Return Radius and volume of recrystallized material in Ostwald Ripening

Within the framework of the LSW theory of Ostwald ripening the amount of volume of the second (solid) phase that is newly formed by recrystallization is investigated. It is shown, that in the late stage, the portion of the newly generated volume formed within an interval from time $t_0$ to $t$ is a certain function of $t/t_0$ and an explicit expression of this volume is given. To achieve this, we introduce the notion of the {\it return radius} $r(t,t_0)$, which is the unique radius of a particle at time $t_0$ such that this particle has – after growing and shrinking – the same radius at time $t$. We derive a formula for the return radius which later on is used to obtain the newly formed volume. Moreover, formulas for the growth rate of the return radius and the recrystallized material at time $t_0$ are derived.

💡 Research Summary

This paper presents a theoretical framework for calculating the volume of material newly formed via recrystallization during the late stages of Ostwald ripening, based on the Lifshitz-Slyozov-Wagner (LSW) mean-field theory. The primary motivation stems from experimental challenges in accurately measuring this newly formed volume in geological contexts, such as calcite recrystallization, using radioactive tracer methods. Existing experimental interpretations can be confounded by changes in particle surface area, which affect tracer uptake and release. This work aims to provide a precise theoretical quantification of the net new volume, separate from surface effects.

The analysis is conducted within the standard LSW theory, which assumes a dilute system of spherical particles, conservation of the total volume of the dispersed solid phase, and growth kinetics governed either by diffusion-limited (DL) or attachment-limited (AL) mechanisms. The growth law for a particle radius R(t) is given in terms of the critical radius R_c(t). In the late stage, the dynamics simplify, leading to a scaling law for R_c(t) ~ t^(1/γ), where γ=3 for DL and γ=2 for AL ripening.

The central theoretical innovation is the introduction of the “return radius,” denoted r(t, t0). It is defined as the unique radius such that a particle having this radius at the initial time t0 will have the exact same radius at a later time t, after potentially undergoing periods of growth and dissolution. To compute this, the authors employ dimensionless variables z = R/R_c and τ = ln(R_c(t)/R_c(0)). The condition for the return radius translates into the equality α(z(t)) = α(z(t0)), where α(z) = ln z + τ(z) and τ(z) is an explicit function given by the integrated growth law (Eq. 11). Since α(z) has a single maximum at z=1, for any initial scaled radius z0 > 1, there exists a unique scaled radius z < 1 satisfying this equality. This defines a function z = ρ(z0).

Using this function ρ, the return time t(r) (the time when a particle of initial radius r returns to its original size) is derived as t(r)/t0 = (z0/ρ(z0))^γ (Eq. 21). Consequently, the return radius r(t, t0) itself is shown to depend only on the ratio t/t0. The growth rate of the return radius at the initial time is explicitly calculated as ṙ(t0) = R_c(t0)/(2γ t0).

The volume fraction Φ_new(t, t0) of material newly formed between t0 and t is then expressed as an integral over the particle size distribution. Only particles with an initial radius larger than the return radius r(t, t0) contribute to this new volume, as they have experienced net growth. By utilizing the known asymptotic scaled size distribution functions h(z) from LSW theory (Eqs. 13, 14), the new volume fraction can be computed in terms of z0 and ρ(z0). The analysis concludes that, like the return radius, this newly formed volume fraction in the late stage is also a function solely of the scaled time t/t0. The paper provides explicit formulas for this volume and its initial rate of formation, thereby offering a fundamental theoretical tool for quantifying recrystallization kinetics and understanding the universal scaling behavior in Ostwald ripening systems.