Fractal dimension of dried foods : a correlation between microstructure and porosity

The quality of dried food is affected by a number of factors including quality of raw material, initial microstructure, and drying conditions. The structure of the food materials goes through deformations due to the simultaneous effect of heat and mass transfer during the drying process. Shrinkage and changes in porosity, microstructure and appearance are some of the most remarkable features that directly influence overall product quality. Porosity and microstructure are the important material properties in relation to the quality attributes of dried foods. Fractal dimension (FD) is a quantitative approach of measuring surface, pore characteristics, and microstructural changes [1]. However, in the field of fractal analysis, there is a lack of research in developing relationship between porosity, shrinkage and microstructure of different solid food materials in different drying process and conditions [2-4]. Establishing a correlation between microstructure and porosity through fractal dimension during convective drying is the main objective of this work.

Quantitative characterization of kaolinite dispersibility in styrene–butadiene rubber composites by fractal dimension

A fractal method was introduced to quantitatively characterize the dispersibility of modified kaolinite (MK) and precipitated silica (PS) in styrene–butadiene rubber (SBR) matrix based on the lower magnification transmission electron microscopic images. The fractal dimension (FD) is greater, and the dispersion is worse. The fractal results showed that the dispersibility of MK in the latex blending sample is better than that in the mill blending samples. With the increase of kaolinite content, the FD increases from 1.713 to 1.800, and the dispersibility of kaolinite gradually decreases. There is a negative correlation between the dispersibility and loading content. With the decrease of MK and increase of PS, the FD significantly decreases from 1.735 to 1.496 and the dipersibility of kaolinite remarkably increases. The hybridization can improve the dispersibility of fillers in polymer matrix. The FD can be used to quantitatively characterize the aggregation and dispersion of kaolinite sheets in rubber matrix.

A quasi-likelihood method for fractal-dimension estimation

We propose a simple method of constructing quasi-likelihood functions for dependent data based on conditional-mean-variance relationships, and apply the method to estimating the fractal dimension from box-counting data. Simulation studies were carried out to compare this method with the traditional methods. We also applied this technique to real data from fishing grounds in the Gulf of Carpentaria, Australia

The unfinished landscape fractal geometry and the aesthetics of ecological design

During the late 20th century it was proposed that a design aesthetic reflecting current ecological concerns was required within the overall domain of the built environment and specifically within landscape design. To address this, some authors suggested various theoretical frameworks upon which such an aesthetic could be based. Within these frameworks there was an underlying theme that the patterns and processes of Nature may have the potential to form this aesthetic — an aesthetic based on fractal rather than Euclidean geometry. In order to understand how fractal geometry, described as the geometry of Nature, could become the referent for a design aesthetic, this research examines the mathematical concepts of fractal Geometry, and the underlying philosophical concepts behind the terms ‘Nature’ and ‘aesthetics’. The findings of this initial research meant that a new definition of Nature was required in order to overcome the barrier presented by the western philosophical Nature¯culture duality. This new definition of Nature is based on the type and use of energy. Similarly, it became clear that current usage of the term aesthetics has more in common with the term ‘style’ than with its correct philosophical meaning. The aesthetic philosophy of both art and the environment recognises different aesthetic criteria related to either the subject or the object, such as: aesthetic experience; aesthetic attitude; aesthetic value; aesthetic object; and aesthetic properties. Given these criteria, and the fact that the concept of aesthetics is still an active and ongoing philosophical discussion, this work focuses on the criteria of aesthetic properties and the aesthetic experience or response they engender. The examination of fractal geometry revealed that it is a geometry based on scale rather than on the location of a point within a three-dimensional space. This enables fractal geometry to describe the complex forms and patterns created through the processes of Wild Nature. Although fractal geometry has been used to analyse the patterns of built environments from a plan perspective, it became clear from the initial review of the literature that there was a total knowledge vacuum about the fractal properties of environments experienced every day by people as they move through them. To overcome this, 21 different landscapes that ranged from highly developed city centres to relatively untouched landscapes of Wild Nature have been analysed. Although this work shows that the fractal dimension can be used to differentiate between overall landscape forms, it also shows that by itself it cannot differentiate between all images analysed. To overcome this two further parameters based on the underlying structural geometry embedded within the landscape are discussed. These parameters are the Power Spectrum Median Amplitude and the Level of Isotropy within the Fourier Power Spectrum. Based on the detailed analysis of these parameters a greater understanding of the structural properties of landscapes has been gained. With this understanding, this research has moved the field of landscape design a step close to being able to articulate a new aesthetic for ecological design.

Fractal and multifractal properties of a family of fractal networks

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al (2007 Proc. Nat. Acad. Sci. USA 104 7746). In this fractal network model, there is a parameter e which is between 0 and 1, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, the dependence relationship of the fractal dimension and the multifractal parameters on parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al (2006 Nature Phys. 2 275). Then from the shape of the τ(q) and D(q) curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension 〈D(1)〉 and the parameter e.

