D

Interface and pore structure are fractal

aD: general fractal dimension.

bd: Euclidean dimension of embedding space (i.e. 1, 2 or 3 respectively in one-, two- or three-dimensional systems).

aD: general fractal dimension.

bd: Euclidean dimension of embedding space (i.e. 1, 2 or 3 respectively in one-, two- or three-dimensional systems).

As an example, consider the situation where a structure is overlaid by a grid. The underlying sites can be categorized as mass sites (occupied sites), surface sites (occupied sites with adjacent empty sites) or pore (empty) sites (Figure 1.2). In order to determine the fractal dimension, it is necessary to quantify the number of mass, surface or pore sites that fall within a distance R from a fixed site (Figure 1.3) [15]. R must always be much greater than the distance between nearest neighbor sites. The quantity Msites(R) refers to the 'mass' in a sphere of radius R. It grows with increasing R [16]:

Each type of site gives rise to a mass-radius relationship of the following form:

As D approaches Euclidean dimensions, the three types of fractal become indistinguishable.

Figure 1.2 Two-dimensional representations of mass fractal (a), surface fractal (b) and pore fractal (c) structures [14]. Reproduced with permission of John Wiley & Sons, Ltd.
Figure 1.3 Illustration of Msites(R). The sites are pictured as adsorbed molecules on a surface, but equally well could be monomers in a polymer chain, primary particles of a colloidal aggregate, etc. [15]. Reproduced with permission of John Wiley & Sons, Ltd.
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