Abstract. Dissolved surface active species, or surfactants, have a tendency to partition to solution surface and thereby decrease solution surface tension. Activating cloud droplets have large surface-to-volume ratios, and the amount of surfactant molecules in them is limited. Therefore, unlike with macroscopic solutions, partitioning to the surface can effectively deplete the droplet interior of surfactant molecules. Surfactant partitioning equilibrium for activating cloud droplets has so far been solved numerically from a group of non-linear equations containing the Gibbs adsorption equation coupled with a surface tension model and an optional activity coefficient model. This can be a problem when surfactant effects are examined by using large-scale cloud models. Namely, computing time increases significantly due to the partitioning calculations done in the lowest levels of nested iterations. Our purpose is to reduce the group of non-linear equations to simple polynomial equations with well known analytical solutions. In order to do that, we describe surface tension lowering using the Szyskowski equation, and ignore all droplet solution non-idealities. It is assumed that there is only one surfactant exhibiting bulk-surface partitioning, but the number of non-surfactant solutes is unlimited. It is shown that the simplifications cause only minor errors to predicted bulk solution concentrations and cloud droplet activation. In addition, computing time is decreased at least by an order of magnitude when using the analytical solutions.
Analytical Solutions of Non-Linear Equations of Power-Law Fluids of Second Grade over an Infinite Porous Plate
