Computer simulation of fractal dimensions of fat crystal networks

Abstract

The rheological properties of fat-structured products are determined by the microstructure of their fat crystal networks, which can be quantified by using microscopical and rheological techniques. Of particular interest to this study is the quantification of the fractal dimension of the network using these two techniques. Fractal dimensions determined by polarized light microscopy include box-counting, particle-counting, and Fouriertransform fractal dimensions, whereas the fractal dimensions determined by small deformation dynamic rheology exploit the dependence of the storage modulus on the solids' volume fraction. This work reveals that different microscopy fractal dimensions are sensitive to different microstructural factors within the fat crystal network, and thus have different physical meanings. The boxcounting fractal dimension, D b , increases with increases in crystal size and area fraction of the fat crystals, whereas the particlecounting fractal dimension, D f , is sensitive to the radial distribution pattern of fat crystals; and the Fourier-transform fractal dimension, D FT , decreases with increasing crystal size. In the studies on the macroscopic physical properties of fat crystal networks, it is necessary to find the determining structural characteristics and then use the fractal dimensions that are most closely related.In food products containing solid fat as the continuous phase, the fat crystals interact with each other to form a 3-D fat crystal network, which percolates throughout the system. The shape, size, number of fat crystals, and the spatial distribution pattern of the fat crystals, which constitute the microstructure of the fat crystal network, are key factors determining the rheological properties of the network and hence the food product. According to the fractal model developed for fat crystal networks by our group (1-3), the microstructure of fat crystal networks is related to their rheological properties through the fractal dimension of the fat crystal networks, D, as: [1] where G´is the shear storage modulus of the fat samples, Φ is the volume fraction of solids [(SFC/100), where SFC = solid fat content] of the samples, D is the fractal dimension of the fat crystal networks, and λ is the pre-exponential factor, which depends on the size of the microstructural element within the network and the nature of the intermolecular forces.The fractal dimensions of fat crystal networks can be obtained from rheology experiments (termed rheology fractal dimensions) and from 2-D image analysis of the polarized light microscopy of fat samples (termed microscopy fractal dimensions). To show that the change in the microstructure of the fat crystal networks causes the change in the rheological properties of the fat samples, it is necessary to have consistency between the rheology fractal dimensions and the microscopy fractal dimensions. Furthermore, several methods have been used to calculate the microscopy fractal dimensions including the extensively used box-counting and particle-counting methods...

Structural and Mechanical Behavior of Tristearin/Triolein-rich Mixtures and the Modification Achieved by Interesterification

“…As Tang et al reported, one of the factors influencing D box is area fraction of crystalline mass. 32 It is evident from Table 4 that there is more filled area in the CI blends compared with the EI blends. Moreover, our previous paper showed there is a correlation between area fraction and D box .…”

“…D box is less sensitive to area fraction when the size of crystals becomes large and it is not sensitive to crystal size when the area fraction is high. 32 Figure 11 shows that D box increases with increasing SFC (P < 0.0001). However, the sensitivity of this method decreases at high concentrations of hard stock.…”

“…This could be due to several factors including a higher proportion of filled space, larger crystals, and a higher order in the spatial distribution of mass. 32 At this point, it is valuable to revisit the meaning of the rheology fractal dimension (D r ) and its relationship to structural features within the network. In the weak-link rheological regime, crystal cluster size is an important factor influencing the rheological properties of fat crystal networks.…”

Fractal Dimensions of Simulated and Real Fat Crystal Networks in 3D Space

“…Tang and Marangoni [12] studied how each microstructural factor individually affected D b , D f , and D FT separately in 2D space through computer simulation. In this paper, we built on our previous simulation study and used computer simulation to show how each of the microstructural factors, such as crystal size, area fraction (AF) and distribution orderliness (randomly distribution or even distribution) of fat crystals affect the fractal dimensions of the fat crystal networks in 3D space.…”

Modeling the Rheological Properties of Fats: A Perspective and Recent Advances

“…17,24 The fractal dimensions determined by microscopy describe the combined effects of morphology and spatial distribution patterns of the fat crystals within the network and are sensitive to the dimensionality of the embedding space. [25][26][27] In the study on the rheological properties of colloidal gel networks, Shih et al 28 considered a three-dimensional colloidal network as being composed of interconnected flocs, and the fractal dimension was used to quantify the relationship between the average floc size and the particle concentration of the colloidal network. Based on the relative value of the elastic constant of the interfloc links to that of the flocs, two regimes were identified: the strong link regime, where the elastic constant of the interfloc links is larger than that of the flocs, and weak link regime, where the elastic constant of the interfloc links is smaller than that of the flocs.…”