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
計畫類別： 個別型計畫 計畫編號： NSC91-2112-M-110-006- 執行期間： 91 年 08 月 01 日至 92 年 07 月 31 日 執行單位： 國立中山大學物理學系(所) 計畫主持人： 姜一民 報告類型： 精簡報告 報告附件： 出席國際會議研究心得報告及發表論文 處理方式： 本計畫可公開查詢 中 華 民 國 92 年 10 月 31 日Melting of Mesoscopic Lattices of Magnetic Fluid Thin Film
Subjected to Perpendicular
Fields
I.M. Jiang
1, C.Y. Wang
1, Y.S. Lin
1, M.S. Tsai
2, H.E. Horng
31. Department of Physics, National Sun Yat-sen University, Kaohsiung, Taiwan, R.O. C. 2. Department of Applied Physics, National Chiayi University, Chaiyi, Taiwan, R.O. C. 3. Department of Physics, National Taiwan Normal University, Taipei, Taiwan, R.O. C.
E-mail: [email protected]
Abstract
The high quality magnetic fluid thin film subjected to a perpendicular field, a separation of nano-particles from the liquid matrix will occur, leading to a phase transition with a phase that is concentrated in particles separating from a dilute phase. The concentrated phase makes up the cylindrical columns that can be arranged to a two-dimensional lattice at appropriate conditions. This kind of artificial lattices is a novel mesoscopic system suited for studying two-dimensional melting. We explore the melting evolution of the lattice by varying the applied field. The ordering of these extraordinary lattices is analyzed with translational and bond -orientation correlation functions. Our analysis ascertains the novel two-dimensional melting theory.
1. Introduction
There has been considerable interest in two-dimensional (2D) melting both in theoretical and experimental condensed matter physics [1,2], since Kosterlitz and Thouless (KT) constructed an elegant phase transition theory in two-dimensional systems [3]. They claim that the 2D XY model provides an unusual type of phase transition driving by vortices. The KT transition of superfluidity in two dimensions is the unbinding of vortex pairs of dilute gas. This novel transition is topological defect-mediated melting, and can be continuous. Subsequently, Halperin, Nelson [4,5] and Young [6] have extended the ideas of the KT mechanism to the 2D melting problem in contrast to the conventional order-disorder phase transition. They proposed a new scenario with two successive continuous phase transitions with an intermediate hexatic phase instead of a single direct first order transition. The bond orientational order (BOO) parameter of the 2D crystal is long-range order (LRO), but the translational order parameter possesses only quasi-long-range order (QLRO). The melting of a 2D crystal transits into hexatic phase, which is driven by the breakup of thermally generated bound dislocation pairs at a temperature Tm. Then
the increase of the density of free dislocations above Tm will result in an exponential
decay of the translational order parameter. However, the orientation order persists, in the sense the BOO decays only algebraically and displays QLRO. Such transition is a continuous phase transition. Subsequently, the spontaneous breakup of free dislocations into their constituent disclinations will drive a second continuous transition to an isotropic fluid from the intermediate hexatic phase. Experimental systems for studying the novel KTHNY 2D melting theory have been explored by some researchers. These include electrons on helium [7], noble gases physisorbed on
substrates [8], liquid crystals [9], polystyrene colloids [10], magnetic bubble arrays [11], and vortex arrays in high temperature superconductors [12].
In this study we find an artificial 2D lattices for the 2D melting study, which is made of magnetic fluid. When a field is applied to the thin film of high-quality magnetic fluid perpendicularly, the nano-particles will separate from the liquid matrix. We observe a phase that is concentrated in particles separates from a dilute phase. The concentrated phase makes up the cylindrical columns. The columns can form 2D lattices in an appropriate range of applied fields [13]. We explore these lattices composed of columns with microscope and pick up images with CCD. The melting evolution of the 2D lattice is explored by varying the applied field. We find that the transition from the high-field phase characterized by bond -orientation long-range order to a low-field disordered phase can imitate that of the low-temperature ordered phase to a high-temperature disordered phase. We examine the digitized image of extraordinary lattices of the novel mesoscopic 2D system. Making analysis of the ordering of these extraordinary lattices with translational and bond-orientation correlation functions, we demonstrate the phenomena of the exotic two-dimensional melting theory in this study.
