An examination of Köhler theory resulting in an accurate expression for the equilibrium radius ratio of a hygroscopic aerosol particle valid up to and including relative humidity 100%

Abstract

The equilibrium hygroscopic behavior of an aqueous solution drop is investigated using the Köhler model to relate the radius ratio ξ ≡ r/rdry, where rdry is the volume‐equivalent dry radius, and the fractional relative humidity h. The Köhler equation is derived and results obtained from it are presented for three situations: when the effect of surface tension can be neglected, for h = 1, and for cloud‐drop activation. The exact solution to this equation is presented, as is an accurate approximate solution for h < 1 that yields insight into the dependences of the equilibrium radius on relative humidity, surface tension, and dry radius. The approximations made in the derivation of the Köhler equation are examined, errors in quantities obtained from this equation are quantified, and the so‐called Debye approximation is introduced which allows accurate parameterization of these errors as a function of rdry. Errors in the radius ratio at activation obtained from the Köhler equation are up to 20% for ammonium sulfate solution drops of the size that typically form cloud drops. Attempts to extend the Köhler model to higher concentrations are examined, and it is seen that the primary cause of inaccuracy in the model is the assumption that the practical osmotic coefficient is unity. On the basis of this analysis, a simple two‐parameter expression is presented for the equilibrium radius ratio as a function of h and rdry that is accurate over a wide range of rdry and for h up to and including unity.