Point defect dynamics in bcc metals

We present an analysis of the time evolution of self-interstitial atom and vacancy (point defect) populations in pure bcc metals under constant irradiation flux conditions. Mean-field rate equations are developed in parallel to a kinetic Monte Carlo (kMC) model. When only considering the elementary processes of defect production, defect migration, recombination and absorption at sinks, the kMC model and rate equations are shown to be equivalent and the time evolution of the point defect populations is analyzed using simple scaling arguments. We show that the typically large mismatch of the rates of interstitial and vacancy migration in bcc metals can lead to a vacancy population that grows as the square root of time. The vacancy cluster size distribution under both irreversible and reversible attachment can be described by a simple exponential function. We also consider the effect of highly mobile interstitial clusters and apply the model with parameters appropriate for vanadium and $\alpha-$iron.Comment: to appear in Phys. Rev.

Experimental Characterization and Population Balance Modelling of the Dense Silica Suspensions Aggregation Process.

Concentrated suspensions of nanoparticles subjected to transport or shear forces are commonly encountered in many processes where particles are likely to undergo processes of aggregation and fragmentation under physico-chemical interactions and hydrodynamic forces. This study is focused on the analysis of the behavior of colloidal silica in dense suspensions subjected to hydrodynamic forces in conditions of destabilization. A colloidal silica suspension of particles with an initial size of about 80 nm was used. The silica suspension concentration was varied between 3% and 20% of weight. The phenomenon of aggregation was observed in the absence of any other process such as precipitation and the destabilization of the colloidal suspensions was obtained by adding sodium chloride salt. The experiments were performed in a batch agitated vessel. The evolution of the particle size distributions versus time during the process of aggregation was particularly followed on-line by acoustic spectroscopy in dense conditions. Samples were also analyzed after an appropriate dilution by laser diffraction. The results show the different stages of the silica aggregation process whose kinetic rates depend either on physico-chemical parameters or on hydrodynamic conditions. Then, the study is completed by a numerical study based on the population balance approach. By the fixed pivot technique of Kumar and Ramkrishna [1996. On the solution of population balance equations by discretization—I. A fixed pivot technique. Chemical Engineering Science 51 (8), 1311–1332], the hypothesis on the mechanisms of the aggregation and breakage processes were justified. Finally, it allows a better understanding of the mechanisms of the aggregation process under flowing conditions

Mechanism of Formation of Monodispersed Colloids by Aggregation of Nanosize Precursors

It has been experimentally established in numerous cases that precipitation of monodispersed colloids from homogeneous solutions is a complex process. Specifically, it was found that in many systems nuclei, produced rapidly in a supersaturated solution, grow to nanosize primary particles (singlets), which then coagulate to form much larger final colloids in a process dominated by irreversible capture of these singlets. This paper describes a kinetic model that explains the formation of dispersions of narrow size distribution in such systems. Numerical simulations of the kinetic equations, with experimental model parameter values, are reported. The model was tested for a system involving formation of uniform spherical gold particles by reduction of auric chloride in aqueous solutions. The calculated average size, the width of the particle size distribution, and the time scale of the process, agreed reasonably well with the experimental values.Comment: 38 pages in plain TeX and 7 JPG figure

Population balance modelling of polydispersed particles in reactive flows

Polydispersed particles in reactive flows is a wide subject area encompassing a range of dispersed flows with particles, droplets or bubbles that are created, transported and possibly interact within a reactive flow environment - typical examples include soot formation, aerosols, precipitation and spray combustion. One way to treat such problems is to employ as a starting point the Newtonian equations of motion written in a Lagrangian framework for each individual particle and either solve them directly or derive probabilistic equations for the particle positions (in the case of turbulent flow). Another way is inherently statistical and begins by postulating a distribution of particles over the distributed properties, as well as space and time, the transport equation for this distribution being the core of this approach. This transport equation, usually referred to as population balance equation (PBE) or general dynamic equation (GDE), was initially developed and investigated mainly in the context of spatially homogeneous systems. In the recent years, a growth of research activity has seen this approach being applied to a variety of flow problems such as sooting flames and turbulent precipitation, but significant issues regarding its appropriate coupling with CFD pertain, especially in the case of turbulent flow. The objective of this review is to examine this body of research from a unified perspective, the potential and limits of the PBE approach to flow problems, its links with Lagrangian and multi-fluid approaches and the numerical methods employed for its solution. Particular emphasis is given to turbulent flows, where the extension of the PBE approach is met with challenging issues. Finally, applications including reactive precipitation, soot formation, nanoparticle synthesis, sprays, bubbles and coal burning are being reviewed from the PBE perspective. It is shown that population balance methods have been applied to these fields in varying degrees of detail, and future prospects are discussed

Boundary value for a nonlinear transport equation emerging from a stochastic coagulation-fragmentation type model

We investigate the connection between two classical models of phase transition phenomena, the (discrete size) stochastic Becker-D\"oring, a continous time Markov chain model, and the (continuous size) deterministic Lifshitz-Slyozov model, a nonlinear transport partial differential equation. For general coefficients and initial data, we introduce a scaling parameter and prove that the empirical measure associated to the stochastic Becker-D\"oring system converges in law to the weak solution of the Lifshitz-Slyozov equation when the parameter goes to 0. Contrary to previous studies, we use a weak topology that includes the boundary of the state space (\ie\ the size $x=0$) allowing us to rigorously derive a boundary value for the Lifshitz-Slyozov model in the case of incoming characteristics. The condition reads $\lim_{x\to 0} (a(x)u(t)-b(x))f(t,x) = \alpha u(t)^2$ where $f$ is the volume distribution function, solution of the Lifshitz-Slyozov equation, $a$ and $b$ the aggregation and fragmentation rates, $u$ the concentration of free particles and $\alpha$ a nucleation constant emerging from the microscopic model. It is the main novelty of this work and it answers to a question that has been conjectured or suggested by both mathematicians and physicists. We emphasize that this boundary value depends on a particular scaling (as opposed to a modeling choice) and is the result of a separation of time scale and an averaging of fast (fluctuating) variables.Comment: 42 pages, 3 figures, video on supplementary materials at http://yvinec.perso.math.cnrs.fr/video.htm