An accurate numerical model for calculating the equilibration rate of a hanging-drop experiment

A numerical model of the equilibration of a hanging-drop experiment has been developed and tested. To obtain accurate calculations with a given precipitant, the vapor pressure of water over water/precipitant solutions must be known for various concentrations of the precipitant. The calculations of the model are in excellent agreement with all available experimental data on hanging-drop equilibration when the necessary vapor pressures are known (ammonium sulfate and sodium chloride). By varying each of the relevant rate constants in the model, the rate-limiting step in the equilibration of a hanging drop is determined. This analysis clearly shows that the rate-limiting step is the diffusion of water vapor from the drop to the reservoir, which agrees with experimental findings. Since the diffusion of water vapor is the rate-limiting step, there is virtually no precipitant concentration gradient in the drop during equilibration. As a result, there is no gravity-induced convection owing to the equilibration. Thus, whereas gravity might have an effect during crystal growth, gravity does not affect the equilibration rate of a hanging-drop experiment to a significant extent, and the diffusion of water vapor will remain the rate-limiting step in the absence of gravity. Finally, the effects of several of the parameters, such as initial drop volume, drop-to-reservoir distance and temperature, are considered quantitatively. The equilibration rate was found to vary nearly linearly with drop volume. The equilibration rate decreases roughly by a factor of three as the temperature decreases from 293 to 276 K. This decrease in the equilibration rate is greater than would be expected when just considering the change in the diffusion coefficient of water vapor in air. This large dependence can, however, be attributed to the change in water-vapor pressure. Most surprisingly, a linear dependence on drop-to-reservoir distance is found, a result that agrees very well with experiment.