A theoretical interpretation of self-similar right-skewed particle size distributions in Ostwald ripening of cementite in ferrite

A multi-particle approach of Ostwald ripening in two-phase systems based on direct interactions is developed which successfully explains the stationary right-skewed shape of the particle size distribution found in coarsening experiments of cementite in ferrite at a volume fraction of about 0.07. To reproduce such an evidence the mean field hypothesis of the classical LSW theory has been replaced by a topological framework where any particle exchanges solutes with all its neighbours within an interaction volume proportional to its size. Then, an effective diffusion distance depending on the current average size in the system and on volume fraction of the second phase has been introduced.     

Effects of internal stresses and intermediate phases on the coarsening of coherent precipitates: A phase-field study

Phase stability, topology and size evolution of precipitates are important factors in determining the mechanical properties of crystalline materials. In this article, the Cahn-Hilliard type of phase-field model was coupled to elasticity equations within a mixed-order Galerkin finite element framework to study the coarsening morphology of coherent precipitates. The effects of capillarity, particle size and fraction, compositional strain, and inhomogeneous elasticity on the kinetics and kinematics of coherent precipitates in a binary dual phase crystal admitting a third intermediate stable/meta-stable phase were investigated. The results demonstrated the ability of the model to simulate coarsening under the concomitant action of Ostwald ripening and mismatch elastic strain mechanisms. Using a phenomenological coarsening power law, coarsening rates were determined to depend on precipitate size and volume fraction, compositional strain, and strain mismatch between precipitates and the matrix. Results also showed that the necking incubation time between two neighboring precipitates depends inversely on the precipitate’s initial sizes; however, under fixed volume fraction of precipitates, any increase in the initial sizes of the precipitates mitigates the coarsening. Meanwhile, the compositional strain and the growth of the intermediate stable/meta-stable phase leads to substantial enhancements of precipitate coarsening.     

Coarsening of cracks in a uniaxially strained solid

We discuss the stress relaxation in a uniaxially strained solid due to the coarsening of a system of parallel cracks. We emphasize similarities and differences of this process to Ostwald ripening in a first order phase transition. A conventional mean-field approximation breaks down and several independent length scales have to be taken into account. Strong elastic interactions between the cracks determine the growth behavior. We derive scaling laws for the coarsening of the different length scales involved and the time evolution of stress relaxation, finally leading to the equilibrium state of a fractured body. The characteristic size of the cracks grows linearly in time which is much faster than in usual Ostwald ripening.     

Qualitative evaluation of the effect of morphology,spatial distribution and coherent interfaces on the coarsening kinetics and size distribution of particles during Ostwald ripening

ABSTRACT

By exploiting the simple structure of the recently developed Multi-Particle model of Ostwald ripening based on pairwise interactions, the effect of extra factors affecting the coarsening process, such as the morphology of particles, the topology of their spatial distribution and the presence of elastic fields surrounding coherent precipitates, is analysed from a qualitative viewpoint.

Two observables disclose the presence of deviations with respect to the behaviour of a system of spherical and incoherent precipitates: the absolute value of the kinetic coarsening constant, together with its evolution with the volume fraction, and the shape of the asymptotic particle size distribution (PSD) for a given volume fraction of second phase.

By using a parametrised ‘rounded cube’ ideal solid it is shown that, as the shape of precipitates flattens, the coarsening constant decreases reaching in some condition values as low as about 20% that of a system of spheres with the same fraction.

Consequently, the curve of the relative kinetic constant with the volume fraction has a slower increase above 0.7 compared to spherical particles and this permits to explain accurately the experimental data on dense systems in the whole range of volume fractions.

It is also shown how the shape of the PSD is influenced by regular arrangements of particles in the space and by interfacial stresses in the matrix surrounding coherent precipitates. In both cases the model predicts a sharpening of the PSD, whose shape tends to become more symmetrical and somehow similar to that of the classical LSW theory.     

