Mass Fractals Research Articles

A visualized analysis of small-angle neutron scattering intensity: concentration fluctuation in alcohol–water mixtures

Small-angle neutron scattering measurements have been performed on tert-butyl alcohol–water mixtures with alcohol concentrations from 0.05 to 0.30 mole fractions at 298, 313 and 328 K. Concentration fluctuations of the mixtures are analysed in terms of fractals. The structure of the concentration fluctuation is visualized by means of a large-scale reverse Monte Carlo technique. Percolation analysis of the visualized structure shows that the concentration fluctuation is characterized by polydisperse mass fractals, as found for 1-propanol–water mixtures. It seems that polydisperse mass fractals are a common structural characteristic in various alcohol–water mixtures.

Particle-size distribution in soils: A critical study of the fractal model validation

Particle size distributions (PSDs) have been analyzed in the past with power-law functions relating cumulative number of particles to diameter and mass of particles to diameter, and the exponents interpreted as fragmentation fractal dimensions ( D). More recently, a mass–time scaling model was proposed assuming a fractal mass–size distribution for particles settling in a liquid. The objective of this study was to test the mass–time and the mass–size relations in soil particle distributions settling in water, and to estimate the range of particle diameters where power-law scaling was applicable. Five soil samples from Argentina, with clay contents ranging from 8% to 32%, were used in the experiments. Particle size analysis was done with the Bouyoucos hydrometer method. Results for D ranged from 2.434 to 2.712 with R 2 ranging from 0.922 to 0.998. The power-law scaling was applicable in the range of particle diameters from 1 to 50 μm, to four of the soils, and from 1 to 35 μm to the fifth one. Besides, a previous set of soil PSD from Argentina, with particle diameters ranging from 1 to 40 μm was also studied. The obtained D values were from 2.427 (8% clay) to 2.928 (75% clay), with R 2 from 0.983 to 0.997. Power laws relating mass to size or mass to time of soil particles settling in a liquid were good models to describe particle size distributions in the silt particle range sizes.

Quantification of the fractal nature of mycelial aggregation in Aspergillus niger submerged cultures

BackgroundFractal geometry estimates have proven useful in studying the growth strategies of fungi in response to different environments on soil or on agar substrates, but their use in mycelia grown submerged is still rare. In the present study, the effects of certain important fermentation parameters, such as the spore inoculum level, phosphate and manganese concentrations in the medium, on mycelial morphology of the citric acid producer Aspergillus niger were determined by fractal geometry. The value of employing fractal geometry to describe mycelial structures was examined in comparison with information from other descriptors including classic morphological parameters derived from image analysis.ResultsFractal analysis of distinct morphological forms produced by fermentation conditions that influence fungal morphology and acid production, showed that the two fractal dimensions DBS (box surface dimension) and DBM (box mass dimension) are very sensitive indexes, capable of describing morphological differences. The two box-counting methods applied (one applied to the whole mass of the mycelial particles and the other applied to their surface only) enabled evaluation of fractal dimensions for mycelial particles in this analysis in the region of DBS = 1.20–1.70 and DBM = 1.20–2.70. The global structure of sufficiently branched mycelia was described by a single fractal dimension D, which did not exceed 1.30. Such simple structures are true mass fractals (DBS = DBM = D) and they could be young mycelia or dispersed forms of growth produced by very dense spore inocula (108–109 spores/ml) or by addition of manganese in the medium. Mycelial clumps and pellets were effectively discriminated by fractal analysis. Fractal dimension values were plotted together with classic morphological parameters derived from image analysis for comparisons. Their sensitivity to treatment was analogous to the sensitivity of classic morphological parameters suggesting that they could be equally used as morphological descriptors.ConclusionStarting from a spore, the mycelium develops as a mass fractal and, depending on culture conditions, it either turns to a surface fractal or remains a mass fractal. Since fractal dimensions give a measure of the degree of complexity and the mass filling properties of an object, it may be possible that a large number of morphological parameters which contribute to the overall complexity of the particles, could be replaced by these indexes effectively.

