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.     

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.     

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     

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.     

A model to describe the settling behavior of fractal aggregates

A model to predict fractal dimension from sedimentating fractal aggregates has been successfully developed. This model was developed using the settling rate and size data of fractal aggregates. In order to test the validity of the model, a purpose-built settling rig, equipped with lens with magnification of 1200x, which can capture images of particles/flocs down to 2 microm in diameter was used. The performance and technique of the settling rig were validated by comparing the measured settling rates of 30- and 50.7-microm standard particles with their theoretical settling rates calculated using Stokes' law. The measured settling rates were within 10% agreement with the calculated Stokes' velocities. The settling rates and sizes of the particles/flocs were analyzed using image analysis software called WiT 5.3. The maximum temperature gradient across the settling column was 0.1 degrees C, which effectively eliminated convective currents due to temperature differences in the settling column. A total of 1000 calcium phosphate flocs were analyzed. Calcium phosphate flocs with fractal dimensions varying from 2.3 to 2.8 were generated via orthokinetic aggregation. Measurements of fractal dimensions, using light scattering, were done simultaneously with the settling experiments and they were found to be constant. The fractal dimensions calculated using the model agreed with those obtained by light scattering to within 12%.     

Structure of polydispersed colloids characterised by light scattering and electron microscopy

The dynamics of growth and aggregation of colloidal silver iodide particles was followed by the static light scattering method. The particles were treated as spheres and they were stable in size in the defined time interval. This approach enabled the use of the Zimm plots in order to determine the radii of gyration and the radii of spherical particles. Stable AgI colloids, either positively or negatively charged, showed the usual Zimm diagrams, while the diagrams were untypical when the stability of the colloids decreased. The untypical Zimm diagrams showed 'curves' with envelopes and 'curves' with minima in the unstable domain and in the domain where the most rapid nucleation occurs, respectively. Satisfactory agreement of particle sizes within the limits of accuracy, determined using static--and dynamic light scattering data and of the values obtained from the electron microscopic images was shown. Fitting the theoretical and experimental data, P(theta) functions showed that the particle shapes approach the theoretical model for spheres and thin discs. The colloid stability of polydispersed aggregates was also explained using the second virial coefficient, its negative sign implying interaction of particles in the solution, its positive value indicating formation of new particles from the supernatant solution. In addition, the colloid stability can be characterised by the mass fractal dimension. For positive stable colloids, Dm = 2.70 +/- 0.26, it can be related to the reaction controlled processes, whereas for negative stable colloids, Dm = 1.97 +/- 0.19, it was attributed to the diffusion controlled processes.     

A simple model for the structure of fractal aggregates

Structural properties of small aggregates containing up to 100 particles have been studied through detailed Monte Carlo cluster-cluster aggregation simulations in both diffusion-limited and reaction-limited conditions. First, the radius of gyration, the radius of the smallest sphere encompassing the cluster, and the particle-particle correlation function, g(r), have been computed based on the positions of all the particles in the cluster, and their fractal scaling has been analyzed. Then, an empirical model has been developed to simulate the g(r) function for aggregates of any size and used to determine the corresponding structural properties and scattering structure factors. Finally, in order to illustrate the application of the structural properties thus computed, two experiments on diffusion-limited aggregation have been performed, and the average scattering structure factors have been measured as a function of time using a small-angle light-scattering device. The obtained average scattering structure factors have been simulated using the Smoluchowski population balance equations, using the single aggregate structural properties and scattering structure factor predicted by the developed empirical g(r) model.     

Structural Interpretations of Static Light Scattering Patterns of Fractal Aggregates

A method based on static light scattering by fractal aggregates is introduced to extract structural information. In this study, we determine the scattered intensity by a fractal aggregate calculating the Structure and the Form factors noted, respectively, S(q) and F(q). We use the approximation of the mean field Mie scattering by fractal aggregates (R. Botet, P. Rannou, and M. Cabane, appl. opt. 36, 8791, 1997). This approximation is validated by a comparison of the scattering and extinction cross sections values calculated using, on the one hand, Mie theory with a mean optical index n) and, on the other hand, the mean field approximation. Scattering and extinction cross sections values differ by about 5%. We show that the mean environment of primary scatterers characterized by the optical index n(s) must be taken into account to interpret accurately the scattering pattern from fractal aggregates. Numerical simulations were done to evaluate the influence of the fractal dimension values (D(f)>2) and of the radius of gyration or the number of primary particles within the aggregates (N=50 to 250) on the scatterers' mean optical contrast (n(s)/n). This last parameter plays a major role in determining the Form factor F(q) which corresponds to the primary particles' scattering. In associating the mean optical index (n) to structural characteristics, this work provides a theoretical framework to be used to provide additional structural information from the scattering pattern of a fractal aggregate (cf. Part II). Copyright 2000 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.     

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.     

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.     

