Theory of the intermediate stage of crystal growth with applications to insulin crystallization

Abstract

A theory for the intermediate stage of crystal growth, where two defining equations one for population continuity and another for mass-balance, is used to study the kinetics of the supersaturation decay, the homogeneous nucleation rate, the linear growth rate and the final distribution of crystal sizes for the crystallization of bovine and porcine insulin from solution. The cited experimental reports suggest that the crystal linear growth rate is directly proportional to the square of the insulin concentration in solution for bovine insulin and to the cube of concentration for porcine. In a previous work, it was shown that the above mentioned system could be solved for the case where the growth rate is directly proportional to the normalized supersaturation. Here a more general solution is presented valid for cases where the growth rate is directly proportional to the normalized supersaturation raised to the power of any positive integer. The resulting expressions for the time dependent normalized supersaturation and crystal size distribution are compared with experimental reports for insulin crystallization. An approximation for the maximum crystal size at the end of the intermediate stage is derived. The results suggest that the largest crystal size in the distribution at the end of the intermediate stage is maximized when nucleation is restricted to be only homogeneous. Further, the largest size in the final distribution depends only weakly upon the initial supersaturation.