a matter-transport path
For a powder to sinter, atoms have to physically move from where there is too much material to where there is too little — from the bulging surfaces of particles into the hollow necks and shrinking pores. But atoms in a solid have several different roads they can travel, and a matter-transport path is simply one of those roads: a specific route, with its own speed and its own consequences, by which material is carried during sintering. The whole science of sintering is, at bottom, a competition among these paths.
There are six classic paths, and the deepest way to sort them is by where the matter comes from — the source — because that decides whether the part shrinks. Three paths draw matter from the particle surface and deliver it to the neck: surface diffusion (atoms creeping along the outside), lattice diffusion originating at the surface, and evaporation-condensation (atoms taking to the vapour and re-landing on the neck). All three fatten the neck and round the pores, but because they only shuffle material around the outside, the particle centres never approach — so these are non-densifying, coarsening paths. The other three draw matter from between the particles: grain-boundary diffusion (matter flowing out of the boundary in the neck plane), lattice diffusion originating at the grain boundary, and plastic or viscous flow. Because these remove material from the region separating the particle centres, the centres slide together and the body shrinks — these are the densifying paths.
This source-based split is the single most important idea in sintering, because it explains a paradox beginners always trip over: a powder can bond firmly, grow beautiful necks, and coarsen visibly while densifying not at all. Every path runs at once; they all draw on the same surface-energy driving force; and which one dominates depends on temperature, particle size, and material. Surface diffusion tends to lead at low temperature and in fine powders (it scales steeply with small size), so heating slowly can waste the driving force on coarsening; grain-boundary and lattice diffusion take over higher up and do the densifying work. Reading a sinter therefore means asking, for the conditions at hand, which path is winning — and steering the schedule so a densifying path stays ahead.
Two identical alumina compacts, one heated straight to 1600 degrees C, the other held long at 1100 degrees C first: the slow-heated one ends up less dense, because the long low-temperature soak let surface diffusion — a non-densifying path — coarsen the pores before the densifying paths could switch on.
Six paths, one rule: matter from the surface coarsens; matter from the boundary densifies.
All the transport paths lower the same surface energy, so all run simultaneously — you cannot switch one off. You can only tilt the balance with temperature, particle size, and dopants so a densifying path outpaces the coarsening ones.