Effect of initial particle size distribution on the dynamics of transient Ostwald ripening: A phase field study Junjie Li, Chunwen Guo, Yuan Ma, Zhijun Wang and Jincheng Wang ⇑ State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, PR China Received 26 January 2015; revised 14 February 2015; accepted 16 February 2015 Available online 3 March 2015 Abstract—The coarsening of polydisperse particles with different initial particle size distributions (PSD) is studied using a quantitative phase field model in two dimensions with emphasis on the transient behavior before reaching steady state. The coarsening rate constant, scaled PSD, scaled evolution law and radial distribution function were systematically examined and compared with available theoretical and experimental results. It is found that the length of transient regime is directly correlated with the extension of the initial scaled PSD toward the large particle size region, i.e. the so called tail, rather than the width of scaled PSD which can be described in terms of standard deviation. Initial distributions with short tails evolve rapidly to the steady state even though the initial width of PSD is large. Whereas, after long time coarsening with a factor of 4–6 change in average radius, initial distributions with long tails still deviate from the steady-state form, yet may appear stable due to the slowly changing rate. Some inconsistencies about the transient coarsening behaviors in previous studies are clarified based on the present results. Moreover, the validity of circular shape assumption is certified when the particle volume fraction is not larger than 0.4. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ostwald ripening; Phase field model; Phase transformation 1. Introduction Ostwald ripening, or particle coarsening, is a common kinetic phenomenon which occurs in the polydisperse two-phase mixture via a diffusive mass flow from small par- ticles to large ones. This process results in a substantial increase in the average particle size and a corresponding decrease in the particle number density. A better under- standing of this process is of great fundamental and practi- cal interest. Since the first quantitative analysis by Lifshitz and Sly- ozov [1] and by Wagner [2] (LSW), numerous modified the- ories [3–22] have been developed to describe the kinetics of Ostwald ripening process. A collective objective of all these theories is to find a proper kinetic equation describing the averaged growth rate of a particle with a given size. Once this is determined, the dynamics of coarsening process can be completely defined by combining with the continuity equation for particle size distribution and the mass conser- vation law. The kinetic equation in LSW theory is only valid in the case of vanishing volume fraction (f v ). Taking into account the effect of interparticle diffusional interac- tion at nonzero volume fraction has been the major objec- tive of modern coarsening theory. The existing approaches to address this issue can be classified into two categories: (i) mean-field description based on some ad hoc assumptions [3–8] or taking into account diffusion screening [9–12]; (ii) microscopic description that represents the mass flow from each particle as a monopole source or sink, employing sta- tistical mechanical methods [13–16] or computer simulation [17–22] to determine the averaged quantity of interest. All these theories carried out analysis strictly only in the long time limit, where the system approaches to a scaling regime (steady state). In this regime, the particle size distribution (PSD) is time independent, or self-similar, when scaled by the average radius. The average radius, <R>, grows with time, t, according to: hRðtÞi 3 hRðt 0 Þi 3 ¼ Kt ð1Þ where <R(t 0 )> is the average radius at initial time t 0 and K is the coarsening rate constant depending on the volume fraction of particles and material parameters. All of the aforementioned theories except those employ- ing numerical simulations are strictly valid only in the long time limit. Their applicability to practical coarsening pro- cess with finite time is hinged on the assumption that the evolution to steady state in experiments is rapid rather than taking infinite long time. This assumption has been widely accepted since the scaled PSD seemed to be self-similar within finite experimental time. However, this viewpoint has been discredited by a series of careful experiments car- ried out by Voorhees and coworkers in microgravity envi- ronments of space shuttle [23–25], which is the most http://dx.doi.org/10.1016/j.actamat.2015.02.030 1359-6462/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Tel.: +86 29 88460650; fax: +86 29 88491484; e-mail: jchwang@nwpu.edu.cn Available online at www.sciencedirect.com ScienceDirect Acta Materialia 90 (2015) 10–26 www.elsevier.com/locate/actamat