Temporal evolution of mesoscopic structure and dynamical scaling of the structure factor of some non-Euclidean systems

Real time kinetics of hydration of calcium silicates with light and heavy water has been investigated at the mesoscopic length scale with the ultra small-angle neutron scattering technique. The scattering data could not be interpreted in terms of a linear theory of phase formation. The predictions of nonlinear theories on the dynamics of phase formation have been examined. The validity of the dynamical scaling hypothesis for phase formation has also been explored. For the real time hydration of silicates with light water, reasonable agreement has been observed with a dynamical scaling hypothesis with a different measure of the characteristic length. The temporal evolution of the characteristic length does not follow a power-law relation with time. It increases with time and reaches a plateau. The mesoscopic structure of the hydrating pastes could not be described in terms of a classical porous medium with a well-defined specific inner surface. In the case of hydration with light water, the hydrating mass exhibits a mass fractal nature throughout hydration, with the mass fractal dimension increasing with time and reaching a plateau. But, in the case of hydration with heavy water, no agreement has been observed with the scaling hypothesis. For hydration with heavy water, the microstructure of the hydrating mass undergoes a transition from mass fractal to surface fractal and subsequently to mass fractal. The qualitative and quantitative features of the kinetics of hydration, as measured in scattering experiments, are strikingly different for hydration with light and heavy water.

Modeling the Primary Drainage Curve of Prefractal Porous Media

Fractal models for the soil water retention curve have largely ignored hysteresis. Such models generally require that all pores be connected to the atmosphere via a network of similar‐sized or larger pores. In random mass fractals and natural porous media, however, not all pores of a given size class empty during drainage due to incomplete pore connectivity and/or the presence of surrounding smaller pores. A modification of Rieu and Sposito's (1991a) prefractal water retention equation is proposed to accommodate this hysteresis during monotonic drying. The resulting expression is identical in form to the empirical Brooks and Corey (1964) model and contains three physically‐based parameters: the proportion of nondraining pores (Pd), the scale factor (b), and the apparent fractal dimension of the primary drainage curve (Dd), which is related to the underlying mass fractal dimension (D) of the porous medium by Dd = D + [log(Pd)]/[log(b)]. Model testing consisted of fitting the modified equation to previously published water retention data for 30 randomized Sierpinski carpets and seven soils. Error sums of squares ranged from 0.001 to 0.093 for the carpets, and from 0.015 to 0.047 for the soils. In contrast, the unmodified Rieu and Sposito (1991a) equation performed very poorly. Best fit estimates of Dd ranged from 0.295… to 1.640… for the two‐dimensional carpets, and from 2.467… to 2.902… for the three‐dimensional soils. Prediction of D, based on estimates of Dd derived from the primary drainage curve, will require additional research on how to obtain Pd and b. Based on the analytical approach outlined here, it should also be possible to model the primary wetting curve, thus providing a more complete fractal description of soil water hysteresis.

DYNAMICS OF FRACTAL SOLIDS

We describe the fractal solid by a special continuous medium model. We propose to describe the fractal solid by a fractional continuous medium model, where all characteristics and fields are defined everywhere in the volume but they follow some generalized equations which are derived by using fractional integrals of fractional order. The order of fractional integral can be equal to the fractal mass dimension of the solid. Fractional integrals are considered as an approximation of integrals on fractals. We suggest the approach to compute the moments of inertia for fractal solids. The dynamics of fractal solids are described by the usual Euler's equations. The possible experimental test of continuous medium model for fractal solids is considered.

WAVE EQUATION FOR FRACTAL SOLID STRING

We use the fractional integrals to describe fractal solid. We suggest to consider the fractal solid as special (fractional) continuous medium. We replace the fractal solid with fractal mass dimension by some continuous model that is described by fractional integrals. The fractional integrals are considered as approximation of the integrals on fractals. We derive fractional generalization of the action functional and the Euler–Lagrange equation for the fractal solid string. The solution of wave equation for fractal solid string is considered.

Conceptual and Practical Implications of Breast Tissue Geometry: Toward a More Effective, Less Toxic Therapy

