Diffusion Models for Hot Pressing with Surface Energy and Pressure Effects as Driving Forces

TLDR

The rate equations for hot pressing cannot be integrated because pore size depends on density in an unknown way. The study develops models for initial, intermediate, and final densification stages that explicitly incorporate surface energy and applied pressure as driving forces. The models modify the surface energy term to γ+PaR/π, adapt Nabarro‑Herring and Coble creep equations with a driving force of (Pa/D+γk), and introduce a concentric‑shell diffusion model for the final stage to better capture densification above 95 %. The initial‑stage equations reproduce earlier surface‑energy‑driven results, and the analysis shows that only the (Pa/D) effective stress relation satisfies self‑consistency for steady‑state diffusion models.

Abstract

Models for initial-, intermediate-, and final-stage densification under pressure have been developed, which explicitly include both the surface energy and applied pressure as driving forces. For the initial stage, the time dependences and size effects given by the integrated equations are identical to those reported earlier for surface energy (alone) as the driving force. The only modification is that the surface energy (γ) is expanded into (γ+PaR/π), where Pa is the applied pressure and R is the particle radius. For the intermediate stage of the process, the Nabarro-Herring and Coble creep models may be adapted to give approximate (∼4×) densification rates for lattice and boundary diffusion models, respectively. In these cases the complex driving force is written as: (Pa/D+γk), where D is the relative density, and k is the pore surface curvature. At the final stage of the process those models are invalid; an alternate model is developed based on diffusive transport between concentric spherical shells which will give a better assessment of the time dependence of densification high density (>95%); the driving force is (Pa/D+γk) in this case also. Because of the fact that the pore size is some unknown function of density, the rate equations cannot be integrated without further information. It is shown that of the various relations which have been assumed in development of models for hot pressing, for the effective stress in relationship to the applied stress and the porosity, (Pa/D) is the only form which satifies the criteria demanded by self-consistency in generation of steady-state diffusion models.