Fractals

Meakin, Paul

doi:10.1002/0471440264.pst611

Encyclopedia of Polymer Science and Technology 2013

DOI: 10.1002/0471440264.pst611

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Fractals

Paul Meakin¹

Abstract: Fractal structures have been observed in a variety of polymers and polymer composites. Their fractal character is associated with the scale invariance of polymer morphology or other properties. This article introduces basic aspects of fractal theory and models of polymer properties such as the percolation model for gelation. Methods to measure fractal dimensionalities and practical applications of fractal polymers are also presented.

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Cited by 2 publications

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Three‐dimensional roughness of stylolites in limestones

Renard

2004

J. Geophys. Res.

102

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[1] Stylolites are dynamic roughly planar surfaces formed by pressure solution of blocks of rocks into each other. The three-dimensional geometry of 12 bedding-parallel stylolites in several limestones was measured using laser and mechanical profilometers, and statistical characteristics of the surfaces were calculated. All the stylolites analyzed turn out to have self-affine fractal roughness with a well-characterized crossover length scale separating two self-affine regimes. Strikingly, this characteristic length scale falls within a very narrow range for all the stylolites studied, regardless of the microstructure sizes. To explain the data, we propose a continuous phenomenological model that accounts for the development of the measured roughness from an initially flat surface. The model postulates that the complex interface morphology is the result of competition between the long-range elastic redistribution of local stress fluctuations, which roughen the surface, and surface tension forces along the interface, which smooth it. The model accounts for the geometrical variability of stylolite surfaces and predicts the dependence of the crossover length scale on the mechanical properties of the rock.

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Three‐dimensional roughness of stylolites in limestones

Renard

2004

J. Geophys. Res.

102

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Nature of inclined growth in thin-layer electrodeposition under uniform magnetic fields

et al. 2012

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Electrochemical deposition (ECD) in thin cells in a vertical position relative to gravity, subject to an external uniform magnetic field, yields a growth pattern formation with dense branched morphology with branches tilted in the direction of the magnetic force. We study the nature of the inclined growth through experiments and theory. Experiments in ECD, in the absence of magnetic forces, reveal that a branch grows by allowing fluid to penetrate its tip and to be ejected from the sides through a pair of symmetric vortices attached to the tip. The upper vortices zone defines an arch separating an inner zone ion depleted and an outer zone in a funnel-like form with a concentrated solution through which metal ions are carried into the tip. When a magnetic field is turned on, vortex symmetry is broken, one vortex becoming weaker than the other, inducing an inclination of the funnel. Consequently, particles entering the funnel give rise to branch growth tilted in the same direction. Theory predicts, in the absence of a magnetic force, funnel symmetry induced through symmetric vortices driven by electric and gravitational forces; when the magnetic force is on, it is composed with the pair of clockwise and counterclockwise vortices, reducing or amplifying one or the other. In turn, funnel tilting modifies particle trajectories, thus, growth orientation.

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Fractals

Cited by 2 publications

References 106 publications

Three‐dimensional roughness of stylolites in limestones

Three‐dimensional roughness of stylolites in limestones

Nature of inclined growth in thin-layer electrodeposition under uniform magnetic fields

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