B. Sapoval

Design of peripheral airways for efficient gas exchange

Peripheral airways combine branched tubes for ventilation with the gas exchanging alveoli in the pulmonary acini, defined as the complex of airways supplied by one first order respiratory or transitional bronchiole. In this part, the replenishment of oxygen at the alveolar surface occurs by a combination of convective air flow with diffusion of oxygen in the air. The transition between convection and diffusion depends on the morphometric properties of the airways. The design of the peripheral airways in the acinus of the human lung is described quantitatively on the basis of measurements obtained on casts of the acinar airways. Comparable data for rat and rabbit are also discussed. On the basis of this morphometric information, a typical path model for human acinar airways is derived. These studies also form the basis for advanced modeling studies of gas exchange and ventilation. In particular the problems occurring because of diffusional screening and the design conditions for minimizing this effect are discussed.

Fractal electrodes and constant phase angle response: exact examples and counter examples

Abstract The ac response of a fractal electrode has been proposed recently to be of the constant phase angle type (CPA) in the approximate model of a hierarchical lumped circuit. In that model the phase angle exponent η is related to the fractal dimension D of the surface by the relation η = (3-D). We present special fractal electrodes for which the calculation can be made without model approximations. This permits to distinguish two categories of electrodes. In the first category with a fractal dimension larger than 5 2 , the response of the system is indeed of the CPA type but the above relation is not verified. In the second category ( D 5 2 ) the above relation is shown to be verified exactly for a finite size electrode while an infinite electrode is shown to be a counter example to the same relation.

The role of the anions in the growth speed of fractal electrodeposits

We report in this note that the growth speed of metal electrodeposits in the case of electrodeposition without supporting electrolyte is governed by the speed at which the anions are withdrawn from the deposit

A simplified model for glass dissolution in water

Numerical simulations of the water dissolution of a random ternary solid are presented. The three elements represent silica, soluble oxides (alkalis and boron) and quasi-insoluble oxides (Al2O3, ZrO2, Fe2O3,...). The soluble species are dissolved immediately when they are in contact with the solution. Their proportion is kept below the percolation threshold. For the other species, one introduces a model of dissolution-recondensation. It is shown that the dissolution rate constants should be dependent on the bonding environment in order to include surface tension. The condensation fluxes are proportional to the concentration of each species in solution. In the dynamic regime (no recondensation), one observes the congruent dissolution of silica and soluble species, after a short initial phase of selective extraction of the soluble species. The common rate of dissolution decreases with the proportion of insoluble species and increases sharply with that of soluble species. This is mainly due to the formation of a porous hydrated layer whose active surface area increases markedly with the proportion of soluble species. In the static regime (finite solution volume), the equilibrium solubility of silica decreases with the proportion of insoluble species and is practically independent of the proportion of soluble species. The porous hydrated layer is rearranged and almost free of soluble species. The ripening of the surface layer makes it protective and inhibits further extraction of the soluble species. These results are in general agreement with the experimental observations on the dissolution of durable glasses.

Acoustical properties of irregular and fractal cavities

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 contributes to increase the losses. The same considerations should apply to acoustical waveguides with irregular cross section.

Self-Stabilized Fractality of Seacoasts through Damped Erosion

Erosion of rocky coasts spontaneously creates irregular seashores. But the geometrical irregularity, in turn, damps the sea waves, decreasing the average wave amplitude. There may then exist a mutual self-stabilization of the wave amplitude together with the irregular morphology of the coast. A simple model of such stabilization is studied. It leads, through a complex dynamics of the earth-sea interface, to the appearance of a stationary fractal seacoast with a dimension close to 4/3. Fractal geometry here plays the role of a morphological attractor directly related to percolation geometry.

Dynamics of the creation of fractal objects by diffusion and 1/f noise

Abstract We present a video picture which shows the dynamics of diffusion or intercalation on a 2D square lattice. The images of that film are obtained from a simulation of diffusion in a simple lattice gas model. The static images show the fractal geometry of the diffusion or intercalation front. Different magnifications of the diffusion front exhibit the self-similarity which is characteristic of the fractal geometry. The visual observation of the dynamics of diffusion exhibits a new effect : the dynamic changes of the topography of the objects are erratic : strong changes in the geometry occur at random times. These dynamical changes are produced on a semi-macroscopic scale. The existence of a new effect, intercalation noise, is predicted. The same physical process could be a source of 1/f α noise in non-homogeneous electronic devices.

Shape-dependency of current through non-linear irregular electrodes

We describe a simple way to understand the non-linear response of an irregular resistive electrode in d=2. It is based on the concept of an active zone in the Laplacian transfer to and across irregular interfaces. It applies to arbitrary electrode geometry and permits to compute the flux across an irregular electrode from its geometry without solving the Laplace problem. The simplifying arguments that are used are tested numerically on prefractal models of the geometrical irregularity. One finds that, for electrodes following a local Butler–Volmer response, the Tafel slope depends on the geometry. It is shown that the measure of the cell impedance leads to the determination of the actual active potential. It also gives the mean size of a part of the electrode with a surface impedance equal to the electrolyte bulk resistivity.

An anatomical and functional model of the human tracheobronchial tree

The human tracheobronchial tree is a complex branched distribution system in charge of renewing the air inside the acini, which are the gas exchange units. We present here a systematic geometrical model of this system described as a self-similar assembly of rigid pipes. It includes the specific geometry of the upper bronchial tree and a self-similar intermediary tree with a systematic branching asymmetry. It ends by the terminal bronchioles whose generations range from 8 to 22. Unlike classical models, it does not rely on a simple scaling law. With a limited number of parameters, this model reproduces the morphometric data from various sources (Horsfield K, Dart G, Olson DE, Filley GF, Cumming G. J Appl Physiol 31: 207-217, 1971; Weibel ER. Morphometry of the Human Lung. New York: Academic Press, 1963) and the main characteristics of the ventilation. Studying various types of random variations of the airway sizes, we show that strong correlations are needed to reproduce the measured distributions. Moreover, the ventilation performances are observed to be robust against anatomical variability. The same methodology applied to the rat also permits building a geometrical model that reproduces the anatomical and ventilation characteristics of this animal. This simple model can be directly used as a common description of the entire tree in analytical or numerical studies such as the computation of air flow distribution or aerosol transport.

Linear and non-linear behavior of fractal and irregular electrodes

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.