Ostwald ripening in porous materials

Abstract

The process of coarsening of an ensemble of clusters is investigated for the case that elastic strains due to matrix - cluster interactions change the process qualitatively as compared with dependencies established theoretically first by Lifshitz and Slezov. Such a qualitatively different behavior occurs always when the energy of elastic deformation in cluster growth increases more rapidly than linear with the volume of a cluster. Analytic solutions, for limiting cases, as well as numerical solutions, for the general case of coarsening in an ensemble of pores with a given pore size distribution, are presented. Possible applications are discussed

On the elastic contribution to crystal growth in complex environments

“…If the elastic contribution, denoted by y m , enters via the boundary condition, the Gibbs-Thomson relation (Eq. (3)) can be modified as [14,15]:…”

“…m e , being positive, or in auxetic systems, negative, is a dimensionless and system-dependent elastic parameter. [14,15], but for an external diffusion field, feeding the crystal. The two first cases ( 1 2 m = , ) are our proposal to model surface nucleation phenomena in the spirit of Burton -Cabrera-Frank type rationale [1,6,8], mostly in protein crystal growth [7,16,17].…”

“…However, in all three cases, we propose to consider a mass-convection regime, where the mass-convection is modeled either in a deterministic (this section) or in a stochastic manner (next section). Typically a 2D nucleation mechanism, especially for biomolecular crystals, is considered most frequently [7,16,17], but specifically one may also expect more exceptional situations, such as a 1D nucleation by molecular rows, taking place on kinks [6], arising on the crystal boundary [18,19], or a 3D nucleation [14,15], in which the full possibility (3D interspace) is explored near the crystal surface by a penetrating (locally diffusive) external field [9]. We have to stipulate that…”

“…L are the circumferences of the nucleus at time t and 0 t respectively, 1 e is a constant, which de-pends upon the elastic as well as the shear moduli of the crystal, see [1,7,15,19] for many types of non-Kossel crystals. In the case of (ideal) spherical symmetry we can write that 1…”

“…/ . Finally, a full three dimensional mechanism (3D case) can be of interest [14,15]. If the origin of the evolution of the elastic strains in crystal growth consists of the difference between the average volume per number of particles in the two phases, crystal and solution, then the total energy of elastic deformation can be characterized by the Young's modulus and Poisson's ratio of the matrix (the environment in which growth takes place) and the crystal.…”