the sintering driving force

Why should a heap of loose powder, held at a hot temperature but never melted, knit itself into a solid block? Nothing pushes on it from outside. The answer is that a powder is enormously wasteful of energy: every particle is coated in surface, and a surface is a place where atoms have fewer neighbours than they would like, so each square metre of surface carries an energy cost. A fine powder hides a vast hidden area — grams of micron-sized particles pack in a few square metres of surface. Nature, given enough heat to let atoms move, spends that surface away. The sintering driving force is simply the system's urge to lower its total interfacial energy, the same urge that pulls a splash of water into a round bead.

There are two ways to shrink the surface bill, and they set up the whole drama of sintering. The system can grow the features — fewer, bigger particles and pores have less area than many small ones — which is coarsening. Or it can replace expensive solid-vapour surface with cheaper solid-solid contact, welding particles together at grain boundaries and squeezing the pores out — which is densification. A grain boundary between two grains still costs energy (gamma_gb), but usually much less than the two free surfaces it replaces (2 gamma_sv), so trading surface for boundary is a net saving. Locally the force shows up as curvature: under a curved surface the chemical potential of an atom is raised or lowered by delta-mu = gamma times Vm times (1/r1 + 1/r2), where Vm is the molar volume and r1, r2 the radii of curvature. Atoms flow from the convex, high-potential humps of the particles toward the sharply concave, low-potential necks between them.

Be honest about the size of this force. For a one-micron powder with a surface energy near 1 J/m^2, the stored excess is only about 100 J/mol — a trifle next to the tens of kilojoules per mole it would take to melt the same material. That is exactly why sintering is slow, needs high temperature to give atoms mobility, and is easily beaten by a coarse powder: halve the particle size and you roughly double the driving force. It is also why fine, uniform powders are the sinterer's best friend, and why an applied pressure (hot pressing, HIP, SPS) is so valuable — an external 30 MPa can dwarf the few MPa of capillary stress that curvature alone provides.