In recent years it has become common to estimate nucleation and growth kinetics for the crystallization of pharmaceuticals, amino acids, and proteins from induction times measured in droplets, typically within microfluidic devices (e.g., Galkin & Vekilov, 1999; Talreja et al, 2005; 2007). A novel stochastic model that has been developed to determine nucleation kinetics assumes the crystallization system to be perfectly mixed, i.e., non-existence of spatial variations (L.M. Goh et al, 2007). If the crystallization system is perfectly mixed, the nucleation location is not affected by spatial variations and there is equal chance for nucleation to take place at any position within the system. This presentation evaluates the accuracy of this assumption. Although the analysis and simulation carried out in this study are applicable to any crystallization within droplets, for specificity the system under consideration is an evaporation-based microfluidic crystallization platform (He et al, 2006; Talreja et al, 2005).

Several researchers have argued that the concentration gradients within drops are negligible due to mixing caused by buoyancy and/or surface-tension-driven convection, i.e., Marangoni convection. For example, Grant and Saville (1991) performed a quasi-steady state analysis which indicated that, with natural convection due to buoyancy, the solution concentration at the crystal surface is essentially equal to the bulk concentration, so that the growth rate is completely controlled by surface kinetics and not limited by mass transport. On the other hand, a later study by the same authors that considered the growth of lysozyme crystals indicated that mass transport limitations were important (Grant & Saville, 1995). A computational simulation by Savino and Monti (1996) within droplets of different configurations showed that concentration gradient within the hanging droplet is small due to buoyancy and Marangoni convection. However, Kawaji (2003) reported that no natural convection was detected. Yet, in another report, the existence of natural convection was proven physically for the ethanol-water system (Kang et al, 2003). The first part of this study discusses an order-of-magnitude analysis to assess whether perfect mixing can be assumed. It is found that convective time scale computed using the Grashof number is several orders smaller than all other time scales including the process time of forming one crystal (the induction time) which indicates that the solution is well-mixed throughout the induction period. Both buoyancy and Marangoni convection (Deen 1998; Ostrach 1982) are considered.

The second part of the study computes the three-dimensional solution concentration field within evaporating droplets for aqueous glycine solution. The purpose of this study is to assess the accuracy of order-of-magnitude scaling analysis and to validate the assumption of perfect mixing used in the studies modeling the nucleation kinetics in evaporation-based microfluidic crystallization platforms (Goh & Braatz, 2007). The governing equations are closely related to those of Savino and Monti (1996), which assumes the Boussinesq approximation and the quasi-stationary approximation, i.e., that the effects of the moving liquid-gas interface due to evaporation are negligible. Axial symmetry is assumed with respect to drop axis. The coupled system of nonlinear distributed parameter equations is solved using Femlab. The final section discusses the analysis and simulation results.

References:

Deen, W.M. (1998). Analysis of Transport Phenomenon. Oxford University Press, Inc.

Galkin, O., Vekilov, P.G. (1999) Direct determination of the nucleation rates of protein crystals. J. Phys. Chem. B, 103, 10965-10971.

Grant, M.L., Saville, D.A. (1991). The role of transport phenomena in protein crystal-growth. J. Crystal Growth, 108, 8-18.

Grant, M.L., Saville, D.A. (1995). Long-term studies on tetragonal lysozyme crystals grown in quiescent and forced-convection environments. J. Crystal Growth, 153, 42-54.

He, G., Bhamidi, V., Tan, R.B.H., Kenis, P.J.A., Zukoski, C.F. (2006). Determination of critical supersaturation from microdroplet evaporation experiments. Crystal Growth & Design, 6, 1175.

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Goh, L.M., Chen, K.J., He, G.H., Bhamidi, V., Kenis, P.J.A., Zukoski, C.F., Braatz, R.D.

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Talreja, S., Kenis, P.J.A., Zukoski, C.F. (2007). A kinetic model to simulate protein crystal growth in an evaporation-based crystallization platform. Langmuir, 23, 4516-4522.

Talreja, S., Kim, D.Y., Mirarefi, A.Y., Zukoski, C.F., Kenis, P.J.A. (2005) Determination of critical supersaturation from microdroplet evaporation experiments. J. Appl. Cryst., 38, 988-995.