The theory of Ostwald ripening

Developments in the theory of Ostwald ripening since the classic work of I. M. Lifshitz and V. V. Slyozov (LS) are reviewed and directions for future work are suggested. Recent theoretical work on the role of a finite volume fraction of coarsening phase on the ripening behavior of two-phase systems is reformulated in terms of a consistent set of notation through which each of the theories can be compared and contrasted. Although more theoretical work is necessary, these theories are in general agreement on the effects of a finite volume fraction of coarsening phase on the coarsening behavior of two-phase systems. New work on transient Ostwald ripening is presented which illustrates the broad range of behavior which is possible in this regime. The conditions responsible for the presence of the asymptotic state first discovered by LS, as well as the manner in which this state is approached, are also discussed. The role of elastic fields during Ostwald ripening in solid-solid mixtures is reviewed, and it is shown that these fields can play a dominant role in determining the coarsening behavior of a solid-solid system.     

Ostwald ripening of beta-carotene nanoparticles

Ostwald ripening, the interfacial-energy-driven dissolution and reprecipitation of solutes, becomes an increasingly significant problem for nanoparticle formulations. We present the first quantitative study of Ostwald ripening for nanoparticle dispersions. The Lifshitz-Slyozov-Wagner (LSW) theory of particle growth driven by diffusion is applied to study beta-carotene nanoparticles with sizes of O(100 nm) formed by our block-copolymer protected Flash Nanoprecipitation process. A numerical implementation of the LSW theory that accounts for the original particle size distribution is presented. The predicted particle sizes from the numerical simulation are compared with the experimental results measured by dynamical light scattering. The results show quantitative agreement with no adjustable parameters. The addition of antisolvent results in the reduction of the ripening rate by dramatically decreasing bulk solubility.     

Domain formation due to Ostwald ripening in bistable systems far from equilibrium

We develop growth rates ford-dimensional domains in bistable reaction-diffusion systems. The growth of the domain is restrained by an inhibitory nonlocal interaction and leads to stable stationary inhomogeneous states as known from many applications in nonequilibrium systems. The mathematical analysis is quite similar to problems of Ostwald ripening in van-der-Waals gases though the physical mechanism is different. We show that the nonlocal interaction leads to a competition between several domains. Finally our approach is applied to optical bistability, thermokinetic systems and to nonlinear semiconductors.     

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.     

Ostwald ripening with kinetically limited bond formation

We present Monte Carlo simulations of the ripening of 2D islands in the case when the formation of the monomer-monomer bonds is kinetically limited. The results obtained indicate that such limitations may modify the early stage of the kinetics. Asymptotically, the ripening is described by the Lifshitz-Slyozov law. Received 29 August 2000     

1D Cahn-Hilliard dynamics: Ostwald ripening and application to modulated phase systems

We use a family of stationary solution of the Cahn-Hilliard dynamics in order to describe the coalescence during a first order phase transition. With this analytical ansatz, we compute the characteristic time for one step of period doubling in Langer's self similar scenario for Ostwald ripening. As an application, the same ansatz is also used to compute the thermodynamically stable period of a 1D modulated phase pattern, described by a Cahn-Hilliard dynamics with long range interaction terms.     

Ostwald ripening under conditions of mixed-type diffusion

The Ostwald ripening in metallic alloys and quasi-zero-dimensional semiconductor structures is investigated under conditions of mixed dislocation-matrix diffusion, when none of the terms j _d and j _v in the expression for the total diffusion flux can be ignored. The size distribution functions and time dependences of the maximum size r _g and critical (average) size r _k(〈r〉) of particles for different ratios x between the fluxes are obtained. It is demonstrated that mixed dislocation-matrix diffusion can occur in the course of the formation of CdS quantum dots prepared through chemical deposition.     

Hydrogen-induced Ostwald ripening at room temperature in a Pd nanocluster film

The structural and morphological changes occurring in an ensemble of vapor deposited palladium nanoclusters have been studied after several hydrogenation cycles with x-ray diffraction, extended x-ray-absorption fine structure spectroscopy, Rutherford backscattering spectrometry, and STM. Initial hydrogenation increased the cluster size, a result that is attributed to hydrogen-induced Ostwald ripening. This phenomenon originates from the higher mobility of palladium atoms resulting from the low sublimation energy of the palladium hydride as compared to that of the palladium metal. The universality of this phenomenon makes it important for the application of future nanostructured hydrogen storage materials.     

