Prediction of three-dimensional fractal dimensions using the two-dimensional properties of fractal aggregates

Fractal dimension analysis using an optical imaging analysis technique is a powerful tool in obtaining morphological information of particulate aggregates formed in coagulation processes. However, as image analysis uses two-dimensional projected images of the aggregates, it is only applicable to one and two-dimensional fractal analyses. In this study, three-dimensional fractal dimensions are estimated from image analysis by characterizing relationships between three-dimensional fractal dimensions (D(3)) and one (D(1)) and two-dimensional fractal dimensions (D(2) and D(pf)). The characterization of these fractal dimensions were achieved by creating populations of aggregates based on the pre-defined radius of gyration while varying the number of primary particles in an aggregate and three-dimensional fractal dimensions. Approximately 2000 simulated aggregates were grouped into 33 populations based on the radius of gyration of each aggregate class. Each population included from 15 to 115 aggregates and the number of primary particles in an aggregate varied from 10 to 1000. Characterization of the fractal dimensions demonstrated that the one-dimensional fractal dimensions could not be used to estimate two- and three-dimensional fractal dimensions. However, two-dimensional fractal dimensions obtained statistically, well-characterized relationships with aggregates of a three-dimensional fractal characterization. Three-dimensional fractal dimensions obtained in this study were compared with previously published experimental values where both two-dimensional fractal and three-dimensional fractal data were given. In the case of inorganic aggregates, when experimentally obtained three-dimensional fractal dimensions were 1.75, 1.86, 1.83+/-0.07, 2.24+/-0.22, and 1.72+/-0.13, computed three-dimensional fractal dimensions using two-dimensional fractal dimensions were 1.75, 1.76, 1.77+/-0.04, 2.11+/-0.09, and 1.76+/-0.03, respectively. However, when primary particles were biological colloids, experimentally obtained three-dimensional fractal dimensions were 1.99+/-0.08 and 2.14+/-0.04, and computed values were both 1.79+/-0.08. Analysis of the three-dimensional fractal dimensions with the imaging analysis technique was comparable to the conventional methods of both light scattering and electrical sensing when primary particles are inorganic colloids.     

Special fractal growth of dendrite copper using a hydrothermal method

Special fractal dendrite Cu nanostructures have been synthesized through a simple hydrothermal method, and the effects of the volume ratio between glycerol and water and the concentration of H₃PO₃ on the morphologies of dendrite Cu have been studied in detail. The Field emission scanning electron microscopy (FESEM), Transmission electron microscopy (TEM) and X-ray diffraction (XRD) have been used to characterize these Cu products. The results indicate that rhombic diamond and different morphologies of fractal dendrite were prepared because of the accumulation of Cu nuclei based on the diffusion-limited aggregation (DLA) and the nucleation-limited aggregation (NLA) model. Fortunately, symmetrical leaf-like dendrite Cu nanostructures different from Cu dendrites reported before have been obtained. Additionally, an explanation for the growth of fractal dendrite Cu has been discussed carefully.     

Model of bound water formation in fractal clusters resulting from reversible aggregation of colloidal particles

A model is proposed for the formation of bound water in fractal clusters that arise in the course of the reversible aggregation of colloidal particles. The dependence of the specific content of bound (nonfreezing) water on the concentration of a colloidal solution is theoretically determined. It is found that this value is related to the concentration of the colloidal solution via a power law. It is shown that the exponent is determined by the total surface area and morphology of fractal clusters, as well as by the mechanism of their aggregation and disintegration.     

Determination of total heterogeneity and fractal dimensions of high-temperature superconductors

Using thermo-analytical and sorptometric methods physicochemical properties and especially surface heterogeneity of HgBa₂Ca₂Cu₃O₈₊, (Hg-1223) was investigated. The desorption energy distribution was derived from mass loss Q-TG and differential mass loss Q-DTG curves of thermodesorption in quasi-isothermal conditions of pre-adsorbed n-octane and water vapour. It is shown that the superconducting Hg-1223 phase is highly sensitive to water vapours. The mechanism of water adsorption depends largely on the activation time. By water vapour saturation in a period of 90 min, physisorption takes place. Prolonged periods result in a chemical decomposition. From nitrogen ad- and desorption isotherms the fractal dimension of superconductors were calculated. A new approach is proposed to calculate fractal dimension from Q-TG curves.This revised version was published online in November 2005 with corrections to the Cover Date.     

