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.     

Formation and structure of pachyman aggregates in dimethyl sulfoxide containing water

The aggregation of pachyman, β-(1 → 3)-D -glucan (M_w = 1.68 × 10⁵) from the Poria cocos mycelia, was investigated using static and dynamic laser light scattering (LLS) in dimethyl sulfoxide (DMSO) containing about 15% water, which leads to large aggregates. Both the time dependence of hydrodynamic radius and the angle dependence of the scattering intensity were used to calculate the fractal dimension (d_f) of the aggregates. The aggregation rate and average size of aggregates increase dramatically with increasing the polymer concentration from 1.7 × 10⁻⁴ g/mL to 8.6 × 10⁻⁴ g/mL, and with the decrease of the solvent quality, that is, water content from 13 to 15%. In the cases, the fractal dimensions change from 1.94 to 2.43 and from 1.92 to 2.54, respectively, suggesting that transforms of aggregation processes: a slow process called reaction-limited cluster aggregation (RLCA) to a fast process called diffusion-limited cluster aggregation (DLCA) in different polymer concentrations and water content. The fractal dimensions above 2 of the fast aggregation is larger than the 1.75 predicted for the ideal DLCA model, suggesting that the aggregation involves a restructuring process through the interchain hydrogen bonding interaction. There are no aggregates of pachyman in DMSO without water, but aggregates formed in the DMSO containing 15% water at 25°C as a compact structure. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 3201–3207, 1999     

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.     

Disruption of polystyrene latex aggregates in capillary flow

 Disruption of polystyrene latex aggregates, formed in 1 M citric acid/phosphate buffer solution at pH 3.8 through diffusion-limited colloid aggregation (DLCA) and in 0.2 M NaCl solution at pH 5.5 through reaction-limited colloid aggregation (RLCA), was studied with respect to aggregate size and fractal nature. This was achieved using small-angle laser scattering in conjunction with a specially designed sampling method, which brought about the elimination of the disruption of the aggregates caused by a commercial stirrer sample unit. Aggregations were carried out in a mixture of deuterium oxide and water instead of water alone as a solvent to minimise sedimentation resulting from the differences in density between the latex particles and the electrolytes. An initial “steady state” in terms of aggregate size and fractal dimension was found to occur after around 20 min and 2 days for DLCA and RLCA aggregates, respectively, at 25 °C. No aggregate disruption was detected for DLCA and RLCA aggregates after their passing through a capillary tube for shear rates up to 1584 and 2694 s⁻¹, respectively. At higher shear rates, significant decreases in the aggregate volume-mean diameter, D[4, 3], occurred after shearing. The degree of reduction in D[4, 3] was larger for DLCA aggregates in comparison to RLCA aggregates. The results would suggest that DLCA aggregates were more subject to disruption during shearing. A high degree of disruption was observed in turbulent flow for both aggregates. Received: 30 June 1999 Accepted in revised form: 11 November 1999     

Further insights into the universality of colloidal aggregation

Dynamic light scattering (DLS) performed at various scattering wave vectors provides detailed information about the aggregation kinetics and the cluster mass distribution (CMD) in colloidal dispersions. Detailed modeling of the aggregation kinetics with population balance equations requires a quantitative connection between the CMD and measurable quantities such as the angle dependent hydrodynamic radii obtained by DLS. For this purpose we evaluate and compare various models for the structure factor of fractal aggregates. Additionally, we introduce a simple scattering model that accounts for the contribution of internal cluster dynamics of fractal clusters to the first cumulant of the dynamic structure factor. We show that this contribution allows to quantitatively describe previously measured experimental data on the scattering wave vector dependence of the hydrodynamic radius in diffusion limited cluster-cluster aggregation (DLCA), which was shown to exhibit some kind of universality behavior (master curve). Using the same scattering model, we analyze a similar set of experimental data but in reaction limited cluster-cluster aggregation (RLCA). We find that in this case the crossover from RLCA to DLCA and gravitational settling both have a significant influence on the CMD and consequently on the scattering wave vector dependent DLS data. Only when accounting for both these effects they temporarily compensate each other and a satisfactory representation of the aggregation master curve is possible for the RLCA data at longer times. Indeed, we find that either crossover from RLCA to DLCA or gravitational settling, when present individually, causes the loss of a master curve for aggregation.     