Mathematics provides greater understanding of the complex process of tumorigenesis. Based on the Gompertzian phenomenon and the Norton-Simon hypothesis, enhanced cell kill can be obtained through a greater chemotherapy dose rate. Results from the 1995 Bonadonna et al. study and the CALGB/Intergroup C9741 study demonstrated that patients in the dose-dense arms had significantly longer disease-free survival and overall survival. Because of the demonstrated applicability of Gompertzian kinetics, attention has been turned to the etiology of the Gompertzian curve. Breast tumor dimensions, as with all tissue dimensions in biology, can be calculated by fractals. A less cell-dense tissue usually has a lower fractal dimension than a tissue with more cells (i.e., a higher cell density is usually due to a higher fractal dimension). Density is the number of cells divided by the tissue volume. When allowed to grow, the density of a tissue with a lower fractal dimension drops quickly. However, a tumor, since it has a higher fractal mass dimension, maintains a high density as it grows bigger, resulting in a more rapid growth rate and a larger final size. Fractal dimensions of infiltrating ductal adenocarcinomas of the breast are high (i.e., 2.98), which results in a very dense tissue compared with normal breast tissue (with a fractal dimension of about 2.25). As expected, the higher fractal dimension results in a high rate of growth. The reason for this high fractal dimension is that breast cancer can be considered as a conglomerate of many small Gompertzian tumors, each of which has a high cell density and hence ratio of mitosis to apoptosis. In mathematical terms, each component of the conglomerate can be considered a small metastasis in itself. Thus, the primary tumor is composed of multiple self-metastases that form around a seed from the tumor to itself. Conventional thinking is that cancers metastasize because they are large, but in fact it may be that they are large because they are self-metastatic. Many genes are associated with the biology of metastasis; these include: A) obligatory cancer genes (most of which regulate mitosis and mitotic rate); B) genes relating to self-metastasis and growth of tumors at local sites, conferring the ability to invade and grow with high cell density; and C) genes that relate to the ability of the cancer to metastasize to distant areas. Additionally, fibroblasts may send out abnormal growth signals causing abnormal breast tissue growth. Consequently, we are not only dealing with abnormal cancer cells, but also with the tissue that surrounds them, or the microenvironment, that is, the "Smith-Bissell" model. These new insights may lead us to change the thrust of our attack from genes involved in mitosis to those involved in metastasis, including metastasis to self, and to use and further improve dose-dense regimens.

Fractal nature of chromatin organization in interphase chicken erythrocyte nuclei: DNA structure exhibits biphasic fractal properties

Arrangement of chromatin in intact chicken erythrocyte nuclei was investigated by small angle neutron scattering. The scattering spectra have revealed that on the scales between 15 nm and 1.5 μm the interior of the nucleus exhibited properties of a mass fractal. The fractal dimension of the protein component of cell nucleus held constant at approximately 2.5, while the DNA organization was biphasic, with the fractal dimension slightly higher than 2 on the scales smaller than 300 nm and approaching 3 on the larger scales.

Dynamical Scaling of the Structure Factor of Some Non-Euclidean Systems

Predictions of nonlinear theories on dynamics of new phase formation have been examined for the hydration of calcium silicates with light water and heavy water. In the case of hydration with light water, reasonable agreement has been observed with dynamical scaling hypothesis with a new measure of the characteristic length. The characteristic length does not follow a power law relation with time. Hydrating mass is found to be mass fractal throughout hydration, with mass fractal dimension increasing with time. But, in the case of hydration with heavy water, no agreement has been observed with the scaling hypothesis. Hydrating mass undergoes transition from mass fractal to surface fractal and finally again to mass fractal. The qualitative features of the kinetics of hydration, as measured in small-angle scattering experiments, are strikingly different for hydration with light water and heavy water.

Mass-fractal clustering and power-law decay of cluster size in 1-propanol aqueous solution.

Mesoscale structure of 1-propanol aqueous solutions with propanol mole fraction xp ranging from 0.1 to 0.33 has been studied by means of small angle neutron scattering (SANS) and large-scale reverse Monte Carlo (RMC) technique. Analysis of the SANS intensities in terms of a fractal model shows that the fractal dimension df of mesoscale structure of the solution is about 1.8-1.9 for water-rich solution and about 1.5 for propanol-rich solution. Percolation analysis on the RMC results reveals that the water molecules and the propanol molecules cluster, respectively, as a mass fractal, the dimension dM of which is about 2.3-2.5 for both clusters for water-rich solution. Furthermore, the distribution of the cluster size is expressed by a simple power law with an exponent tau of about 1.35-1.5 for the propanol clusters and 1.05-1.2 for the water clusters. These results imply that the current solution is characterized by polydisperse mass fractals. In fact, a theoretical relation for polydisperse system of mass fractals, df = dM(2-tau), holds well in the current solution. The characteristic change in df from 1.8-1.9 to 1.5 described above is attributed to the crossover between the water-rich regime and the propanol-rich regime. Most of the water molecules and the propanol molecules are located on the interface between clusters, and the water molecules form thin layers of about 10 A thick irrespective of 1-propanol content studied.

