液晶及磁流體之非線性及偏極光學特性研究(II)The Study of Nonlinear and Polarization Optics of Liquid Crystals and Magnetic Fluid Thin Films (II)

行政院國家科學委員會專題研究計畫 成果報告

液晶及磁流體之非線性及偏極光學特性研究(Ⅱ)

中 華 民 國 93 年 10 月 28 日

液晶及磁流體非線性及偏極光學特性研究(Ⅱ)

姜一民

摘要

以低頻磁場垂直加在磁流體薄膜，引致相分離成濃與淡磁粒子區

域，柱狀濃磁粒子區形成二維格子，如此有序的磁流體格子應用於光

學元件上極具潛力。我們的研究有序磁流體格子的動態形成方式。

NSC 92-2112-M-110-010

The study of nonlinear and polarization optics of liquid crystals and magnetic fluid thin films II

(Dynamic formation of columnar lattices in magnetic fluid thin films subjected to oscillating perpendicular magnetic fields)

I.M. Jiang

The application of a low-frequency oscillating magnetic field perpendicular to a magnetic fluid thin film leads to the separation of a phase that is concentrated in particles from a dilute phase. The concentrated phase forms cylindrical columns that construct two-dimensional lattices. The ordered structure of magnetic fluid thin films is the basis for potential optical applications. We investigate the dynamical ordering formation of columnar lattices in magnetic fluid thin films subjected to oscillating perpendicular magnetic fields.

I. Introduction

The versatile structures of magnetic fluid thin film subjected to an external field have attracted much interest because of the induced optical properties and potential applications1-9. It is well known that the application of a magnetic field perpendicular to the magnetic fluid thin film leads to a phase separation—the separation of the concentrated portion in magnetic particles from the liquid matrix of magnetic fluid. The concentrated phase forms cylindrical columns that construct two-dimensional lattices. Henceforth, the ordered structure of magnetic fluid thin films constitutes the basis for potential optical applications. It is noted that remarkable magneto-optical effects in magnetic fluids, such as birefringence10-15, field-dependent transmission12,15, and magnetochromatics,16,17 have been investigated. Nonetheless, it takes hours to establish stable ordered structures by applying a direct current magnetic field to a magnetic fluid thin film. D. Wirtz and M. Fermiger18 filled a narrow cell with magnetic fluid, and then they applied an oscillating magnetic field from the long side with a frequency below 10 Hz to study the one-dimensional stripes pattern. Promislow and Gast19 have studied the suspensions structures of ellipsoidal aggregates of a magnetic fluid by applying a pulsed field. Shaking the columnar structure by using of an oscillating field is recommended to produce a stable ordered structure naturally. It is amazing that besides the shaking effect, applying only a low-frequency oscillating magnetic field to some intensity can also produces a two-dimensional ordered columnar lattice structure as the result in applying dc magnetic field. Specifically, one may conveniently employ commercial electricity of 60 Hz. Hence, it will be a marvelous improvement on forming ordered structures in magnetic fluid thin films subjected to an oscillating force for potential optical applications. In this study, we explore the dynamical ordering formation of columnar lattices in magnetic fluid thin films subjected to low-frequency oscillating

perpendicular magnetic fields.

II. Experiments and Analysis

The homogeneous magnetic fluid consists of colloidal magnetic (MnFe2O4)

particles, which are dispersed in a continuous kerosene carrier1-5. The average diameter of the dispersed particles ranges from 5 to 10 nm. The coprecipitation technique allows magnetic monodomain particles to be coated with a surfactant layer. The steric repulsion of surfactant covering makes the suspensions homogeneous. Its saturated magnetization is 8.7 emu/g in this study. To form a magnetic fluid thin film, we seal the magnetic fluid in a rectangularglass cell of 6μm thickness. Typically, a spatially uniform magnetic field is generated perpendicularly on the thin film by a pair of coils, which are cooled by circulating water in a copper tube. The sample is kept at room temperature. In the absence of an applied field, thermal energy leads the particles in Brownian motion, whereas the magnetic nanoparticles phase separate into droplet form as they overcome the thermal agitation when a magnetic field is applied over some critical intensity. We are, first of all, able to observe a disordered pattern at lower magnetic fields. In an intense magnetic field, the droplet aggregation divides into similar-sized cylindrical columns that have similar magnitude of magnetic moment. The size of the column, and the separation between columns vary with the magnitude of the applied terminal magnetic field, the rising rate of the field, and the concentration of the magnetic particles in the fluid, etc.5 Through the dipolar interaction among columns, the condensed columns of the magnetic fluid thin film form an ordered two-dimensional hexagonal lattice at higher fields.

