Mass Fractal Dimension Research Articles

Simultaneous static and dynamic light scattering approach to the characterization of the different fibrin gel structures occurring by changing chloride concentration

The structure of fibrin fibers has been investigated by simultaneous elastic and dynamic light scattering measurements. This approach allows the recovery of reliable structural parameters of the fibrin fibers while checking for the reliability of the necessarily ab initio assumptions on some nonexperimentally accessible parameters. The number of protofibrils per fiber section, N, related to the fiber diameter, and the gel mass fractal dimension Dm are obtained. Since the fiber size is largely controlled by ions in the gelling solution, a detailed characterization of the fiber structure formed by changing Cl− concentration is given. While N values decrease from 6000 to 1.5 protofibrils per fiber section, going from 0 to 300 mM in CCl−,Dm increases from 1 to 1.8. This means that the overall gel structure is affected by Cl−, being the fiber size decreased and branch points per unit volume increased.

Effect of forage cropping treatments on soil structure and relationships with fractal dimensions

Effects of six different forage cropping treatments, perennial ryegrass (RG), alfalfa (AL), bromegrass (BR), small burnet (SB), subterranean clover (SC) and purple crownvetch (CV), on soil structural parameters such as, mean weight diameter (MWD), aggregate stability (AS), dispersion ratio (DR), bulk density (BD) and penetration resistance (PNTR) were determined. Forage cropping decreased the proportion of microaggregates in the fractions < 0.5 mm in size and increased the proportion of macroaggregates in the fractions > 1.00 mm in size. All forage cropping significantly increased organic carbon (OC) content (1.34–1.84%), MWD (0.89–1.15 mm), AS (71.76– 80.01%) and decreased DR (15.30– 9.76%), BD (1.45–1.28 g cm − 3) and PNTR (2.94– 1.89 MPa) compared with the fallow control treatment. The most positive effects on soil structural parameters were generally obtained with BR treatment while the lowest effects on that were obtained with RG treatment amongst the forages. Fragmentation ( D) and mass ( D m) fractal dimensions were decreased with increasing numbers of macroaggregates of the clay soil due to forage cropping. The lowest values of D and D m were found as 2.059 and 1.582 for BR treatment while the highest values of them were found as 2.683 and 2.085 for the control treatment, respectively. Both fractal dimensions gave positive relationships with OC content, MWD, AS, and negative relationships with DR, BD and PNTR. Development of macroaggregates in clay soil by the forage cropping was represented by the fractal dimension better than mass fractal dimension and MWD.

Asphaltenic aggregates are polydisperse oblate cylinders

Small-angle neutron scattering (SANS) has proven to be very useful for deducing the sizes and morphologies of asphaltenic aggregates in solution. A wide variety of intra-particle structure factors have previously been applied to SANS scattering spectra, but the studies often provided limited information concerning the quality of the fits and the Q range over which the models were applied. Selection of an appropriate form factor that closely approximates the structure of asphaltenic aggregates is important for determining the properties of asphaltenic aggregates, such as the radius of gyration ( R G ), molar mass, and apparent fractal dimension. This study evaluates various mono- and polydisperse intra-particle structure factor models as applied to four asphaltene scattering spectra. Agreement of the model fit parameters ( I 0 and R G ) with those obtained from Guinier analyses suggests that such a form factor model is physically reasonable. Reduced χ 2 values for each non-linear least squares fit indicates how well a given model fits to the entire Q range studied for the scattering intensity distribution. In the polydispersity analyses, an analytical function is introduced to model the scattering behavior of oblate cylinders with a Schultz distribution of radii. Results indicate that the polydisperse radius oblate cylinder model best approximates the shape of asphaltenic aggregates.

Forward light scattering to characterise structure of flocs composed of large particles

Abstract Forward light scattering has been a popular technique for measuring the fractal dimension of aggregated or flocculated particles. The method relies on the measurement of the slope of scattered intensity versus scattering vector ( q ) on a log–log plot. In the Rayleigh–Gans–Debye approximation, the negative of the slope has the value of the mass fractal dimension of the aggregates. In this work, simulation results were used to test the idea that Fraunhofer diffraction can be used to describe scattering from fractal aggregates. Polystyrene latex of diameter 4.9 μm was aggregated in a forward light scattering instrument, and the fractal dimension was determined independently by the volume obscuration method. Comparison of the experimental data with simulations showed poor agreement. It is suggested that multiple scattering within the aggregate structure, ignored in the usual Fraunhofer formulation, is significant.

