Fractal dimension network

Polycondensation reactions (eqs. 3 and 4), continue to occur within the gel network as long as neighboring silanols are close enough to react. This increases the connectivity of the network and its fractal dimension. Syneresis is the spontaneous shrinkage of the gel and resulting expulsion of Hquid from the pores. Coarsening is the irreversible decrease in surface area through dissolution and reprecipitation processes. 

Fig. 9. Relation between relaxation exponent n and fractal dimension d for a three-dimensional network. In case of complete screening of excluded volume, values of 0 < n < 1 are possible if d is chosen between 1.25 and 2.5...

Lenormand R, Zarcone C (1985) Invasion percolation in an etched network Measurement of fractal dimension. Phys Rev Lett 54 2226-2229... 

Hydrolysis of tetraalkoxysilanes in pure water is usually incomplete, but it is more effectively carried out in an alcohol-water mixture and can be catalyzed by H+ or weak bases such as ammonia. Polymerization occurs in the range pH 2-7 under neutral conditions, the rate of polymerization is limited by the slowness of hydrolysis, whereas in acidic media hydrolysis is complete before polymerization begins. The nanostructure of the gel (density of particles, fractal dimensions of particle clusters, and degree of cross-linking of particles to form a network) is controlled by these conditions. 

The fractal dimension of a microstructural network can be determined Theologically by diluting a fat with an oil that does not appreciably dissolve the fat under the test conditions (preferably at a low temperature and crystallized rapidly to prevent fractionation). The exact range of dilutions... 

Our work suggests that the fractal dimension of a network is a measure of the order in the spatial distribution of the solid mass in the network as well as the degree of fill of such space. High fractal dimensions are associated with more ordered distributions and higher degrees of fill. 

The fractal dimension of a crystal network is an important parameter in terms of its relation to mechanical strength. However, the values of the pre-exponential term, A, (and the solid fat content) are equally important. For spherical microstructures (Narine and Marangoni, 1999c Marangoni, 2000 Marangoni and Rogers, 2003),... 

The fractal dimension of the fat crystal network in milk fat decreased from 2.5 to 2.0 when the cooling rate was increased. Concomitantly, the particle-related constant, A, increases. These results demonstrate how a faster cooling rate leads to a less ordered spatial distribution of mass within the microstructural network, which would result in a lower value of D, and a decrease in the average particle diameter, which would result in a higher value of A, as predicted by our model. These microstructural changes were correlated with a much higher yield force value for the rapidly cooled milk fat (64.1 3.3N versus 33.0 3.9N for the samples cooled at 5.0°C/min and 0.1°C/min, respectively). 

The fractal dimension D is used to quantify the micro structure of the fat crystal networks, where d is the Euclidean dimension, x is the backbone fractal dimension that is estimated between 1 and 1.3. The backbone fractal dimension describes the tortuosity of the effective chain of stress transduction within a cluster of particles yielding under an externally applied stress (Shih et al. 1990 Kantor and Webman 1984). 

The shear storage modulus of the network is proportionally related to the force constant K. Thus, G is also related to the particle volume fraction via the fractal dimension D) of the network. 

In practice, the fractal dunensions of fat crystal networks, are measured by determining the shear storage modulus, G, of the fat crystal networks within the LVR measured by small deformation rheology. The fractal dimension of the fat crystal network may be calculated from the slope of the curve as > = 3-1/slope of logarithm of G at different SFC, O. 

A relatively new method to determine the fractal dimension is by using oil permeability measurements. In this method, the permeability coefficient, B, is measured for fat samples containing different SFC. This physical fractal dimension, the permeability fractal dimension. Dp, links the volumetric flow rate of liquid oil penetrating a colloidal fat crystal network with its SFC as (Bremer et al. 1989) ... 

Bremer et al. (1989) described a detailed experimental method used to measure the permeability fractal dimension of a fat crystal network. Similarly to the rheology fractal dimension, the permeability fractal dimension, Dp, could be obtained from the nonlinear regression between Q and O as shown in Figure 17.23 (Tang and Marangoni 2005). 

Figure 17.24. General scheme used to measure the microscopy fractal dimension of a colloidal fat crystal network.

In addition two-dimensional imaging of polarized light micrographs of fat samples may be employed to determine the fractal dimension. The general scheme to measure the microscopy fractal dimension of a colloidal fat crystal network is shown in Figure 17.24. 

Table 17.7 summarizes the effects of the microstructural factors on the microscopy fractal dimensions, Dj, y, and Zlpr- Different fractal dimensions reflect different aspects of the microstructure of the fat crystal networks and thus have different meanings. It is necessary to define which structural characteristic is most closely related to the macroscopic physical property of interest (mechanical strength, permeability, diffusion) and then use the fractal dimension that is most closely related to the particular structural characteristic in the modeling of that physical property. 

Tang, D., and Marangoni, A.G. (2006). Quantitative study on the mierostrueture of colloid fat crystal networks and fractal dimensions. Advanees in Colloid and Interface Science. 

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