Novel SiGe island coarsening kinetics: ostwald ripening and elastic interactions

Real-time light scattering measurements of coherent island coarsening during SiGe/Si heteroepitaxy reveal unusual kinetics. In particular, the mean island volume increases superlinearly with time, while the areal density of islands decreases at a faster-than-linear rate. Neither observation is consistent with standard considerations of Ostwald ripening. Modification of the standard theory to incorporate the effect of elastic interactions in the growing island array reproduces the observed behavior.     

Coarsening and self-organization in dilute diblock copolymer melts and mixtures

This paper explores the evolution of a sharp interface model for phase separation of copolymers in the limit of low volume fraction. Particles both exchange material as in usual Ostwald ripening, and migrate because of an effectively repulsive nonlocal energetic term. Coarsening via mass diffusion only occurs while particle radii are small, and they eventually approach a finite equilibrium size. Migration, on the other hand, is responsible for producing self-organized patterns.We construct approximations based upon an ansatz of spherical particles similar to the classical LSW theory to derive finite dimensional dynamics for particle positions and radii. For large systems, kinetic-type equations which describe the evolution of a probability density are constructed. For systems larger than the screening length, we obtain an analog of the homogenization result of Niethammer & Otto [B. Niethammer, F. Otto, Ostwald ripening: The screening length revisited, Calc. Var. Partial Differential Equations 13-1 (2001) 33-68]. A separation of timescales between particle growth and migration allows for a variational characterization of spatially inhomogeneous quasi-equilibrium states.     

粗化过程中颗粒界面形状演化的三维多相场法研究

利用多相场模型对液-固两相体系中固相颗粒的粗化过程进行了三维模拟,对粗化过程中的界面形状分布进行了统计分析,研究了不同固相体积分数下颗粒连接状态对界面形状演化及粗化速率的影响.模拟结果表明:当颗粒间存在大量连接时,粗化速率随固相分数的变化速率比颗粒无连接时变缓,且随着粗化进行,高曲率的双曲形界面所占比例不断降低,低曲率的椭球形界面所占比例逐渐增多;无论固相颗粒间是否发生连接,界面形状演化经历一定阶段后,三维界面形状分布均呈现自相似性,但随着固相体积分数的增加,界面形状分布呈现自相似性所需的时间延长.     

Dynamics and stability of nanostructures on metal surfaces

Ultrathin metal films consisting of two-dimensional clusters are typically unstable: the cluster ensemble has the tendency to reduce its total free energy via Ostwald ripening or dynamic coalescence of mobile clusters. In this paper we give an overview of recent model experiments addressing these coarsening mechanisms. The experiments have been performed using STM on ensembles consisting of adatom or vacancy clusters with typical diameters in the nanometer range on fcc(111)-metal surfaces. Agreement with and deviations from conventional theories are discussed. Received: 29 March 1999 / Accepted: 17 August 1999 / Published online: 30 September 1999     

界面润湿性及固相体积分数对颗粒粗化动力学影响的相场法研究

利用多相场模型模拟了液-固两相体系中固相颗粒的粗化过程, 分析了界面润湿性及固相体积分数对粗化指数、粗化速率及颗粒尺寸分布的影响.结果表明, 不同固相体积分数下粗化指数基本不变, 但粗化速率常数及尺寸分布与固相体积分数及界面润湿性密切相关.在完全润湿条件下, 随着固相体积分数的增加, 粗化速率常数逐渐增大; 而非完全润湿条件下, 随着固相体积分数的增加, 粗化速率常数增大速度变缓, 且当润湿性较低、 固相分数较大时, 粗化速率常数还将随体积分数的增加而下降. 此外, 模拟结果表明各种润湿条件下颗粒的尺寸分布均随着固相分数增加而变宽, 分布峰值降低, 但非完全润湿条件下峰值下降变缓.模拟结果为理解不同实验观测结果之间的分歧提供了依据. 关键词： 粗化 相转变 相场法 润湿性     

Ostwald ripening in nanoalloys: when thermodynamics drives a size-dependent particle composition

Ostwald ripening has been broadly studied because it plays a determinant role in the evolution of cluster size during both chemical and physical synthesis of nanoparticles. This thermoactivated process causes large particles to grow, drawing material from the smaller particles, which shrink. However, this phenomenon becomes more complex when considering the coarsening of metallic alloy clusters. The present experimental and theoretical investigations show that the relative composition of CoPt nanoparticles can be strongly modified during high temperature annealing and displays a size-dependent behavior. This compositional change originates from the higher evaporation rate of Co atoms from the nanoparticles. More importantly, this effect is expected in all alloy clusters containing species with different mobilities.     

