Fractal analysis of lyotropic lamellar liquid crystal textures

doi:10.1016/S0378-4371(03)00026-8

Physica A: Statistical Mechanics and its Applications

Volume 323, 15 May 2003, Pages 107-123

https://doi.org/10.1016/S0378-4371(03)00026-8 Get rights and content

Abstract

We apply fractal analysis to study the birefringence textures of lyotropic lamellar liquid crystal system (water/cethylpyridinium chloride/decanol). Birefringence texture morphologies are important as they provide information on the molecular ordering as well as defect structures and therefore has been adopted as a standard method in characterizing different phases of liquid crystals. The system under consideration shows a gradual morphological transition from mosaic to oily streak structures and then to maltese cross texture when the water content is increased. Since these textures are the characteristic fingerprints for the lyotropic lamellar phases, it is necessary to have robust techniques to obtain image quantifiers that can characterize the morphological structure of the textures. For this purpose we employ three different approaches namely the Fourier power spectrum for monofractal analysis, the generalized box-counting method for multifractal analysis and multifractal segmentation technique for estimating the space-varying local Hurst exponents. The relationships between estimated image parameters such as the spectral exponent, the Hurst exponent and the fractal dimension with respect to patterns observed in the birefringence textures are discussed.

Introduction

Textural analysis has contributed significantly in the characterization of complex disordered materials, especially with the continuing efforts to link the physical properties of interest to those textural parameters that are extracted from the images [1]. Fractal geometry supplies a versatile and powerful theoretical framework to describe complex surfaces [2]. Fractal models have been successfully applied to the quantitative description of microstructures such as surface roughness [3], grain boundaries in disordered metals and alloy, classification of shape and textures [4], [5], characterization of thin film surfaces [6], [7]. Even though there are increasing number of studies on various fractal growth phenomena in soft condensed materials, only few have discussed on the applications of fractal analysis to the birefringence patterns observed in optical polarized microscopy experiments for phase characterization of liquid crystals systems. Self-assembly of amphiphilic molecules in aqueous solutions usually in the presence of co-surfactants or short chain aliphatic molecules form a vast variety of complex structures [8], [9]. In this study, we shall restrict the analysis to lamellar L_α phase, which is one of the most studied lyotropic liquid crystal phases [10]. These phases are modelled as infinite parallel bilayer membranes consist of one or more amphiphilic molecules periodically stacked in space separated by a solvent. Lamellar phases exhibit long range smectic order and show characteristic birefringent textures when observed under polarizing light microscope. The morphological features of the liquid crystal textures are unique with respect to their phases and are often used for phase recognition purposes. Nevertheless, successful identification of complex mixtures of phases is often difficult and requires vast experiences. It is therefore, desirable to develop robust quantitative techniques for computer aided texture analysis for efficient characterization of the surface morphologies.

Fractal-based techniques for texture analysis offer many advantages especially for surfaces that exhibit self-similar properties. For example, the fractal dimension is directly linked to surface roughness. Futhermore, the underlying scale invariance that is observed in most complex surfaces can often be translated into power-law behaviors in certain image variables. Spectral methods focus on the properties of Fourier spectra hereby capturing global information about the image ‘energy’ distribution across scales or wavenumbers. Power-law spectral behavior implies scale invariance or simply indicates fractal characteristic. If the scaling exponent of the power-law behavior remains a constant over a certain range of scales, then the object is said to be globally self-similar. Self-similarity can be characterized using various geometrically related parameters such as the Hurst exponent H and spectral exponent α, which can be linked to fractal dimension using a particular model or stochastic process as the generator of such morphologies. Hurst exponent and the fractal dimension are useful quantifiers for analyzing surface roughness and texture segmentation. However, a single fractal dimension is insufficient for describing a surface that exhibits anisotropy or inhomogeneous scaling properties. Under such circumstances, one has to generalize the concept of monofractal to multifractal to cater for the heterogeneity of the surfaces that may be formed by the coexistence of many fractal sets of different fractal dimensions. Other technique such as the gray-level coocurrence matrix (GLCM) [11] also offers the possibility of characterizing anisotropy.

