The origin of anomalous enhancement of electromagnetic fields in fractal aggregates of metal nanoparticles

It is shown that the anisotropy of the environment of metal nanoparticles with plasmon absorption in fractal aggregates is the most important and universal characteristic underlying their unique electrodynamic properties. It is noted that it is this morphological feature, but not the fractal distribution of particles in aggregates as such, that plays the dominant role in the manifestation of the enhancement of a local field. In this case, fractal aggregates possess the ability to enhance local electromagnetic fields only owing to their inherent local anisotropy; macroscopic characteristics of aggregates do not markedly influence their electrodynamic interactions with the external field. The quantitative characteristic of local anisotropy is introduced. Statistical correlation between the factor of local anisotropy and fractal dimension D of aggregates is established within the range 1.6 < D < 2.8. It is disclosed that the local anisotropy is independent of the fractal dimension within the wide range (1.6 < D < 2.5) except for the range D > 2.5 corresponding to aggregates with close-packed particles where the factor of local anisotropy tends toward zero. Strong correlation in the spatial arrangement of particles with the largest local anisotropy of the environment in aggregates and the strength of the local electromagnetic field is established using aggregates of silver nanoparticles as an example for the spectrum in the visible range; a polarization dependence of this correlation is revealed. It is noted that parameters of local anisotropy can be used to determine the degree of imperfection of colloidal crystals via optical methods.     

Viscosity of liquid suspensions with fractal aggregates: Magnetic nanoparticles in petroleum colloidal structures

New physical model is presented resulting in a simple formula for the dependence of viscosity η of colloidal liquid solution on the shear rate G applicable to a wide variety of systems including complex natural liquids like petroleum. The principal point of the model is the fractal nature of colloid particle aggregates present in the liquid. Such aggregates are experimentally detected now in non-Newtonian liquids. The model is based on calculation of energy loss on colloidal particle aggregate of fractal structure localized in the flow of liquid with shear rate. We have performed the viscosity measurement experiments which confirmed successfully the developed physical model. Also, we demonstrate experimentally that petroleum colloidal particles and magnetic iron oxide nanoparticles can form composite fractal-like aggregates in natural petroleum materials. Our model can explain both the non-Newtonian properties of petroleum and sensitivity of petroleum viscosity to external magnetic fields.     

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     

Small-angle X-ray scattering study on the microstructure evolution of zirconia nanoparticles during calcination

Zirconia nanoparticles have been synthesized from zirconium hydroxide precipitates followed by a supercritical CO₂ extraction. The microstructure evolution of these zirconia nanoparticles during the calcination at the moderate temperature has been investigated. Assisted by the analyses of TEM and XRD, small-angle X-ray scattering (SAXS) study offers possibilities to a comprehensive and quantitative characterization of the structural evolution on the nanometer scales. The as-synthesized zirconia sample exhibits a mass fractal structure constructed by the surface fractal particles. Such a structure can be preserved up to 300 °C. After calcination at 400 °C, considerable structural rearrangement occurs. In the interior of nanoparticles zirconia nanocrystallites emerge. It is the scattering from such zirconia nanoparticles that give rise to the broadened crossover in the ln[J(q)] vs. ln q plot and the scattering peak in the ln[q³J(q)] vs. q² plot. With a further increase in the calcination temperature, the power-law region at large-q in ln J(q) vs. ln q plot expands, and the peak in ln[q³J(q)] vs. q² plot shifts towards lower q values, indicating size increases in both the nanocrystallites and nanoparticles. Besides, the mass fractal structure constructed by zirconia nanoparticles can be largely preserved during the moderate temperature calcination.     

Deposition of spherical particles onto cylindrical solid surfaces. I. Numerical simulations

The permeability of fractal porous aggregates with realistic three-dimensional structure is investigated theoretically using model aggregates composed of identical spherical primary particles. Synthetic aggregates are generated by several techniques, including a lattice-based method, simulation of aggregation by differential settling and turbulent shear, and the specification of simple cubic structures, resulting in aggregates characterized by the number of primary particles, solid fraction, characteristic radius, and fractal dimension. Stokesian dynamics is used to determine the total hydrodynamic force on and the distribution of velocity within an aggregate exposed to a uniform flow. The aggregate permeability is calculated by comparing these values with the total force and velocity distribution calculated from the Brinkman equation applied locally and to the entire aggregate using permeability expressions from the literature. The relationship between the aggregate permeability and solid fraction is found to be best predicted by permeability expressions based on cylindrical rather than spherical geometrical elements, the latter tending to underestimate the aggregate permeability significantly. The permeability expressions of Jackson and James or Davies provide good estimates of the force on and flow through porous aggregates of known structure. These relationships are used to identify a number of general characteristics of fractal aggregates.     

