Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces

In this, the first of a series of papers, we lay the foundations for appreciation of chemical surfaces as D-dimensional objects where 2≤D<3. Being a global measure of surface irregularity, this dimension labels an extremely heterogeneous surface by a value far from two. It implies, e.g., that any monolayer on such a surface resembles three-dimensional bulk rather than a two-dimensional film because the number of adsorption sites within distance l from any fixed site, grows as lD. Generally, a particular value of D means that any typical piece of the surface unfolds into mD similar pieces upon m-fold magnification (self-similarity). The underlying concept of fractal dimension D is reviewed and illustrated in a form adapted to surface-chemical problems. From this, we derive three major methods to determine D of a given solid surface which establish powerful connections between several surface properties: (1) The surface area A depends on the cross-section area σ of different molecules used for monolayer coverage, according to A∝σ(2−D)/2. (2) The surface area of a fixed amount of powdered adsorbent, as measured from monolayer coverage by a fixed adsorbate, relates to the radius of adsorbent particles according to A∝RD−3. (3) If surface heterogeneity comes from pores, then −dV/dρ∝ρ2−D where V is the cumulative volume of pores with radius ≥ρ. Also statistical mechanical implications are discussed.