Dina Farin

Chemistry in noninteger dimensions between two and three. II. Fractal surfaces of adsorbents

The concept of fractal dimension D of surfaces, advanced as natural measure of surface irregularity in part I of this series, is shown to apply to a remarkable variety of adsorbents: graphites, fume silica, faujasite, crushed glass, charcoals, and silica gel. The D values found for these examples vary from two to almost three (for smooth and very irregular surfaces, respectively), thus covering the whole possible range. They quantify the intuitive picture that surface inhomogeneities are minor, e.g., in graphites, but dominant, e.g., in charcoal. The analysis is based on adsorption data, with main focus on adsorbates of varying molecular cross section. They include N2, alkanes, polycyclic aromatics, a quaternary ammonium salt, and polymers. The straight‐line plots so obtained confirm also a number of reported on‐surface conformations of specific adsorbates. The converse method to get D from varying the size of adsorbent particles is exemplified for fume silica and crushed glass.

Surface geometric irregularity of particulate materials: The fractal approach

Abstract Surface geometric irregularities of a wide variety of particulate materials are determined by applying the fractal theory of surface science (P. Pfeifer and D. Avnir, J. Chem. Phys. 79, 3558 1983). Reanalysis and reinterpretation of previously published experimental data, revealed the following surface-fractal dimensions, all falling in the expected range 2.0 to 3.0, for the following materials: periclase—1.95; a number of quartzes—2.02 to 2.21; iron oxide—2.59; porous silica gel—3.0; coal dusts—2.33 and 2.53; six carbonate rocks—2.16 to 2.97; seven types of soils—2.19 to 2.99; a number of crushed rocks from nuclear test site—2.7 to 3.0. We describe in detail the method by which these fractal dimensions were determined, i.e., studying the dependence of monolayer values (n) on particle radii (R): n ∝ RD-3. Interestingly, several authors had presented their experimental results as log n vs log R, yet no explanation was offered to the straight lines obtained; the notion of self-similarity (fractal surfaces) fits nicely in all these studies. The various features of our approach are described in connection with the analyzed examples, e.g., we exemplify estimation of particle diameters by this approach. Attention is given to the question of range of self-similarity. It is suggested that materials with fractal surfaces were formed in general by an iterative mechanism. Based on the massive list of fractal surfaces found (D. Avnir, D. Farin, and P. Pfeifer, Nature (London) 308, 261, 1984), this is probably the operative mechanism for the majority of natural and synthetic materials.

Ideally irregular surfaces, of dimension greater than two, in theory and practice

Abstract We apply Mandelbrots [1] concept of fractal (noninteger) dimension D to surfaces: We describe how surfaces can provide environments (e.g. for adsorbates) with 2⩽D 2 are “ideally irregular” (self-similar) surfaces; and D→3 obtains in the limit where a surface visits every point of a volume. We present three major methods to get D from experiment. Case studies give ample instances for well-defined D>2. Some eminent consequences are implied.

The carotenoid pigments in the juice and flavedo of a mandarin hybrid (Citrus reticulata) cv michal during ripening

Abstract The carotenoid pigments ofthe mandarin hybrid ( Citrus reticulata ) cv Michal, in the juice and flavedo were characterized at three ripening stages, before, during and after colour break. During ripening the characteristic mandarin pattern was formed in the juice, which contained cryptoxanthin as the principal pigment. In the flavedo the chloroplast carotenoid pattern of the green fruit changed into the characteristic pattern of C. reticulata with a high level of citraurin which, together with cryptoxanthin, imparts an intensive reddish tint to the hybrid. The flavedo contained an unusual C 30 apocarotenoid, β-citraurinene (8′-apo-β-caroten-3-ol). The flavedo carotenoids of this accidental hybrid were compared with the carotenoids of the presumed parents Dancy tangerine and Clementine. The hybrid resembles more the second parent, from which it inherited the ability to biosynthesize a higher amount of citraurin as well as citraurinene. Citraurinene, considered a Citrus hybrid-specific pigment, was found for the first time in a Citrus variety. A possible biosynthetic pathway of the Citrus C 30 -apocarotenoids is proposed.

