A Bounded Supersaturation Model for Nonclassical Crystal Growth: from Impurity Pinning to Macrostep-Dominated Recovery

Abstract Nonclassical crystal growth in impurity-containing systems frequently exhibits growth arrest, macrostep-assisted growth recovery, and hysteresis, which cannot be adequately described by conventional impurity-pinning theories alone. In this work, we develop the pinning-activated growth macrostep transition model (PAGMTM), a bounded phenomenological kinetic model that provides a unified description of impurity-controlled crystal growth through the coupled processes of elementary-step pinning, macrostep activation, growth recovery, and reverse macrostep deactivation. The model introduces two characteristic driving-force thresholds associated with elementary-step depinning and macrostep activation. Growth suppression is represented through an impurity-controlled reduction function, whereas growth recovery is described by a bounded macrostep activation function that progressively restores the intrinsic crystal growth kinetics without exceeding the clean-surface growth limit. The inverse form of the activation function further enables estimation of characteristic kinetic activation levels (e.g., 10%, 50%, and 90%), providing a quantitative measure of the transition from elementary-step-dominated to macrostep-assisted growth. Complementary formulations based on supersaturation and temperature (supercooling) are developed, allowing the framework to be applied to both isothermal and cooling crystallisation systems. The model is validated against literature data for potassium dihydrogen phosphate (KDP) and K2SO4 crystal growth in the presence of Fe(III) and Cr(III) impurities, successfully reproducing growth suppression, progressive macrostep-assisted recovery, and hysteresis during both forward and reverse growth. The results demonstrate that impurity-mediated crystal growth can be quantitatively interpreted using a unified kinetic framework that links experimentally measurable growth-rate behavior with effective interfacial transition parameters, providing a physically interpretable approach for analyzing impurity-controlled nonclassical crystal growth.