Formation of fractal porous silica by hydrolysis of TEOS in a bicontinuous microemulsion

Mesoporous silica particles have been prepared by hydrolysis of TEOS (Si(OC₂H₅)₄) in bicontinuous microemulsions containing polyoxyethylene (POE) dodecylether, isooctane and water. TEOS was dissolved in a continuous water phase and hydrolyzed by the dispersed water at around the phase inversion temperature (60°C). Undulating solid materials with layered mesostructures were produced from middle-phase microemulsions in the three phase region (o/w=0.2–0.7). On the other hand, the solids obtained from the lower aqueous phase in the three phase region were found to have a heterogeneous disordered structure. Measurements of the fractal dimensions were performed in the macropore region using a box-counting method for the outline of the SEM texture. We found that the macropore size distribution in the particles prepared from the middle-phase microemulsion follows the fractal rule with a dimension of 1.7. From the results of nitrogen adsorption/desorption curves on the silica, a steep increase in the adsorption amounts was observed at a relative pressure below 0.2, and adsorption/desorption hysteresis was also observed at a relative pressure between 0.3 and 0.5. These studies suggest that the silica synthesized in the bicontinuous microemulsion mesostructure has a very broad size range from micro to macropores with a fractal distribution.     

Formation process and fractal growth model of HCFC-141b refrigerant gas hydrate

The microscopic visualization experiment on the formation process of HCFC-141b refrigerant gas hydrate has been investigated, and the morphological photos of hydrate formation process have been obtained. The results show that gas hydrate originally nucleated on the interface of refrigerant HCFC-141b and water under the condition of supercooling, then the hydrate grows continually due to the inducement of formed nucleation and diffusion of refrigerant. The formation of gas hydrate presents an arboreous phenomenon. The fractal dimension of the hydrate formation morphology on different stages was calculated. The calculating results indicate that the initial stage of the hydrate formation belongs to fractal growth, and the dimension is about 1.52. Based on the fractal theory, an RIN-DLA (random inducement nucleation-diffusion limited aggregation) model for the HCFC-141b hydrate growth was developed. The hydrate growth process was simulated with the developed model, and the fractal dimension for the simulated     

Electrochemistry at fractal surfaces

Rough and partially active electrodes are frequently modelled as fractals. These models are surveyed with special attention to the time dependence of the diffusion controlled current towards fractal interfaces. By means of a real system (rough platinum) it is shown how an electrochemical surface characterization procedure can be elaborated by using fractal concepts.     

Spontaneous formation of fractal structures on triglyceride surfaces with reference to their super water-repellent properties

Alkylketene dimer (AKD), a kind of wax, has been known to form fractal surfaces spontaneously and show super water-repellency. Such formation of water-repellent and fractal surfaces was also found in this work for triglycerides. Since the crystal phase transitions of these waxes were well studied, we studied the formation of their fractal surfaces through contact angle measurements, differential scanning calorimetry (DSC), and X-ray diffraction (XRD). From time-dependent contact angle measurements, it was found that the formation of super water-repellent surfaces with fractal structures occurred spontaneously also on the triglyceride surfaces at different temperatures. The freshly solidified triglyceride surfaces were almost transparent, and their initial contact angles of water were close to 110 degrees. The surfaces then became rough and clouded after being incubated for a certain time at a specified temperature. The super water-repellent surfaces were quite rough and showed fractal structures with the dimension of ca. 2.2 calculated from the scanning electron microscopic (SEM) images by the box-counting method. The phase transformation from a metastable state to a stable cystalline one after the solidification from the melt of triglycerides was clearly observed by DSC and XRD measurements. The fractal crystalline structures and the super water-repellency resulted from this phase transformation and the crystal growth. Ensuring the initial sample solidified into the metastable state and curing the surface at an appropriate temperature are key factors for the successful preparation of fractal triglyceride surfaces by the solidification method.     

Characterising latex particles and fractal aggregates using image analysis

The fractal nature of latex particles and their aggregates was characterised by image analysis in terms of fractal dimensions. The one- and two-dimensional fractal dimensions, D ₁ and D ₂, were estimated for polystyrene latex aggregates formed by flocculation in citric acid/phosphate buffer solutions. The dimensional analysis method was used, which is based on power law correlations between aggregate perimeter, projected area and maximum length. These aggregate characteristics were measured by image analysis. A two-slopes method using cumulative size distributions of aggregate length and solid volume has been developed to determine the three-dimensional fractal dimension (D ₃) for the latex aggregates. The fractal dimensions D ₁, D ₂ and D ₃ measured for single latex particles in distilled water agreed well with D ₁ = 1, D ₂ = 2 and D ₃ = 3 expected for Euclidean spherical objects. For the aggregates, the fractal dimension D ₂ of about 1.67 ± 0.04 (±standard deviation) was comparable to the fractal dimension D ₃ of approximately 1.72 ± 0.13 (±standard deviation), taking the standard deviations into account. The measured three-dimensional fractal dimension for latex aggregates is within the fractal dimension range 1.6–2.2 expected for aggregates formed through a cluster-cluster mechanism, and is close to the D ₃ value of about 1.8 indicated for cluster formation via diffusion-limited colloidal aggregation. Received: 28 September 1998 Accepted: 29 October 1998     

