Crystal Formation (Nucleation) under Kinetically and Diffusion Controlled Growth Conditions
Ingo H. Leubner, The Journal of Physical Chemistry, 91:6069-6073 (1987)
This paper describes the full derivation of the balanced nucleation and growth (BNG) model for nucleation under diffusion and kinetically controlled growth conditions. This derivation includes and exceeds the initial derivation of the model.
A model of crystal formation for homogeneous nucleation of sparingly soluble compound was derived, which takes into account the balance between crystal growth and classical nucleation (BNG model). The model quantitatively relates the number of stable crystals formed to the precipitation conditions, and to the growth mechanism of the crystals. This model is free of arbitrary adjustable parameters.
In the present work, the general model was derived and solved for kinetically controlled growth conditions. The results were compared with those for a previously derived model under diffusion –controlled growth conditions. Both models show the same functional dependence of crystal size, Z, on addition rate, R, solubility, Cs, and temperature, T.
Two significant differences are as follows:
(1) the diffusion-controlled model depends on the diffusion constant, D, of the solutes in solution, while the kinetic model is dependent on the surface integration constant, Ki.
(2) In the diffusion-controlled model, the crystal number is constant throughout the precipitation after a relatively short transition period, while in the kinetically controlled case the crystal number decreases as the crystal size, r, in creases, so that Z*r is constant.
The latter is in agreement with published data for the initial stages of an AgI precipitation at 35C, indicating that in that case crystal growth was kinetically controlled. From the transition point from kinetically to diffusion –controlled growth, the kinetic surface integration constant could be estimated to about 56 cm4/s mol. For AgCl (cubic), AgBr (cubic and octahedral), and AgI (at 70C), the crystal numbers were constant after the first minute of the precipitation. This is in agreement with diffusion –controlled growth processes. While the present nucleation model was confirmed for silver halide precipitations, it is expected to be applicable to other systems, which rely on homogeneous nucleation.