Collision rate constant for non-spherical particles moving under shear flow

Highlights

• Rate constant of particle collision in a shear flow.

• Particles with simple shapes as spheroids, discs and needles.

• Monte Carlo calculations and analytical expressions.

Abstract

Taking into account the morphology of the particles in the aggregation dynamics is addressed theoretically in this paper. The rate constant or kernel of collision between particles with simple shapes (spheroids oblate and prolate, disc, needle) has been calculated from a Monte Carlo algorithm simulating shear aggregation. The corresponding data were used to build a model, which comes in the form of an empirical expression linking the rate constant of collision, the shear rate and three shape parameters describing each particle. This concise expression represents very well all the data issued from Monte Carlo calculation. Statistical analysis of the Monte Carlo calculations and the proposed model as well has been carefully achieved.

Introduction

Aggregation of fine particles is a phenomenon frequently met during the synthesis of particles by precipitation or crystallization, in suspensions containing a precursor of ceramics, in aerosols⋯Aggregation of particles in a fluid depends on several phenomena. The collision between two primary particles can arise from the Brownian motion, the fluid and particle velocity fields. So, Von Smoluchowski has calculated the collision rate for spheres in the case of Brownian motion and particles moving in a shear flow [1], [2]. This approach has been extended to spherical particles moving in a turbulent flow [3] or a rarefied atmosphere [4]. As a consequence, a collision rate constant has been calculated for each mechanism. In its early history, aggregation has been strongly related to colloids science where physico-chemical aspects are prevailing. So, Van der Waals and Electric Double Layer interactions have been identified and constitute the basis of DLVO theory [5], [6]. This one addresses the stability of the particle suspension. Fuchs [7] established the link between aggregation rate constant and inter-particle interaction by introducing a multiplicative factor into the expression of the collision rate constant. By a similar way, Spielman [8] and Zeichner et al. [9] introduced the effect of the hydrodynamic resistance on aggregation rate. Later refinements have included retarded Van der Waals interaction, non DLVO forces, roughness parameters, non-wetting effect [10]. Therefore, aggregation kernel is now written as the product of the collision rate constant and the aggregation efficiency. At the scale of the colliding primary particles, the impact of the particle asphericity has not been rigorously treated. The considered geometrical parameter is only the radius of the volume equivalent sphere. Brownian aggregation is the predominant mechanism for nanoparticles in aqueous suspension whereas shear aggregation is the one for micro-particles. The two mechanisms occur at the same time for particles sized within the [0.1–1 μm] range [11]. The experimental validation of the mentioned models and theories is based on the study of the first instants of aggregation, for whom only doublets of primary particles are formed [12].

At the later instants of aggregation, clusters of primary particles are formed. Simulations, e.g. Monte-Carlo simulations, make it possible their computational formation [13], [14], [15], [16]. Statistical analysis of the clusters formed by basic mechanisms shows that the larger ones have a fractal-like structure, i.e. porous objects with self-similar spatial ordering. Collision between aggregates has to take account of their permeability as well [17], [18]. However, the comparison between these simulations and experiments suggests that internal mechanisms as restructuring, sintering between primary particles and fragmentation occur [19], [20], [21], [22], [23], [24], [25], [26]. The resulting aggregates are more compact and elongated. Consideration of such phenomena in simulations is yet under study.

Moreover the dynamics of the population of interacting and merging aggregates may be modeled by solving a population balance equation. Aggregates are roughly depicted by one internal variable (the volume of matter or the radius of the volume equivalent sphere or the radius of the spherical hull) and, possibly, another one (porosity or fractal dimension). In the special case of homogeneous kernel the population density reaches a self-preserving shape [27], [28], [29]. In the other cases the complexity and the large variety of phenomena affecting the aggregation kinetics and the morphology changes restrict complete and realistic description of the aggregate population to date.

On the other hand, the synthesis of particles with controlled morphology has developed since the pioneering work of Matijevic [30], [31]. The authors strove to control the shape and size of the particles and to understand the mechanisms of synthesis. However, few studies have been dedicated to the aggregation of non-spherical primary particles. Unfortunately they do not correspond to shear aggregation [32] or consider only the late stage of aggregation in relation with the suspension rheology [33]. Quantitative studies about kinetics often consider the amount of matter and therefore the radius of the volume equivalent sphere as the relevant parameter involved in the modelling of the aggregation. This approximation has also been applied to dense aggregates with spherical hull. The objective of this paper is to evaluate the effect of the non-sphericity on the rate constant of aggregation. We have specifically focused our study on the collision rate constant, i.e. assuming an aggregation efficiency coefficient equal to one. This follows the classical methodology being to distinguish collision dynamics and attachment kinetics. We will consider particles with a simple shape: sphere, discs, needles and spheroids. They have been selected to represent the whole of the precipitated particles and eventually lead to analytical calculations. We restrict ourselves to the collision induced by shear flow, which is the case for primary particles in particulate systems with Peclet number larger than 1, e.g. aqueous suspension of particles whose largest dimension is greater than 1 μm. The ultimate goal of the article is to propose an approximate expression to the rate constant of collision; this relation will enable the numerical solving of a population balance equation, the particles being made up of objects with simple shapes, real or representative of more complex morphologies [34].

The paper is organized as follows: Section 2 describes the tools and methods used to calculate the rate constant of collision. Section 3 presents the results corresponding to the collision of objects having the same shape, but homothetically different. Section 4 presents the results corresponding to the collision of particles with different shapes. Section 5 will discuss these results and conclude the paper.