This research focuses on modeling the transport phenomena within the hanging drop caused by the evaporation, as this information is required to determine first-principles nucleation kinetics from the measured induction times. The solute concentration increases locally near the liquid-air interface resulting in a diffusive flux toward the center of the hanging drop. A density gradient is associated with the concentration gradient, which induces natural convection within the droplet. A third transport phenomenon is Marangoni (surface tension-driven) convection which results from the dependency of surface tension on local concentration. Although the focus of this presentation is on the pharmaceutical application, the analyses can be applied to the crystallization of other organic molecules such as proteins and amino acids.
Klupsch et al (2003) provided an analytical solution to a pure diffusion problem involving a drop undergoing evaporation by Langmuir kinetics. A computational simulation involving combined diffusion, natural convection, and Marangoni convection was reported by Savino and Monti (1996). The analytical solution for an isothermal system undergoing solely Marangoni convection was presented by Hu and Larson (2005), and Ristenpart et al (2007) indicated that the direction and magnitude of Marangoni flow is related to the ratio of thermal conductivities between the substrate and the droplet.
This presentation describes a mix of exact and approximate analytical solutions for the solution concentration within the drop in the case of diffusion, as well as computational solutions for the cases of diffusion and diffusion combined with natural convection. Analytical expressions are derived for the maximum concentration difference within the drop, which is important for assessing the accuracy of the commonly made assumption of well-mixedness. These expressions characterize crystal-solvent systems and operating conditions for which the drop is not well-mixed. Comparisons are made to the analytical and computational models developed by other researchers who address similar problems.
Hu, Larson, Langmuir, 21, 3972-3980, 2005.
Klupsh, Mühlig, Hilgenfeld, Colloids & Surface Science A, 231, 85-102, 2003.
Ristenpart, Kim, Domingues, Wan, Stone, Physics Review Letters, 99, 234502, 2007.
Savino, Monti, J Crystal Growth, 165, 308-318, 1996.