In this paper, we investigate the scaling behaviors of the image textures of lyotropic lamellar liquid crystal system in gray-level as well as black and white image formats. Optical studies of the birefringence patterns of liquid crystal offer useful information on the defect structures and thus it has been adopted as a standard method in characterizing different phases [12]. Birefringent textures in liquid crystals are due to the spatial variation of the director field which are visualized through the optical contrast between regions with different orientations of this axis with respect to a particular reference plane. The molecules in liquid crystals usually have characteristic anisotropic optical properties. Thus by passing polarized light through a liquid crystal specimen, one can detect the alignment of the molecules. Maximum birefringence is observed when the angle between the specimen principal plane and the analyzer permitted electric vector vibrational direction overlap. Interference between the recombining white light rays in the analyzer vibration plane often produces a spectrum of color, which is due to residual complementary colors arising from destructive interference of white light. In order to perform quantitative studies on the texture images, the original color images are converted into gray-level [0-255] format with the brighter regions corresponding to the constructive interference patterns in the birefringence textures.

We have chosen the lyotropic lamellar liquid crystal quasi-ternary system (water/cethylpyridinium chloride/decanol) that shows a gradual morphological transition from mosaic to oily streak structures and then to maltese cross texture upon increasing the water content. We use the monofractal analysis based on two-dimensional (2D) Fourier spectral technique on gray-level images to investigate the power-law scaling behavior. This is followed by the estimation of the generalized box-counting dimension using the multifractal analysis on black and white images. The black pixels correspond to textures with gray-level pixel values from 77 and above (based on 30% thresholding). An alternative approach to studying multifractality or anistropic local scaling in the gray-level images based on multifractional Brownian motion (MBM) with space-varying Hurst exponents is also given.

Section snippets

Experimental method and image preprocessing

The lyotropic liquid crystal under consideration is a quasi-ternary system consisted of cationic surfactant cetylpyridinium chloride (Fluka, 98%), decanol (Fluka, 99.5%) and doubly distilled and deionized water (resistivity $18.2 MΩ$ ). The samples are prepared by mixing the weighed components thoroughly in screw-capped glass tubes at room temperature (25°C). The surfactant to decanol ratio is fixed at 1:1 for all the samples such that changes in the lamellar mesophases would only be due to

Fractional Brownian sheet

Many natural and man-made surfaces exhibit self-similarity at different scales or resolutions in the form of [2] $M(ar)=a^{f(D)} M(r),$ where M(r) represents any measurable property of the fractal object (for example perimeter, surface area, mass), r represents the length of the measured metric property, a is the scaling factor, and f(D) is a linear function of the fractal dimension.

In modelling natural or man-made textures, various models for generating fractal curves and surfaces have been suggested

Generalized box-counting dimension

The main characteristic of fractal objects is their self-similar scaling property—the invariance under magnification for a wide range of temporal or length scales. For uniform fractal, the scaling is uniquely described by a single scaling exponent or fractal dimension D_B defined as $D_{B} = lim ε→0 ln N(ε) ln (1/ε),$ where ε is the lattice constant acting as non-overlapping covers of the structure under investigation, and N(ε) is the number of the cubes contained in the minimal cover.

The gray-level images

Discussions and conclusions

The surface textures of birefringence patterns of liquid crystal reveal a great deal of information about molecular ordering in different phases. Image analysis based on fractal and multifractal techniques are becoming an integral part of many material surface characterization software. Fractal dimension and the related scaling exponents such as Hurst exponent have been shown to be useful measurable image parameters that can be linked to physical properties.

In this work, we have investigated

Acknowledgements

We thank the Malaysian Ministry of Science, Technology and Environment for the research grants IRPA 09-02-02-0092 and IRPA 09-02-02-0091, Ismail Bahari of UKM for the microscopy facility and the Microscopy Unit of National University of Singapore for assistance with the freeze fracture study.

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Fractal analysis of lyotropic lamellar liquid crystal textures

Abstract

Introduction

Section snippets

Experimental method and image preprocessing

Fractional Brownian sheet

Generalized box-counting dimension

Discussions and conclusions

Acknowledgements

Physica A

The Fractal Geometry of Nature

J. Roy. Statist. Soc. B

IEEE Trans. Pattern Anal. Mach. Intell.

Fractals

Fractal Concepts in Surface Growth

Fresenius J. Anal. Chem.

Liquid Crystals

The Physics of Liquid Crystal

J. Phy. II Fr.