB6 + GnB7 + HnB8 + InB9 + JnB10 (30) yn(u) ) Pn(u) + an(u)B + bn(u)B2 + cn(u)B3 + dn(u)B4 + en(u)B5 + fn(u)B6 + gn(u)B7 + hn(u)B8 + in(u)B9 + jn(u)B10

(23)

Subtracting eqs 30 and 31 into eq 28 and equating B, B2, ..., B9, and B10 to zero, we can obtain a set of 10 differential equations. These equations are solved by expanding an(u), bn(u), cn(u), dn(u), en(u), fn(u), gn(u), hn(u), in(u), and jn(u) in terms of Legendre functions. This mathematical treatment is too complicated to write, and all procedures are eliminated to shorten this paper. The coefficients of the expansion are subjected to the condition.

(24)

∫-1+1 yn2 du ) 2n 2+ 1

(22)

The solution g(t) can be given by

g(t) ) (constant)exp(-λΘt)

(27)

(31)

The right side of eq 21 is a function of t, and the left side is a function of u. Hence the value of the quantity to which each side is equal must be a constant, which is termed as -λ. Accordingly, eq 21 may be written as two differential equations such as

1 ∂g ) -λg Θg ∂ t d dh (1 - u2) - 2(1 - u2)γhu + λg ) 0 du du

γu2 2

into eq 25 and using eq 17, we can find that yn(u) must satisfy the equation

(20)

Introducing eq 20 into eq 18 and dividing by g(t) h(u), we have

{

yn(u) ) hn(u) exp -

(17)

By using eq 14 and 16, we have

(26)

as long as hn and hm are finite at u ) +1 and -1. Equation 26 indicates that hn exp (-γu2) and hm exp (-γu2) are mutually orthogonal and that λn and λm are the eigenvalues. Making the substitution

(15) where a is an average half-length of CFs and p is an average aspect ratio. η0 is the viscosity of the CF dispersed PVA solution. τ is given by

∫-1+1 hnhm exp(-γu2) du ) 0

The equation for h(u) cannot be integrated analytically. However, eq 23 is transformed into the self-adjoint form by multiplying it by exp(-γu2). That is,

B3 ,

(32)

In the present paper, we retained all terms up to B10. It is to be noted that An, Cn, En, Gn, and In become zero. Here, the general solution of eq 21 is given by

Carbon Fiber Axes in Polymer Solutions

J. Phys. Chem. C, Vol. 112, No. 40, 2008 15619

∞

ω(u, t) )

∑ Wnhn exp(-λnΘt)

n)0

( )

∞

)

2

∑ Wnyn exp γu2

n)0

exp(-λnΘt)

(33)

where Wn are the expansion coefficients and they are determined so as to satisfy the initial condition (t ) 0) as follows: ∞

ω(u, t) )

( )

∞

2

∑ Wnhn ) ∑ Wnyn exp γu2

n)0

n)0

)

1 4π

(34)

From eq 34 and the orthogonality of yn, we can obtain

Wn )

2n + 1 8π

( ) 2

∫-1+1 yn exp - γu2

du

(35)

The solution of eq 25 for λ ) 0 gives h0, and it is found that

ω(u, ∞) ) W0h0 )

exp(γu2) 2π

∫-1+1 exp(γu2) du

(36)

Here it may be noted that the second-order orientation factor j (t) as a function of time can be F200(t) and the average angle θ obtained by using eq 33, as follows:

F200(t) )

[

] [

3〈cos2 θ〉 - 1 3〈u2 〉 - 1 ) 2 t 2 ) θ(t) )

]

t

∫-1+1 3u 2- 1 ω(u, t) du (37) 2

∫0π θω(cos(θ, t)) sin θ dθ

(38)