Characterization of fractal particles using acoustics, electroacoustics, light scattering, image analysis, and conductivity

Fractals are aggregates of primary particles organized with a certain symmetry defined essentially by one parameter-a fractal dimension. We have developed a model for the interpretation of acoustic data with respect to particle structure in aggregated fractal particles. We apply this model to the characterization of various properties of a fumed silica, being but one example of a fractal structure. Importantly, our model assumes that there is no liquid flow within the aggregates (no advection). For fractal dimensions of less than 2.5, we find that the size and density of aggregates, computed from the measured acoustic attenuation spectra, are quite independent of the assumed fractal dimension. This aggregate size agrees well with light-scattering measurements. We applied this model to the interpretation of electroacoustic data as well. A combination of electroacoustic and conductivity measurements yields sufficient data for comparing the fractal model of the particle organization with a simple model of the separate primary particles. Conductivity measurements provide information on particle surface conductivity reflected in terms of the Dukhin number (Du). Supporting information for the zeta potential and Du can also be provided by electroacoustic measurements assuming thin double-layer theory. In comparing values of Du from these two measurements, we find that the model of separate solid particles provides much more consistent results than a fractal model with zero advection. To explain this, we first need to explain an apparent contradiction in the acoustic and electroacoustic data for porous particles. Although not important for interpreting acoustic data, advection within the aggregate does turn out to be essential for interpreting electrokinetic and electroacoustic phenomena in dispersions of porous particles.     

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.     

Experimental manifestations of the correlation between the local structure of silver nanoparticle aggregates and their absorption spectra

Experimental data testifying in favor of the correlation between electrodynamic characteristics of fractal aggregates and the structure of the local environment of particles are discussed. One of the possible variants of the control of local structure of disordered silver nanoparticle aggregates placed into a polymer matrix via a significant decrease in its volume is proposed and realized. An indirect method for the detection of the variations in the local structure with the use of plasmon absorption spectra is proposed and substantiated. The evolution of local anisotropy of the environment of the particles of loose aggregates during their compaction is studied. Differences in the absorption spectra of silver nanoaggregates in hydrosols with various concentrations of water-soluble polymer that determine the properties of the local structure of aggregates are discussed, these differences being manifested upon the formation of fragments with quasi-ordered arrangement of particles and in the absence of such fragments.     

T型混合反应器内共沉淀法制备吡啶硫酮铜

在T型混合反应器内通过喷射共沉淀方法合成了吡啶硫酮铜(CPT).利用扫描电镜和小角度光散射实验手段研究了分散液中吡啶硫酮铜的形貌和大小。通过喷射共沉淀方法获得的初级吡啶硫酮铜颗粒为棒状, 在分散液中这些棒状颗粒易发生团聚, 形成分形维数为2.1的团聚体。用平均回转半径(R_g)表征分散液中团聚体的平均大小, 考察了反应温度和反应物化学计量比(吡啶硫酮钠/硫酸铜)对团聚体大小的影响, 随着反应温度的降低, 团聚体的回转半径逐渐减小;过量的吡啶硫酮钠也有利于降低团聚体的大小, 当吡啶硫酮钠过量量达到~25%时, 进一步增大吡啶硫酮钠的过量量, 团聚体的回转半径不再发生明显变化。     

T型混合反应器内共沉淀法制备吡啶硫酮铜

在T型混合反应器内通过喷射共沉淀方法合成了吡啶硫酮铜(CPT)。利用扫描电镜和小角度光散射实验手段研究了分散液中吡啶硫酮铜的形貌和大小。通过喷射共沉淀方法获得的初级吡啶硫酮铜颗粒为棒状,在分散液中这些棒状颗粒易发生团聚,形成分形维数为2.1的团聚体。用平均回转半径()表征分散液中团聚体的平均大小,考察了反应温度和反应物化学计量比(吡啶硫酮钠/硫酸铜)对团聚体大小的影响,随着反应温度的降低,团聚体的回转半径逐渐减小;过量的吡啶硫酮钠也有利于降低团聚体的大小,当吡啶硫酮钠过量量达到~25%时,进一步增大吡啶硫酮钠的过量量,团聚体的回转半径不再发生明显变化。     