Fractal mass–size scaling of wetting soil aggregates

Structure is an important factor of soil functioning in ecosystems. Soil aggregate size distributions are commonly used to characterize soil structure. Relationships between density of dry soil aggregates and aggregate size present a different way to use aggregate-related information in soil structure characterization. Those relationships have been simulated assuming soils to be mass fractals. Aggregates in field soil are not air-dry. The relationships between mass and size differ between dry aggregates and wet aggregates because aggregates shrink as water content decreases. Our objective was to find out whether the mass fractal model can be applied to wet aggregates. Aggregates from the plow layer of Greyzem soil were brought to four different levels of water contents, and the kerosene method was used to measure volume of aggregates within diameter ranges of 3–5, 5–7, and 7–10 mm. It appeared that the wetter aggregates were less prone to loosening as the water increased. The mass fractal model was applicable to wet aggregates under the assumption of linear dependence of the fractal dimension D m and the unit size aggregate mass a on the gravimetric water content w (g g −1). Dependencies D m=2.925+0.284 w and a=0.808−0.123 w resulted in R 2=0.9999 for the regression line of simulated versus measured aggregate mass. Fractal modeling of mass–size scaling in wet soil aggregates presents a set of aggregate-based parameters for soil structure that may reflect soil properties and can be explored as an index of soil ability to support functions of ecosystems.

SURFACE CHARACTERIZATION OF REPROCESSED SINGLE-USE MEDICAL CATHETERS BY FRACTAL MASS DIMENSION

Reprocessing of single-use medical devices is an issue of concern and discussion due to infection risk and operation failure. Cleaning procedures and sterilization processes can produce structural changes and relevant topographical alteration of the biomedical device surfaces which may activate mechanisms able to cause unwanted biological response. Atomic Force Microscopy analysis (AFM) to measure Poly(vinyl chloride) (PVC) catheters surface roughness variation along successive sterilization cycles was applied. In this work, the relation between the number of reprocessing cycles and the fractal mass dimension of reprocessed catheter surface microscopic images is studied.

MASS-SIZE SCALING IN SOIL AGGREGATES AS AFFECTED BY AGGREGATE WATER CONTENT AND SOIL COMPACTION

Determining aggregate size distribution is a common way of characterizing soil structure. Information about soil structure can also be derived from examining the aggregate mass or density of aggregate size. There have been several reports indicating that the density-size relationships in air-dry aggregates follow predictions of a model assuming aggregates to be mass fractals. It was demonstrated recently that such model is applicable to wet aggregates if parameters of this model are assumed to be linear functions of gravimetric water contents. The objective of this study was to evaluate sensitivity of fractal parameters to soil compaction caused by wheel traffic. Irrigated and nonirrigated plots were laid out at silty clay Greyzem under fallow, and treatments of one tractor pass and three tractor passes were applied. The volume of individual aggregates for four depths in the plow layer was measured by kerosene method at air-dry water content, at two intermediate water contents between saturation and air-dry, and at saturation. The mass fractal model fitted data in a satisfactory manner within the range of water contents from air-dry to saturation with a R 2 of 0.999. Both the slope and the intercept of the dependence of fractal dimension and reference aggregate mass on water content were more sensitive to compaction than to soil bulk density and aggregate size distributions. Parameters of fractal scaling showed promise for diagnosing compaction in studied soil.

Logarithmic corrections to scaling in critical percolation and random resistor networks.

We study the critical behavior of various geometrical and transport properties of percolation in six dimensions. By employing field theory and renormalization group methods we analyze fluctuation induced logarithmic corrections to scaling up to and including the next-to-leading order correction. Our study comprehends the percolation correlation function, i.e., the probability that two given points are connected, and some of the fractal masses describing percolation clusters. To be specific, we calculate the mass of the backbone, the red bonds, and the shortest path. Moreover, we study key transport properties of percolation as represented by the random resistor network. We investigate the average two-point resistance as well as the entire family of multifractal moments of the current distribution.

Fullerene - polymer complexes: fractal crossover in solutions

Poly(vinylpyrrolidone)(PVP)-tetraphenylporphyrin(TPP)-C60-complexes in heavy water at concentration C P=1 wt.% have been studied at 293 K. Starting from PVP+TPP-complex and doping it with C60 up to C~6 wt.%, we recognised complexes' structural evolution. We observed the changes from mass fractals with dimension D M~2.3 at C=0 wt.% to surface fractals, having dimension D S ~2.8 at C~2 wt.%. Then the system returns to mass fractals, D M(C)~2.9-2.8 at C~5-6 wt.%. These structural peculiarities we explain as a result of cross linking action of fullerene, forming a microgel above a critical concentration C*~1 wt.%. Meanwhile, the network of over-doped complex (C~5 wt.%) is degraded, indicating a strong influence of fullerene on the water structure.

