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

Abstract

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.

Liposome—glass interaction. Influence of the lipid bilayer phase state

“…Recent work [17][18][19] has shown large differences between the smooth and nitrogen (from BET measurements) area of solid surfaces for a large variety of nonporous solids, reflecting the surface roughness on a molecular scale. The latter can be quantified using the concept of surface fractal dimension [17][18][19].…”

“…The latter can be quantified using the concept of surface fractal dimension [17][18][19]. It rationalizes that on a molecularly rough surface, the surface area available for saturated adsorption depends on the size of adsorbing molecules.…”

Fabrication and Fractality of Fe2O3-CeO2/ZSM-5 Composites for High-Temperature Desulfurization

“…In recent decades, fractal analysis has been widely applied in a few fields of scientific researches to characterize the geometric and structural properties of fractal surfaces and pore structures [33,34]. Fractal dimension is often adopted to quantitatively evaluate the irregularities of the fractal surface and pore structures.…”

Chaos theory in chemistry and chemometrics: a review

“…The second two publications of this series were a single two-part paper devoted to the exploration of the application of fractal dimensionality as a diagnostic tool for heterogeneous surfaces, providing evidence that many different types of surface adsorbents have a sufficient capability to produce simple monolayers to maintain the theoretical basis for using an experimental fractal analysis based on monolayer coverage [22,23]. This paper is also the first in this series of four publications in which the mathematical relationship between D and r is given a rigorous equation:…”