In situ measurement of fractal dimension using focussed beam reflectance measurement

Abstract Focussed beam reflectance measurement is a technique that shows some promise as a technique for the online characterisation of floc size and structure. A scanning laser is focused into process fluid and the reflected intensity is measured as a function of time. This light intensity signal is normally processed to return chord length measurements which contain combined information on particle size and shape. This study shows that the unprocessed signal can be expected to encode information about the texture of the particle being scanned in addition to size and shape. For the purpose of characterising floc structure, a power law exponent can be recovered from the Fourier transform of the unprocessed signal and this in turn is related to the mass fractal dimension of flocs. The technique is demonstrated with computer generated flocs and with flocculated polystyrene latex of a primary particle size of 5 μm and a mass fractal dimension of Dm = 2.5 as determined by the volume obscuration method. Results obtained through simulation studies into scanning of computer generated flocs demonstrate that a correlation exists between the power law exponent and the mass fractal dimension. Loose structures with mass fractal dimension of Dm = 1.64 returned a power law exponent of β = −l.77 whilst a tighter structure of mass fractal dimension of Dm = 2.34 returned a power law exponent of β = −1.92. Results obtained through scanning of polystyrene latex flocs of mass fractal dimension of Dm = 2.49 returned a power law exponent of β = −2.00 which appears not to be in quantitative agreement with the simulation results.

Power-law scaling in dimension-to-biomass relationship of fish schools

Motivated by the finding that there is some biological universality in the relationship between school geometry and school biomass of various pelagic fishes in various conditions, I here establish a scaling law for school dimensions: the school diameter increases as a power-law function of school biomass. The power-law exponent is extracted through the data collapse, and is close to 3 5 . This value of the exponent implies that the mean packing density decreases as the school biomass increases, and the packing structure displays a mass-fractal dimension of 5 3 . By exploiting an analogy between school geometry and polymer chain statistics, I examine the behavioral algorithm governing the swollen conformation of large-sized schools of pelagics, and I explain the value of the exponent.

Reaction‐limited aggregation in presence of short‐range structural forces

AbstractA geometrically discretized sectional population balance model for reaction‐limited aggregation of colloidal suspensions is presented. The two important model parameters are collision frequency factor and collision efficiency factor. The collision frequency factor is derived from physically realistic arguments proposed for collision of fractal aggregates. The collision efficiency factor is computed as a function of total interaction energy between particles, including short‐range structural repulsion forces. The irregular and open structure of aggregates is taken into account by incorporating their mass fractal dimension. The characteristic time constant of reaction‐limited aggregation, derived from dynamic scaling of mean aggregate size‐aggregation time data, is found to correlate with electrolyte concentration. The population balance model is tested with published experimental data for aggregation of γ‐alumina suspensions in the presence of different electrolytes. It is shown that the slow kinetics of aggregation under certain conditions of pH and electrolyte concentration require inclusion of short‐range structural repulsion forces along with van der Waals attraction and electrical double layer repulsion forces in an extended DLVO theory. The model predictions are in good agreement with experimental data for time evolution of mean aggregate diameter in the reaction‐limited aggregation regime. © 2005 American Institute of Chemical Engineers AIChE J, 2005

SANS study of fractal microstructure and pore morphology in porous titania

High pure titania powder was prepared by solution based hydroxide precipitation technique using chloride precursor and ammonia as precipitating agent. Effect of sintering temperature on fractal microstructure and pore size distribution in pressed compacts of titania with graded porosity obtained by firing at various temperatures have been investigated by small-angle neutron scattering technique. The results showed that the powder granules are aggregates and composed of fine primary particles of average size 2.5 nm. The granules possess a fractal structure of mass fractal dimension 2.4. With increase in sintering temperature the granules gradually lost their fractal microstructure and the intra- and inter-granular pore size distributions in the compacts shifted towards higher radius size with a decrease in porosity. A power law size distribution for the intra-granular fine pores and lognormal size distribution for the inter-granular coarse pores were used for fitting the model.

Monitoring simultaneously the growth of nanoparticles and aggregates by in situ ultra-small-angle x-ray scattering

Ultra-small-angle x-ray scattering can provide information about primary particles and aggregates from a single scattering experiment. This technique is applied in situ to flame aerosol reactors for monitoring simultaneously the primary particle and aggregate growth dynamics of oxide nanoparticles in a flame. This was enabled through the use of a third generation synchrotron source (Advanced Photon Source, Argonne IL, USA) using specialized scattering instrumentation at the UNICAT facility which is capable of simultaneously measuring nanoscales to microscales (1nmto1μm). More specifically, the evolution of primary-particle diameter, mass-fractal dimension, geometric standard deviation, silica volume fraction, number concentration, radius of gyration of the aggregate, and number of primary particles per aggregate are measured along the flame axis for two different premixed flames. All these particle characteristics were derived from a single and nonintrusive measurement technique. Flame temperature profiles were measured in the presence of particles by in situ Fourier transform infrared spectroscopy and thermophoretic sampling was used to visualize particle growth with height above the burner as well as in the radial direction.