Accessibility of irregular (fractal) surfaces of porous silicas to silylation reactions

Abstract The effect of surface irregularity on parameters such as surface concentration, area per ligand and interligand distance is discussed with special emphasis on silylation reactions of silica. For the proper evaluation of these parameters, it is suggested that either the effective area for the ligand (and not the nitrogen-BET value) or the fractal (adsorptive or reactive) dimension of the surface be used as a basic parameter, Experimental examples were analysed according to this approach.

Crystallite size effects in chemisorption on dispersed metals

In the elucidation of the crystallite size, 2R, of metals dispersed on inert supports, an assumption is made that np α R2, where np is the chemisorption capacity of one particle. We have found that in many chemisorption studies the relation between np and R is more accurately given by np α RDc, ~1.8 < Dc < ~2.2. It is shown that some basic models of area-size relations can account for such Dc deviations from 2. These include small-size effects, size-dependent chemisorption stoichiometry, and the size-dependent ratio of the crystallographic planes.

The effects of the fractal geometry of surfaces on the adsorption conformation of polymers at monolayer coverage part I. The case of polystyrene

Abstract Surface geometry effects on the adsorption conformations of polymers under plateau conditions are found and analyzed by using fractal geometry considerations. The re-analysis of many adsorption studies, with special emphasize on polystyrene, shows that while on a flat surface a dense packing is formed, distorting the solution conformation into a prolate shape, on highly porous objects the solution conformation changes only slightly upon adsorption. These general trends show a surprisingly low sensitivity to the chemical nature of the polymer—solvent—surface system, emphasizing the crucial role of environmental geometry in determining adsorption conformations.

Fractal analysis of size effects and surface morphology effects in catalysis and electrocatalysis.

Fractal geometry tools are used in order to analyze several related problems in surface science, catalysis, and electrocatalysis. The effects of complex morphologies of adsorbents, catalysts, and electrodes on various molecular processes with these materials are determined both theoretically and experimentally. It is shown that fractal geometry provides a convenient and natural tool for the elucidation of geometry-performance relations in heterogeneous chemistry. Issues covered are particle size effects in physisorption and chemisorption; morphology effects on a variety of catalytic processes with unsupported catalysts (including coal liquefaction, alkene polymerizations, oxidations, dehydrogenations, and esterifications); surface accessibility effects on molecular interactions in an Eley-Rideal mechanism; surface patterning effects on concentration profiles near the surface; and electrode-morphology effects on a variety of electrochemical and electrocatalytic processes. The domains of applicability of the fractal approach to these problems is discussed.

The Fractal Nature of Molecule-Surface Chemical Activities and Physical Interactions in Porous Materials

Abstract We report that fractal scaling laws relate the details of the complex reactive and non-reactive geometries of porous materials with a wide variety of chemical and physical molecule-surface interactions. These scaling laws relate the following property/scale pairs: physisorption monolayer value/particle size of the adsorbent; chemisorption capacity/particle size; catalytic activity of supported metal catalysts/dispersion or catalyst particle size; adsorbate surface reaction rate/particle size; monolayer value/physiorbed adsorbate size; monolayer of adsorbed polymer/the molecular weight of the polymer; the amount of derivatizing agent necessary for full surface grafting/size of the reverse-phasing reagent; In all these cases the property/scale relation has the form of a power-law in which the exponent is a simple function of the effective fractal dimension of the process. Well over one hundred case analyses show the generality and applicability of fractal scaling laws in heterogeneous chemistry.

Photoprocesses on Fractal Surfaces

Publisher Summary This chapter discusses photoprocesses on fractal surfaces. The initial interest in fractal geometry stemmed mainly from the striking similarity between computer-generated objects, as formed by applying the rules of this geometry, and “real” objects as found in many natural or man-made objects. The algorithms used for creating the fractal objects are usually quite simple and involve an iterative construction procedure. Because of this iterative procedure, the objects thus obtained have the property of being self-similar—that is, the various magnifications of the object look similar. The chapter describes the use of molecules as yardsticks. The first yardstick for the resolution analysis discussed, is the molecule interacting with the surface. For an irregular surface, the smaller the molecule is, the finer are the surface details it can probe and the larger will be the apparent surface area, A, as determined from monolayer coverage. Rather than measuring area with a set of molecules, one can perform an equivalent resolution analysis—namely, to take only one yardstick and to measure the area of the object as a function of its size.