An analysis of fractal geometry of macromolecular gelation

With fractal geometry theory and based on experiments, an analysis of fractal geometry behavior of gelation of macromolecules was carried out. Using the cross-linking copolymerization of styrene-divinylbenzene (DVB) as an example, through the determinations of the evolution of the molecular weight, size and the dependence of scattering intensity on the angle of macromolecules by employing laser and synchrotron small angle X-ray scattering, respectively, this chemical reaction was described quantitatively, its fractal behavior was analyzed and the fractal dimension was also measured. By avoiding the complex theories on gelation, this approach is based on modern physical techniques and theories to perform the analysis of the behavior of fractal geometry of macromolecular gelation and thus is able to reveal the rules of this kind of complicated gelation more essentially and profoundly.     

Estimation of the dynamic size of fractal aggregates

A model is presented for an aggregation act occurring between two aggregates of any mass and fractal dimension. The kinetics of aggregation is also analyzed, as well as some previous works concerning the structure of fractal aggregates. As a result, a generalized curve is derived describing the normalized dynamic radius of clusters of spherical character as dependent on both the aggregate fractal dimension and the space dimension. It is shown how the curve may be utilized to determine the dynamic size of anisotropic aggregates. The obtained dependence can be used to estimate the dynamic size of fractal aggregates, to evaluate the prefactor in mass–radius relation and to model the aggregation kinetics.     

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

We have established experimental strategies on how to create a Menger sponge-like fractal body and how to control its fractal dimension. The essence was to utilize alkylketene dimer (AKD), which spontaneously forms super-water-repellent fractal surface. We prepared "fractal AKD particles" with fractal surface structure as templates of pores in fractal body. The fractal body was synthesized by filling the remained space between the packed template particles with a tetramethyl orthosilicate solution, solidifying it by the sol-gel process, and removing the template by calcinations. We have succeeded in systematically creating fractal bodies of silica with different cross-sectional fractal dimensions D(cs)=1.87, 1.84, and 1.80 using "fractal template particles" compressed under the ratio=1.0, 2.0, and 3.0, respectively. We also discussed the possibilities of their fractal geometries in comparison with mathematical models. We concluded that the created fractal bodies were close to a Menger sponge and its modified one. Our experimental strategy allows us to design fractality of porous materials.     

An experimental study of gas adsorption on fractal surfaces

The validity of the fractal versions of the FHH and BET theories for describing the adsorption of butane and nitrogen on a variety of partially dehydroxylated silica surfaces has been tested. The fractal dimensions obtained from adsorption data have been compared with those obtained completely independently using SAXS. It was found that the fractal dimensions obtained from butane adsorption isotherms, using both the fractal FHH and fractal BET theories, agreed well with the corresponding values obtained from SAXS over overlapping length scales. However, in general, a systematic deviation between the fractal dimension obtained from nitrogen adsorption and that obtained from SAXS was observed. The fractal dimensions obtained from nitrogen adsorption were consistently larger than those obtained from SAXS, which is the opposite of what has often been found in the literature. It has been suggested that the differences in the suitability of the adsorption theories tested to describe butane and nitrogen adsorption is due to the significant difference between the interaction strengths of these two different molecules with silica surfaces. A modified theory that can account for the discrepancy between the fractal dimensions obtained from nitrogen adsorption and SAXS has been proposed. The implications of the new theory for the accuracy of nitrogen adsorption BET surface areas for silicas are discussed.     

Excitation energy transport in fractal aggregates

A theory is developed to describe excitation energy transport in fractal aggregates. These fractal structures have densities which scale as a fractional power of the radius. Because of this, each donor is in a different environment with respect to donor and/or trap density. Consequently, ensemble averages cannot be performed using centro-symmetric coordinates. This problem has great similarities to the case of excitation energy transport in finite volumes. In this work the theory developed by Ediger and Fayer for the finite volume case is adapted to the fractal case. Förster limit (donor-trap transfer) and one-component (donor-donor transfer) systems are considered. Specific examples of Förster transfer are treated for the infinite size fractal, two-dimensional fractals and transfer from the aggregate to the excluded volume. A simple formalism for deriving expressions for the mean relaxation time of excitation is presented.     