Image processing of the fractal aggregates composed of nanoparticles

The image-processing algorithms are described for chain branched aggregates composed of spherical particles. The processing procedure is based on setting off circles corresponding to the primary particles (spherules) in the images, followed by the construction of a digital model representing a set of diameters and coordinates of all spherules and their centers. The computational procedures are considered in which the digital model is used to calculate a series of parameters characterizing the aggregate’s structure and morphology, including fractal dimension for an individual aggregate and average dimension for a set of aggregates. The “clearance radius” of the aggregate is calculated as a half of geometric mean of the aggregate’s length L and width W. To determine L and W, the algorithm based on searching for the minimum-area (LW) rectangle circumscribing the aggregate’s contour is proposed.     

Experimental and modeling study of breakage and restructuring of open and dense colloidal aggregates

In this work we present experimental and simulation analysis of the breakage and restructuring of colloidal aggregates in dilute conditions under shear. In order to cover a broad range of hydrodynamic and interparticle forces, aggregates composed of primary particles with two sizes, d(p) = 90 and 810 nm, were generated. Moreover, to understand the dependence of breakage and restructuring on the cluster structure, aggregates grown under stagnant and turbulent conditions, having substantially different initial internal structures with fractal dimension d(f) equal to 1.7 and 2.7, respectively, were used. The aggregates were broken by exposing them to a well-defined elongational flow produced in a nozzle positioned between two syringes. To investigate the evolution of aggregate size and morphology, respectively, the mean radius of gyration, , and d(f) were monitored during the breakup process using light scattering and confocal laser scanning microscopy. It was found that the evolution of aggregates' fractal dimension during breakage is solely controlled by their initial structure and is independent of the primary particles size. Similarly, the scaling of the steady-state vs the applied hydrodynamic stress is independent of primary particle size, however, depends on the history of aggregate structure. To quantitatively explain these observations, the breakage process was modeled using stokesian dynamics simulations incorporating DLVO and contact interactions among particles. The required flow-field for these simulations was obtained from computational fluid dynamics. The complex flow pattern was simplified by considering a characteristic stream line passing through the zone with the highest hydrodynamic stress inside the nozzle, this being the most critical flow condition experienced by the clusters. As the flow-field along this streamline was found to be neither pure simple shear nor pure extensional flow, the real flow was approximated as an elongational flow followed by a simple shear flow, with a stepwise transition between them. Using this approach, very good agreement between the measured and simulated aggregate size values and structure evolution was obtained. The results of this study show that the process of cluster breakup is very complex and strongly depends on the initial aggregate structure and flow-field conditions.     

Self‐Assembly Fractal Microstructures of HPAM/Chromium Acetate and HPAM/Phenolic Aldehyde Colloidal Dispersion Gel

Abstract

HPAM (partially hydrolyzed polyacrylamide)/chromium acetate and HPAM/phenolic aldehyde colloidal dispersion gels (CDGs) were investigated microscopically using atomic force microscope. The results show that the colloidal dispersion gels eventually form self‐assembly branch‐like fractal structures over a scanning range of micrometers. The fractal aggregates of single twigs formed by compact assembly of nanometer particles were observed over a smaller scanning range regardless of the concentration of HPAM and the crosslinking reagent. This indicated an HPAM‐dependence for the formation of the fractal structure and the crosslinking reagent independence of the geometrical morphology of the gel. Also, the results demonstrated that the elastic modulus (G′) of the fractal structure formed by the smaller (nanometer‐sized) colloidal particles was one order of magnitude higher than obtained for the micrometer‐sized particles. The elastic modulus (G′) and the dynamic stability of the gels increased with decreasing particle diameter.     

