Measurement of Fractal Aggregates of Polydisperse Particles Using Small-Angle Light Scattering

The effect of primary particle polydispersity on the structure of fractal aggregates has been investigated through the salt-induced, diffusion-limited aggregation of mixtures of hematite. The fractal dimension was determined experimentally using three independent methods: q dependence of static light scattering, kinetic scaling, and correlation of aggregate mass and linear size both determined from Guinier scattering. The fractal dimensions D(f) obtained were 1.75+/-0.03, 1.76+/-0.03, and 1.70+/-0.05, respectively. The use of a previously derived fractal mean particle size was validated in allowing data collapse to master curves for the aggregation kinetics data. The fractal mean particle size is shown to have general utility by taking a number weighting to describe polydisperse aggregation kinetics and a mass weighting to describe small q scattering behavior. Copyright 2000 Academic Press.     

Analysis of the structure of very large bacterial aggregates by small-angle multiple light scattering and confocal image analysis

This work aims at developing a more accurate measurement of the physical parameters of fractal dimension and the size distribution of large fractal aggregates by small-angle light scattering. The theory of multiple scattering has been of particular interest in the case of fractal aggregates for which Rayleigh theory is no longer valid. The introduction of multiple scattering theory into the interpretation of scattering by large bacterial aggregates has been used to calculate the fractal dimension and size distribution. The fractal dimension is calculated from the form factor F(q) at large scattering angles. At large angles the fractal dimension can also be computed by considering only the influence of the very local environment on the optical contrast around a subunit. The fractal dimensions of E. coli strains flocculated with two different cationic polymers have been computed by two techniques: static light scattering and confocal image analysis. The fractal dimensions calculated with both techniques at different flocculation times are very similar: between 1.90 and 2.19. The comparison between two completely independent techniques confirms the theoretical approach of multiple scattering of large flocs using the Mie theory. Size distributions have been calculated from light-scattering data taking into account the linear independence of the structure factor S(q) relative to each size class and using the fractal dimension measured from F(q) in the large-angle range or from confocal image analysis. The results are very different from calculations made using hard-sphere particle models. The size distribution is displaced toward the larger sizes when multiple scattering is considered. Using this new approach to the analysis of very large fractal aggregates by static light multiple scattering, the fractal dimension and size distribution can be calculated using two independent parts of the scattering curve.     

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.     

Growth of Fractal Aggregates in the Intermediate Regime of Particle Accommodation

The growth dynamics of fractal aggregates was studied within the framework of continuum model in the self-consistent mean field approximation. The regime that is intermediate between the diffusion-limited aggregation and reaction-limited aggregation was considered. The dependence of aggregate fractal dimension on the attachment probability of particles during their collisions with an aggregate was obtained. In the limiting cases, the values of fractal dimension coincide with those determined earlier. The domain of the values of attachment probability was revealed where several regions characterized by their own values of fractal dimension were specified in the structure of growing cluster. Physical nature of the emergence of various regions in the aggregate structure was discussed.     

Structures of DNA-linked nanoparticle aggregates

The room-temperature structure of DNA-linked gold nanoparticle aggregates is investigated using a combination of experiment and theory. The experiments involve extinction spectroscopy measurements and dynamic light scattering measurements of aggregates made using 60 and 80 nm gold particles and 30 base-pair DNA. The theoretical studies use calculated spectra for models of the aggregate structures to determine which structure matches the observations. These models include diffusion-limited cluster-cluster aggregation (DLCA), reaction-limited cluster-cluster aggregation (RLCA), and compact (nonfractal) cluster aggregation. The diameter of the nanoparticles used in the experiments is larger than has been considered previously, and this provides greater sensitivity of spectra to aggregate structure. We show that the best match between experiment and theory occurs for the RLCA fractal structures. This indicates that DNA hybridization takes place under irreversible conditions in the room-temperature aggregation. Some possible structural variations which might influence the result are considered, including the edge-to-edge distance between nanoparticles, variation in the diameter of the nanoparticles, underlying lattice structures of on-lattice compact clusters, and positional disorders in the lattice structures. We find that these variations do not change the conclusion that the room-temperature structure of the aggregates is fractal. We also examine the variation in extinction at 260 nm as temperature is increased, showing that the decrease in extinction at temperatures below the melting temperature is related to a morphological change from fractal toward compact structures.     

