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 bee n developed and tested. To obtain accurate calculations with a given precip itant, the vapor pressure of water over water/precipitant solutions must be known for various concentrations of the precipitant. The calculations of t he model are in excellent agreement with all available experimental data on hanging-drop equilibration when the necessary vapor pressures are known (a mmonium sulfate and sodium chloride). By varying each of the relevant rate constants in the model, the rate-limiting step in the equilibration of a ha nging drop is determined, This analysis clearly shows that the rate-limitin g step is the diffusion of water vapor from the drop to the reservoir, whic h agrees with experimental findings. Since the diffusion of water vapor is the rate-limiting step, there is virtually no precipitant concentration gra dient in the drop during equilibration. As a result, there is no gravity-in duced convection owing to the equilibration. Thus, whereas gravity might ha ve an effect during crystal growth, gravity does not affect the equilibrati on rate of a hanging-drop experiment to a significant extent, and the diffu sion of water vapor will remain the rate-limiting step in the absence of gr avity. Finally, the effects of several of the parameters, such as initial d rop volume? drop-to-reservoir distance and temperature, are considered quan titatively, The equilibration rate was found to vary nearly linearly with d rop volume. The equilibration rate decreases roughly by a factor of three a s the temperature decreases from 293 to 276 K. This decrease in the equilib ration rate is greater than would be expected when just considering the cha nge in the diffusion coefficient of water vapor in air. This large dependen ce 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.