Mass fractal dimension and the compactness of proteins

Vibrational dynamics and energy flow in a protein are related by Alexander-Orbach theory to the protein's mass fractal dimension D and spectral dimension d . Proteins: Struct., Funct. Bioinf. 55, 529 (2004)] recently proposed a relation between d and protein size based on their computational analysis of a set of proteins ranging from about 100 to several thousand amino acids. We report here values for D computed for 200 proteins from the Protein Data Bank (PDB) ranging from about 100 to over 10 000 amino acids and examine variation of D with protein size. The average D is found to be 2.5, significantly smaller than a completely compact three-dimensional collapsed polymer. Indeed, we find that on average a protein in its PDB configuration fills about three-quarters of the volume within the protein surface. Protein mass is also found to scale with radius of gyration with an exponent of 2.5 for this set of proteins.

Mass fractal characteristics of silica sonogels as determined by small-angle x-ray scattering and nitrogen adsorption

A sample series of silica sonogels was prepared using different water-tetraethoxysilane molar ratio $({r}_{w})$ in the gelation step of the process in order to obtain aerogels with different bulk densities after the supercritical drying. The samples were analyzed by means of small-angle x-ray-scattering (SAXS) and nitrogen-adsorption techniques. Wet sonogels exhibit mass fractal structure with fractal dimension $D$ increasing from $\ensuremath{\sim}2.1\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}\ensuremath{\sim}2.4$ and mass-fractal correlation length $\ensuremath{\xi}$ diminishing from $\ensuremath{\sim}13\phantom{\rule{0.3em}{0ex}}\mathrm{nm}\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}\ensuremath{\sim}2\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, as ${r}_{w}$ is changed in the nominal range from 66 to 6. The process of obtaining aerogels from sonogels and heat treatment at $500\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$, in general, increases the mass-fractal dimension $D$, diminishes the characteristic length $\ensuremath{\xi}$ of the fractal structure, and shortens the fractal range at the micropore side for the formation of a secondary structured particle, apparently evolved from the original wet structure at a high resolution level. The overall mass-fractal dimension $D$ of aerogels was evaluated as $\ensuremath{\sim}2.4$ and $\ensuremath{\sim}2.5$, as determined from SAXS and from pore-size distribution by nitrogen adsorption, respectively. The fine structure of the ``secondary particle'' developed in the obtaining of aerogels could be described as a surface-mass fractal, with the correlated surface and mass-fractal dimensions decreasing from $\ensuremath{\sim}2.4\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}\ensuremath{\sim}2.0$ and from $\ensuremath{\sim}2.7\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}\ensuremath{\sim}2.5$, respectively, as the aerogel bulk density increases from $0.25\phantom{\rule{0.3em}{0ex}}({r}_{w}=66)$ up to $0.91\phantom{\rule{0.3em}{0ex}}\mathrm{g}∕{\mathrm{cm}}^{3}\phantom{\rule{0.3em}{0ex}}({r}_{w}=6)$.

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.

Monte Carlo simulation of aggregate morphology for simultaneous coagulation and sintering

A model for simulation of the three-dimensional morphology of nano-structured aggregates formed by concurrent coagulation and sintering is presented. Diffusion controlled cluster–cluster aggregation is assumed to be the prevailing coagulation mechanism which is implemented using a Monte–Carlo algorithm. Sintering is modeled as a successive overlapping of spherical primary particles, which are allowed to grow as to preserve overall mass. Simulations are characterized by individual ratios τ of characteristic collision to fusion time. A number of resulting aggregate-structures is displayed and reveals structure formation by coagulation and sintering for different values of τ. These aggregates are described qualitatively and quantitatively by their mass fractal dimension Df and radius of gyration. The fractal dimension increases from 1.86 for pure aggregation to ≈ 2.75 for equal characteristic time scales. As sintering turns out to be more and more relevant, increasingly compact aggregates start to form and the radius of gyration decreases significantly. The simulation results clearly reveal a strong dependence of the fractal dimension on the kinetics of the concurrent coagulation and sintering processes. Considering appropriate values of Df in aerosol process simulations may therefore be important in many cases.