Aerogelation Process Simulation by a Cluster-Cluster Aggregation Algorithm

A ballistically-limited cluster-cluster aggregation (BLCA) model was developed to simulate aerogelation processes. In the model, the clusters move along linear paths, in random directions, in a finite box. When two aggregates contact each other, they are combined irreversibly to form a larger aggregate. As expected, the simulations show that the aggregation time is much shorter than that obtained with diffusion-limited cluster-cluster aggregation (DLCA) models. The minimum concentration, c_g, required for gel formation scales as L^D–3, where L is the length of the sides of the box and D is the fractal dimension of the aggregates (D 1.95). For a concentration c larger than c_g, the mean free path of the aggregating clusters, , scales as c^–1.1. The pair correlation function g(r) and its Fourier transform S(q) were determined for the single large aggregates formed at the end of the simulations. These functions indicate that there is a characteristic length which scales as c^1/(D–3). As observed previously for the DLCA model, there is a discrepancy between the fractal dimensions obtained from g(r) and S(q).     

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.     

Universal occurrence of soot superaggregates with a fractal dimension of 2.6 in heavily sooting laminar diffusion flames

Using small-angle light scattering we show that a new phase of soot with size ca. 10 microm and a fractal dimension of D approximately equal to 2.6 exists in laminar diffusion flames for a wide range of heavily sooting fuels. This new phase appears to be a supramicrometer extension of the well-known submicrometer, D approximately equal to 1.8 phase of soot formed via diffusion-limited cluster aggregation (DLCA). The occurrence of this new soot phase correlates with an empirical sooting index for fuels. This supports a creation scenario in which these supramicrometer aggregates are created via a percolation of the submicrometer, D approximately equal to 1.8 aggregates.     

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.     

Two-dimensional colloidal aggregation: concentration effects

Extensive numerical simulations of diffusion-limited (DLCA) and reaction-limited (RLCA) colloidal aggregation in two dimensions were performed to elucidate the concentration dependence of the cluster fractal dimension and of the different average cluster sizes. Both on-lattice and off-lattice simulations were used to check the independence of our results on the simulational algorithms and on the space structure. The range in concentration studied spanned 2.5 orders of magnitude. In the DLCA case and in the flocculation regime, it was found that the fractal dimension shows a linear-type increase with the concentration phi, following the law: d(f)=d(fo)+aphi(c). For the on-lattice simulations the fractal dimension in the zero concentration limit, d(fo), was 1.451+/-0.002, while for the off-lattice simulations the same quantity took the value 1.445+/-0.003. The prefactor a and exponent c were for the on-lattice simulations equal to 0.633+/-0.021 and 1.046+/-0.032, while for the off-lattice simulations they were 1.005+/-0.059 and 0.999+/-0.045, respectively. For the exponents z and z', defining the increase of the weight-average (S(w)(t)) and number-average (S(n)(t)) cluster sizes as a function of time, we obtained in the DLCA case the laws: z=z(o)+bphi(d) and z'=z'(o)+b'phi(d'). For the on-lattice simulations, z(o), b, and d were equal to 0.593+/-0.008, 0.696+/-0.068, and 0.485+/-0.048, respectively, while for the off-lattice simulations they were 0.595+/-0.005, 0.807+/-0.093, and 0.599+/-0.051. In the case of the exponent z', the quantities z'(o), b', and d' were, for the on-lattice simulations, equal to 0.615+/-0.004, 0.814+/-0.081, and 0.620+/-0.043, respectively, while for the off-lattice algorithm they took the values 0.598+/-0.002, 0.855+/-0.035, and 0.610+/-0.018. In RLCA we have found again that the fractal dimension, in the flocculation regime, shows a similar linear-type increase with the concentration d(f)=d(fo)+aphi(c), with d(fo)=1.560+/-0.004, a=0.342+/-0.039, and c=1.000+/-0.112. In this RLCA case it was not possible to find a straight line in the log-log plots of S(w)(t) and S(n)(t) in the aggregation regime considered, and no exponents z and z' were defined. We argue however that for sufficiently long periods of time the cluster averages should tend to those for DLCA and, therefore, their exponents should coincide with z and z' of the DLCA case. Finally, we present the bell-shaped master curves for the scaling of the cluster size distribution function and their evolution when the concentration increases, for both the DLCA and RLCA cases.     