On the theory of structuring magnetic suspensions

The results of the three-dimensional computer and analytical simulation are presented for the kinetics of chain-shaped aggregate growth in suspensions of magnetizable non-Brownian particles. The results of the computer experiment show that, when the volume fraction of particles is no larger than 2–3%, chain-shaped aggregates are formed in the suspensions under the action of a field. The dependence of average number <n> of particles in a chain on time t is adequately described by the power law <n> = Ct ^k . The experiment indicates that, in contract to the common power approximations, in which exponent k is considered to be a universal constant parameter, it depends on the concentration of particles and their interactions with walls bounding a suspension. At concentrations noticeably exceeding 2–3%, dense bulk aggregates are formed in suspensions. The kinetics of their growth depends on the sizes of a suspension-containing vessel.     

Scale invariance of viscosity and cluster structure of epoxy formulations

The concentration dependences of the relative viscosity of epoxydiane resins (M _n = 400–650) in solutions of monoglycidyl esters, Cardura and Laproxide, and of the dynamic viscosity of filled epoxy formulations in a shear flow were studied. The optimal concentrations of the formulation components and structural parameters of the fractal aggregates of oligomers and clusters of pigment (filler) particles were calculated in the approximation of the cluster lattice model.     

Mass-mobility characterization of flame-made ZrO2 aerosols: Primary particle diameter and extent of aggregation

Gas-borne nanoparticles undergoing coagulation and sintering form irregular or fractal-like structures affecting their transport, light scattering, effective surface area, and density. Here, zirconia (ZrO₂) nanoparticles are generated by scalable spray combustion, and their mobility diameter and mass are obtained nearly in situ by differential mobility analyzer (DMA) and aerosol particle mass (APM) measurements. Using these data, the density of ZrO₂ and a power law between mobility and primary particle diameters, the structure of fractal-like particles is determined (mass-mobility exponent, prefactor and average number, and surface area mean diameter of primary particles, d_va). The d_va determined by DMA-APM measurements and this power law is in good agreement with the d_va obtained by ex situ nitrogen adsorption and microscopic analysis. Using this combination of measurements and above power law, the effect of flame spray process parameters (e.g., precursor solution and oxygen flow rate as well as zirconium concentration) on fractal-like particle structure characteristics is investigated in detail. This reveals that predominantly agglomerates (physically-bonded particles) and aggregates (chemically- or sinter-bonded particles) of nanoparticles are formed at low and high particle concentrations, respectively.     

Novel phthalocyanine crystals as a conductive filler in crosslinked epoxy materials: Fractal particle networks and low percolation thresholds

Novel nanosized crystals of aquocyanophthalocyaninatocobalt (III) (Phthalcon 11) were used as a conductive filler in crosslinked epoxy materials. The crosslinked composite materials had a very low percolation threshold (φ_c ≈ 0.9 vol %). The relationship between the volume conductivity and the filler fraction follows the scaling law of the percolation theory and suggests that the conducting particle networks were formed by random percolation of primary aggregates. The occurrence of the low φ_c can be explained by the presence of a fractal Phthalcon 11 particle network formed from fractal aggregates during crosslinking. The position of the percolation threshold and the volume conductivity of these crosslinked materials were found to depend heavily on the processing conditions applied. These dependencies are explained in terms of specific particle–matrix interactions and the particle–particle interactions and by taking into account different mechanisms of particle network formation. © 2005 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 33–47, 2006     

Fractal structure in natural gels: effect on carbon sequestration in volcanic soils

Allophanic soils are interesting in terms of environmental properties especially because of their potentialities as sinks for “greenhouse gases” by the way of C sequestration. These volcanic soils contain amorphous clays (allophanes) and exhibit higher organic carbon content than the one measured in other clay soils. We measured the C content of a set of allophanic soils and showed that the C content is positively correlated to the allophane content. We also measured the part of organic matter transformed into CO₂ during a respiration experiment and showed that the decomposition is lowered as the soils allophane content increases. Allophane aggregates are very close to the synthetic gels: high specific surface area large pore volume, fractal structure, large water content and important irreversible shrinkage during drying. In this work we characterized by Small Angle X-Ray Scattering (SAXS) the fractal structure of the allophane aggregates at the nano scale. We hypothesized that the peculiar structure and the associated low accessibility of the allophanic soils could explain the high organic carbon content and the associated poor transformation into CO₂. The tortuous structure of the allophane aggregates plays the role of a labyrinth which fix and traps soil organic carbon.