Figure 9 shows the orientation distribution functions calculated at the indicated values of Θt for the indicated four CF

contents dispersed in 10% PVA solution. The theoretical calculations for the indicated dispersion solutions were done by using the same γ value associated with each equilibrium state in Figure 7. The termination error up to the 10th-order perturbation did not give any problem for the numerical calculation, and the calculated distribution functions become almost stable as shown in Figure 9. The curve profile calculated at the indicated CF contents becomes sharper with increasing Θt and closes to the corresponding saturated curve in Figure 7. Namely, the calculated curve profile with a maximum value at θ ) 0° become sharper with elapsing time. This means that the orientation of CFs tends to orient gradually with elapsing time. Here we must emphasize that the estimation of CF magnetic orientation in terms of the distribution function is very important in comparison with the average estimations represented by eqs 37 and 38. Because the distribution function represented by eq 33 only reveals actual orientation behavior. Here it should be noted that the γ value represented by eq 17 is essentially independent of the CF content, and then the saturation time must be the same. Also the value of the rotational diffusion constant Θ must be almost the same, since the admixture of CFs up to 2.5% was confirmed not so sensitive to the change in viscosity ηo of the solution. As discussed before, the large difference at the 2.5% CF content is probably thought to be due to the collision between adjacent CFs under the magnetic field. In contrast, the CFs can be oriented without any disturbance between adjacent CFs at 0.3, 0.5, and 1.0% contents. This indicates that, at 2.5% content, the disturbance by the collision is very serious to promote the magnetic orientation degree. In Figure 9, t is the period relating to Θt and the value is calculated by eq 15. In the calculation, the average half-length ao of CFs and the average aspect ratio p are given as 22.5 µm and 9.0, respectively. The viscosities η0 at 10 and 13% PVA

Figure 9. 2πω(cos θ) of CF axes in the composites prepared from 10% PVA solution containing the indicated CF contents. Open circles are plotted values predicted by eq 9, and solid curves are calculated on the basis of eq 33.

15620 J. Phys. Chem. C, Vol. 112, No. 40, 2008

Nakano and Matsuo

Figure 10. 2πω(cos θ) of CF axes in the composites prepared from 10 and 13% PVA solutions with 1.0% CF content. Open circles are plotted values predicted by eq 9, and solid curves are calculated by eq 33.

Figure 11. (a) Second-order orientation factor calculated by eq 37 and (b) the average angle for the composites prepared from 10% PVA solution containing the indicated CF contents.

solutions are given as 2.241 and 7.457 P, respectively. The values of t to reach their saturation state at 0.3, 0.5, and 1.0% CF are calculated to be 53, 62, and 82 s, respectively. This indicates that the calculated saturation period becomes longer with increasing CF content, although the γ value differences among the three CF contents are very small. The CF content dependence of the saturation period may be rigorous because of fewer possibility of the collision between neighboring CFs under the magnetic orientation. Parts a and b of Figure 10 show the orientation distribution functions calculated at the indicated values of Θt. The calculations were carried out for 10 and 13% PVA solutions containing 1.0% CF content. Panel a, appeared already in Figure 9c, is shown again to compare the two curve profiles to each other. Of course, the calculation was done for the indicated dispersion solutions by using the same γ value, indicating the equilibrium state in Figure 8. The curve at 13% PVA concentration is duller than that at 10% PVA. This result also indicates that the high viscosity of solution due to an increase in PVA content hampers the magnetic orientation of CFs. Judging from Figure 1, the magnetic orientation of CFs in 13% PVA solution becomes poorer than that in 10% PVA solution in spite of the same CF content (1%). This is attributed to the decrease of Θ proportional to the reciprocal value of viscosity ηo as represented in eq 15. In Figure 10, the predicted saturation time of the orientation of CF axes at 13% PVA solution becomes longer than that at 10% PVA solution. It takes more than 1 h because of very high viscosity of the solution, as shown in Figure 1. Parts a and b of Figure 11 show the second-order orientation j (t) respectively for 0.3, factor F200(t) and the average angle θ 0.5, 1.0, and 2.5% CF contents dispersed in 10% PVA solution.