Effect of polymer adsorption on PEO-coated latex particles during salt-induced aggregation

The salt-induced aggregation of polystyrene particles in dilute aqueous solutions has been studied by means of dynamic light scattering measurements and the hydrodynamic radius of the resulting aggregates has been evaluated during the time evolution of the whole process. Poly(ethylene oxide) (PEO) polymer adsorbed on the particle surface at different amounts has been used to modify the inter-particle interactions resulting in the formation of clusters of increasing size or in the stabilization of the suspension, depending on the polymer molecular weight. The aggregation regime, i.e. a diffusion limited cluster aggregation (DLCA) occurring in the polymer-free latex suspension, is partially modified according to the polymer percentage adsorbed on the particle surface. At high polymer content, the polystyrene latex undergoes a complete steric stabilization. The deviation from a DLCA regime has been observed for different polymer contents and for polymers of different molecular weights, from 1.5 to 2000 kD. The alterations of the aggregation rates, induced by the polymer interactions, are presented and briefly discussed.     

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.     

Mesoscopic structure formation in colloidal aggregation and gelation

Recent small-angle light scattering experiments have revealed that diffusively aggregating spherical particles develop structure on a mesoscopic length scale (∼ tens of particles). The mesoscopic structural length scale persists even when the aggregation proceeds to the formation of a space-spanning network (a gel). We review the technique of small-angle light scattering, survey the experimental evidence for mesoscopic structure formation, discuss attempts at understanding these experimental observations by computer simulation of irreversible and reversible diffusion-limited cluster aggregation (DLCA), and propose a coherent picture for the understanding of non-equilibrium aggregation in the context of phase transitions.     

高磺化度聚苯胺体系中的分形结构研究

通过透射电镜的观察研究发现磺化聚苯胺的胶体聚集体和胶粒内部结构都具有分形体的特征 ,从而将分形的概念及其数学模型引入共轭导电聚合物体系之中 .磺化聚苯胺胶体的聚集体为很不均匀的分支状开放结构 ,其形成过程可用扩散控制集团聚集模型 (DLCA)进行模拟 ,计算机模拟生成的图形及其分形维数都与实验观测结果相当吻合 .胶粒由于是在分散介质所形成的平均化场中生成 ,屏蔽效应减弱 ,是比由它组成的聚集体要致密的球形结构 ,该结构的生成可用随机雨点模型模拟且结果相近 .     

Thermal restructuring of fractal clusters: the case of a strawberry-like core-shell polymer colloid

Thermal restructuring of fractal styrene-acrylate copolymer clusters dispersed in water has been investigated experimentally in the temperature range between 313 and 363 K. The particles constituting the clusters are of strawberry-like core-shell structure with a soft core and a rigid shell grafted on the core polymer chains. Due to the incomplete coverage of the core, the rather soft core may "flow out" through the open areas of the shell, leading to coalescence with the neighboring particles. The clusters were generated under diffusion-limited cluster aggregation conditions, and the restructuring kinetics was monitored by small-angle light scattering. Two sets of thermal restructuring experiments have been performed at various temperatures: (1) restructuring of growing clusters during aggregation and (2) restructuring of preformed clusters in the absence of aggregation. It is found that restructuring occurs only at temperature values above 323 K. In the absence of aggregation, restructuring leads to an increase of the fractal dimension and a decrease of the radius of gyration of the clusters. At sufficiently long times, both quantities reach a plateau value due to the presence of the grafted rigid shell, which constrains the coalescence of the soft core. A simple model, based on coalescence theory of liquid droplets and accounting for the incomplete coalescence and its dependence on temperature, has been developed to interpret the restructuring kinetics in the absence of aggregation. It is found that the proposed model can represent the measured experimental data well.     

