Abstract
For an unexpected variety of solids, the surface topography from a few up to as many as a thousand angstroms is very well described by fractal dimension,D. This follows from measurements of the number of molecules in surface monolayers, as function of adsorbate or adsorbent particle size. As an illustration, we present a first case, amorphous silica gel, whereD has been measured independently by each of the two methods. (The agreement, 3.02±0.06 and 3.04±0.05, is excellent, and the result is modeled by a “heavy” generalized Menger sponge.) The examples as a whole divide into amorphous and crystalline materials, but presumably all of them are to be modeled as random fractal surfaces. The observedD values exhaust the whole range between 2 and 3, suggesting that there are a number of different mechanisms by which such statistically self-similar surfaces form. We show that fractal surface dimension entails interfacial power laws much beyond what is the source of theseD values. Examples are reactive scattering events when neutrons of variable flux pass the surface (this is of interest for locating fractal substrates that may support adlayer phase transitions); the rate of diffusion-controlled chemical reactions at fractal surfaces; and the fractal implementation of the traditional idea that the active sites of a catalyst are edge and apex sites on the surface.
Similar content being viewed by others
References
D. Avnir and P. Pfeifer,Nouv. J. Chim. 7:71 (1983).
P. Pfeifer, D. Avnir, and D. Farin,Surf. Sci. 126:569 (1983).
P. Pfeifer and D. Avnir,J. Chem. Phys. 79:3558 (1983);80:4573 (1984).
D. Avnir, D. Farin, and P. Pfeifer,J. Chem. Phys. 79:3566 (1983).
D. Avnir, D. Farin, and P. Pfeifer,Nature 308:261 (1984).
P. Pfeifer,Appl. Surf. Sci. 18:146 (1984).
D. Avnir, D. Farin, and P. Pfeifer,J. Colloid Interface Sci., in press.
B. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
R. S. Sayles and T. R. Thomas,Nature 271:431 (1978); M. V. Berry and J. H. Hannay,Nature 273:573 (1978); P. A. Burrough,Nature 294:240 (1981).
H. G. E. Hentschel and I. Procaccia,Phys. Rev. A 28:417 (1983).
S. Lovejoy,Science 216:185 (1982).
E. Jakeman,Nature 307:110 (1984).
L. Niemeyer, L. Pietronero, and H. J. Wiesmann,Phys. Rev. Lett. 52:1033 (1984).
D. Paumgartner, G. Losa, and E. R. Weibel,J. Microscopy 121:51 (1981).
J. E. Avron and B. Simon,Phys. Rev. Lett. 46:1166 (1981).
M. Suzuki,Progr. Theor. Phys. 69:65 (1983).
A. Kapitulnik, A. Aharony, G. Deutscher, and D. Stauffer,J. Phys. A 16:L269 (1983); D. C. Hong and H. E. Stanley,J. Phys. A 16:L475 (1983), and references cited therein.
S. Havlin and D. Ben-Avraham,Phys. Rev. A 26:1728 (1982).
I. Magid, Z. Djordjevic, and H. E. Stanley,Phys. Rev. Lett. 51:143 (1983).
T. A. Witten and P. Meakin,Phys. Rev. B 28:5632 (1983); H. Gould, F. Family, and H. E. Stanley,Phys. Rev. Lett. 51:686 (1983); F. Family,Phys. Rev. Lett. 51:2112 (1983); P. Meakin,Phys. Rev. Lett. 51:1119 (1983); M. Kolb, R. Botet, and R. Jullien,Phys. Rev. Lett. 51:1123 (1983).
P. Bak,Rep. Progr. Phys. 45:587 (1982).
D. J. Thouless and Q. Niu,J. Phys. A 16:1911 (1983); R. E. Prange, D. R. Grempel, and S. Fishman,Phys. Rev. B 28:7370 (1983), and references cited therein.