The flocs generator reactor-FGR: a new basis for flocculation and solid–liquid separation

This work presents the design and the basic development of an innovative in-line mixing helical (coiled) reactor for the flocculation and solid–liquid separation of suspended particles. The device system was named flocs generator reactor (FGR), a compact flocculation system whereby the flocculation of particles is assisted by the kinetic energy transfer from the hydraulic flow through the reactor. The flow mixing, characterized as a plug flow regime with a slow dispersion, promoted the required hydrodynamics conditions to disperse the polymer flocculant (cationic polyacrylamide) and also to generate “strong” flocs of Fe(OH) 3 (colloidal suspension model system) at pH 7.5. The solid–liquid separation of the flocs was then achieved either by settling or by flotation with microbubbles (30–70 μm). Studies were carried out with different models of FGRs (varying the length/volume ratio), and process efficiency (mainly kinetics) was evaluated as a function of polymer concentration and operating parameters, namely, feed flow rate, air-to-solids ratio and residence time. Results showed that the aerated flocs are readily floated with rising rates of the order of 120 m h −1, values higher than those obtained for the settling of the nonaerated flocs (18–24 m h −1). The FGR requires short residence times and generates dense, well-structured flocs (with a mass fractal dimension, d F of about 3) which withstand high shear forces. Due to the high process efficiency and high loading capacity shown, it is believed that the FGR has a good potential as an in-line flocculation (or flotation) separator device in applications requiring high rate solid–liquid separations.

Hard Structured Particles: Suspension Synthesis, Characterization, and Compressibility

Hard interactions are developed on three grades of fumed silica by eliminating interparticle forces and sterically stabilizing the particles by attaching an organic coating to the surface of the particles, suspending them in an index-matching solvent and screening the electrostatics. These hard-structured particles are studied to understand the effects of the particle's microstructure on suspension properties without the influence of interparticle forces other than volume exclusion, Brownian, and hydrodynamic interactions. Light and X-ray scattering studies of low-volume-fraction suspensions suggest that the fumed silicas consist of primary particle of radius of gyration R(g1) approximately equals 16 nm and aggregate size R(g2) approximately 50 nm and mass fractal dimension D(f) approximately equals 2.2. Osmotic compressibilities of these suspensions are measured as a function of particle concentration exploring the packing mechanism of fumed silica. While there is minimal detectable change in the primary particle size, R(g2) varies by approximately 15%, providing insight into how suspension properties are related to particle size. As expected of hard particles with the same microstructure, the concentration dependence on the osmotic pressure superimposes with volume fraction of solids. The comparison of fumed-silica-suspension measurements to the known behavior of hard-sphere suspensions demonstrates the effects of particle geometry on suspension properties with indications of interpenetration of the fumed silica due to their open geometry.

Flow properties of hard structured particle suspensions

A class of structured particles is developed such that the effects of particle interactions are adjusted enabling suspension mechanics to be studied without the influence of interparticle forces other than volume exclusion, Brownian and hydrodynamic interactions. Three grades of these near-hard structured particles are developed from fumed silica. These particles have the same structure as measured by the mass fractal dimension. Stress dependent viscosities and diffusivity measurements are reported as a function of particle concentration, providing an opportunity to determine the scaling behavior of these properties with the two characteristic sizes: the primary particle and aggregate sizes. Except at stresses exceeding the critical value demarking the onset of shear thickening, flow curves superimposed with each other and with hard sphere suspensions compared at the same effective packing fraction, φ/φm.

Effects of Particle Shape on the Unipolar Diffusion Charging of Nonspherical Particles

Unipolar diffusion charging of nonspherical particles was investigated for various particle shapes. We researched with TiO2 agglomerates produced by the thermal decomposition of titanium tetraisopropoxide (TTIP) vapor. TTIP was converted into TiO2 in the furnace reactor and was subsequently introduced into the sintering furnace. Increasing the temperature in the sintering furnace, aggregates were restructured into higher fractal dimensions. The aggregates were classified according to their mobility using a differential mobility analyzer. The projection area and the mass fractal dimension of particles were measured with an image-processing technique performed by using transmission electron microscope (TEM) photographs. The selected aggregates were charged by the indirect photoelectric charger, and the average number of charges per particle was measured by an aerosol electrometer and a condensation particle counter. For particles of the same mobility diameter, our results showed that the particle charge qua...