As shown in panel a, the factor becomes unity for perfect orientation of the CF axes with respect to the film normal direction (magnetic field direction), while it becomes zero for the random orientation. The factor increases drastically with increasing Θt and tends to level off. This tendency is almost the same as the behavior of the dispersed solutions with 0.3, 0.5, and 1.0% CF contents. On the other hand, the factor for the 2.5% CF content increased gradually. This indicates that the low value of F200(t) and slow saturation time are due to the drastic collision between adjacent CFs under magnetic orientation and the magnetic field used in the present work is too low to enhance the orientation of the CF axes in the dispersion solution containing 2.5% CF content. j (t) becomes lower with increasing Θt The average angle θ and tends to level off for 0.3, 0.5, and 1.0% CF contents dispersed in 10% PVA solution, indicating the progression of the orientation of the CF axes. On the other hand, the orientation for the 2.5% CF content is not so significant. Panels a and b of Figure 12 shows the estimation secondj (t) order orientation factor F200(t) and the average angle θ respectively for 10 and 13% PVA solution containing 1.0% CF. The results in panels a and b indicate that the increase in PVA concentration causes drastic increase in viscosity and the orientation of CF axes becomes less pronounced. The second-order orientation factor and the average angle in Figures 11 and 12 provide simple representation to facilitate easy understanding. Even so, the detailed phenomenon must be evaluated in terms of the orientation distribution function. j (t) shown The saturation value of Θt estimated by F200(t) and θ in Figures 11 and 12 are almost 0.25 for the 0.3, 0.5, and 1.0% CF contents, while the corresponding values of Θt estimated

Carbon Fiber Axes in Polymer Solutions

Figure 12. (a) Second-order orientation factor calculated by eq 37 and (b) the average angle for the composites prepared from 10 and 13% PVA solutions with 1.0% CF content.

by ω(cos θ) shown in Figures 9 and 10 are 0.26, 0.30, and 0.40, respectively. This means that the estimation by ω(cos θ) is much better for pursuing precise estimation for the orientation behavior of CF axes under magnetic field. j (t) is close to 10° at the saturated state of the CF orientation, θ but the most probable density of CF axes is not maximum at 10° and it must be 0° as shown in Figures 7-10. To avoid such a contradiction, the orientation function must be adopted. As discussed elsewhere,22-24,44 however, the estimation of the orientation function by X-ray diffraction is very difficult for dispersed solution and liquid crystals under electric and magnetic fields. Most of the estimations have been carried out by birefringence measurements associated with the second-order orientation factor. In the present work, of course, it was very difficult to evaluate the orientation of CF axes in solution under the magnetic field by X-ray diffraction technique, and then the estimation was done for the composites on the basis of the assumption that the same orientation degree of CF axes in the dispersed solution is maintained in the bulk specimen prepared by gelation/crystallization and then by evaporation of the solvent from the gels. Namely, any change in the orientation degree of the CFs under the magnetic field does not occur under the evaporating process of the solvent. Finally we shall refer to the physical meaning of γ, reflecting the profile of the equilibrium orientation distribution function under magnetic field. In the present experiment, the volume V of a CF and the applied magnetic flux density B in eq 13 are known parameters. Returning to Figures 7 and 8, the γ value can be determined to be almost 16.8, if the orientation of CFs was assumed to be the equilibrium state in the dispersion solution, the PVA content being 10% against the solvent,