Fractal and complex network analyses of protein molecular dynamics

Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h(q)h(q) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h(2)h(2) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H≈0.5H≈0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h(2)h(2) of MF-DFA on the time series, exponent λλ of the exponential degree distribution and fractal dimension dBdB of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between 〈h(2)〉〈h(2)〉 (from MF-DFA on time series) and 〈dB〉〈dB〉 of the converted HVGs for different energy, pressure and volume.

Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent 〈λ〉 increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and 〈λ〉 can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension 〈dB〉 of recurrence networks decreases with the Hurst index H of the associated FBMs, and their dependence approximately satisfies the linear formula 〈dB〉≈2-H, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index H=0.5 possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension 〈D(1)〉 and the average correlation dimension 〈D(2)〉 on the Hurst index H can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic and the fractal nature of the associated FBM series.

Multifractal characterisation and analysis of complex networks

Complex networks have been studied extensively due to their relevance to many real-world systems such as the world-wide web, the internet, biological and social systems. During the past two decades, studies of such networks in different fields have produced many significant results concerning their structures, topological properties, and dynamics. Three well-known properties of complex networks are scale-free degree distribution, small-world effect and self-similarity. The search for additional meaningful properties and the relationships among these properties is an active area of current research. This thesis investigates a newer aspect of complex networks, namely their multifractality, which is an extension of the concept of selfsimilarity. The first part of the thesis aims to confirm that the study of properties of complex networks can be expanded to a wider field including more complex weighted networks. Those real networks that have been shown to possess the self-similarity property in the existing literature are all unweighted networks. We use the proteinprotein interaction (PPI) networks as a key example to show that their weighted networks inherit the self-similarity from the original unweighted networks. Firstly, we confirm that the random sequential box-covering algorithm is an effective tool to compute the fractal dimension of complex networks. This is demonstrated on the Homo sapiens and E. coli PPI networks as well as their skeletons. Our results verify that the fractal dimension of the skeleton is smaller than that of the original network due to the shortest distance between nodes is larger in the skeleton, hence for a fixed box-size more boxes will be needed to cover the skeleton. Then we adopt the iterative scoring method to generate weighted PPI networks of five species, namely Homo sapiens, E. coli, yeast, C. elegans and Arabidopsis Thaliana. By using the random sequential box-covering algorithm, we calculate the fractal dimensions for both the original unweighted PPI networks and the generated weighted networks. The results show that self-similarity is still present in generated weighted PPI networks. This implication will be useful for our treatment of the networks in the third part of the thesis. The second part of the thesis aims to explore the multifractal behavior of different complex networks. Fractals such as the Cantor set, the Koch curve and the Sierspinski gasket are homogeneous since these fractals consist of a geometrical figure which repeats on an ever-reduced scale. Fractal analysis is a useful method for their study. However, real-world fractals are not homogeneous; there is rarely an identical motif repeated on all scales. Their singularity may vary on different subsets; implying that these objects are multifractal. Multifractal analysis is a useful way to systematically characterize the spatial heterogeneity of both theoretical and experimental fractal patterns. However, the tools for multifractal analysis of objects in Euclidean space are not suitable for complex networks. In this thesis, we propose a new box covering algorithm for multifractal analysis of complex networks. This algorithm is demonstrated in the computation of the generalized fractal dimensions of some theoretical networks, namely scale-free networks, small-world networks, random networks, and a kind of real networks, namely PPI networks of different species. Our main finding is the existence of multifractality in scale-free networks and PPI networks, while the multifractal behaviour is not confirmed for small-world networks and random networks. As another application, we generate gene interactions networks for patients and healthy people using the correlation coefficients between microarrays of different genes. Our results confirm the existence of multifractality in gene interactions networks. This multifractal analysis then provides a potentially useful tool for gene clustering and identification. The third part of the thesis aims to investigate the topological properties of networks constructed from time series. Characterizing complicated dynamics from time series is a fundamental problem of continuing interest in a wide variety of fields. Recent works indicate that complex network theory can be a powerful tool to analyse time series. Many existing methods for transforming time series into complex networks share a common feature: they define the connectivity of a complex network by the mutual proximity of different parts (e.g., individual states, state vectors, or cycles) of a single trajectory. In this thesis, we propose a new method to construct networks of time series: we define nodes by vectors of a certain length in the time series, and weight of edges between any two nodes by the Euclidean distance between the corresponding two vectors. We apply this method to build networks for fractional Brownian motions, whose long-range dependence is characterised by their Hurst exponent. We verify the validity of this method by showing that time series with stronger correlation, hence larger Hurst exponent, tend to have smaller fractal dimension, hence smoother sample paths. We then construct networks via the technique of horizontal visibility graph (HVG), which has been widely used recently. We confirm a known linear relationship between the Hurst exponent of fractional Brownian motion and the fractal dimension of the corresponding HVG network. In the first application, we apply our newly developed box-covering algorithm to calculate the generalized fractal dimensions of the HVG networks of fractional Brownian motions as well as those for binomial cascades and five bacterial genomes. The results confirm the monoscaling of fractional Brownian motion and the multifractality of the rest. As an additional application, we discuss the resilience of networks constructed from time series via two different approaches: visibility graph and horizontal visibility graph. Our finding is that the degree distribution of VG networks of fractional Brownian motions is scale-free (i.e., having a power law) meaning that one needs to destroy a large percentage of nodes before the network collapses into isolated parts; while for HVG networks of fractional Brownian motions, the degree distribution has exponential tails, implying that HVG networks would not survive the same kind of attack.

