Pore size distribution in porous glass: fractal dimension obtained by calorimetry

By differential Scanning Calorimetry (DSC), at low heating rate and using a technique of fractionation, we have measured the equilibrium DSC signal (heat flow) J _q ⁰ of two families of porous glass saturated with water. The shape of the DSC peak obtained by these techniques is dependent on the sizes distribution of the pores. For porous glass with large pore size distribution, obtained by sol-gel technology, we show that in the domain of ice melting, the heat flow J_q is related to the melting temperature depression of the solvent, ΔT _m , by the scaling law: J _q ⁰∼ΔT _m ^{- (1 + D)}. We suggest that the exponent D is of the order of the fractal dimension of the backbone of the pore network and we discuss the influence of the variation of the melting enthalpy with the temperature on the value of this exponent. Similar D values were obtained from small angle neutron scattering and electronic energy transfer measurements on similar porous glass. The proposed scaling law is explained if one assumes that the pore size distribution is self similar. In porous glass obtained from mesomorphic copolymers, the pore size distribution is very sharp and therefore this law is not observed. One concludes that DSC, at low heating rate ( q? 2^°C/min) is the most rapid and less expensive method for determining the pore distribution and the fractal exponent of a porous material. Received 23 July 1999 and Received in final form 16 February 2001     

On singularity of energy measures on self-similar sets

We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089.Mathematics Subject Classification (2000): 28A80 (60G30, 31C25, 60J60)     

Spectral self-similarity in fractal one-dimensional photonic structures

Transmission spectra of one-dimensional fractal multilayer structures are found to exhibit self-similar properties. Self-similarity manifests itself in the shape of a transmission envelope (map of transmission dips) rather than in the map of resonance transmission peaks, as is commonly the case with spectra of quasiperiodic systems. To observe the self-similarity, one needs to apply a power transformation to the transmittance in addition to the usual frequency scaling. The values of this power as well as the scaling factor have been calculated analytically and found to depend on the geometrical parameters of the structure.     

Fractals and fractal approximation in structural mechanics

The present paper presents several new applications of the theory of fractals in structural mechanics. Until now most of the existing applications of the theory of fractals concern the calculation of fractal dimensions in physical phenomena, especially with respect to dynamical systems. The present paper deals with several other aspects of the theory of fractals which, from the standpoint of mechanics, seem to be of greater importance. Indeed the methods of fractal analysis permit the formulation and solution of difficult or yet unsolved mechanical problems or their treatment from an entirely new point of view. This paper is a first attempt towards this direction.A first version of this paper was presented at the International Meeting New Developments in Structural Mechanics in Catania, Italy, 4–6 July 1990. The meeting was dedicated to the memory of Professor Manfredi Romano.     

Multidimensional infinitely divisible cascades

The framework of infinitely divisible scaling was first developed to analyse the statistical intermittency of turbulence in fluid dynamics. It also reveals a powerful tool to describe and model various situations including Internet traffic, financial time series, textures ... A series of recent works introduced the infinitely divisible cascades in 1 dimension, a family of multifractal processes that can be easily synthesized numerically. This work extends the definition of infinitely divisible cascades from 1 dimension to d dimensions in the scalar case. Thus, a class of models is proposed both for data analysis and for numerical simulation in dimension d≥1. In this article, we give the definitions and main properties of infinitely divisible cascades in d dimensions. Then we focus on the modelling of statistical intermittency in turbulent flows. Several other applications are considered.     

Kolmogorov dispersion for turbulence in porous media: A conjecture

We will utilise the self-avoiding walk (SAW) mapping of the vortex line conformations in turbulence to get the Kolmogorov scale dependence of energy dispersion from SAW statistics, and the knowledge of the effects of disordered fractal geometries on the SAW statistics. These will give us the Kolmogorov energy dispersion exponent value for turbulence in porous media in terms of the size exponent for polymers in the same. We argue that the exponent value will be somewhat less than for turbulence in porous media.     

Diffusion-limited aggregates grown on nonuniform substrates

In the present paper, patterns of diffusion-limited aggregation (DLA) grown on nonuniform substrates are investigated by means of Monte Carlo simulations. We consider a nonuniform substrate as the largest percolation cluster of dropped particles with different structures and forms that occupy more than a single site on the lattice. The aggregates are grown on such clusters, in the range the concentration, p$p$, from the percolation threshold, _pc$p_c$ up to the jamming coverage, _pj$p_j$. At the percolation threshold, the aggregates are asymmetrical and the branches are relatively few. However, for larger values of p$p$, the patterns change gradually to a pure DLA. Tiny qualitative differences in this behavior are observed for different k$k$ sizes. Correspondingly, the fractal dimension of the aggregates increases as p$p$ raises in the same range _pc≤p≤_pj$p_c\le p\le p_j$. This behavior is analyzed and discussed in the framework of the existing theoretical approaches.     

Essential self adjointness of laplacians on riemannian manifolds with fractal boundary

We stutly the problem of the essential self-adjointness of Laplacians on Riemannian Manifolds. We show that a sufficient, condition is if the boundary of a manifold M. by which we mean M/M, where M is the metric completion of M, is almost polar set. Two other results concern special cases. when the boundary is some sort of fractal set or even a manifold. It is shown that the boundary is almost polar set in the cases under certain dimensional restrictions.     

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space X$X$, our paper studies elements of vector analysis, _Lp$L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on X$X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets.