Menger sponge-like fractal body created by a novel template method

The universal relationship between sample dimensions and cooperative phenomena: effects of fractal dimension on the electronic properties of high-TC cuprate observed using electron spin resonance

As demonstrated by the unique electronic properties of nanostructured materials, which are qualitatively different from the bulk properties of the same materials, there should be a general relationship between the dimensions of the sample structures and the physical/chemical properties. However, it is demanding to examine this intriguing problem experimentally, because one cannot prepare a series of samples based on the same material with systematically varying and noninteger dimensions. This problem is solved by considering the fractal dimensions of samples. The electronic structures of a series of powder samples of the high-T_C superconductor YBa₂Cu₃O_7-δ with the fractal dimensions (D_f) of 2.5-3.0 were investigated using the electron spin resonance (ESR) spectra at 293 K, to determine a general and quantitative relationship between the electronic properties and fractal dimensions of the samples. The observed D_f-dependences of ESR parameters, such as g-values and linewidths, were quantitatively consistent with those of the critical temperatures, critical current densities, and lower critical magnetic fields, all of which exhibited anomalies at D_f ∼ 2.9. Considering the geometrical features of the fractal models, a hypothesis for explaining the observed D_f-dependences of the cooperative phenomena has been proposed, suggesting a universal structural instability at particular fractal dimensions, which affects all the physical and chemical properties of the samples.

Structural Properties of Vicsek-like Deterministic Multifractals

Deterministic nano-fractal structures have recently emerged, displaying huge potential for the fabrication of complex materials with predefined physical properties and functionalities. Exploiting the structural properties of fractals, such as symmetry and self-similarity, could greatly extend the applicability of such materials. Analyses of small-angle scattering (SAS) curves from deterministic fractal models with a single scaling factor have allowed the obtaining of valuable fractal properties but they are insufficient to describe non-uniform structures with rich scaling properties such as fractals with multiple scaling factors. To extract additional information about this class of fractal structures we performed an analysis of multifractal spectra and SAS intensity of a representative fractal model with two scaling factors—termed Vicsek-like fractal. We observed that the box-counting fractal dimension in multifractal spectra coincide with the scattering exponent of SAS curves in mass-fractal regions. Our analyses further revealed transitions from heterogeneous to homogeneous structures accompanied by changes from short to long-range mass-fractal regions. These transitions are explained in terms of the relative values of the scaling factors.

The structure of deterministic mass and surface fractals: theory and methods of analyzing small-angle scattering data

Small-angle scattering (SAS) of X-rays, neutrons or light from ensembles of randomly oriented and placed deterministic fractal structures is studied theoretically.

Structural characterization of chaos game fractals using small-angle scattering analysis

Small-angle scattering (SAS) technique is applied to study the nano and microstructural properties of spatial patterns generated from chaos game representation (CGR). Using a simplified version of Debye formula, we calculate and analyze in momentum space, the monodisperse scattering structure factor from a system of randomly oriented and non-interacting 2D Sierpinski gaskets (SG). We show that within CGR approach, the main geometrical and fractal properties, such as the overall size, scaling factor, minimal distance between scattering units, fractal dimension and the number of units composing the SG, can be recovered. We confirm the numerical results, by developing a theoretical model which describes analytically the structure factor of SG. We apply our findings to scattering from single scale mass fractals, and respectively to a multiscale fractal representing DNA sequences, and for which an analytic description of the structure factor is not known a priori.

Scattering from surface fractals in terms of composing mass fractals

It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.

Techniques for direct experimental evaluation of structure-transport relationships in disordered porous solids

Determining structure–transport relationships is critical to optimising the activity and selectivity performance of porous pellets acting as heterogeneous catalysts for diffusion-limited reactions. For amorphous porous systems determining the impact of particular aspects of the void space on mass transport often requires complex characterization and modelling steps to deconvolve the specific influence of the feature in question. These characterization and modelling steps often have limited accuracy and precision. It is the purpose of this work to present a case-study demonstrating the use of a more direct experimental evaluation of the impact of pore network features on mass transport. The case study evaluated the efficacy of the macropores of a bidisperse porous foam structure on improving mass transport over a purely mesoporous system. The method presented involved extending the novel integrated gas sorption and mercury porosimetry method to include uptake kinetics. Results for the new method were compared with those obtained by the alternative NMR cryodiffusometry technique, and found to lead to similar conclusions. It was found that the experimentally-determined degree of influence of the foam macropores was in line with expectations from a simple resistance model for a disconnected macropore network.

Wanted: a positive control for anomalous subdiffusion

Anomalous subdiffusion in cells and model systems is an active area of research. The main questions are whether diffusion is anomalous or normal, and if it is anomalous, its mechanism. The subject is controversial, especially the hypothesis that crowding causes anomalous subdiffusion. Anomalous subdiffusion measurements would be strengthened by an experimental standard, particularly one able to cross-calibrate the different types of measurements. Criteria for a calibration standard are proposed. First, diffusion must be anomalous over the length and timescales of the different measurements. The length-scale is fundamental; the time scale can be adjusted through the viscosity of the medium. Second, the standard must be theoretically well understood, with a known anomalous subdiffusion exponent, ideally readily tunable. Third, the standard must be simple, reproducible, and independently characterizable (by, for example, electron microscopy for nanostructures). Candidate experimental standards are evaluated, including obstructed lipid bilayers; aqueous systems obstructed by nanopillars; a continuum percolation system in which a prescribed fraction of randomly chosen obstacles in a regular array is ablated; single-file diffusion in pores; transient anomalous subdiffusion due to binding of particles in arrays such as transcription factors in randomized DNA arrays; and computer-generated physical trajectories.