It is noted that when we apply an oscillating magnetic field to a magnetic fluid thin film, the condensed concentrated portion of the magnetic fluid will not disperse at low-frequency cases. In the present study, we amplify the oscillating signal of a function generator with a constant current amplifier (NF Electronic Instruments

TA-250) to produce the oscillating magnetic field. The current is measured to adjust the employed voltage of the function generator in order to control the intensity of the applied oscillating magnetic field. We find that when we employ appropriate intensity of low-frequency oscillating magnetic fields, the ordered two-dimensional lattices can be sustained in a magnetic fluid thin film. These two-dimensional hexagonal patterns are visualized with an Olympus transmission microscope (BX-51). A charge coupled device (CCD) video camera is used to acquire the image.

Kosterlitz, and Thouless20, Nelson, and Halprin21,22, and Young23, have constructed a novel two-dimensional melting theory. This theory20-25 proposes that the two-dimensional solid phase is characterized by a constant bond-orientation correlation function, although the translational correlation function presents slow algebraically decay due to intrinsic defects. The translational correlation function, G(r), gives the possibility of finding another point at a relative distance r apart and can be written as:

) ( ) ( ) (r r0 r' G = ρ ρ

where the average is taken over all reference points r0 and r’ with r −r =r

' 0 .

Concerning the bond orientation related to that r distance, the bond-orientation correlation function, G6(r), is defined as:

) ( ) ( ) ( * ' 6 0 6 6 r r r G = φ φ

where )φ6(ri is the local orientation order parameter and is defined as:

∑

where is the angle between a fixed reference axis and the bond linking particles i

and j. The summation is taken over all the bonds of nearest neighbors. If the nearest neighbors consist of an exact hexagon of equal-divided angles, then the modulus is 1.

ij

It decides the surrounding bond-orientation quality of each lattice point, whereas the bond-orientation correlation function, G6(r), measures the extent to which the

six-fold orientation order persists for separations comparable to r. It classifies the long-ranged ordering.

Charge coupled device images are digitized with a threshold chosen to render the pattern faithfully. They display distinct black and white images. We can locate the center of each magnetic column of digital images. By using of the digital data, we calculate the local orientation order parameter, φ₆(r_i), translational correlation function, G(r), and bond-orientation correlation function, G6(r)

8,24

.The ordering properties of two-dimensional lattices in magnetic fluid thin films subjected to the oscillating perpendicular fields are analyzed in this study. The useful ordering classification is then provided for the high potential of optical applications.

III. Results and Discussion

We obtain an image with a CCD video camera five minutes after a magnetic fluid thin film is applied with an oscillating perpendicular magnetic field of root-mean-square intensity 300 Oe. Figure 1 shows the lattice structures subjected to the oscillating field at frequencies 10, 50, 60, 70, 80, and 90 Hz, respectively. It is noted that in a low-frequency situation, the condensed concentrated portion of the magnetic fluid does not disperse in time when the field is reversed. The slow diffusion of magnetic particles may be the reason for the fact that the surface tension is still able to bundle the concentrated column. Inspecting the variation, we find that columnar lattice points disappear when the frequency of the oscillating field is raised to 80 Hz. Shown in figure 2 is the number of columnar lattice points versus frequency in the same-sized viewing window. In general, the evolution of lattices with applied frequencies of oscillating fields is as follows: lower

thinner and the lattice points of the ordered structure will disappear, when the frequency of the oscillating field is raised to some higher value; the transition occurs abruptly within a narrow frequency range. It is also noted that a lower frequency could sustain the ordered structure to the higher critical intensity of the applied oscillating field.

We then select a fixed intensity to explore the dynamical variation of columnar lattices. It is noted that at medium intensity of an oscillating magnetic field of root-mean-square value 100 Oe, the equilibrium lattice spacing is almost the same with an increasing frequency, unless the lattice collapses at about 500 Hz. The lattice spacing is about 3.1μm. Figure 3 displays the time series of the Delaunay

triangulation plot of 50, 100, 300, 1000, 1500, and 1900 seconds, when the magnetic fluid thin film is applied with an oscillating perpendicular magnetic field of frequency 100 Hz and of root-mean-square intensity 100 Oe. We obtain the digitized image of 480 x 480 pixels to draw the Delaunay triangulation plot, in which there are usually six coordinates for each lattice point. The hatched region shows defects with more or less than six coordinates. In the plot of 1900 sec, it is noted that the lines connecting lattice points are almost parallel to one another and few defects appear. The observed structure demonstrates a well-ordered hexagonal lattice. Figure 4 displays the fast Fourier transform (FFT) of the time series images in figure 3. At 50 sec, the FFT shows a blurred circle at first. As time elapses, six spots appear around the central one. At 1900 sec, they become six distinct bright spots. This addresses the evolution of the lattice transformation from disorder to an ordered state. As is verified by the evolution of the Delaunay triangulation plot, the lattice structure is stabilized after 1900 sec.

