The Definition of “Critical Radius” for a Collection of Nanoparticles Undergoing Ostwald Ripening

Abstract

An improved, exact analysis of surface Ostwald ripening of a collection of nanoparticles is presented in an effort to redefine the critical radius involved in the kinetic models of ripening. In a collection of supported particles of different sizes, the critical radius is the size of the particle that is in equilibrium with the surrounding adatom concentration. Such a particle neither grows nor shrinks due to Ostwald ripening, whereas larger particles grow and smaller particles shrink. We show that previous definitions of critical radius are applicable only for limiting regimes where the Kelvin equation has been linearized. We propose a more universally applicable definition of critical radius that satisfies the constraints of mass balance.

Ostwald ripening in a Pt/SiO2 model catalyst studied by in situ TEM

“…Thus, although the size-dependency reflected by eq. 1 is generally accepted to be valid for larger particles, it is likely insufficient for the smallest particle sizes [9,13,22,49]. The size-dependent ripening of the smallest particles has been subject to debate: Datye et al emphasized that, even within the Gibbs-Thomson model, eqs.…”

“…2 and 3 are imprecise for small particles because the equations rely on a first order Taylor expansion of eq. 1 [9]. The models, expressed by eqs.…”

“…In this approximation r* equals the mean arithmetic radius of the particle ensemble [8]. For sufficiently small particles, the Taylor expansion is inapplicable and r* deviates from the mean radius [9].…”

Effect of Particle Morphology on the Ripening of Supported Pt Nanoparticles

“…α' describes the system specific parameters, such as the diffusion energy barrier for atomic species to cross the metalsupport interface as well as the metal surface energy, the atomic volume and the temperature. R* is the critical radius of curvature corresponding to the particle size which is in equilibrium with the mean-field concentration of diffusing atomic species 6,20 . Since R* equals the mean radius of curvature 33 , this parameter is equaled to the arithmetic mean of R for the particle ensemble in the simulations.…”

“…A vast amount of the published coarsening studies has focused on the evolution of the nanoparticle sizes as a function of time and process conditions [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . A prototypical nanocatalyst system for studying coarsening consists of Pt nanoparticles supported on an Al 2 O 3 material, which is relevant as an oxidation catalyst in diesel and lean-burn engine exhaust aftertreatment technologies.…”