2. Experiments and Analysis
Magnetic fluids typically consist of colloidal magnetic particles, dispersed in a continuous carrier phase [14,15,16,17]. The average diameter of the dispersed particles ranges between 5-10 nm. The co-precipitation technique let the magnetic particles coated with a surfactant layer. The steric repulsion of surfactant coats makes the suspensions homogeneous. The magnetic fluid displays soft magnetic
behavior as applied magnetic fields. Its magnetization is 8.2 emu/g. The magnetic fluid sample is sealed in a rectangularglass cell of 6μm thickness to form thin film. And the sample keeps in room temperature. The applied perpendicular magnetic fields are generated by Helmholtz coils, which are cooled by circulating water in copper tube. The magnetic field is uniform, because the microscope observation limits the exploration in a small region of the film. The dispersed particles attract or repel one another through dipolar interaction depending on their positions. In the absence of an external magnetic field, thermal energy leads the particles in Brownian motion. Whereas, with application of an external magnetic field, the magnetic particles tends to align with the field, partially overcoming the thermal agitation. As the field reaches some critical point, the violation of thermodynamically stability occurs and the condensation initiates. The concentrated magnetic particles aggregates come into droplet form. Both the size and the number of the droplet tend to increase with the magnitude of the applied field at this stage. As we intensify the magnetic field furthermore, the larger droplet aggregates split into parts of appropriate size due to the repulsive force among parallel domains is getting significant, and the surface tension is unable to bundle them together. In an intense magnetic field, the droplet aggregates divides into similar-sized columns that have similar magnitude of magnetic moment. The size of the column, and the separation between columns depend on the magnitude of the applied terminal magnetic field, the raising rate of the field, and the concentration of the magnetic particles in the fluid, etc. [18].
Through the dipolar interaction among columns, the columnar lattices of high quality magnetic fluid in a thin film subjected to perpendicular magnetic fields can form 2D lattices. The disordered pattern will appear at lower magnetic fields first,
and will arranged to be ordered 2D hexagonal lattices at higher fields. Patterns are visualized with an Olympus transmission microscope. CCD images are digitized with a threshold chosen to render the pattern faithfully recorded . They display high contrast black and white images. Digital images can be used to locate the center of each magnetic column.
In order to analyze the 2D lattice of the system, we use the digital data to calculate both translational and bond-orientation correlation functions [19]. The translational correlation function, G(r), is defined as the possibility of finding another point at 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 points r ’
with r −r' =r
0 .
The bond-orientation correlation function G6(r) relating bond-orientation at relative distance r apart is defined as follows:
) ( ) ( ) ( * ' 6 0 6 6 r r r G = φ φ
where φ6(ri) is the local orientation order parameter and is defined as follows:
∑
= . . 6 6 6 1 ) ( n n i i ij e r θ φwhere θij 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. The bond-orientation correlation function measures the extent to which sixfold orientational order persisits for separations comparable to r.
We then analyze the ordering of 2D lattices forming with magnetic fluid subjected to perpendicular fields in terms of the translational correlation function, G(r), and the bond-orientation correlation function, G6(r), respectively.
3. Results and discussion
We acquire the image of the thin film of magnetic fluid subjected to a perpendicular magnetic field. Figure 1 displays the typical image of ordered phase at field of 500 and 170 Oe, respectively. There are about 1340 centers in the image. The insertion figure is the fast Fourier transformation (FFT) image. Because it shows six distinct bright spots around the center one, which indicates a fairly ordered hexagonal structure, the columns of concentrated magnetic fluid form a 2D hexagonal lattice is evident. The distance between columns can be worked out by directly counting the number of image pixels from the location of the columns or calculating with the formula d=2π/k from the k-space distance. Thus the average distance between columns in this case is about 3.4μm. Figure 1(c) and (d) display the Delaunay triangulation plot of the image of Figure 1(a) and (b), respectively. The lines connecting lattice points almost parallel one another. Except in the small-hatched area, almost every lattice point has six coordinates. This also demonstrates that the observed structure is a well-ordered hexagonal lattice at high field region.
While we lower down the applied field, the dislocation appearing in the ordered lattice will abrupt increase below 130 Oe. Figure 2 displays the images of 130 and 120 Oe of the applied field around occurring the change. The hatched region of the Delaunay triangulation plot displays the appearing of defects in the lattice. Shown in the Figure 3 is the concentration of the regular six-bond lattice points and the
defective five- and seven-bond lattice points. In the ordered state, the perfect
six-bond lattice points are dominant, and less than 1% lattice points are occupied by the defect. The transition occurs at 120 Oe, where there exists a steep rising of the concentration of the defective lattice points.
In order to measure the ordering, we analyze with translational and
ensemble average of 40 pictures taken at the specific magnetic field. We inspect the ordered phase first. Apply the perpendicular magnetic field of 500 Oe on the film for two hours, the system well establish the equilibrium ordered phase. We then lower down the applied field in 2 Oe step to explore the variation of the columnar lattice structure. In each step, the field is quickly decreased and we wait 5 minutes for equilibrium. Afterward we pick up 40 pictures in two minutes. There is no significant variation of the ordered lattice in the high field region. The diagram of the
translational correlation function, G(r), and of the bond-orientation correlation
function, G6(r), of typical ordered lattices at 170 Oe is shown in Figure 4(a) and (b).