Dynamic scaling properties of TeO2-based gels

Homogeneous, transparent, and mechanically rigid gels have been successfully synthesized in the tellurium isopropoxide-isopropanol-citric acid and water system. The sol to gel transition and the gels microstructure have been studied by using small angle X-ray scattering (SAXS) experiments. For any value of the two key synthesis parameters, which are the citric acid ratio and the alkoxide concentration, very small Te-rich elementary particles, about 1-1.5 nm in radius, form immediately when the water is added, leading to colloidal sols. During gelation, these elementary particles stick progressively together to build up fractal aggregates by a pure hierarchical aggregation process which has been identified as a reaction-limited cluster aggregation (RLCA) mechanism. The SAXS curve analysis, based on scaling concepts, shows that the gelling network exhibits a time and length scale invariant structure factor characterized by self-similarity. This self-similarity is also displayed for a wide range of chemical compositions and the gel microstructures only differ in their fractal aggregate size according to the tellurium isopropoxide concentration as well as the citric acid ratio.     

Structure of Hydrous Ferric Oxide Aggregates

The small angle light scattering behavior of hydrous ferric oxide flocs is examined here and found to provide useful insights into the nature of the aggregates formed despite the large size of these aggregates at later times. The flocs appear to exhibit fractal properties over a significant size range though the aggregates appear to be easily disrupted through mixing effects resulting in breakup and/or restructuring to denser assemblages. Background electrolyte concentrations also have some impact on floc structure but mixing effects and apparent destabilization by ferric ions limit the effect of added electrolytes on the stability and structure of ferric oxyhydroxides. Similar estimates of fractal dimensions of these hydrous ferric oxide flocs are obtained both by static light scattering analysis and by a cluster mass scaling approach. The choice of density distribution cutoff function has some impact on derived size and structure parameters and further refinement in this area is needed. Copyright 2000 Academic Press.     

Micro- to nanoscale structure of biocompatible PLA-PEO-PLA hydrogels

We observe large-scale structures in hydrogels of poly(l-lactide)-poly(ethylene oxide)-poly(l-lactide) (PLLA-PEO-PLLA) ranging in size from a few hundred nanometers to several micrometers. These structures are apparent through both ultra-small angle scattering (USAS) techniques and confocal microscopy. The hydrogels showed power law scattering in the USAS regime, which is indicative of scattering from fractal structures. The fractal dimension of the scattering from hydrogels revealed that the gels have large size aggregates with a mass fractal structure over the nanometer-to-micrometer length scales. The aggregates also seem to become more "dense" with an increase in the molecular weight of crystalline PLLA domains. Visualization through confocal microscopy confirms that the gels have a microstructure of interspersed micrometer-sized polymer inhomogeneities with water channels running between them. The presence of micrometer-sized water channels in the hydrogels has very important implications for biomedical applications.     

Light Absorption by the Clusters of Colloidal Gold and Silver Particles Formed During Slow and Fast Aggregation

Spectra of absorption (400–800 nm) by the aggregates of colloidal gold (5, 15, and 30 nm in diameter) and silver (20 nm in diameter) particles were studied experimentally and theoretically. It was revealed that, during fast aggregation corresponding to the diffusion-limited cluster aggregation (DLCA), the pattern of spectra is dependent on the size of primary particles. Spectra with the additional absorption maximum in the red region are observed for 15 and 30 nm gold and 20 nm silver particles, while the absorption spectrum for 5 nm particles is characterized by only one maximum shifted to the red region. The slow aggregation resulted in a decrease in plasmon absorption peak with no marked shift to the red region and to the broadening of long-wave absorption wing. From data on electron microscopy, typical branched DLCA-clusters were formed during fast aggregation, whereas small compact aggregates and a noticeable number of single particles were observed in a system during slow aggregation. The computer model of the diffusion-limited cluster-cluster aggregation was used to explain these results. Optical properties of aggregates were calculated using coupled dipole method (CDM or DDA) and the exact method of a multipole expansion. Corrections for the size effect were introduced into the bulk optical constants of metals for nanosized particles. It was shown that a modified version of DDA (Markel et al.,Phys. Rev. B, 1996, vol. 53, no. 5, p. 2425) allows us to explain the pattern of experimental spectra of DLCA-aggregates and their dependence on a monomer size. The exact method was applied to calculate the extinction cross sections of metallic aggregates demonstrating strong electrodynamic interaction between particles. The number of higher multipoles that are required to adequately describe this interaction is much larger than the number of terms of an ordinary Mie series and is the main obstacle to the exact calculation of the spectra of metallic aggregates with a large number of particles.     

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.