Mass fractal, porosity

The porosity can be controlled in two ways. The first method is based on the scaling of mass Mf and size tf of the mass fractal particles. Since density equals mass/volume, the density pf of a mass fractal object varies in three-dimensional space as ... 

Thus the porosity of a mass fractal decreases with its size and when complete interpenetration is avoided (which requires Df > 1.5) the porosity can be controlled by the size of the branched specimen during drying. Examples of this procedure are given by Brinker et al. [42]. This discussion reveals a dilemma to obtain the smallest pores, interpenetration should be large and is obtained by D < 1.5 and low condensation rates. This leads however to low porosities. 

Besides the determination of the porosity, pore size distribution and surface area, the (surface or mass) fractal dimension of dry gels may be of interest. To this purpose, small-angle X-ray scattering can be used nitrogen sorption and mercury porosimetry also offer possibilities to extract this structural information (see, for example, Blacher et al. (2000)). 

Table I. Refractive indices and vol% porosities (calculated from the Lorentz-Lorenz relation [18]) for B2 and AAB films as a function of aging time normalized by the gelation time. Mass fractal dimension values are for sols aged for comparable normalized aging times.

The values of the mass fractal dimension (obtained from SAXS [16]) show how the structure of the silicate polymers in the sol varies with the different routes these sols take to the same final dilution, H20/Si ratio and pH. The mass fractal dilnensions are considerably different. We would expect AAB, which is right at the borderline of D = 1.5 for Equation 2, to exhibit different behavior than B2 with D = 2.3. However, Table I shows that both sols exhibit an increase in porosity with sol age (once the pore-plugging species are burned out of B2). The reason for this is the tradeoff between cluster porosity and cluster interpenetration. B2 clusters are less porous but pack less efficiently than AAB clusters. This tradeoff is illustrated schematically in Figure 1. 

Fumed silica is "glass soot" made by burning silicates in flames. It is known to be mass-fractal particles, i.e. submicron-size aggregates composed of random, weakly branched strings of 100 A silica spheres, with a great deal of internal porosity. 

Thus, for mass fractals (0 < D < 3), the porosity of an individual object increases with its size, whereas for uniform objects (D = 3), porosity is invariant with size. 

During film formation by dip-coating, primary sol species are rapidly aggregated by evaporation of solvent. The porosity of the secondary aggregate structure depends on the extent of interpenetration of the primary species. The extent of interpenetration is inversely related to the mean number of intersections Mi 2 of two polymers of radius rc and mass fractal dimension D1 and D2 confined to the same region of three-dimensional space (2) ... 

Equation 3 assumes every point of intersection results in immediate and irreversible attachment, chemically equivalent to an infinite condensation rate. Finite condensation rates mitigate the criterion for mutual transparency, i.e. since every point of intersection does not result in immediate and irreversible "condensation", interpenetration may occur for structures characterized by D > 1.5. The value of equation 3 is that it provides a qualitative understanding of the effect of structure on the extent of polymer interpenetration and thus porosity smaller polymer sizes and lower mass fractal dimensions favor denser films, and larger polymer sizes and greater mass fractal dimensions favor more porous films. In all cases increased condensation rates reduce the tendency toward interpenetration, favoring more porous films. 

The Values of KWW Exponent v, Fractal Dimension Dp, Porosity 4>m Obtained from the Relative Mass Decrement (A, B, C, and D Glasses) and BETA (E, F, and G Glasses) Measurements and Average Porosity (4>p) Estimated from Dielectric Spectra for Porous Glasses Samples. 

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