[Paper Review] Evolution of vacancy pores in bounded particles

Vladimir Yanovsky, M. I. Kopp+1 more|arXiv (Cornell University)|Sep 18, 2018

Advanced Mathematical Modeling in Engineering11 references3 citations

TL;DR

This paper derives a nonlinear system of ordinary differential equations describing the diffusion-driven evolution of vacancy pores within spherical nanoparticles under quasi-stationary diffusion conditions. It shows that unlike in unbounded matrices, pores in bounded spherical particles always shrink and migrate toward the center due to the absence of a critical pore size, with distinct behaviors for small and large pores confirmed analytically and numerically.

ABSTRACT

In the present work, the behavior of vacancy pore inside of spherical particle is investigated. On the assumption of quasistationarity of diffusion fluxes, the nonlinear equation set was obtained analytically, that describes completely pore behavior inside of spherical particle. Limiting cases of small and large pores are considered. The comparison of numerical results with asymptotic behavior of considered limiting cases of small and large pores is discussed.

Motivation & Objective

• To understand the dynamics of vacancy pores in bounded spherical nanoparticles, which are common in meso- and nanosystems.
• To derive a complete analytical description of pore evolution, including radius, matrix radius, and center-to-center distance, under quasi-stationary diffusion fluxes.
• To investigate the absence of a critical pore size in bounded particles, contrasting with the growth/dissolution dichotomy in unbounded matrices.
• To establish limiting behaviors for small and large pores and validate them against numerical simulations.
• To provide a reference framework for comparing with numerical modeling of pore dynamics in nanoscale systems.

Proposed method

• Formulation of a system of nonlinear ordinary differential equations based on quasi-stationary diffusion fluxes and equilibrium vacancy concentration at pore and matrix surfaces.
• Use of asymptotic expansion in terms of the small parameter ε = 1 − r/rₛ to analyze limiting cases of small and large pores.
• Derivation of evolution equations for pore radius r, matrix radius rₛ, and center-to-center distance L, up to second-order terms in ε.
• Numerical solution of the approximate equations (58)–(60) for validation, with initial conditions r(0)=1, rₛ(0)=1.5, L(0)=0.15, and A=10⁻¹.
• Comparison of numerical solutions of the approximate system with exact solutions to verify accuracy and assess error order (O(ε³)).
• Incorporation of vacancy flux contributions from both pore and matrix surfaces, with renormalization of diffusion coefficients to account for material-specific effects.

Experimental results

Research questions

• RQ1 How does the evolution of a vacancy pore inside a spherical nanoparticle differ from that in an unbounded matrix?
• RQ2 What determines the absence of a critical pore size in bounded spherical particles, and how does this affect pore stability?
• RQ3 How do the dynamics of small and large pores differ in terms of their shrinkage rate and centerward motion?
• RQ4 To what extent do asymptotic approximations for small and large pores accurately describe the full evolution of the pore?
• RQ5 How can the derived analytical equations serve as a benchmark for numerical modeling of pore behavior in nanoparticles?

Key findings

• Pores in spherical nanoparticles always shrink and move toward the center, with no critical size separating growth and dissolution modes, unlike in unbounded matrices.
• For small pores near the center, the pore radius decreases linearly with time, indicating a simple, predictable healing process.
• For small pores near the matrix boundary, the shrinkage rate is proportional to the square root of time, indicating a diffusion-limited process with a distinct scaling behavior.
• An explosive mode of pore size reduction is predicted in the limit of very small pores, indicating a rapid final stage of dissolution.
• The numerical solution of the approximate equations (58)–(60) agrees well with the exact solution, with discrepancies only at O(ε³), validating the asymptotic approach.
• The hydrodynamic approximation used in this work remains valid for weakly anisotropic shapes, though strong anisotropy or faceting may alter behavior significantly.