A multiscale study of silty soil structure

SummaryDependency of soil properties on scale is a crucial issue in soil physics. In this paper, fractal approaches are used in two case studies in France and Australia, respectively, to study how measured physical soil properties change with the sample spacing and the scale of observation. At a scale of 10–1000 m (104 to 106 mm), fractals were applied to sample data from a linear transect, while at the 10−6 to 102 mm scale, fractals were applied in two dimensions to analyse both soil micro‐ and macrostructure, based on thin section samples. Porosity was characterized by short‐range spatial variations using sample spacings of 0.5 and 5 m (from the transect data), and a sample spacing of 1 cm (from the thin section analysis). The size of the representative elementary volume (REV) or representative elementary area (REA), required to represent statistically the elementary soil structure, was identified in three ways: (i) by the correlation length of a representative interconnected pore network, (ii) by the upper limit of the non‐linear increase with observation scale of mean porosity (upper limit of the solid mass fractal domain), and (iii) by the non‐linear decrease with observation scale of the coefficient of variation, CV, of mean porosity. Two embedded REAs were identified: the first (0.1–0.4 mm) related to the soil microstructure whereas a second (11–44 mm) related to the soil macrostructure. The solid mass fractal dimensions of the two embedded structural domains showed that hierarchical heterogeneity of soil structure was more pronounced for microstructures than for macrostructures. The mean area ratio of microstructural matrix/total surface and the CV of mean microporosity both scale similarly at observation scales smaller than the REA size. Their scaling exponents were both related to the fractal dimension of microstructural matrix. This preliminary study shows that the theory of fractals applied to soil structures at a specific scale range cannot be directly applied to predict soil physical properties at another scale range. This is because there are different interdependent structuring processes operating at different scales resulting in fractal dimensions being consistent only over particular domain limits.

Determination of branch fraction and minimum dimension of mass-fractal aggregates

Particles of micrometer to nanometer size often aggregate to form branched structures. Such materials include metals and metal oxides as well as biological and polymeric materials (considering the persistence length as a primary unit). Characterization of such structures is difficult since they typically display disordered, irregular features in three dimensions. Branched aggregates display two limiting size scales: that of the primary particle, R1 and that of the aggregate, R2. The mass-fractal model is often used to describe such structures where the aggregate mass, z=M2/M1, is related to the aggregate size, r=R2/R1, through a scaling relationship z=alpha r (d(f)), where the lacunarity alpha is close to 1 and may depend on the growth mechanism. Scattering of x rays, light and neutrons yields a direct measure of the mass-fractal dimension since I(q) approximately q(-d(f)) for 1/R2<q<1/R1 using scaling arguments. For linear, monodisperse aggregates with convoluted chain paths, analytic functions describing both the scaling and larger-size aggregate scattering regimes have been reported. For example, the Debye function for linear, Gaussian coils describes scattering when d(f)=2. Real, mass-fractal aggregates, however, can display variability from the linear chain, monodisperse model. Often the branch content is of vital importance to understanding both the growth of aggregates and their physical properties, especially dynamic properties. An approach is presented for the analysis of aggregate branching from static small-angle scattering. Comparison is made with analytic, simulation, and experimental results from the literature.

Fractal Analysis of Soil Water Desorption Data Collected on Disturbed Samples with Water Activity Meters

We combined water activity meters and fractal modeling to facilitate rapid, physically based characterization of the soil water retention curve. Desorption data (6 points per sample) were collected in the tension (h) range 2.0 × 102 to 1.5 × 105 kPa using the gravimetric method to measure water content (w) and water activity meters to measure h Thirty‐two disturbed samples from a long‐term nitrogen fertilization and tillage comparison study on a silt loam soil were analyzed. A new form of an established fractal equation was derived: w = ahD−3 − ρw/ρs, where a is a compound parameter including the bulk density (ρb) and air entry tension (ha), D is the mass fractal dimension, ρw is the density of water, and ρs is the particle density. This model was fitted to the measured water retention curves by nonlinear regression analysis. The a and D parameters were estimated, while ρw and ρs were fixed at 1.00 and 2.65 Mg m−3, respectively. Convergence was always achieved and the equation fitted the data extremely well; residual sums of squares ranged from 1.2 × 10−6 to 6.7 × 10−5, with a median value of 2.2 × 10−5 Estimates of a (0.62–0.74) and D (2.948–2.963) were physically reasonable, and sensitive to soil management practices. The a parameter increased, r = 0.80 (P &lt; 0.01), (signifying decreasing ρb and/or ha) whereas D decreased, r = −0.75 (P &lt; 0.01), (signifying more rapid capillary drainage) with increasing soil carbon content. Additional research is needed to test this approach on other soil types, and to assess the influence of soil disturbance and variations in ρs on the model's performance.