They display succession of spikes, and therefore the 2D lattice ascertains ordered phase. It is noted that the spike height of the bond-orientation correlation function almost keep constant, while that of the translational correlation function decays
slowly. This demonstrates that the bond-orientation is more likely to persist despite of the appearing of defects.
Shown in Figure 5 is the evolution of translational correlation functions G(r) of 170, 130, 120, 110 and 100 Oes. Series variation is presented. Shown in Figure 6 is the evolution of bond-orientation correlation functions G6(r) of 170, 130,120, 110 and 100 Oes. They display more distinctive series variation than G(r). Defects are rare in the high field region; the bond-orientation correlation function shows constant value. As we lower down the applied magnetic field, the hexagonal lattice is getting disorder. Even defects increase to a significant amount, the bond-orientation correlation is persisted, and the peak height of oscillating bond-orientation correlation function decay algebraically slows. The exponent of the fitting algebraic function, ~ r−α , versus the magnetic field is shown in Figure 7. As the field decreases further, the applied magnetic field will not be able to hold the system ordered structure. The ordered lattice melts through a quick transition. We notice that the exponent of the fitting algebraic function increases rapidly at the melting threshold. The theoretical threshold exponent of 2D system is the value of 1/4 [3,20]. Inspecting the experimental result, the exponent goes beyond 1/4 when
the applied magnetic field decreases to 120 Oe ascertains the 2D melting occurs. We also note that the bond-orientation correlation function presents exponentially rapid decaying behavior below 120 Oe, which means that the observed lattice loses its bond-orientation correlation.
As for the exponential fitting of the correlation functions, ~ exp(-r/ξ), we examine the behavior of correlation length. Figure 8 shows the correlation length ξ and ξ6 of translational and bond-orientation correlation functions versus the
applied fields. Long correlation length addresses that the build of long-range order from 120 Oe. It is noted that the bond-orientation correlation function displays longer correlation length. This also demonstrates that the bond-orientation is more likely to persist. At 120 Oe both correlation lengths steeply decrease to the order of lattice spacing. Thus we ascertain the 2D lattice is melting as the applied magnetic field decreases to 120 Oe, with the mesoscopic lattice demonstrating randomly isotropic distribution.
4. Conclusion:
Apply magnetic field on the magnetic fluid thin films perpendicularly, the concentration portion separated from the carrier matrix will arrange to ordered lattice under appropriate conditions. The applied field can be manipulated to simulate the temperature control processes. Henceforth, the disturbed ordering lattice can occur the order-disorder transition. Since this kind of order-disorder transition displays distinguished features of an equilibrium phase transition, it will be suitable to mimic 2D phase transition. The 2D mesoscopic crystal is essentially a defected crystal with dislocations destroying the translational symmetry. Nevertheless the bond-orientation order will be sustained, thus the nonconventional order will be kept in the system. We study the ordering of the extraordinary 2D magnetic fluid thin film lattice with the bond-orientation correlation function as well as the translational
correlation function. The quantitative measurements reveal the following features of the transition. In a narrow range of magnetic fields, the correlation length ξ and ξ6 falls from a value comparable to the size of the system to a small value of lattice
spacing. In the transition from the high field phases characterized by constant bond-orientation correlation function to a low-field disordered phase, the long-range orientation order in the lattice decreases substantially. Examining the algebraic fitting of the bond-orientation correlation function, we find the power-law exponent shows abrupt increase and at 120 Oe passes through 1/4, which is the theoretical threshold for 2D melting. The analysis ascertains the exotic 2D melting phenomena in the artificial mesoscopic lattices of magnetic thin films in this study.
Acknowledgements
We would like to acknowledge the National Science Council of Taiwan, R. O. C. for financial support.
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Figure captions:
perpendicular magnetic field of 500 and 170 Oe, respectively. The insertion is the fast Fourier transformation (FFT) of the image. (c) and (d) Delaunay triangulation plots for above images.
Fig. 2 (a) and (b) Typical images at field of 130 and 120 Oe, respectively. The insertion is the fast Fourier transformation (FFT) of the image. (c) and (d) Delaunay triangulation plots for the above images.
Fig. 3 The concentration of the regular six-bond lattice points and the defective five- and seven-bond lattice points versus the magnetic field.
Fig. 4 (a) and (b) are the translational correlation function, G(r), and the bond-orientation correlation function, G6(r), of the typical image of ordered phase at field of 170 Oe.
Fig. 5 Evolution of translational correlation functions G(r) of 170, 130, 120, 110, and 100 Oe.
Fig. 6 Evolution of bond-orientation correlation functions G6(r)of 170, 130, 120, 110, and 100 Oe.
Fig. 7 The exponent of the fitting algebraic function versus the magnetic field.
Fig. 8 Correlation length ξ and ξ6 of translational and bond-orientation correlation
functions versus the magnetic field.
