Pseudo-equilibrium equations for calcium phosphate precipitation with multi-unit particles

In the reaction of calcium phosphate precipitation from a near-neutral solution, an amorphous phase and hydroxyapatite nanoparticles appear successively, and the reaction system containing either of the two kinds of precipitates is in a non-equilibrium state. Here, we report an extension of our previous study on the relationship between a precipitate and the corresponding ion product in pseudo-equilibrium states. We adopted two reaction series, collected samples at various stages, and analyzed the solution chemistry data on the basis of a simplified reaction model. We derived two types of pseudo-equilibrium equations from the two series, respectively. These equations reveal the presence of multiple structural units in a precipitate particle and correlate the ion product with the average value of the surface proportion per structural unit (m). The m value decreased as the number of structural units in a particle (u) increased. Besides, we determined the free energy differences (ΔG) from the pseudo-equilibrium constants (Kd), which explains the observations in the crystal ripening process well. Notably, the two types of pseudo-equilibrium equations take the common form of “Kd = ion product” if u→∞. Overall, our findings provide new insights into the study of precipitation reactions. The relationship between the ion product and the structural unit of a precipitate particle is not only fundamental in chemistry, but may also be applicable to certain natural or biological systems at non-equilibrium, such as marine sedimentation and human vascular calcification.