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We present a detailed numerical test of the coarse-graining method, proposed by Sapoval to compute the flux through an irregular interface in the case where the local response is inhomogeneously distributed. It is shown, through comparison with detailed finite elements simulations, that this method permits to deduce the flux across an irregular interface from its topography only, as for example in the case of non-uniform polarisability in electrochemistry. The interest of the method lies in its computational simplicity. It then constitutes an essential step towards the understanding of the flux across irregular interfaces in non-linear regimes. |
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Acoustical properties of irregular cavities described by fractal shapes are investigated numerically. Geometrical irregularity has three effects. First, the low frequency modal density is enhanced. Second, many of the modes are found to be localized at the cavity boundary. Third, the acoustical losses, computed in a boundary layer approximation, are increased proportionally to the perimeter area of the resonator and a mathematical fractal cavity should be infinitely damped. We show that localization induces even higher losses. The same considerations should apply to acoustical waveguides with irregular cross-section. |
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Vibrations of 2d systems with irregular or fractal free boundaries are studied on specific examples. The eigenmodes are calculated numerically using the analogy between Helmholtz and diffusion equations. We discuss the influence of the fractal boundary on the low-frequency part of the spectrum and on waveforms. The density of states is increased by the irregularity and exhibits oscillations at special frequencies which depend on the geometry. Surprisingly, many states are found to be confined at the fractal boundary. Increasing the perimeter fractality induces increased confinement. |
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We discuss the effects of geometrical irregularities on the properties of quantum states in irregular wells. This may apply to small crystallites, quantum dots or wires. We show that geometrical irregularity may play an essential role in the electro-optical properties of porous silicon. The fundamental state is found to be essentially confined in the inner free volume of the irregular structure. The consequence is a screening of the fundamental wave-function from the surface. This leads to an enhancement of the quantum confinement effect hence an increase of the effective band gap and modifications in the density of states in the near band gap region. The screening also lowers the coupling with surface states, a fact which may contribute to explain the large luminescence yield found in this material. We show that geometrical irregularities may have a surprising passivation effect for the fundamental state. In the same way the acoustic phonons have a qualitative tendency to be localized near the surface so that the electron-phonon coupling could be also decreased, for the fundamental energy state, by the existence of geometrical irregularity. |
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The behavior of electron and phonon excitations in systems with irregular boundaries is discussed on the specific example of prefractal shapes. We show that both electron states and acoustical phonons exhibit localization properties. This effect is stronger when the fractality of the shape is increased. In consequence the electron-phonon interaction in small crystallites should be strongly dependent on their shape or roughness. This localization could play a role in the thermal properties of glasses where internal partial crytallization has no reason to build pseudo-crytalline entities which should be smooth. |
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We describe a simple way to compute the response of an irregular resistive interface to a Laplacian field in d=2. It permits to find the linear response of electrodes with an arbitrary geometry from the image only of the electrode. It also allows to compute the non-linear response of self similar electrodes. This method applies in principle to arbitrary irregular geometry in d=2 and it permits to predict generally that the slope of the Tafel plot is divided by the fractal dimension. |
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In contrast with standard percolation where criticality is reached only for a particular value pc of the driving parameter p, gradient percolation exists without the precise tuning of a percolation parameter. For this reason it may be a common physical situation. Very generally, gradient percolation will appear in a uniform system whenever there exists a local random response to an excitation which varies in space. We show that such a situation exists in the example of photographic imaging, due to the random aspect of the photographic process. In this case gradient percolation may be used as a filter for recovering fuzzy images. This filter has the advantage of self-adjusting and to be neutral in regards to the size of the objects. In particular it could be used to increase artificially the depth of focus on photographs that are partially fuzzy. |
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The damping of the vibrations of very irregular discretized systems embedded in a viscous fluid is studied in the particular case of the vibrations of percolation clusters. We develop a formal description for the "regularity" of a vibrational mode. This permits us to measure numerically how the local fluctuations in the vibration amplitude contribute to the viscous damping. The fact that the regularity is found to be larger that that of a single localized state on a linear chain is indicative of the very structure of the percolation cluster made of blobs and red bonds. |
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We apply results from the study of fractal electrodes to diffusion across an irregular passive membrane with finite permeability. Diffusion efficiency is more specially discussed and seems to explain semi-quantitatively the final step of respiratory physiology i.e. the diffusion and capture of oxygen in the acinus region of the lung. We define a "best possible acinus" for which the acinus cut by a plane has a length equal to the ratio of the diffusivity to the membrane permeability. The observed anatomy of the acinus of several animals corresponds roughly to this optimized geometry. This close relation between morphometry and transport parameters like diffusivities and permeabilities could help to understand the design of biological organisms. |
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The damping of the vibrations of very irregular discretized systems embedded in a viscous fluid is studied.in the particular case of the vibrations of a percolation cluster. We develop a formal description for the "regularity" of a vibrational mode. This permits us to measure numerically how the local fluctuations in the vibration amplitude increase the viscous damping. We find that a major part of the dissipation is localized in a small fraction of the next nearest neighbors pairs. This is a qualitative indication of the existence of a "localization within localization" of the dissipation. The fact that the regularity is found to be larger than that of a single localized state on a linear chain is qualitatively indicative of the structure of the percolation cluster made of blobs and red bonds. |
