the dihedral angle
Where a grain boundary buried inside a solid runs up to a pore or a free surface, it does not just stop bluntly — the two solid surfaces meeting the boundary tilt to form a little valley, and the angle of that valley is the dihedral angle. Picture two soap films meeting a third along a line: they pull on each other and settle at a definite angle where the tensions balance. In a ceramic the same balance, at the groove where a grain boundary meets a pore's surface, fixes a characteristic angle that quietly governs whether pores shrink, whether a liquid soaks into the boundaries, and what shape the last pores take.
The angle is set by a tug-of-war between surface tensions. At the bottom of the groove, the two solid-vapour surfaces (each pulling with tension gamma_sv) must balance the grain boundary (pulling with tension gamma_gb) trying to zip the groove shut. The balance gives 2 times gamma_sv times cos(psi/2) = gamma_gb, where psi is the dihedral angle. So a high grain-boundary energy (relative to the surface energy) makes the boundary pull hard and opens the groove to a small psi; a low boundary energy gives a large psi near 180 degrees, a barely-there groove. For a wetting liquid at a boundary the same equation reads cos(psi/2) = gamma_gb / (2 times gamma_sl): if the liquid's solid-liquid energy is low enough, psi drops to zero and the liquid penetrates all the way along the grain boundaries, wrapping every grain in a film.
The dihedral angle matters because it sets the equilibrium shape and stability of the pores that must be removed in the final stage. A pore is really a little polyhedron bounded by the surfaces meeting the boundaries at angle psi; whether that pore's surfaces are net concave (so it wants to shrink and disappear) or net convex (so it is stable and refuses to close) depends on the interplay of psi with how many grains surround the pore. Pores with few surrounding grains and a favourable dihedral angle shrink away; pores with many neighbours can reach a stable shape and stubbornly survive, capping the density. In liquid-phase sintering the dihedral angle is just as decisive: a small psi means the liquid wets and spreads along the boundaries (good for fast densification, but it leaves a continuous glassy grain-boundary film that weakens high-temperature strength), while a large psi means the liquid balls up at grain corners instead of penetrating.
Thermally etch a polished alumina and tiny grooves appear exactly where grain boundaries meet the surface: measure the groove angle and you have the dihedral angle, from which the ratio of grain-boundary energy to surface energy for that ceramic follows directly.
The dihedral angle is the groove angle where a grain boundary meets a pore — a balance of boundary and surface tensions.
The dihedral angle decides whether the last pores can shrink at all: a pore surrounded by enough grains reaches an equilibrium shape and stops shrinking, so a body can hit a density ceiling set by geometry, not by running out of time.