Shown in Figure 5 are φ maps of local order parameter for the time series ₆ images in figure 3. Each projection point in the φ map represents a local 6

points are randomly distributed. The diffusion of φ projections reveals that there is 6

no long-range bond-orientation order on the columnar lattice. As time elapses, the projection points of the φ map gradually become concentrated. Further progress, 6

the φ map at 1900 sec, shows a localized spot. The distinct ordered feature means ₆ that every lattice point displays behaviors of similar bond-orientation. It is noted that at 1500 sec, another localized spot in the φ map emerges besides the main ₆

localized spot. The Delaunay triangulation plot in this condition is divided into two portions. The bond-orientation in the smaller portion on the left side displays a

different tilting behavior. The appearance of two localized spots in φ map indicates 6

there are two domains within the observation range.

The series dynamical variation of the ordered lattices of various frequencies is also explored. Figure 6 displays the time series bond-orientation correlation function of columnar lattices of magnetic fluid thin films subjected to a 100 Oe oscillating field of frequencies at 10, 60, 100, and 160 Hz, respectively. We use the

bond-orientation correlation function to characterize the order of the lattice. A fast oscillating field induces a fast evolution. Constant peak heights appear at 7200, 5500, 1900, and 700 sec, respectively. When an oscillating field with the same intensity of 100 Oe is applied, the 160 Hz case is significantly faster in obtaining the ordered structure than that in the other cases. The trend indicates that fast oscillating field induces order transformation more quickly.

We trace the movement of each lattice column. The displacement of the lattice point, which moves every 100 seconds, is measured. And the speed of every second in vibration is averaged in each period. Figure 7(a) shows the displacement of the lattice column versus time, with applied oscillating fields at frequencies 10, 60, 100, and 160 Hz, respectively, and figure 7(b) shows the vibration speed of the lattice column versus time, with applied oscillating fields at frequency 10, 60, 100, and 160 Hz,

respectively. We find the higher frequency case displays a fast vibration speed and a more active motion with a longer displacement in each 100-second period. Therefore, the lattice column is more active to adjust the position with the application of a higher frequency oscillating magnetic field, so that the lattice structure of the magnetic fluid thin film is able to form an order more quickly. It is noted that distinct different orientation domains are usually found in the dc applied field case due to a retarded variation of the domain orientation. Because the ordering structure forms slowly, a higher field is needed in the dc case to nucleate different orientation domains to a single one. In the use of an oscillating magnetic field, a smaller intensity is required to establish the ordered structure, and the shaking effect could form an order more quickly. The advantage of applying an oscillating magnetic field to form the ordered structure is thus significant.

IV. Conclusion

We have demonstrated that the nano-magnetic particles phase separates from the liquid matrix of magnetic fluid with the use of a low-frequency oscillating magnetic field. An oscillating magnetic field applied perpendicularly to a magnetic fluid thin film can form ordered columnar lattice structures. The ordering is characterized in terms of the fast Fourier transformation, the local order parameter map, and the bond-orientation correlation function. We have verified that a low-frequency oscillating magnetic field can transform columnar lattices into an ordered structure more efficiently. We have then proposed a significant method to form ordered structures of grating optical devices with magnetic fluid thin films for the high potential of optical applications.

Acknowledgements

Science Foundation of Taiwan, R. O. C. under grant NSC 92-2112-M-110-010, and from YCL Electronics Co., Ltd.

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Fig. 1 The lattice structures of magnetic fluid thin film subjected to the oscillating magnetic field, H = 300 Oe, at f = 10, 50, 60, 70, 80, and 90 Hz, respectively. Fig. 2 Number of lattice points versus frequency.

Fig. 3 The Delaunay triangulation plots of images taken from columnar lattices subjected to oscillating magnetic field: f = 100 Hz and H = 100 Oe, at 50, 100, 300, 1000, 1500, and 1900 sec, respectively.

Fig. 4 The fast Fourier transform (FFT) of the time series images in figure 3. Fig. 5 φ maps of local order parameter of the time series images in figure 3. ₆

Fig. 6 The time series bond-orientation correlation function of columnar lattices subjected to oscillating field H = 100 Oe and f = 10, 60, 100 and 160 Hz, respectively.

Fig. 7 (a) The displacement of the lattice column versus time, and (b) the vibration speed of the lattice column versus time. The applied oscillating field: f = 10, 60, 100, and 160 Hz, respectively.

0 20 40 60 80 100 0 500 1000 1500 2000 2500

N o

f colum

n

frequency(Hz)

0 1000 2000 3000 4000 5000 6000 7000 8000 0 2 4 6 8 10 12 14 16 (a) Hz=160 Hz=100 Hz=60 Hz di s p lacement (pi xe l) (b) =10 time(sec) 0 1000 2000 3000 4000 5000 6000 7000 8000 0.8 0.9 1.0 1.1 1.2 spe e d( p ixel /s e c ) time(sec) Hz=160 Hz=100 Hz=60 Hz=10 Jiang-JAP-Fig. 7