Deterministic fractals: extracting additional information from small-angle scattering data

The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)q(D) is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q(D) in the momentum space is related to the log-periodicity of the quantity g(r)r(3-D) in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.

Spreading behavior of water droplets on fractal agar gel surfaces

Agar gels with hierarchical rough surfaces, referred to as "fractal agar gels," were prepared to model biological surfaces coated with mucus. Agar gels with rough surfaces of fractal dimension D = 2.2 were synthesized by transferring a fractal surface structure of alkylketene dimer (AKD). The rough structure accelerated the spreading of water droplets and induced the appearance of a wicking front. The mechanism of acceleration of the spreading on fractal surfaces based on a semiquantitative theoretical model was also clarified.

Scattering from generalized Cantor fractals

A fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set, is considered. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from zero to one in one dimension and from zero to three in three dimensions. The intensity profile of small-angle scattering from the generalized Cantor fractal in three dimensions is calculated. The system is generated by a set of iterative rules, each iteration corresponding to a certain fractal generation. Small-angle scattering is considered from monodispersive sets, which are randomly oriented and placed. The scattering intensities represent minima and maxima superimposed on a power law decay, with the exponent equal to the fractal dimension of the scatterer, but the minima and maxima are damped with increasing polydispersity of the fractal sets. It is shown that, for a finite generation of the fractal, the exponent changes at sufficiently large wave vectors from the fractal dimension to four, the value given by the usual Porod law. It is shown that the number of particles of which the fractal is composed can be estimated from the value of the boundary between the fractal and Porod regions. The radius of gyration of the fractal is calculated analytically.

Monitoring phase behavior of sub- and supercritical CO2 confined in porous fractal silica with 85% porosity

Phase behavior of CO(2) confined in porous fractal silica with volume fraction of SiO(2) phi(s) = 0.15 was investigated using small-angle neutron scattering (SANS) and ultrasmall-angle neutron scattering (USANS) techniques. The range of fluid densities (0 < (rho(CO(2)))(bulk) < 0.977 g/cm(3)) and temperatures (T = 22 degrees C, 35 and 60 degrees C) corresponded to gaseous, liquid, near critical and supercritical conditions of the bulk fluid. The results revealed formation of a dense adsorbed phase in small pores with sizes D < 40 A at all temperatures. At low pressure (P < 55 bar, (rho(CO(2)))(bulk) < 0.2 g/cm(3)) the average fluid density in pores may exceed the density of bulk fluid by a factor up to 6.5 at T = 22 degrees C. This "enrichment factor" gradually decreases with temperature, however significant fluid densification in small pores still exists at temperature T = 60 degrees C, i.e., far above the liquid-gas critical temperature of bulk CO(2) (T(C) = 31.1 degrees C). Larger pores are only partially filled with liquid-like adsorbed layer which coexists with unadsorbed fluid in the pore core. With increasing pressure, all pores become uniformly filled with the fluid, showing no measurable enrichment or depletion of the porous matrix with CO(2).

Fractal heterogeneous media

A method is presented for generating compact fractal disordered media by generalizing the random midpoint displacement algorithm. The obtained structures are invasive stochastic fractals, with the Hurst exponent varying as a continuous parameter, as opposed to lacunar deterministic fractals, such as the Menger sponge. By employing the detrending moving average algorithm [A. Carbone, Phys. Rev. E 76, 056703 (2007)], the Hurst exponent of the generated structure can be subsequently checked. The fractality of such a structure is referred to a property defined over a three-dimensional topology rather than to the topology itself. Consequently, in this framework, the Hurst exponent should be intended as an estimator of compactness rather than of roughness. Applications can be envisaged for simulating and quantifying complex systems characterized by self-similar heterogeneity across space. For example, exploitation areas range from the design and control of multifunctional self-assembled artificial nanostructures and microstructures to the analysis and modeling of complex pattern formation in biology, environmental sciences, geomorphological sciences, etc.

Characterization and structural investigation of fractal porous-silica over an extremely wide scale range of pore size

We have succeeded in creating Menger sponge-like fractal body, i.e., porous-silica samples with Menger sponge-like fractal geometries, by a novel template method utilizing template particles of alkylketene dimer (AKD) and a sol-gel synthesis of tetramethyl orthosilicate (TMOS). We report here the first experimental results on characterization and structural investigations of the fractal porous-silica samples prepared with various conditions such as calcination temperature and packing condition of the template particles. In order to characterize the fractal porous-silica samples, pore volume distribution, porosity and specific surface area were measured over an extremely wide scale from 1 nm to 100 microm by means of mercury porosimetry, (1)H NMR cryoporometry, nitrogen gas adsorption experiments together with direct evaluations of cross-sectional fractal dimension D(cs), and size limits of D(cs). We have found that the pore volume distribution and specific surface area of the fractal porous-silica samples can be discussed in terms of different fractal porous structures at different scale regions.