Rigid aromatic fractal polyamides

The preparation of rigid aromatic, highly branched polyamides is described. Owing to the method of preparation and the chosen ratio of difunctional to trifunctional monomers, these entities are highly porous and not dendrimeric in nature. They better conform with the fractal model and are therefore called fractal polyamides (FPs). The effects of variations in the polymerization procedure, in total monomer concentration, in the ratio of amine to carboxyl groups and in the duration of the polycondensation reaction are investigated. Some characterization was performed and the results are presented and briefly discussed.     

The fractal nature of Escherichia coli biological flocs

The fractal structures of Escherichia coli biological flocs were characterised in terms of fractal dimension, which is a measurement of how the bacteria in the flocs occupy space. The dimensional analysis methods, based on power law correlations between floc perimeter, projected area and maximum length, were used to determine the one- and two-dimensional fractal dimensions (D(1) and D(2)) of E. coli flocs formed by flocculation in chitosan solution with a concentration of 10.0 mg chitosan per g dry cell weight (DCW), giving D(1)=1.07+/-0.06 and D(2)=1.70+/-0.08 (+/-S.D.). The three-dimensional fractal dimension (D(3)) of the E. coli flocs was determined by the two-slopes method, using cumulative size distributions of floc length and solid volume, to be 1.99+/-0.08 (+/-S.D.), which is close to the value of D(3)=2.14+/-0.04 (+/-S.D.) measured by the small angle light scattering method. The results demonstrate that E. coli flocs flocculated with chitosan have a fractal nature, as their fractal dimensions D(1), D(2) and D(3) differ from the values of 1, 2 and 3 expected for the spherical Euclidean object, respectively.     

Self-organized formation of chainlike silver nanostructure with fractal geometry

A novel silver nanostructure with a chainlike fractal network was self-organized synthesized via a simple solvothermal process by using poly(vinyl pyrrolidone) (PVP) as the coordination reagent. X-ray diffraction indicated that the chainlike fractal networks are of face centered cubic structure. Transmission electron microscopy shows that the diameters of the chain are around 100–150 nm. A Cayley fractal tree model was proposed to explain the formation of fractal geometry. It can be expected that the solvothermal method would be also appropriate for the preparation of such nanostructures for other metals.     

Nonlinear optics of metal fractal clusters

Linear and nonlinear optical properties of metal fractal clusters are studied. Experimental evidence is presented proving that clusters possess unique optical nonlinearities, characterized by rapid response, broadbandness and enormously high values of nonlinear susceptibility. A 10⁶ times enhancement of the DFWM signal has been observed experimentally in silver particles aggregated into clusters. For the first time frequency-and polarization-selective photomodification of fractal silver clusters has been realized.     

Multiple rate-determining steps for nonideal and fractal kinetics

We show that the kinetic model of a single rate-determining step in a reaction mechanism can be extended to systems with multiple overall reactions for which the elementary reactions obey nonideal or fractal kinetics. The following assumptions are necessary: (1) The system studied is either closed or open, but no constraints exist preventing the evolution toward equilibrium. (2) Elementary reactions occur in pairs of forward and backward steps. (3) The kinetics of the elementary steps are either nonideal or fractal and are compatible with equilibrium thermodynamics. (4) The number of reaction routes is identical with the number of rate-determining steps. If these hypotheses are valid, then the overall reaction rates can be explicitly evaluated: they have a form similar to the kinetic equations for the elementary reactions and the apparent reaction orders and fractal coefficients can be expressed analytically in terms of the kinetic parameters of the elementary reactions. We derive a set of relationships which connect the equilibrium constants of the reaction routes, the corresponding overall rate coefficients, and the stoichiometric numbers of the rate-determining steps. We also derive a set of generalized Boreskov relations among the apparent activation energies of the forward and backward overall processes, the corresponding reaction enthalpies, and the stoichiometric coefficients of the rate-determining steps. If the elementary reactions obey fractal kinetics, the same is true for the rate-determining steps. The fractal exponents of the forward and backward overall reactions are linear combinations of the fractal exponents of the fractal elementary reactions. Similar to the theory of single rate-determining steps, our approach can be used for selecting suitable reaction mechanisms from experimental data.     

Hydrodynamic radius of fractal clusters

The Kirkwood-Riseman theory has been used to derive an analytical formula for the evaluation of the hydrodynamic radii of fractal clusters. The proposed relation is based on knowledge of the particle-particle correlation function and can be applied to clusters containing any number of particles larger than 4. The calculated values of the hydrodynamic radius are compared with the results of other theoretical approaches proposed in the literature, as well as several experimental data. Finally, the developed relation has been used in connection with a population balance equation model that computes the clusters' mass distribution to estimate the average hydrodynamic radius for a population of fractal clusters of different sizes. The obtained results have been compared to suitable experimental data for a silica colloidal suspension aggregating under reaction-limited conditions.