The distribution of a macromolecular solute within an evaporating drop: An exact analytical solution

A space- and time-dependent concentration profile of a low-diffusive solute within an evaporating droplet will be formed due to the forced inward motion of the solute at the confining, inward-moving free surface (solute entrainment), if the solvent evaporates and any convection within the droplet can be neglected. To study this effect theoretically, a generalized diffusion equation including the solute entrainment is derived. An exact analytical solution for spherical and semi-spherical droplet geometries is presented assuming a constant solute diffusion coefficient (linear problem) and an ideal evaporation kinetics of the Langmuir type. Analytical approximations are outlined, and conditions are given to apply the results in practice where the ideal evaporation kinetics is disturbed. Starting from a constant solute concentration, a new spherical-symmetrical concentration profile with maximum at the droplet boundary is approached during evaporation, which is scaled by the time-dependent droplet radius only, and which is independent of the starting droplet radius for small deviations from the Langmuir evaporation kinetics. A concentration enhancement at the droplet boundary by a factor of about 2 compared with the mean concentration level is possible for macromolecular solutes under conditions as used, e.g., in evaporating droplet arrangements for protein crystal growth. In addition, the solute entrainment can act to increase a starting concentration variation parallel to the droplet surface if the droplet kinetics exceeds a certain threshold value. Since any convective flux within the droplet has to be excluded in order to apply our results, the convection-free state has to be proved by observations taking into account that convective instabilities in evaporating droplets are hardly to predict.