J. Phys. Chem. C, Vol. 112, No. 40, 2008 15621 containing CFs with 0.3, 0.5, and 1.0% contents. If this is the case, the value of anisotropic diamagnetic susceptibility χa is calculated to be 7.57 × 10-10. The value is much lower than the established values (10-5-10-4) reported for several kinds of intrinsic CF.9,10,14,20,21 When the established values were adopted in the present system, the orientation of CFs must be perfect with respect to the magnetic field direction because of a very large volume of a CF used in this experiment. Judging from the gelation time of ca. 2 h estimated for the dispersion solution at 30 °C, it may be considered that the orientation of the CFs did not reach the equilibrium state within 2 days at room temperature in the summer season (27-33 °C). Furthermore, the poor orientation of the CF axes was attributed to the preferential uniplanar orientation of CFs on the film surface under no magnetic fields (see WAXD patterns on the left side in Figure 3) to ensure the stable state. Hence the preferential orientation of the CFs under the magnetic field, whose direction is perpendicular to the film surface, must be lower in comparison with the case that the magnetic field is applied parallel to the film surface, since the latter case is due to the orientation with respect to the magnetic field direction by rotation of CFs on the two-dimensional plane. The former case obviously provided lower γ value and consequently the value of χa estimated in the present experiment becomes lower than the intrinsic values, indicating that the orientation of CFs in the three-dimensional space is the inverse behavior to minimize the potential energy, assuring the stable orientation state of CFs. In spite of very difficult estimation,45 further studies must be taken into consideration in terms of the orientation distribution of graphite crystallites with respect to the CF axis, since theoretical curves sensitive to the value of χa are broader than the predicted curves as shown in Figures 7-10. Incidentally, electric conductivities of the present PVA-CF composite films are hardly affected by the magnetic orientation of CFs, and the values were ca. 1011 S/cm indicating the original conductivity of PVA films. The orientation of a large amount of CFs must be carried out under high magnetic field to prepare PVA-CF composites with anisotropic electric conductivity. 4. Conclusion The magnetic orientation of CF axes within PVA matrix was estimated in terms of the orientation distribution function by using the orientation function of the reciprocal lattice vector of the (002) plane measured by X-ray diffraction technique. Under magnetic field, the composite was prepared by gelation/ crystallization from dispersed PVA-CF solution and by evaporation of solvent from the gel under magnetic field. The estimation of the orientation function of CF axes within the composite film were obtained by a series of expansion of the function of the reciprocal lattice vector of the (002) plane, and the predicted function of CF axes was analyzed in terms of the magnetic field energy at the equilibrium state. The predicted function was in good agreement with the theoretical function calculated by choosing the value of experimental parameter γ, which is represented by the anisotropic diamagnetic susceptibility of the CF, applied magnetic flux density, and the volume of the CF. Under the magnetic orientation process, the orientation function of the CF axes in the solution was calculated as a function of time up to the equilibrium state by using the diffusion equation. The detailed analysis of γ values provided that the orientation of CFs did not reach the equilibrium state because of a drastic increase in viscosity by gelation of the solvent. As further analysis, the function could be obtained as a function of time and rotational diffusion constant associated with the

15622 J. Phys. Chem. C, Vol. 112, No. 40, 2008 rotation of CF axes in addition to the γ value. In comparison between the experimental (predicted) and theoretical results, it turned out that the effective magnetic orientation of CF axes must be higher to overcome the collision of neighbor CFs before the formation of stiff gels. Namely, it was evident that the orientational degree of CF axes at the equilibrium state becomes lower and the period up to the equilibrium becomes longer with increasing viscosity of the solution. Acknowledgment. The authors are indebted to Prof. T. Kimura of Kyoto University who has developed magnetic orientation of polymer chains. References and Notes (1) Beaugnon, E.; Tournier, R. Nature (London) 1991, 349, 470. (2) Berry, M. V.; Geim, A. K. Eur. J. Phys 1997, 18, 307. (3) Kitamura, N.; Makihara, M.; Hamai, M.; Sato, T.; Mogi, I.; Awaji, S.; Watanabe, K.; Motokawa, M. Jpn. J. Appl. Phys., Part 2 2000, 39, L324. (4) Weiss, A.; Witte, H. Magnetochemie; Verlag Chemie GmbH: Weinheim, Germany, 1973. (5) Timbrell, V. J. Appl. Phys. 1972, 43, 4839. (6) Matthews, M. J.; Dresselhaus, M. S.; Dresselhaus, G.; Endo, M.; Nishimura, Y.; Hiraoka, T.; Tamaki, N. Appl. Phys. Lett. 1996, 69, 430. (7) Fujiwara, M.; Chidiwa, T.; Tokunaga, R.; Tanimoto, Y. J. Phys. Chem. B 1998, 102, 3417. (8) Fujiwara, M.; Fukui, M.; Tanimoto, Y. J. Phys. Chem. B 1999, 103, 2627. (9) Fujiwara, M.; Oki, E.; Hamada, M.; Tanimoto, Y.; Mukouda, I.; Shimomura, Y. J. Phys. Chem. A 2001, 105, 4383. (10) Kimura, T.; Ago, H.; Tobita, M.; Ohshima, S.; Kyotani, M.; Yumura, M. AdV. Mater. 2002, 14, 1380. (11) Casavant, M. J.; Walters, D. A.; Schmidt, J. J.; Smalley, R. E. J. Appl. Phys. 2003, 93, 2153. (12) Fischer, J. E.; Zhou, W.; Vavro, J.; Llaguno, M. C.; Guthy, C.; Haggenmueller, R.; Casavant, M. J.; Walters, D. E.; Smalley, R. E. J. Appl. Phys. 2003, 93, 2157. (13) Takahashi, T.; Yonetake, K.; Koyama, K.; Kikuchi, T. Macromol. Rapid Commun. 2003, 24, 763. (14) Takahashi, T.; Suzuki, K.; Awano, H.; Yonetake, K. Chem. Phys. Lett. 2007, 436, 378. (15) Sujiyama, J.; Chanzy, H.; Maret, G. Macromolecules 1992, 25, 4232.