Morphology of dry lignins and size and shape of dissolved kraft lignin particles by X-ray scattering

Lignin is a highly branched polymer consisting of phenylpropane units, and it is one of the ingredients of the supporting matrix in plant cell walls. The morphology of several lignins extracted from plant cell walls using different methods was studied by small-angle and ultra-small-angle X-ray scattering. A power-law type intensity was observed for the dry lignins, but on the basis of the power-law exponent the fractal approach often applied to lignins is not fully justified. However, the intensity of kraft lignin did show a power law with surface fractal dimension D(s) = 2.7 +/- 0.1. The specific surface area of the lignins ranged from about 0.5 to 60 m(2)/g with 20% relative accuracy. The radius of gyration was determined from small-angle X-ray scattering data for aqueous solutions of kraft lignin. The shape of the particles in NaCl and NaOH solutions was found to be elongated. The particles were about 1-3 nm thick, while the length (5-9 nm) depended on the solvent and on the lignin concentration. The size of these primary particles was approximately the same as the size of the pores in the fractal aggregates of the dry kraft lignin. Their size was determined to be about 3.5 nm.     

Influence of natural organic matter and ionic composition on the kinetics and structure of hematite colloid aggregation: implications to iron depletion in estuaries

The stability and aggregation behavior of iron oxide colloids in natural waters play an important role in controlling the fate, transport, and bioavailability of trace metals. Time-resolved dynamic light scattering experiments were carried out in a study of the aggregation kinetics and aggregate structure of natural organic matter (NOM) coated hematite colloids and bare hematite colloids. The aggregation behavior was examined over a range of solution chemistries, by adjusting the concentration of the supporting electrolyte-NaCl, CaCl2, or simulated seawater. With the solution pH adjusted so that NOM-coated and bare hematite colloids were at the same zeta potential, we observed a significant difference in colloid stability which results from the stability imparted to the colloids by the adsorbed NOM macromolecules. This enhanced stability of NOM-coated hematite colloids was not observed with CaCl2. Aggregate form expressed as fractal dimension was determined for both NOM-coated and bare hematite aggregates in both NaCl and CaCl2. The fractal dimensions of aggregates formed in the diffusion-limited regime indicate slightly more loosely packed aggregates for bare hematite than theory predicts. For NOM-coated hematite, a small decrease in fractal dimension was observed when the solution composition changed from NaCl to CaCl2. For systems in the reaction-limited regime, the measured fractal dimensions agreed with those in the literature. Colloid aggregation was also studied in synthetic seawater, a mixed cation system to simulate estuarine mixing. Those results describe the important phenomena of iron oxide aggregation and sedimentation in estuaries. When compared to field data from the Mullica Estuary, U.S.A., it is shown that collision efficiency is a good predictor of the iron removal in this natural system.     

Fractal structure of asphaltene aggregates

A photographic technique coupled with image analysis was used to measure the size and fractal dimension of asphaltene aggregates formed in toluene-heptane solvent mixtures. First, asphaltene aggregates were examined in a Couette device and the fractal-like aggregate structures were quantified using boundary fractal dimension. The evolution of the floc structure with time was monitored. The relative rates of shear-induced aggregation and fragmentation/restructuring determine the steady-state floc structure. The average floc structure became more compact or more organized as the floc size distribution attained steady state. Moreover, the higher the shear rate is, the more compact the floc structure is at steady state. Second, the fractal dimensions of asphaltene aggregates were also determined in a free-settling test. The experimentally determined terminal settling velocities and characteristic lengths of the aggregates were utilized to estimate the 2D and 3D fractal dimensions. The size-density fractal dimension (D(3)) of the asphaltene aggregates was estimated to be in the range from 1.06 to 1.41. This relatively low fractal dimension suggests that the asphaltene aggregates are highly porous and very tenuous. The aggregates have a structure with extremely low space-filling capacity.     

Hydrodynamic forces and critical stresses in low-density aggregates under shear flow

The distribution of stresses in rigid colloidal aggregates under a shear flow was investigated numerically for particle-cluster and cluster-cluster aggregates with fractal dimensions ranging from 1.7 to 2.3. stokesian dynamics was used to calculate the hydrodynamic force on each monomer, while the internal intermonomer interactions were calculated by applying force and torque balances on each primary particle. Although the hydrodynamic forces act mainly on the periphery of the clusters, their filamentous structure propagates and accumulates internal stresses toward the inner region of the aggregates, where consequently the most loaded intermonomer bonds are located. The spatial stress distribution, when scaled by the proper power of the radius of gyration, is independent of aggregate size and fractal dimension. This feature has made it possible to identify the most probable locations of bond failure in the structure and to estimate the relationship between shear rate and particle size for the occurrence of restructuring and of breakage.     

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 fragmentation behavior of colloidal aggregates on their fractal structure

The fragmentation dynamics of aggregate of non-Brownian particles in shear flow is investigated numerically. The breakup behaviors of aggregates having the same connectivity but the different space-filling properties are examined. The Lagrangian particle simulation in a linear flow field is performed. The effect of surrounding fluid on the motion of multiple particles is estimated by Stokesian dynamics approach. The inter-particle force is calculated from the retarded van der Waals potential based on the Lifshitz theory. The results obtained in this work indicate that the fragmentation behavior of colloidal aggregates depends on their fractal structure. However, if the resultant aggregate size is smaller than the critical one, the fragmentation behavior shows the universality regardless of their original structure. Furthermore, the restructuring of aggregate in shear flow and its effect on the fragmentation process are also discussed.     

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.