Scattering Behavior of Restructured Aggregates: A Simulation Study

Simulations of diffusion-limited cluster-cluster aggregation (DLCA) with no restructuring, full restructuring, and partial restructuring have been performed. The scattering patterns produced from these aggregates have been simulated using the Rayleigh-Gans-Debye approximation. Pure DLCA aggregates produced a scattering pattern with the slope of the fractal region being about -1.8. In contrast, the slope of the fractal region of the scattering pattern for fully restructured aggregates was about -2.1, indicating an increase in fractal dimension. Partial restructuring at large length scales produced an upward turn in the scattering pattern at low qr(o), while at high qr(o) the fractal section of the pure DLCA aggregate was retained. This last result was expected and is consistent with the results and postulations of several other workers. This simulation shows that the type of scattering pattern often obtained from orthokinetic or sheared aggregation can be produced by restructuring of aggregates at large length scales. Copyright 2001 Academic Press.     

Evidence of Shear Rate Dependence on Restructuring and Breakup of Latex Aggregates

Small-angle static light scattering has been used to probe the evolution of aggregate size and structure in the shear-induced aggregation of latex particles. The size of aggregates obtained from the particle-sizing instrument (Coulter LS230) was compared with the size of those obtained with another approach utilizing the Guinier equation on the scattering data. Comparison of the two methods for studying the effects of mixing on the evolution of the aggregate size with time revealed similar trends. The aggregate structures were quantified in terms of their fractal dimensions on the grounds of the validity of Rayleigh-Gans-Debye scattering theory for the fractal aggregates. Analysis of the scattering patterns of aggregates verified that restructuring of the aggregates occurred as the aggregates were exposed to certain shear environments, resulting in a scale-dependent structure that could not be quantified by a fractal dimension. The effect of restructuring on aggregate size was particularly noticeable when the aggregates were exposed to average shear rates of 40 to 80 s(-1), whereas no significant restructuring occurred at lower shear rates. At 100 s(-1), the fragmentation of aggregates appeared to be more significant than aggregate compac-tion. Copyright 2001 Academic Press.     

Optimization of the Thermal Radiation Extinction of Silicon Carbide in a Silica Powder Matrix

In order to optimize the infrared extinction of a SiC-powder in a silica powder matrix, Mie scattering calculations for spherical SiC-particles have been performed. A single oscillator-model was applied to calculate the optical constants of SiC. Taking into account the particle size distribution of a commercially available SiC-powder, its wavelength dependent extinction coefficient was calculated. The result is in very good agreement with the extinction spectrum of the powder derived by infrared optically measurements. Mie scattering theory also was used to find the optimum mean SiC-particle diameter of a mixture of 20% SiC-powder and 80% silica powder. This revised version was published online in July 2006 with corrections to the Cover Date.     

Solvent entrainment in and flocculation of asphaltenic aggregates probed by small-angle neutron scattering

While small-angle neutron scattering (SANS) has proven to be very useful for deducing the sizes and masses of asphaltenic aggregates in solution, care must be taken to account for solvation effects within the aggregates so as to not err in the characterization of these important systems. SANS measurements were performed on solutions of asphaltenes dispersed in deuterated solvents in which a broad spectrum of solute and solvent chemical compositions was represented. Fits to the scattering intensity curves were performed using the Guinier approximation, the Ornstein-Zernike (or Zimm) model, a mass-fractal model, and a polydisperse cylinder model. The mass-fractal model provided apparent fractal dimensions (2.2-3) for the aggregates that generally decreased with increasing aggregate size, indicating increased surface roughness for larger aggregates. The polydisperse cylinder model provided typical values of the particle thicknesses from 5 to 32 angstroms, the average particle radius from 25 to 125 angstroms, and approximately 30% radius polydispersity. Subsequent calculation of average aggregate molar masses suggested a range of solvent entrainment from 30 to 50% (v/v) within the aggregates that were consistent with previous viscosity measurements. Additional calculations were performed to estimate the proportion of microparticle to nanoparticle aggregates in the solutions. The results indicate that the inclusion of solvation effects is essential for the accurate determination of aggregate molecular weights and fractal dimensions.     

Formation of Small Silica Aggregates by Turbulent Aggregation

Aggregation of silica powder in water has been experimentally studied by turbidimetry. Aggregation was carried out in a stirred tank under physicochemical conditions corresponding to attractive interparticle forces. The effects of different primary particle sizes and stirring rates on aggregation dynamics have been studied. The scattering cross sections of silica aggregates were calculated in the framework of the anomalous diffraction approximation of light scattering theory. Aggregation has been studied by using Kusters's and Brakalov's approaches. By comparison between experimental and theoretical turbidity changes with time it has been shown that aggregates are small and slightly porous. The aggregation process is characterized by a weak fractal dimension D(wf) and an aggregate limit size L.D(wf) is found in the range 2.4-2.5. D(wf) (respectively L) is a weakly increasing (respectively decreasing) function of the stirring rate or of the shear rate. Copyright 2001 Academic Press.     