Nonuniversality in Low-Volume-Fraction Ostwald Ripening

We study Ostwald ripening in the regime of small volume fraction and consider spatially periodic systems whose size is smaller than the screening length. Within the snapshot perspective we obtain an explicit characterization of the leading-order deviation to the classical mean-field theory by Lifshitz, Slyozov and Wagner (LSW). Using this representation, we show that the corrections are not universal, in the sense that the mean value has a strong dependence on geometry, and arbitrarily large fluctuations can happen with finite probability. AMS Subject classification: 35B27, 74N20, 82C26.     

Ostwald Ripening of Concentrated Alkane Emulsions: A preliminary report

Fiber optic dynamic light scattering (FODLS) was used to study the kinetics of aging processes in emulsions of n-alkanes stabilized by a surfactant. The method is particularly useful for this purpose because it permits measurements in concentrated emulsions. Complications that may occur in traditional DLS due to the extreme dilution, such as solubilization of the particles in the medium, can be easily avoided in this way. The experimental results show that the main aging process is Ostwald ripening. The results are in agreement with theoretical predictions based on the Lifshitz-Slyozov-Wagner (LSW) theory as far as it concerns the relation between the molecular solubility of the n-alkanes (in the aqueous phase) and the Ostwald ripening rate. This illustrates that the main rate-determining factor is the molecular diffusion of the alkane molecules through the continuous (aqueous) phase. The Ostwald ripening rate is affected by the concentration of surfactant but in all situations is higher than expected from LSW theory.     

Nucleation, coarsening and kinetic scaling of two-dimensional islands on GaSb(0 0 1)

Using a combination of molecular beam epitaxy and in situ surface X-ray diffraction, we investigate the nucleation and coarsening of monolayer high islands on GaSb(0 0 1) during deposition in real time. We find an activation energy for island nucleation of 1.55 ± 0.16 eV, indicating a stable nucleus size larger than two atoms. For intermediate temperatures where GaSb homoepitaxy is stable, the lateral coarsening of the islands after deposition is described by Ostwald ripening. The average island sizes during coarsening are isotropic, although with different size distributions in different directions. The size distributions do not change during coarsening, implying kinetic scaling.     

An extended hydrodynamic theory for particle coarsening—I: Theory

The asymptotic hydrodynamic approach to the mathematical solution of the kinetics of particle coarsening (Ostwald ripening), first proposed by Lifshitz and Slyozov and Wagner, has been extended and generalized. The coarsening rate equations in dimensionless forms have been separated into asymptotically determinate and indeterminate parts. The generalized evaluation of the asymptotically indeterminate part allows the consideration of coarsening processes that are more complicated than those involving power law type kinetics considered in previous studies. Further, the extended theory allows the determination of particle size distribution function and averaged particle coarsening kinetics even in cases where different physical processes occur at different size regimes within the same distribution of particles.     

Dynamics of gold clusters on amorphous carbon films induced by annealing in a transmission electron microscope

The change of the size distribution of Au clusters induced by annealing was studied in situ by transmission electron microscopy. Starting from statistically distributed Au clusters on a thin amorphous carbon film, “islands” are formed within a few months storage at room temperature, which consist of Au clusters with sizes <4 nm embedded in a thin Au film. These islands cover originally areas with sizes around 25 × 70 nm². If the temperature is increased in the transmission electron microscope two different processes can be clearly distinguished that lead to the coarsening of the cluster size distribution: cluster coalescence and (contactless) Ostwald ripening. The degree and rate of the coarsening are found to depend on the underlying surface (Au film or amorphous carbon) and the exposure to the high-flux high-energy electron beam, which can be estimated to lead to high-temperature excursions in a cluster on a 10⁻¹² s time scale. The experimental findings are confirmed by Monte-Carlo simulations using the many-body Gupta potentials in order to calculate the Au/Au interaction. Moreover, the results of MC simulations suggest an electron-beam induced formation of a “quasi-two-dimensional gas” of small highly mobile Au species on the Au film, which promotes Ostwald ripening.     

