Kinetics of small particle activation in supersaturated vapors

We examine the nucleated (with barrier) activation of perfectly wetting (zero contact angle) particles ranging from bulk size down to one nanometer. Thermodynamic properties of the particles, coated with liquid layers of varying thickness and surrounded by vapor, are analyzed. Nano-size particles are predicted to activate at relative humidity below the Kelvin curve on crossing a nucleation barrier, located at a critical liquid layer thickness such that the total particle size (core + liquid layer) equals the Kelvin radius (Fig. 1). This barrier vanishes precisely as the critical layer thickness approaches the thin layer limit and the Kelvin radius equals the radius of the particle itself. These considerations are similar to those included in Fletcher's theory (Fletcher, 1958) however the present analysis differs in several important respects. Firstly, where Fletcher used the classical prefactor-exponent form for the nucleation rate, requiring separate estimation of the kinetic prefactor, we solve a diffusion-drift equation that is equivalent to including the full Becker-Doering (BD) multi-state kinetics of condensation/evaporation along the growth coordinate. We also determine the mean first passage time (MFPT) for barrier crossing (Wedekind et al., 2007), which is shown to provide a generalization of BD nucleation kinetics especially useful for barrier heights that are considerably lower than those typically encountered in homogeneous vapor-liquid nucleation, and make explicit comparisons between the MFPT and BD kinetic models. Barrier heights for heterogeneous nucleation are computed by a thermo-dynamic area construction introduced recently to model deliquescence and efflorescence of small particles (McGraw and Lewis, 2009). In addition to providing a graphical representation of the activation process that offers new insights, the area construction provides a molecular approach that avoids explicit use of the interfacial tension. Typical barrier profiles for water condensation on particles of different size are shown in Fig. 2. The main emphasis of this paper lies in the computation of these barriers and barrier crossing rates by the various methods described. The present theory is especially relevant to determining the lower size limit for particle detection in condensation particle counters. Recently it has been shown that particles can take on liquid at sizes significantly smaller than predicted from the Kelvin relation in the no-barrier case (Winkler et al., 2008). This implies a detection principle based on the activated nucleation model that is the focus of the present study. Calculations are presented for water, n-propanol, and several other condensable vapors