Buoyancy and surface-tension-driven convection in hanging-drop protein crystallizer

This paper deals with natural and Marangoni convection in hanging (or sitting) drop protein crystallizers. In the pre-nucleation phase the drop is modelled as a mixture of water, precipitating agent and protein, bounded by an undeformable interface with a surface tension exhibiting a linear dependence on the concentrations; axial symmetry is assumed with respect to the drop axis. The post-nucleation phase is modelled assuming a given location of the crystal and appropriate boundary conditions for the concentrations of protein and precipitating agent in the neighbourhood of the crystal and at the drop surface. The final state of the pre-nucleation is used as the initial condition for the post-nucleation phase. The field equations, written in a suitable spherical co-ordinates system, are solved, with appropriate boundary and symmetry conditions, by a numerical algorithm based on finite-difference schemes. The study cases refer to the crystallization of lysozyme in a solution of sodium chloride in water, for two configurations, full-size and half-size geometries. The computations indicate that for these configurations solute transport is dominated by convection and that the convection velocities are one or even two orders of magnitude larger than the characteristic diffusion velocities. In the pre-nucleation phase solute Marangoni effects are negligible for the half-zone geometry but in the full-size geometry they are comparable to buoyancy driven flows. Calculations of buoyancy flows around a growing crystal show that in ground conditions non-uniform concentration gradients may have a detrimental effect on the growth kinetics.