Simulation of ESR spectra of metal nanoparticle aggregates

Aggregates of polydisperse particles characterized by the preset values of the fractal dimensions and prefactor are built with the use of a special algorithm. ESR spectra of such aggregates are calculated in the self-consistent field approximation. It is shown that, even in sufficiently large aggregates with fractal structure, spectra greatly depend on the character of packing of large and small particles in the aggregate.     

Cavity ring-down spectroscopy measurements of single aerosol particle extinction. I. The effect of position of a particle within the laser beam on extinction

A continuous wave distributed feedback diode laser operating in the near infrared at wavelengths close to 1650 nm has been used to measure the extinction of light by single aerosol particles. The technique of optical feedback cavity ring-down spectroscopy (CRDS) was used for measurement of CRDS events at a repetition rate of 1.25 kHz. This very high repetition rate enabled multiple measurements of the extinction of light by single aerosol particles for the first time and demonstrated the dependence of light scattering on the position of a particle within the laser beam. A model is proposed to explain quantitatively this phenomenon. The minimum detectable dimensionless extinction coefficient epsilonmin was determined to be 3x10(-6). Extinction values obtained for single spherical polymer beads from a monodisperse sample of particles of diameter of 4 microm are in near-quantitative agreement with the values calculated by the Mie scattering theory. The deviations from the Mie theory expected for measurement of extinction by CRDS using a continuous wave laser are discussed in the companion paper.     

Degree of aggregation of indocyanine green in aqueous solutions determined by Mie scattering

Indocyanine green in water and aqueous NaCl solutions forms large J-aggregates. The degree of aggregation is estimated by Rayleigh scattering and Mie scattering analysis. The refractive index of the aggregates responsible for scattering is deduced from an aggregate absorption cross-section spectrum using the Kramers-Kronig relations. The average degree of aggregation for indocyanine green in water is found to be of the order of 10⁷ while it rises to 10⁹ for indocyanine green in 0.01 molar aqueous NaCl solution.     

Cluster-cluster aggregation kinetics and primary particle growth of soot nanoparticles in flame by light scattering and numerical simulations

The agglomeration kinetics of growing soot generated in a diffusion atmospheric flame are here studied in situ by light scattering technique to infer cluster morphology and size (fractal dimension D(f) and radius of gyration R(g)). SEM analysis is used as a standard reference to obtain primary particle size D(P) at different residence times. The number N(P) of primary particles per aggregate and the number concentration n(A) of clusters are evaluated on the basis of the measured angular patterns of the scattered light intensity. The major finding is that the kinetics of the coagulation process that yields to the formation of chain-like aggregates by soot primary particles (size 10 to 40 nm) can be described with a constant coagulation kernel beta(c,exp)=2.37x10(-9) cm3/s (coagulation constant tau(c) approximately = 0.28 ms). This result is in nice accord with the Smoluchowski coagulation equation in the free molecular regime, and, vice versa, it is in contrast with previous studies conducted by invasive (ex situ) techniques, which claimed the evidence in flames of coagulation rates much larger than the kinetic theory predictions. Thereafter, a number of numerical simulations is implemented to compare with the experimental results on primary particle growth rate and on the process of aggregate reshaping that is observed by light scattering at later residence times. The restructuring process is conjectured to occur, for not well understood reasons, as a direct consequence of the atomic rearrangement in the solid phase carbon due to the prolonged residence time within the flame. Thus, on one side, it is shown that the numerical simulations of primary size history compare well with the values of primary size from SEM experiment with a growth rate constant of primary diameter about 1 nm/s. On the other side, the evolution of aggregate morphology is found to be predictable by the numerical simulations when the onset of a first-order "thermal" restructuring mechanism is assumed to occur in the flame at about 20 ms residence time leading to aggregates with an asymptotic fractal dimension D(f,infinity) approximately = 2.5.     

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.     

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.     