Ostwald ripening of charged supported metal nanoparticles: Schottky model

Due to high surface area, supported metal nanoparticles are thermodynamically prone to sintering. The experimental studies of this process exhibit sometimes transient bimodal particle size distributions. Such observations may result from the support heterogeneity. Looking retrospectively, one can also find the prediction that in the case of Ostwald ripening this feature can be related to charge of metal nanoparticles. In real systems, this charge is often associated with the metal–support interaction and can be interpreted in the framework of the Schottky model. Using this model, the author shows that the charge redistribution cannot be behind bimodal particle size distributions. Moreover, the corresponding contribution to the driving force for Ostwald ripening is typically much smaller than the conventional one.     

Ostwald ripening, chiral crystallization, and the common-ancestor effect

We introduce an agent-based model for advection-mediated chiral autocatalysis in which we take into account dissolution-crystallization processes through Ostwald ripening. We demonstrate that the latter phenomenon is the key to explaining previously puzzling experimental results whereby complete chiral symmetry breaking is attained from an initially unbiased mixture of seed crystals. We show that this homochirality is achieved by what has been termed the common-ancestor effect: Ostwald ripening removes competing lineages, leaving only one common ancestor for the entire system.     

Surface diffusion of copper on tantalum substrates by Ostwald ripening

In this work, we report on the determination of surface diffusion coefficient of copper on tantalum substrates by Ostwald ripening. It is shown that impurities, such as oxygen, strongly influence the kinetics of dewetting of copper films on tantalum substrates. Two technologically important interfaces with copper were investigated: Cu/β-Ta and Cu/α-Ta. For copper surface diffusion on β-Ta surface, a surface diffusion coefficient of was measured at 550 °C. The temperature dependence of surface diffusion was investigated between 400 °C and 550 °C. Using an Arrhenius relationship, an activation energy of 0.83 ± 0.1 eV and a pre-exponential factor of were calculated. For copper surface diffusion on α-Ta surface, a diffusion coefficient of was measured at 550 °C. We discuss the diffusion mechanism involved during the cluster growth and the origin of the faster surface diffusion of copper on the β-Ta substrate as compared to the α-Ta phase.     

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.     

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.     

Interfacial free energies and solute diffusivities from data on Ostwald ripening

Using recent theoretical modifications of the kinetic constants characterizing Ostwald ripening, it is demonstrated that accurate values of the interfacial free energy, , and solute diffusivities, D, can be obtained from experimental data when the kinetics of particle growth are measured in conjunction with independent measurements of either the decrease of the matrix supersaturation or the increase in volume fraction with aging time. The accuracy of is limited only by the assumption that the matrix phase is an ideal solid solution, and is effectively independent of the influence of equilibrium volume fraction, _e, on the kinetics of coarsening. Analyses of the available data on the coarsening of -type (Ni₃X) precipitates in binary Ni–Al, Ni–Si and Ni–Ti alloys yield values of =6.9±0.3, 10.2±3.0 and 13.0 mJ/m², respectively, assuming ideal solution thermodynamics; a more realistic thermodynamic model for the Ni–Al solid solution raises the value of in Ni–Al alloys to 8.1±0.2 mJ/m². Proportional increases probably obtain in the other two alloys. The accuracy with which D can be evaluated from comparable data depends theoretically on _e. However, analyses of the same data yield values of D in very good agreement with the results of conventional diffusion experiments. This is consistent with the absence of an effect of _e on the kinetics of Ostwald ripening in these alloys over the ranges of _e investigated.     

Ostwald ripening of nanoislands in semiconductor heterosystems and its influence on optical properties

Influence of the Ostwald ripening on character of the size distribution in semiconductor heterosystems with quantum dots as island films consisting of cylindrical, disclike islands of constant height is studied. Size distribution function and formulae for temporal changes of average island size are derived. Calculations are carried out within the LSW theory, in assumption of joint action of two (diffusion and Wagner) growing mechanisms. Comparison of the calculating results with experimental data proves validity of the accepted model approximations. Expectancy of correlation of the distribution function, growing mechanisms and properties of heterosystems is discussed.