Eq 4

where v0 is the heating or cooling rate, Tf is the final sintering temperature, and Ti is the initial temperature. This relationship is verified in heating/quenching experiments for nanocrystalline zirconia (Ref 36). In general, the degree of coarsening occurring during heating in time t to the sintering temperature is approximately equivalent to one third of the grain size obtained for an isothermal anneal (Eq 3) at the sintering temperature for time t (Ref 35).

Pressureless Sintering. Pressureless sintering of nanocrystalline powder occurs generally by surface-assisted and boundary-diffusion-assisted mechanisms. One feature of pressureless sintering is rapid densification of agglomerated regions followed by removal of interagglomerate porosity. Densification rates for the latter stage are associated with sintering of polycrystalline powder whose particle size is associated with the densified agglomerate. The overall effect is rapid initial densification that slows to conventional rates as the agglomerates become fully dense. This effect is mitigated by application of external pressure.

Nanocrystalline powders densify at sintering temperatures significantly lower than conventional powders, as exemplified in Fig. 7 for titania. Generally, densification occurs at 0.2 to 0.4 Tm, compared with 0.5 to 0.8 Tm for conventional powder. To rationalize this decrease in the sintering temperature, different scaling laws have been applied. For particles under 100 nm, Alymov et al. (Ref 37) developed an empirical relationship for the dependence of the sintering onset temperature, Ts, on the mean particle size D:

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