Fractal Morphology and Breakage of DLCA and RLCA Aggregates

Latex aggregates, formed in 1 M McIlvaine buffer solution and 0.2 M NaCl solution, have been characterized in terms of aggregate size distribution and fractal morphology. This was achieved using three sizing techniques (image analysis, laser scattering, and electrical sensing) in which size distributions and fractal properties of the aggregates were measured. Estimates of fractal dimensions were made using the two-slope method based on dimensional analysis and the small-angle light scattering method. Aggregate suspensions were prepared using both water and a mixture of heavy water/ water as the solvent. The latter essentially eliminated sedimentation, which was observed after one day of aggregation when water alone was used as a solvent. Latex aggregates formed by diffusion-limited colloid aggregation (DLCA) and reaction-limited colloid aggregation (RLCA) had fractal dimensions close to 1.8 and 2.1, respectively. As observed through image analysis, DLCA aggregates possessed a loose tenuous structure, whereas RLCA aggregates were more compact. Disruption of both DLCA and RLCA aggregates has been investigated in laminar flow and turbulent capillary flow. The shear forces introduced by a laminar shear device with a shear rate up to 1711 s(-1) were unable to bring about aggregate breakup; shearing facilitates aggregate growth in the case of DLCA. However, latex aggregates were significantly disrupted after passage through a turbulent capillary tube at 95209 s(-1). Copyright 2000 Academic Press.     

Dependence of aggregate strength, structure, and light scattering properties on primary particle size under turbulent conditions in stirred tank

The steady-state size and structure of aggregates produced under turbulent conditions in stirred tank, for primary particle diameter, d(p), equal to 420 nm and 120 nm, were studied experimentally for various values of the volume average shear rate, G, and solid volume fraction, phi, and compared with data for d(p) = 810 nm. To exclusively investigate the effect of dp, polystyrene latexes with same type and similar density of surface charge groups (sulfate) were used. The mass fractal dimension, d(f), obtained by image analysis, was found to be invariant of d(p) and G, with a value equal to 2.64 +/- 0.18. Small-angle static light scattering was used to characterize the cluster mass distributions by means of the root-mean-square radius of gyration, R(g), and the zero-angle intensity of scattered light, I(0), whose steady-state values proved to be fully reversible with respect to G. The absolute values of R(g) obtained for similar phi and G proved to be independent of d(p), and for all studied conditions, R(g) was proportional to G-1/2. At very low phi, a critical aggregate size for breakage was obtained and used to evaluate the aggregate cohesive force, as a characteristic for the aggregate strength. The aggregate cohesive force was found to be independent of aggregate size, with similar values for the investigated dp. Due to large d(p) and high d(f), the effect of multiple light scattering within the aggregates was found to be present, and by relating the scaling of R(g) with I(0) to d(f), the corresponding correction factors were evaluated. By combination of the independently measured aggregate size and structure, it is possible to experimentally determine the relation between the maximum stable aggregate mass and the hydrodynamic stresses independent of the multiple light scattering present for large d(p) and compact aggregates.     

Two-Layer Model of Colloidal Gold Bioconjugates and Its Application to the Optimization of Nanosensors

An optical model of the conjugates of colloidal gold nanoparticles with biopolymers is analyzed in terms of two-layer spherical particle with the gold core and dielectric coating. The Mie theory was used to study the dependence of variations (caused by the adsorption of a biopolymer on the particle surface) in the extinction and light scattering (at 90°) spectra on the gold core diameter (d = 5–200 nm), the shell refractive index and thickness s (ratio s/d = 0–1). Some theoretical results by Templeton et al. (J. Phys. Chem. B, 2000, vol. 104, pp. 564–570) on the two-layer dipole model were corrected. It is shown that the dependence of spectral shifts of the extinction and scattering peaks on the conjugate structure is adequately described by the dipole approximation. In particular, we found the universal dependence of the normalized spectral shift of extinction maximum on the s/d ratio. Having in mind the optimization of conjugate–nanosensors, we studied the problem of what particle size is optimal for the transformation of biopolymer adsorption event into the variations in the spectral parameters of extinction and light scattering. Based on the calculations of extinction maximum values and positions, as well as on calculated differential extinction spectra, we concluded that a maximal conjugate efficiency corresponds to the core diameters of 40–80 nm. We also discussed the principles of conjugate–nanosensors optimization for the polymer shell structure.     

From fractal to nanorod porphyrin J-aggregates. Concentration-induced tuning of the aggregate size

A structural change from fractal to nanorod J-aggregates of tetrakis(4-sulfonatophenyl)porphyrin has been obtained by acting on the intermolecular interaction potential. The size and shape of these self-assembled porphyrin clusters have been monitored under different experimental conditions, by means of polarized and depolarized dynamic light scattering and small and wide angle elastic light scattering. At sufficiently low porphyrin concentration and high ionic strength, the shielded repulsive potential seems to be responsible for the fractal structure of the aggregates. On the contrary, at low ionic strength (nonshielded potential) and high porphyrin concentration, these species self-assemble in a rodlike arrangement. The length of the so-formed rod-shaped aggregates decreases on increasing porphyrin concentration. Moreover, both fractals and rods display a structure-dependent optical activity induced by a chiral template.