Modeling the fractal nature of fat crystal networks

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Tang, Dongming
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University of Guelph

This thesis investigated the microstructural basis of several 2D and 3D microscopy fractal dimensions used to quantify the microstructure of fat crystal networks by using computer simulation. Different microscopy fractal dimensions were found to be sensitive to different microstructural factors of fat crystal networks, and thus have different physical meanings. The box-counting fractal dimension, 'Db', increases with an increase in crystal size and area fraction of fat crystals, while the particle-counting fractal dimension, 'Df,' is sensitive to the radial distribution pattern of fat crystals, and the Fourier-Transform fractal dimension, ' DFT,' decreases with increasing crystal size. The relationship between 2D and 3D fractal dimensions of a self-similar fractal object was addressed through both theoretical analysis and experimental study. The common approach to obtain the 3D fractal dimension of a self-similar fractal object by adding one to its corresponding 2D fractal dimension was found to be inappropriate. A 3D box-counting, particle counting and mass fractal dimension must be determined through image analysis of 3D volume elements of the fractal object. During this work, several computer programs were developed to assist the simulation, and calculating fractal dimensions of the simulation images and the micrographs of real fat samples including 3D-FD, which is used to calculate fractal dimension of a self-similar object in 3D space and became commercially available in 2006. The concept of "heterogeneity of stress distribution" was introduced into the modeling of rheological properties of fat crystal networks. The volume fraction of solids term ('[Phi]') in the original fractal model was replaced by '[Phi]e', the effective volume fraction of solids, in the modified fractal model, which represents the volume fraction of stress-carrying solids. An analytical expression of '[Phi] e' was obtained through probabilistic approach and a modified expression for the scaling relationship between 'G'' and ' [Phi]' is obtained. The modified fractal model fits the experiment data well and successfully explains the non-linear log-log plot between the shear elastic modulus of colloidal networks and their volume fraction of solids.

microscopy fractal dimensions, microstructure, fat crystal networks, computer simulation