A pressed bed of ceramic powder (green body) densifies in the kiln by solid-state diffusion: atoms migrate to the necks between touching particles, grain boundaries advance and the pore network shrinks — no melting involved. The rate is set by an Arrhenius diffusion term, and both neck growth and grain growth share it, which is exactly why kiln schedules matter: firing hotter always densifies faster, but it also coarsens the grains faster, and coarse grains hurt the very strength densification was supposed to buy.
k(T) = exp[ -Q/R (1/T - 1/Tref) ] Arrhenius rate, Q ≈ 380 kJ/mol
(x/a)^5 = C · k(T) · t / a^3 two-sphere neck-growth model
ρ(t) = ρ0 + (ρf-ρ0)[1 - exp(-5(x/a)^2)] densification from neck ratio
d^3 = d0^3 + Kg · k(T) · t cubic grain-growth law
σf = σ0 · exp(-b·P) · sqrt(d0/d) Ryshkewitch porosity term × Hall-Petch grain term
- Sintering temperature — sets k(T). The Arrhenius term is exponential, so a few hundred degrees changes both densification and grain-coarsening rates by an order of magnitude or more.
- Initial particle size — neck growth scales as 1/a³, so finer powders sinter dramatically faster at the same temperature; this is why ceramic processing invests heavily in fine, well-dispersed powders.
- Furnace time speed — how fast simulated kiln time (hours) advances; watch density climb while grains keep coarsening even after most porosity is gone.
- Predicted strength — combines the Ryshkewitch porosity law σf = σ0·exp(-b·P) (residual porosity P is the dominant weakening term) with Hall-Petch-type grain-size strengthening, so strength can actually fall if you keep firing well past the density plateau.
Real-world relevance: this porosity/grain-size trade-off is exactly why industrial ceramic firing schedules hold at a carefully chosen peak temperature and time rather than simply firing as hot as possible — over-firing alumina, zirconia or silicon-nitride parts is a well-known cause of strength loss despite near-zero porosity.