FRACTAL DIMENSIONS AND CORRECTIONS TO SCALING FOR CRITICAL POTTS CLUSTERS

Renormalization group and Coulomb gas mappings are used to derive theoretical predictions for the corrections to the exactly known asymptotic fractal masses of the hull, external perimeter, singly connected bonds and total mass of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality. For q = 4 these include exact logarithmic (as well as log-log) corrections.

Microstructure-controlled magnetic properties of the bulk glass-forming alloy Nd60Fe30Al10

We report a combination of analytical transmission electron microscopy, small angle neutron scattering, and studies of magnetic properties of the glass-forming alloy Nd60Fe30Al10. These investigations show the existence of an in situ formed finely dispersed nanocrystalline Nd-rich phase embedded in a Fe-rich glassy matrix of a bulk sample. The crystalline phase forms an extended network over the whole sample but its volume fraction is small compared to that of compact phase. Small angle neutron scattering data exhibit power law behavior with an exponent of −2.5 indicating the formation of a mass fractal. The microstructure observed may be related to phase separation in the undercooled liquid which induces a microstructure that can explain the hard magnetic behavior of such an intrinsic composite.

Identification of mass fractal in chemically synthesized ZnS quantum dots

Zinc sulphide quantum dots chemically capped with thioglycerol having two different sizes have been synthesized. The particles have a disordered sphalerite structure and are slightly contracted by 1% against the bulk. Small angle X-ray scattering investigations reveal that powders of these nanocrystallites are mass fractals that aggregate via a reaction-limited process to form irregular but rather dense networks with a fractal dimensionality of Df = 2.7 and 2.1, respectively.

REACTION ENGINEERING PRINCIPLES OF PROCESSES CATALYZED BY FRACTAL SOLIDS

Fractals are fascinating objects that can serve as good models for highly porous materials or for materials of high surface areas, and several models of aggregation, leading to fractal objects, resemble the preparation methods of certain classes of catalysts. Evidence of fractal scaling in catalysts has been accumulating for the past two decades. This review focuses on processes that occur on or within fractal porous objects and addresses such issues as the scaling properties of such processes and their superiority over processes in random or nonfractal media. A plethora of reaction engineering problems has been studied. Fractal structures induce unique scaling properties: Diffusion on a fractal media is anomalous and has been extensively investigated; although it may have implications for certain problems of adsorption, surface diffusion, and reaction, and several continuum models for describing it were suggested, it probably does not describe reaction and pore diffusion in a porous pellet. Anomalous temporal or parameter scaling was shown to describe processes of diffusion toward a reactive or adsorbing corrugated fractal surface and, typically, a multifractal description is necessary to characterize the data. Fractional temporal scaling describes the adsorption process from a closed volume. Fractional scaling with respect to the thickness of a stagnant film was derived for instantaneous reaction limited by diffusion through such a film. Fractional scaling with respect to the rate constant (k) was shown to apply for several processes of moderate reaction velocities, such as diffusion toward a diffusion-limited aggregate. Etching of surface and mass fractals also exhibit fractional temporal scaling. Diffusion and reaction in a self-similar pore-fractal network exhibit a new intermediate asymptote, in which the rate is only weakly dependent on k, which lies between the known kinetics-limited (rate ∼ k) and diffusion-limited (k 1/2) asymptotes. Within this domain, the selectivity to an undesired slow side reaction, in a system of two parallel reactions, can be suppressed significantly when its order is higher than that of the main fast reaction. These results may have several technical implications. Fractal surface catalysts may be less or more sensitive to changes in the operating conditions than nonfractal surfaces. Reduced sensitivity to increasing rate constant is achieved, for example, in a corrugated fractal catalyst exposed to a fixed reaction concentration. Comparison of the rates in a pore-fractal catalyst with those in a uniform-pore object showed that the rate in the former is superior in the intermediate k-insensitive domain. Selectivities in a system of two parallel reactions may also be better in the pore fractal as described earlier. Also, the apparent rate of deactivation, when it is a uniform process, can also be suppressed. Future theoretical work should test the validity of these conclusions in stochastic three-dimensional fractals. Experimental verification is still lacking; the superiority of pore fractals for several processes of diffusion and reaction should serve as an incentive for well-designed experiments.