MODELING THE SHAPES OF CRYSTAL SIZE DISTRIBUTIONS FROM CRYSTAL GROWTH MECHANISMS

The evolution of the shape of a crystal size distribution (CSD) can be used to infer growth laws for crystals. For example, the often assumed constant growth rate law (dX/dt=k, where X is crystal diameter, t is time, and k is a constant) is unrealistic in most systems because it causes the variance of the CSD to approach 0 during growth, and because it destroys the shape of commonly occurring lognormal CSDs. Conversely, proportionate growth models (dX/dt=kX) best describe the evolution of CSDs in most systems. This law can be derived from a modified version of the Law of Proportionate Effect (LPE: X(j+1)=X(j) + v(j)e(j)X(j), where j refers to the growth cycle or time interval, v is a function of the proportion of the total volume of material available for each crystal for each growth cycle, and e is a random number that varies between 0 and 1). When v is large with respect to X, as is the case immediately after nucleation, the variance of the CSD increases with mean diameter and a lognormal CSD results. However, as X grows larger with respect to v, the variance remains constant as the mean crystal diameter increases, and the proportionate growth law is approximated. Simulating crystal growth by LPE using the Galoper computer program leads to some surprising conclusions, several of which have been confirmed by experiment: (1) If a small crystal doubles in diameter, a large crystal in the same system also will tend to double in diameter during the same time interval, even though such growth involves adding much more volume to the larger crystal; (2) Crystal growth has a random component (e), which indicates that one can not predict accurately the growth rate for individual crystals, but only for the distribution of crystals. This inherent randomness also accounts for crystal growth dispersion; (3) The relative shape of a CSD generally is determined soon after nucleation, and then is maintained during proportionate growth; (4) Crystal growth appears to occur in nanometer-size jumps, rather than continuously with time; and, (5) Proportionate growth occurs when reactants are supplied to the crystal surface by advection under stirred conditions, whereas constant growth is favored when supply is by diffusion under stagnant conditions.