D. Campbell and H. Rose, eds.,Order in Chaos, Physica 7D (1983), Chap. 4.
C. P. Allen, J. T. Colvin, D. G. Stinson, C. P. Flynn, and H. J. Stapleton,Biophys. J. 38:299 (1982).
R. F. Voss, R. B. Laibowitz, and E. I. Allessandrini,Phys. Rev. Lett. 49:1441 (1982); A. Kapitulnik and G. Deutscher,Phys. Rev. Lett. 49:1444 (1982).
M. S. Wolfe, 57th Colloid and Surface Science Symposium, Toronto, 1983.
D. A. Weitz and M. Oliveria, preprint (1983).
B. H. Kaye,Direct Characterization of Fineparticles (Wiley, New York, 1981), Chapter 10.3; S. Peleg, J. Naor, R. Hartley, and D. Avnir,IEEE Trans. Pattern Anal. Machine Intel!., in press.
Y. Gefen, A. Aharony, and B. B. Mandelbrot,J. Phys. A 16:1267 (1983); J. A. Given and B. B. Mandelbrot,J. Phys. B 16:L565 (1983).
G. Binnig and H. Rohrer,Helv. Phys. Acta 55:726 (1982).
A. W. Adamson,Physical Chemistry of Surfaces, 4th ed. (Wiley, New York, 1982); S. J. Gregg and K. S. W. Sing,Adsorption, Surface Area, and Porosity, 2nd ed. (Academic Press, London, 1982).
D. Farin, D. Avnir, and P. Pfeifer,Langmuir 1, to appear.
U. Welz, H. Wippermann, and P. Pfeifer, unpublished.
P. G. Watson, inPhase Transitions and Critical Phenomena, Vol. 2, C. Domb and M. S. Green, eds. (Academic Press, London, 1972), p. 101.
Y. Gefen, B. B. Mandelbrot, and A. Aharony,Phys. Rev. Lett. 45:855 (1980); Y. Gefen, Y. Meir, B. B. Mandelbrot, and A. Aharony,Phys. Rev. Lett. 50:145 (1983); Y. Gefen, A. Aharony, and B. B. MandelbrotJ. Phys. A 16:1267 (1983).
R. Jackson,Transport in Porous Catalysts (Elsevier, Amsterdam, 1977); R. E. Cunningham and R. J. J. Williams,Diffusion in Gases and Porous Media (Plenum Press, New York, 1980); E. A. Mason and A. P. Malinauskas,Gas Transport in Porous Media: The Dusty-Gas Model (Elsevier, Amsterdam, 1983).
R. J. Good and R. S. Mikhail,Powder Technol. 29:53 (1981).
M. V. Berry,J. Phys. A 12:781 (1979); M. V. Berry and T. M. Blackwell,J. Phys. A 14:3101 (1981).
H. Steinhaus,Coll. Math. 3:1 (1954).
P.-G. de Gennes,C. R. Acad. Sci. Paris, Ser. II 295:1061 (1982).
A. Le Mehaute and G. Crepy,C. R. Acad. Sci. Paris, Ser. II 295:685 (1982); A. Le Mehaute and G. Crepy, inFast Ion Transport in Solids (North-Holland, Amsterdam, 1984) to appear.
F. H. Constable, inHandbuch der Katalyse, Vol. 5, G.-M. Schwab, ed. (Springer, Vienna, 1957), p. 141; D. Kalló, inContact Catalysis, Vol. 1, Z. G. Szabó, ed. (Elsevier, Amsterdam, 1976), p. 306.
H. Kral,Ber. Bunsen-Ges. 75:1114 (1971).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pfeifer, P., Avnir, D. & Farin, D. Scaling behavior of surface irregularity in the molecular domain: From adsorption studies to fractal catalysts. J Stat Phys 36, 699–716 (1984). https://doi.org/10.1007/BF01012933
Issue Date:
DOI: https://doi.org/10.1007/BF01012933