Population balance for aggregation coupled with morphology changes

International audienceIn the past, the kinetics of aggregation has been extensively studied. Aggregation rates were measured and calculated thanks to a population balance. Aggregate morphologies were measured or got by computer simulations. However, the link between the aggregation kinetics and the morphology changes with time is not so clear. The modelling of aggregation may be even more complex as restructuring of aggregates occurs. The aim of this paper is to propose a new formulation taking into account at once kinetics of collision and morphology change rate. We built a bivariate population balance with matter volume and porous volume as internal parameters. The population balance equation contains the standard collision term and a convective term representing the porous volume change. The latter is split into two contributions, which is due to the aggregation process itself and the other one is due to the restructuring. The expressions of the first contribution are determined for Brownian and shear aggregations

Origin and value of a limit size during shear aggregation revisited

During the aggregation of fine particles in a shear flow a limit size value aL for aggregates is reached. Most researchers have related aL to the shear rate by means of a power law. We examine in this paper the different ways in order to model the phenomena leading to a limit size. The main results in the field of drop-drop and bubble-particle systems are briefly reviewed to help us to propose a coherent description of phenomena occurring in particle-particle systems. Kernels for coalescence, aggregation, breakage and erosion are recalled. An improvement of the aggregation kernel in the case of the collision between aggregates is proposed. We show that an analysis of the whole process in term of aggregation-fragmentation competition will be preferred to a collision which would be less efficient between large aggregates. In this framework we present a modelling relating aggregation kernel and fragmentation kernel to a limit size value. As a consequence, the main result is the exponent value of the limit size-shear rate power law

Light scattering cross section as a function of pair distribution density

International audienceThis paper presents a simple method of approximation for calculating the scattering cross section for a random orientated convex particle illuminated by a non polarized electromagnetic wave. This method is proved efficient for sphere and spheroids as the scattering efficiency is smaller than one and as the material is optically either soft or hard

Relationship between the morphology and the light scattering cross section of optically soft aggregates

International audienceThe purpose of this paper is to establish the suitable analytical expressions for the light scattering cross section of aggregates constituted by optically soft primary particles. The used optical approximation is the well-known Rayleigh-Debye-Gans approximation. The geometrical description of the aggregate is based on the pair, or distance, distribution density function describing at once the two spatial scales (primary particle and aggregate). Firstly, a study of the two-particle set is presented. An exact and an approximated formulations are proposed for the distance distribution density and the scattering cross section. Then the corresponding results are extended to the aggregates

Modelling of aggregate restructuring in a weakly turbulent flow

International audienceAn aggregate may undergo a restructuring that leads to a change of its porosity. The cause of this morphology change is not only the colloidal forces between the primary particles but also the interaction between the primary particles and the flowing fluid. We investigate in this paper the restructuring of the aggregates with Van der Waals forces between the primary particles in the presence of the Brownian motion and a weakly turbulent flow. The motion of all the primary particles constituting the aggregate is studied thanks to a 1-D model based on the work of G. Narsimhan [Model for drop coalescence in a locally isotropic turbulent flow field, J. Colloid Interface Sci., 272 (2004) 197-209]. A shrinking rate law, i.e. aggregate porosity versus time, is proposed after a transition to a 3-D model. Its characteristic time depends on the initial porosity, the number of primary particle and a characteristic time at the scale of the primary particle

Quels défis pour la granulométrie optique ? What challenges does optical particle sizing face?

National audienceLes constructeurs de granulomètres proposent des instruments dont la qualité métrologique est indéniable. Cependant, les logiciels d'interprétation du signal optique reposent la plupart du temps sur l'hypothèse de sphéricité des particules. Ce papier passe en revue les différents problèmes à résoudre pour obtenir une information morphologique plus riche et réaliste que celle couramment proposée. On focalisera notre attention sur le problème direct qui est le préalable à toute amélioration : étant donné une particule de morphologie donnée, quelle est sa signature optique ? Manufacturers of particle sizers sell on the market instruments with high quality of the metrology. However the software used for analysing the optical signal are often based on some hypothesis like the sphericity of the particles. This paper reviews the challenges to get more realistic information about the particle morphology. We focus the paper on the direct problem which is the beginning of any improvement of the inverse problem: given a particle morphology, what is its optical response

Approximation of the light scattering cross section for aggregated non absorbent spherical particles

International audienceWe present in this article a comparison of the light scattering cross-section obtained by using exact calculations (GMMcode computed by Xu) and several approximated methods such as the Compact Sphere (CS), Anomalous Diffraction(AD), Effective Refractive Index (ERI) and Percival–Berry–Khlebtsov (PBK). We carry this out, for different shapes ofaggregate (i.e. compact and linear configuration) consisting of different numbers of primary particles (2, 4, 8, 16, 64, 100).We study it as a function of a size parameter of the primary particles in the range from 0 to 10 and as a function of the typeof materials (SiO2, Al2O3, TiO2).We present the validity range of these approximated methods and conclude that ERI is the most promising one for allthe ranges of size parameter (i.e. the average error of this method is, compared to the exact method, around 3% for acompact configuration)

Light Scattering Cross Section of Hydrophilic and Hydrophobic Silica Aggregates

20 pagesInternational audienceThe aggregation dynamics of solid particles in liquid media is currently followed by optical based particle sizing methods. Because it can be used in situ and applied to a wide particle size range, turbidimetry is acknowledged as one of the best methods for this characterization. Although much work has been done about aggregation, some aspects are less known and require additional experimental and theoretical research. This is particularly the case of aggregation of hydrophobic particles. Corresponding aggregates are 3-phase objects (solid-liquid-gas) the morphology and optical properties of which are not known. Present work rests on the turbidimetric study of hydrophilic and hydrophobic silica samples in stirred aqueous solutions. Modelling involves different aspects: aggregate morphology, aggregate optical properties, and aggregation dynamics. This paper particularly emphasizes the second one. Fractal-like models are proved to be representative of the aggregate morphology even at small size. Light scattering cross-section of the aggregates is calculated from their averaged projected area; effective refractive index is proved to be a good parameter for modelling their optical properties both for hydrophilic and hydrophobic aggregates. Classical models of porous aggregate formation (Kusters theory) are used for describing the aggregation dynamics

Approximation of the scattering cross section for aggregated spherical particles

International audienceExact solution for scattering cross section coming from Xu's computations is compared with different approximations

The chord length distribution of a dumbbell shaped aggregate: Analytical expression

International audienceDumbbell shaped aggregates are small particles synthesized in precipitation reactors. Their characterization by optical methods needs the chord length distribution (CLD) of such a shape. We present in this paper the analytical calculation of two CLDs corresponding to two different definitions of CLD. Comparison with Monte-Carlo simulations is presented. Good agreement is found between the exact calculation and simulations