Transport of Mass at the Nanoscale during Evaporation of Droplets: the Hertz−Knudsen Equation at the Nanoscale Marcin Zientara,* ,† Daniel Jakubczyk, † Marek Litniewski, ‡ and Robert Holyst ‡ † Institute of Physics of the Polish Academy of Sciences, al. Lotnikó w 32/46, 02-668 Warsaw, Poland ‡ Institute of Physical Chemistry of the Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland ABSTRACT: The applicability of the Hertz−Knudsen equation to the evolution of droplets at the nanoscale was investigated upon analysis of existing molecular dynamics (MD) simulations ( Holyst; et al. Phys. Rev. Lett. 2008, 100, 055701; Yaguchi; et al. J. Fluid Sci. Technol. 2010, 5, 180−191; Ishiyama; et al. Phys. Fluids 2004, 16, 2899−2906). The equation was found satisfactory for radii larger than ∼4 nm. Concepts of the Gibbs equimolecular dividing surface and the surface of tension were utilized in order to accommodate the surface phase density and temperature profiles, clearly manifesting at the nanoscale. The equimolecular dividing surface was identified as the surface of the droplet. A modification to the Tolman formula was proposed in order to describe surface tension for droplet radii smaller than ∼50 nm. We assumed that the evaporation coefficient for a system in and out of equilibrium may differ. We verified that this difference might be attributed to surface temperature change only. The empirical dependencies of the evaporation coefficient and the surface tension for a flat interface, of liquid Ar in Ar gas at equilibrium, at the nanoscale, upon temperature was taken from existing MD data. Two parametrizations of the Hertz−Knudsen equation were proposed: (i) one using the off-equilibrium condensation coefficient and the effective density and (ii) another one using the effective density and the temperature at the interface. The second parametrization leads to an approximate solution of the Hertz−Knudsen equation requiring no free parameters. Such a solution is suitable for experimental use at the nanoscale if only the temperature of the droplet (core) can be measured. ■ INTRODUCTION The ubiquitous processes of evaporation and condensation arouse continued interest. A detailed modeling of these processes, especially concerning the evaporation of droplets, is of interest, because of their role in the Earth’s ecosystem and in technology. Molecular dynamics (MD) simulations 1 provide valuable data on these processes. However, the large scale, engineering applications of MD are still far from being feasible, while it seems that the continuous-medium descriptions of evaporation/condensation can provide much valuable informa- tion even at the submicroscale (see e.g.; refs 2 and 3). In this work, we examine the extent of applicability of the (kinetic theory of gases) Hertz−Knudsen (HK) model 4,5 to nanoscale evaporation. We used available in literature MD data 6,8,9 to test our hypotheses. The HK model considers free, ballistic motion of vapor particles near the droplet surface and introduces the permeability of the gas−liquid interface to these particles in a form of the accommodation (evaporation, condensation) coefficient as an empirical parameter. The HK model utilizes the concept of geometrical droplet radius and well-localized (bulk) physical properties such as surface tension or density. It works well enough at the microscale though it may not be sufficient to describe droplet evaporation/condensation at the nanoscale where sharp boundaries are lost. The surface region is discernible at the nanoscale (see Figure 1) while it would be invisible at the microscale. If the HK model is to be adapted to the nanoscale, it is necessary to (re)define a few fundamental concepts: where the surface of the droplet is located, where the surface tension is exerted, which values from the density and temperature profiles should be used in transport equations, and so forth. Similarly, the applicability of the Antoine, Kelvin, and Tolman equations at the nanoscale must be verified. We used concepts of the Gibbs equimolecular dividing surface and the surface of tension (see e.g., ref 10). They were introduced by Gibbs and developed independently by Tolman and Koenig, who derived approximate formulas linking the positions of these surfaces. 11,12 The equimolecular (equimolar) dividing surface is a boundary in a hypothetical system of equal total number of molecules, at which the density changes discontinuously from liquid to vapor. 10 The surface of tension is defined as the surface for which the Laplace equation holds exactly for all droplet radii. The equimolecular dividing surface was identified as the surface of the droplet and was used as a reference surface in model investigations. It is worth noticing (see Figure 1) that the droplet temperature is practically constant below the Gibbs surface. Received: September 14, 2012 Revised: November 19, 2012 Published: December 12, 2012 Article pubs.acs.org/JPCC © 2012 American Chemical Society 1146 dx.doi.org/10.1021/jp3091478 | J. Phys. Chem. C 2013, 117, 1146−1150