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We describe a simple way to study the response of an irregular resistive interface to a Laplacian field. Using this method, one can find the response of an arbitrary electrochemical electrode from its geometry alone. The same method applies to the study of the steady-state transfer across irregular membranes with finite permeability. |
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The viscous damping of fracton vibrations in 2d percolation clusters is computed. It is found that the anomalous damping is due to the local irregular distribution of the vibration amplitudes. There exists a localization of the damping within the usual localization of fractons. This effect is strongly enhanced by non linear dissipation. This is the first evidence of the damping power of fractal structures. |
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It is shown on a specific example that fractal boundary conditions drastically alter the properties of wave excitations in space. The low frequency part of the vibration spectrum of a finite range fractal drum is computed using an analogy between the Helmoltz equation and the diffusion equation. The irregularity of the frontier is found to strongly influence the density of states at low frequency. The fractal perimeter generates a specific screening effect. Very near the frontier the decrease of the waveform is related directly to the behavior of the harmonic measure. The possibility of localization of the vibrations is qualitatively discussed and we show that localized modes may exist at low frequencies if the geometrical structure possess narrow paths. Possible application of these results to the interpretation of thermal properties of binary glasses are briefly discussed. |
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It is shown that the iteration of a simple approximate equivalent circuit gives the impedance of a d=2 self-similar electrode. It permits to find geometrically the working set of the electrode or the information fractal. |
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A comprehensive study of the macroscopic transport parameters of self-similar interfaces is presented. The iteration of a simplified equivalent leads to the prediction of a simple mathematical expression for the impedance of fractal electrodes in d=2 and d=3. The same value is predicted by scaling arguments and verified by extended numerical simulations in d=2. Experiments on model electrodes confirm the theoretical prediction. We introduce the approximate concept of information fractal. It gives a very simple access to the theory and a description of the regions of the fractal surface which are really active for the transport. The same result should apply to transport across fractal membranes and to certain Eley-Rideal heterogeneous catalysis processes. |
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The low frequency vibration spectrum of a surface fractal with a rigid fractal boundary presents a smaller density of states than an ordinary resonator. Such vibrations, called fractinos, should be present in partially phase separated glasses if one component is soft. In addition the interfaces between the phase separated regions are fractal diffusion fronts which are mass fractals close to percolation cluster hulls. It is proposed that the fractons which have been claimed to exist in these non fractal materials are the vibrations of these internal interfaces. |
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Fractal boundary conditions drastically alter wave excitations. The low frequency vibrations of a membrane bounded by a rigid fractal contour are observed and localized modes are found. The first lower eigenmodes are computed using an analogy between the wave and the diffusion equations. The fractal frontier induces a strong confinement of the wave analogous to super-localization. The waveforms exhibit singular derivatives near the boundary. |
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It is shown that there exist an exact analogy between the study of the impedance of irregular electrodes, the effective transfer across a irregular membrane and the catalytic activity of catalyst with the same irregular geometry possibly fractal. In this frame one can explain the existence of non-integer reaction order and relate them to reaction dimensions. |
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A new method of simulating the response of an irregular interface was investigated. It is based on an exact mapping between the Laplace equation and the steady state diffusion equation with mixed boundary conditions. Simulations in 2d show that D.L.A. and other self-similar fractal electrodes exhibit the so-called Constant Phase Angle (C.P.A.) behavior. In the case of D.L.A. electrodes the C.P.A. exponent is found by this method to be close to the inverse of the fractal dimension. We show that this is directly related to the fact that in d=2 the admittance is proportional to the overall size of the self-similar electrode. |
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The electrical response of porous electrodes is calculated in several particular cases, which permit to approach the response of a realistic model for a porous interface. The case of non-blocking surfaces and the case of the diffusion impedance of a fractal electrode are also considered. The use of Bode diagrams is shown to provide a very simple means for calculating phase angles and algebraic values for the impedance. It is demonstrated that for a blocking deterministic Sierpinski electrode the impedance presents oscillations around C.P.A.. Various electrochemical regimes, blocking , non-blocking and diffusive are considered giving rise to a variety of exponents. For blocking electrodes it is shown that at a given frequency, the power is dissipated in certain parts of the electrodes having a characteristic size which is a direct function of frequency. The fact that the response of the system is linear permits to relate in general the d.c. response to the phase angle in the blocking regime and to study certain diffusive cases. It also permits to deal with cases very common practically where the response of a flat surface would itself exhibit C.P.A.. In the case of a pure diffusion impedance the response is shown to be related directly to the Minkowski-Bouligand exterior dimension of the interface through the exponent (D-1)/2. This approach can be generalized to any type of irregular electrode independently of its fractal character. If both diffusion and Faradaic impedance play a role, a C.P.A. response exists for a porous electrode with an exponent equal to D-2. We discuss various regimes in which diffusion plays a role together with Faradaic, resistive and capacitive effects. It is shown that there is in general no relation between fractal dimension and constant phase angle except in the case of diffusion. The response of irregular electrodes is shown to be related to the fractal dimension when the electrochemical regime is local. |