Fluctuations provide strong selection in Ostwald ripening

A selection problem that appears in the Lifshitz-Slyozov (LS) theory of Ostwald ripening is reexamined. The problem concerns selection of a self-similar distribution function (DF) of the minority domains with respect to their sizes from a whole one-parameter family of solutions. A strong selection rule is found via an account of fluctuations. Fluctuations produce an infinite tail in the DF and drive the DF towards the "limiting solution" of LS or its analogs for other growth mechanisms.     

The droplet size distribution in the late stage of phase separation

We consider a collection of droplets during the late stage of phase separation in a closed system. Its coarsening is driven by surface energy and leads asymptotically to a linear growth of the mean droplet volume with time (Ostwald ripening). The droplets grow either from the supersaturated uncondensed phase (coalescence) or by collisions with subsequent fusion (coagulation). The combination of both mechanisms leads asymptotically to a self-similar evolution of the size distribution of the droplets when the coagulation kernel is homogeneous with degree zero. We calculate the scaled droplet size distribution for Brownian and constant kernel and compare the effects of coagulation with the effects of correlation and screening discussed in the literature. We compare our results for the asymptotic scaled distribution with computer simulations for the combined coalescence and coagulation processes.     

Analysis of the coarsening of solid dispersed systems

A method is proposed for computer analysis of the kinetics of coarsening of solid dispersed phases due to the Ostwald ripening of microparticles and its associated processes. The proposed method provides useful information on the occurrence of various processes in a dispersed system as it approaches the equilibrium state. In this method, the similarities and differences between the experimental and theoretical size distributions of microparticles are preliminarily determined by comparing the characteristics of the microparticle size distributions with the corresponding moments. The quality and reliability of identifying the density function of the microparticle size distribution are evaluated from an analysis of the relationships between these characteristics.     

A discrete model of Ostwald ripening based on multiple pairwise interactions

A discrete multi-particle model of Ostwald ripening based on direct pairwise interactions is developed for particles with incoherent interfaces as an alternative to the classical LSW mean field theory. The rate of matter exchange depends on the average surface-to-surface interparticle distance, a characteristic feature of the system which naturally incorporates the effect of volume fraction of second phase. The multi-particle diffusion is described through the definition of an interaction volume containing all the particles involved in the exchange of solute. At small volume fractions this is proportional to the size of the central particle, at higher volume fractions it gradually reduces as a consequence of diffusion screening described on a geometrical basis. The topological noise present in real systems is also included. For volume fractions below about 0.1 the model predicts broad and right-skewed stationary size distributions resembling a lognormal function. Above this value, a transition to sharper, more symmetrical but still right-skewed shapes occurs. An excellent agreement with experiments is obtained for 3D particle size distributions of solid–solid and solid–liquid systems with volume fraction 0.07, 0.30, 0.52 and 0.74. The kinetic constant of the model depends on the cube root of volume fraction up to about 0.1, then increases rapidly with an upward concavity. It is in good agreement with the available literature data on solid–liquid mixtures in the volume fraction range from 0.20 to about 0.75.     

Growth mechanism of nanocrystals in solution: ZnO, a case study

We investigate the mechanism of growth of nanocrystals from solution using the case of ZnO. Spanning a wide range of values of the parameters, such as the temperature and the reactant concentration that control the growth, our results establish a qualitative departure from the widely accepted diffusion controlled coarsening (Ostwald ripening) process quantified in terms of the Lifshitz-Slyozov-Wagner theory. Further, we show that these experimental observations can be qualitatively and quantitatively understood within a growth mechanism that is intermediate between the two well-defined limits of diffusion control and kinetic control.