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# Fractal dimension

About: Fractal dimension is a(n) research topic. Over the lifetime, 14764 publication(s) have been published within this topic receiving 329050 citation(s).

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TL;DR: Fractal dimension D as discussed by the authors is a global measure of surface irregularity, which labels an extremely heterogeneous surface by a value far from two, and it implies 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.

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 cov...

887 citations

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TL;DR: In this article, it is shown that the scattering law for a fractal object is given by S(Q), Q−D, where Q is the magnitude of the scattering vector.

Abstract: Fractal structures are characterized by self similarity within some spatial range. The mass distribution in a fractal object varies with a power D of the length R, smaller than the dimension d of the space. When the range of physical interest falls below 1000 A, scattering techniques are the most appropriate way to study fractal structures and determine their fractal dimension D. Small-angle neutron scattering (SANS) is particularly useful when advantage can be taken of isotopic substitution. It is easy to show that the scattering law for a fractal object is given by S(Q), ~ Q−D, where Q is the magnitude of the scattering vector. However, in practice some precautions must be taken because, near the limits of the fractal range, there are important deviations from this simple law. Some relations are derived which can be applied in relatively general situations, such as aggregation and gelation. The effects of polydispersity, important, in particular, in situations described by percolation models, are also shown.

844 citations

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TL;DR: The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of Fractals to nonlinear dynamical systems, and to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor.

Abstract: Fractals arise from a variety of sources and have been observed in nature and on computer screens. One of the exceptional characteristics of fractals is that they can be described by a noninteger dimension. The geometry of fractals and the mathematics of fractal dimension have provided useful tools for a variety of scientific disciplines, among which is chaos. Chaotic dynamical systems exhibit trajectories in their phase space that converge to a strange attractor. The fractal dimension of this attractor counts the effective number of degrees of freedom in the dynamical system and thus quantifies its complexity. In recent years, numerical methods have been developed for estimating the dimension directly from the observed behavior of the physical system. The purpose of this paper is to survey briefly the kinds of fractals that appear in scientific research, to discuss the application of fractals to nonlinear dynamical systems, and finally to review more comprehensively the state of the art in numerical methods for estimating the fractal dimension of a strange attractor.

837 citations

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TL;DR: In this article, asymptotic expansions of these integral forms are used to constrain a unified equation for small-angle scattering, and this approach is extended to describe structural limits to mass-fractal scaling at the persistence length.

Abstract: The Debye equation for polymer coils describes scattering from a polymer chain that displays Gaussian statistics. Such a chain is a mass fractal of dimension 2 as evidenced by a power-law decay of −2 in the scattering at intermediate q. At low q, near q ≃ 2π/Rg, the Debye equation describes an exponential decay. For polymer chains that are swollen or slightly collapsed, such as is due to good and poor solvent conditions, deviations from a mass-fractal dimension of 2 are expected. A simple description of scattering from such systems is not possible using the approach of Debye. Integral descriptions have been derived. In this paper, asymptotic expansions of these integral forms are used to describe scattering in the power-law regime. These approximations are used to constrain a unified equation for small-angle scattering. A function suitable for data fitting is obtained that describes polymeric mass fractals of arbitrary mass-fractal dimension. Moreover, this approach is extended to describe structural limits to mass-fractal scaling at the persistence length. The unified equation can be substituted for the Debye equation in the RPA (random phase approximation) description of polymer blends when the mass-fractal dimension of a polymer coil deviates from 2. It is also used to gain new insight into materials not conventionally thought of as polymers, such as nanoporous silica aerogels.

793 citations

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TL;DR: In this article, the authors studied the topography of various natural rock surfaces from wavelengths less than 20 microns to nearly 1 meter, including fresh natural joints (mode I cracks) in both crystalline and sedimentary rocks, a frictional wear surface formed by glaciation and a bedding plane surface.

Abstract: The mechanical and hydraulic behavior of discontinuities in rock, such as joints and faults, depends strongly on the topography of the contacting surfaces and the degree of correlation between them. Understanding this behavior over the scales of interest in the earth requires knowledge of how topography or roughness varies with surface size. Using two surface profilers, each sensitive to a particular scale of topographic features, we have studied the topography of various natural rock surfaces from wavelengths less than 20 microns to nearly 1 meter. The surfaces studied included fresh natural joints (mode I cracks) in both crystalline and sedimentary rocks, a frictional wear surface formed by glaciation, and a bedding plane surface. There is remarkable similarity among these surfaces. Each surface has a “red noise” power spectrum over the entire frequency band studied, with the power falling off on average between 2 and 3 orders of magnitude per decade increase in spatial frequency. This implies a strong increase in rms height with surface size, which has little tendency to level off for wavelengths up to 1 meter. These observations can be interpreted using a fractal model of topography. In this model the scaling of the surface roughness is described by the fractal dimension D. The topography of these natural rock surfaces cannot be described by a single fractal dimension, for this parameter was found to vary significantly with the frequency band considered. This observed inhomogeneity in the scaling parameter implies that extrapolation of roughness to other bands of interest should be done with care. Study of the increase in rms height with profile length for two extreme cases from our data provides an idea of the expected variation in mechanical and hydraulic properties for natural discontinuities in rock. This indicates that in addition to the scaling of topography, the degree of correlation between the contacting surfaces is important to quantify.

778 citations