Effect of trash on the coagulation and flocculation of sugarcane juice

The processing of juice expressed from whole green sugarcane crop (stalk and trash) leads to poor clarification performance, reduced sugar yield and poor raw sugar quality. The cause of these adverse effects is linked to the disproportionate contribution of impurities from the trash component of the crop. This paper reports on the zeta (ζ) potential, average size distribution (d50) and fractal dimension (Df) of limed juice particles derived from various juice types using laser diffraction and dynamic light scattering techniques. The influence of non-sucrose impurities on the interactive energy contributions between sugarcane juice particles was examined on the basis of Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. Results from these investigations have provided evidence (in terms of particle stability) on why juice particles derived from whole green sugarcane crop are relatively difficult to coagulate (and flocculate). The presence of trash reduces the van der Waals forces of attraction between particles, thereby reducing coagulation and flocculation processes. It is anticipated that further fundamental work will lead to strategies that could be adopted for clarifying juices expressed from whole green sugarcane crop.

Effect of trash on the coagulation and flocculation of sugarcane juice

The processing of juice expressed from whole green sugarcane crop (stalk and trash) leads to poor clarification performance, reduced sugar yield and poor raw sugar quality. The cause of these adverse effects is linked to the disproportionate contribution of impurities from the trash component of the crop. This paper reports on the zeta (?) potential, average size distribution (d50) and fractal dimension (Df) of limed juice particles derived from various juice types using laser diffraction and dynamic light scattering techniques. The influence of non-sucrose impurities on the interactive energy contributions between sugarcane juice particles was examined on the basis of Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. Results from these investigations have provided evidence (in terms of particle stability) on why juice particles derived from whole green sugarcane crop are relatively difficult to coagulate (and flocculate). The presence of trash reduces the van der Waals forces of attraction between particles, thereby reducing coagulation and flocculation processes. It is anticipated that further fundamental work will lead to strategies that could be adopted for clarifying juices expressed from whole green sugarcane crop.

The influence of impurities on calcium phosphate floc structure and size in sugar solutions

Settling, dewatering and filtration of flocs are important steps in industry to remove solids and improve subsequent processing. The influence of non-sucrose impurities (Ca2+, Mg2+, phosphate and aconitic acid) on calcium phosphate floc structure (scattering exponent, Sf), size and shape were examined in synthetic and authentic sugar juices using X-ray diffraction techniques. In synthetic juices, Sf decreases with increasing phosphate concentration to values where loosely bound and branched flocs are formed for effective trapping and removal of impurities. Although, Sf did not change with increasing aconitic acid concentration, the floc size significantly decreased reducing the ability of the flocs to remove impurities. In authentic juices, the flocs structures were marginally affected by increasing proportions of non-sucrose impurities. However, optical microscopy indicated the formation of well-formed macro-floc network structures in sugar cane juices containing lower proportions of non-sucrose impurities. These structures are better placed to remove suspended colloidal solids.

Influence of biodiesel fuel composition on the morphology and microstructure of particles emitted from diesel engines

This study investigates the morphology, microstructure and surface composition of Diesel engine exhaust particles. The state of agglomeration, the primary particle size and the fractal dimension of exhaust particles from petroleum Diesel (petrodiesel) and biodiesel blends from microalgae, cotton seed and waste cooking oil were investigated by means of high resolution transmission electron microscopy. With primary particle diameters between 12-19 nm, biodiesel blend primary particles are found to be smaller than petrodiesel ones (21±2 nm). Also it was found that soot agglomerates from biodiesels are more compact and spherical, as their fractal dimensions are higher, e.g. 2.2±0.1 for 50% algae biodiesel compared to 1.7±0.1 for petrodiesel. In addition, analysis of the chemical composition by means of x-ray photoelectron spectroscopy revealed an up to a factor of two increased oxygen content on the primary particle surface for biodiesel. The length, curvature and distance of graphene layers were measured showing a greater structural disorder for biodiesel with shorter fringes of higher tortuosity. This change in carbon chemistry may reflect the higher oxygen content of biofuels. Overall, it seems that the oxygen content in the fuels is the underlying reason for the observed morphological change in the resulting soot particles.