| We will not enter into the solubility theory and solubility prediction as the reader can find comprehensive syntheses in, but recall some fundamental aspects. Applying the Fick’s law, it is easy to demonstrate that the mass transfer rate of a particulate solid of mass M (composed of particles with an average volume Vp) into a liquid of volume VL is proportional to the solid surface S :
where h is the mass transfer coefficient (generally estimated by h = D/e where D is the diffusion coefficient of the compound in the liquid and e the thickness of the diffusion layer), CS the solid solubility and Cb the solute bulk concentration. This leads to the Noyes-Whitney equation giving the dissolution rate R (defined as the concentration change R = dM/dt / VL):
Supposing that Cb remains very small in comparison with the saturation solubility CS, and as the solid surface area Sp is proportional to Vp 2/3, equation (1), into:
known as the Hixson-Crowell cube root law.
On the contrary, when the initial amount of solid approaches the amount needed for reaching a saturated solution, the following equation is obtained, known as the Negative two-thirds law:
Equations (3) and (4) can be expressed in the case of spherical mono-dispersed particles, showing the dependence of the dissolution rate with the particle diameter. But, as powders are never mono-disperse and rarely spherical, this is of poor help. The knowledge of the particle size distribution may also mislead: A powder sample with a small mean diameter and large size distribution may have a deceptively low dissolution rate due to the presence of big particles at the end of the distribution. It is always better to consider the specific area a rather than the particle size, the more because the particle size information may hide particle reagglomeration that considerably reduces the specific area.
Moreover, it is to be noticed that the solubility CS of solid particles also depends on the particle size, increasing for colloidal suspensions: Particles with diameter below 1 µm possess significantly greater solubility than larger ones; this difference may be attributed to a greater specific surface area and higher surface free energy for fine particles in comparison with their larger counterparts. It was widely observed that the very fine particles have a tendency to dissolve and recrystallize onto the larger ones, producing a shift in particle size distribution until an equilibrium solubility is reached (“Ostwald ripening”). |