phase-boundary-controlled kinetics
Not every solid-state reaction is throttled by diffusion. Sometimes the atoms cross the product layer easily, and the slow step is instead the chemical act at the reaction front itself — breaking old bonds and building the new phase at the moving interface. When that interface reaction is the bottleneck, the process is phase-boundary-controlled (or reaction-controlled), and it follows a completely different rhythm from the diffusion-limited case.
The tell-tale difference is in the geometry of time. If the rate-limiting step is the reaction happening all along the phase boundary, then progress is set by how fast that boundary sweeps inward, at a roughly constant velocity — not by an ever-lengthening diffusion path. For a shrinking spherical particle reacting from the surface inward, this gives the contracting-volume (or shrinking-core) law, in which the unreacted radius shrinks linearly with time, so 1 minus (1 minus f)^(1/3) is proportional to time, where f is the fraction reacted. Crucially, the front advances in proportion to time itself (linear kinetics), not to the square root of time as in the parabolic diffusion-controlled case. That contrast — linear versus parabolic — is how experimenters diagnose which step is in charge.
Which regime rules can change during a single firing. Early on, when the product layer is thin, diffusion across it is easy and the interface reaction is the slow step, so kinetics are phase-boundary-controlled and roughly linear; as the layer thickens, the diffusion path lengthens until crossing it becomes the bottleneck and the process crosses over to parabolic, diffusion-controlled kinetics. Reading which law the data follow tells the ceramist what actually limits a reaction — and therefore whether to attack it with a higher temperature (which speeds both, but helps a reaction-controlled step more if its activation energy is higher) or with finer powder (which mainly helps the diffusion-controlled regime by shortening paths). The honest caveat: real reactions rarely obey one idealised law perfectly, and nucleation of the new phase can add its own slow step on top.
A calcining or reacting powder that follows the shrinking-core law — with the reacted fraction obeying 1 minus (1 minus f)^(1/3) proportional to time — is telling you the rate is set by the reaction sweeping across the phase boundary at constant speed, not by diffusion through a product layer.
Phase-boundary-controlled kinetics: the slow step is the reaction at the moving interface, so a front advances at constant speed (linear in time), unlike the sqrt-of-time parabolic diffusion-controlled case.
Linear (interface-controlled) and parabolic (diffusion-controlled) kinetics are two ends of one story: a reaction can start interface-controlled while its product layer is thin and cross over to diffusion-controlled as that layer thickens. Fitting the wrong law over the whole range misreads the mechanism.