Nakano and Matsuo (16) Revol, J.-F.; Godbout, L.; Dong, X. M.; Gray, D. G.; Chanzy, H.; Maret, G. Liq. Cryst. 1994, 16, 127. (17) Dong, X. M.; Gray, D. G. Langmuir 1997, 11, 3029. (18) Kimura, T.; Kawai, T.; Sakamoto, Y. Polymer 2000, 41, 809. (19) Kimura, T.; Yoshino, M. Langmuir 2005, 21, 4805. (20) Kimura, T.; Yoshino, M.; Koshimizu, W.; Koike, M.; Kawai, T. Langmuir 2000, 16, 858. (21) Kimura, T. Polym. J. (Tokyo, Jpn.) 2003, 35, 823. (22) Duke, R. W.; DuPre´, D. B. Macromolecules 1974, 7, 374. (23) DuPre´, D. B.; Duke, R. W. J. Chem. Phys. 1975, 63, 143. (24) Matsumoto, M.; Watanabe, H.; Yoshioka, K. J. Phys. Chem. 1970, 74, 2182. (25) Bin, Y.; Kitanaka, M.; Zhu, D.; Matsuo, M. Macromolecules 2003, 36, 6213. (26) Bin, Y.; Chen, Q.; Tashiro, K.; Matsuo, M. Phys. ReV. B 2008, 77, 035419. (27) Roe, R. J.; Krigbaum, W. R. J. Chem. Phys. 1964, 40, 2608. (28) Krigbaum, W. R.; Roe, R. J. J. Chem. Phys. 1964, 41, 737. (29) Roe, R. J. J. Appl. Phys. 1965, 36, 202. (30) Matsuo, M.; Hirota, K.; Fujita, K.; Kawai, H. Macromolecules 1978, 11, 1000. (31) Matsuo, M.; Ozaki, F.; Kurita, H.; Sugawara, S.; Ogita, T. Macromolecules 1980, 13, 1187. (32) Nakashima, T.; Xu, C.; Bin, Y.; Matsuo, M. Polym. J. (Tokyo, Jpn.) 2001, 33, 54. (33) Bin, Y.; Koganemaru, A.; Nakashima, T.; Matsuo, M. Polym. J. (Tokyo, Jpn.) 2005, 37, 192. (34) Matsuo, M.; Sawatari, C.; Iwai, Y.; Ozaki, F. Macromolecules 1990, 23, 3266. (35) Bin, Y.; Tanabe, Y.; Nakabayashi, C.; Kurose, H.; Matsuo, M. Polymer 2001, 42, 1183. (36) Matsuo, M.; Bin, Y.; Nakano, M. Polymer 2001, 42, 4687. (37) Matsuo, M.; Sato, R.; Yanagida, N.; Shimizu, Y. Polymer 1992, 33, 1640. (38) Matsuo, M.; Adachi, R.; Jiang, X.; Bin, Y. Macromolecules 2004, 37, 1324. (39) Bin, Y.; Ooshi, K.; Yoshida, K.; Nakashima, T.; Matsuo, M. Polym. J. (Tokyo, Jpn.) 2004, 36, 394. (40) Xi, Y.; Ishikawa, H.; Bin, Y.; Matsuo, M. Carbon 2004, 42, 1699. (41) Bin, Y.; Mine, M.; Koganemaru, A.; Jiang, X.; Matsuo, M. Polymer 2006, 47, 1308. (42) Kaburagi, M.; Bin, Y.; Zhu, D.; Xu, C.; Matsuo, M. Carbon 2003, 41, 915. (43) Matsuo, M.; Kakei, K.; Nagaoka, Y. J. Chem. Phys. 1981, 75, 5925. (44) Broersma, S. J. Chem. Phys. 1960, 32, 1626. (45) Matsuo, M.; Xu, C. Polymer 1997, 38, 4311.

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