the pore-size distribution
Total porosity tells you how much void a ceramic contains, but says nothing about whether that void is one big cavern or a million tiny bubbles. The pore-size distribution fills that gap: it is the record of how the pore volume is spread across pore sizes, usually drawn as a curve or histogram with pore diameter along the bottom and the amount of pore volume (or the number of pores) at each size going up. It is the pore world's version of a class of children sorted by height — the same average can hide a very different spread.
The distribution is measured in several ways, each with a size range. Mercury intrusion porosimetry pushes liquid mercury (which does not wet ceramics) into the open pores under rising pressure; the Washburn relation says a higher pressure is needed to force mercury into a narrower pore, so the pressure-versus-volume curve converts directly into a pore-size curve, good from millimetres down to a few nanometres. Gas adsorption (the BET-type isotherm) reaches the finest nanopores. Direct image analysis of a polished section counts and sizes the pores you can see under the microscope. A distribution can be narrow (all pores about one size, often what you want) or broad, and can even be bimodal — two humps — betraying, say, fine pores between particles plus coarse pores between agglomerates.
Why care about the spread and not just the total? Because in a brittle ceramic the single largest pore usually acts as the strength-limiting flaw, and Griffith's rule ties strength to the square root of that worst flaw's size — so two parts with identical total porosity can differ hugely in strength if one hides a few big pores. In a membrane or filter, the largest pore sets what can slip through and thus the filtration cut-off, while the smallest pores set the flow resistance. The pore-size distribution, not the bare porosity number, is what a designer of filters, catalyst supports, or high-reliability structural parts actually tunes.
Two batches of a structural silicon nitride both test at 2 percent porosity, but batch A fails on average 30 percent weaker. Mercury porosimetry shows batch A's distribution has a small tail of 40 micron pores from soft agglomerates, while batch B's pores are all under 3 micron. The tail, not the total, is the culprit.
It is the biggest pore in the tail of the distribution, not the average, that breaks a brittle part.
Mercury intrusion sees only pores that connect to the surface (open porosity), so it can undercount or miss sealed closed pores. And the pressures involved can crush delicate pore walls, distorting the very distribution you are trying to measure.