The mass accommodation coefficient αmass of water vapor in NaCl solutions has been studied for realistic lower atmospheric conditions. To determine αmass, a combination of experimental data and computational fluid dynamics (CFD) modeling has been applied. Experiments were performed at the Leipzig Aerosol Cloud Interaction Simulator (LACIS), a laminar flow diffusion tube for measurements of both hygroscopic growth and cloud condensation nuclei (CCN) activation. Quasi-monodisperse sodium chloride particles with 54 nm and 108 nm in diameter have been used as condensation nuclei. To characterize particle growth, mean droplet diameters have been measured at the outlet of LACIS with a white-light optical particle spectrometer. Measurements were performed for different saturation ratios in the range between 1.0 and 1.02. Experiments have been modeled using the Computational Fluid Dynamics Code (CFD-Code) FLUENT6 combined with the Fine Particle Model (FPM). For determination of αmass, measured droplet diameters have been compared with calculated ones. The accommodation coefficient in the theoretical calculations was varied to achieve a quantitative comparison with the measurements. Experimental data shown in this study are consistent with αmass > 0.30. Therefore our results support previous studies applying different experimental techniques.
 Atmospheric aerosol particles affect the radiative balance and thus climate on Earth both directly and indirectly. The direct effect is connected with scattering and absorption of incoming radiation by the aerosol particles. On the other hand, aerosol particles act as cloud condensation nuclei (CCN). Cloud properties and precipitation are influenced by number concentration and size distribution of the cloud droplets. This indirect effect of aerosol particles leads to changes of the planetary albedo and the hydrological cycle [Lohmann and Feichter, 2005]. Because of its importance, knowledge about aerosol particle sources, formation and growth and also about CCN activation and droplet growth is important for climate modeling.
 CCN activation and droplet growth is related to the mass transfer of water vapor molecules from the gas phase to the liquid or solid phase of the droplets. Because of the large latent vaporization heat of water, heat transfer is also important during the mass transfer process.
 An important parameter controlling the vapor-liquid interactions is the accommodation coefficient of water vapor on liquid water surfaces. Thereby, mass and thermal (or heat) accommodation coefficients are distinguished. The mass accommodation coefficient (αmass) is the probability for a vapor molecule, which hits the droplet surface to be incorporated into the bulk liquid. The thermal accommodation coefficient (αheat) is the probability, with which an impinging gas molecule comes into thermal equilibrium with the liquid droplet.
 Because of the relevance regarding climate modeling, a large number of investigations with respect to mass (αmass) and heat (αheat) accommodation coefficients and also several reviews [e.g., Marek and Straub, 2001; Davidovits et al., 2006] have been published. In spite of the large number of studies, the published values for αmass varied widely and cover three orders of magnitude from about 0.001 to 1.0 [Marek and Straub, 2001]. In contrast, the values published for αheat are in better agreement and spanned only one order of magnitude from about 0.1 to 1.0 [see, e.g., Li et al., 2001; Winkler et al., 2006].
 The experiments to determine αmass (and also αheat) cover different methods such as droplet trains [e.g., Li et al., 2001] and expansion chambers [e.g., Winkler et al., 2004, 2006]. In some studies the condensation of vapor on droplets [e.g., Li et al., 2001; Winkler et al., 2006] and in others the evaporation of water from droplets [e.g., Cappa et al., 2005; Smith et al., 2006; Jakubczyk et al., 2007] was investigated. The considered environments were also different in these studies. Some experiments were made under low-pressure conditions [e.g., Li et al., 2001], while in others atmospheric pressure was considered [e.g., Jakubczyk et al., 2007]. Also the considered Knudsen number regimes and therefore the investigated mass and heat transfer processes may differ in the different studies. Two recent studies considering free molecular conditions [Cappa et al., 2005] and near continuum conditions [Jakubczyk et al., 2007] published values of αmass clearly smaller than unity, while in a study considering the transitional regime [Winkler et al., 2006] αmass was about unity. Furthermore, using different modeling approaches for describing droplet growth may result in different values for αmass [Jakubczyk et al., 2007]. Further, some of the studies revealed a temperature dependence of αmass. For example Li et al.  observed a negative temperature dependence. Their droplet train flow reactor experiments yielded values from 0.17±0.03 at 280 K up to 0.32±0.04 at 258 K. Other studies [e.g., Winkler et al., 2006], observed no temperature dependence of αmass. Finally, the considered saturation ratios were different in the different studies. To explain the differences between the published values, a recent review [Davidovits et al., 2004] argued, that the phase transition process is not the same in the different studies. They suggest that, the two-step process of gas uptake into the bulk liquid (gas-surface, surface-bulk) suggested by Kolb et al.  changed into a one-step process under fast droplet growth conditions. Already Marek and Straub  suggested a categorization of the experiments according to their timescales. They pointed out, that experiments involving growing drops with continuously renewing surfaces have a tendency to the higher end of the range (αmass > 0.1) while experiments with quasi static interfaces tend to result in lower values (αmass < 0.1, [Marek and Straub, 2001]). Anyway, the controversy is not solved so far.
 With our study we want to determine the mass accommodation coefficient αmass of water vapor on a highly diluted NaCl-water droplet for realistic lower atmospheric conditions. αmass was determined using a combined analysis of experimental data and theoretical growth curves. The method was similar to Winkler et al. . However, in contrast to Winkler et al. , a different experimental technique has been used. Instead of an expansion chamber our measurements were performed by the means of a laminar flow diffusion chamber called Leipzig Aerosol Cloud Interaction Simulator (LACIS [Stratmann et al., 2004]). The maximum saturation ratio inside the tube varied between approximately 1.00 and 1.02, while in the work by Winkler et al.  most experiments were made for saturation ratios between 1.3 and 1.5. In LACIS, the total gas pressure was about 1000 hPa and the temperature inside the tube varied between about 273 K and 295 K. As cloud condensation nuclei (CCN), NaCl particles with 50 nm and 100 nm mass equivalent diameter (in the work by Winkler et al. : 9 nm Ag particles) were used. The interaction time in LACIS was about 1.5 s (in the work by Winkler et al. : 50 μs).
2.1. Experimental Approach
 In our study, we measured droplet growth with the Leipzig Aerosol Cloud Interaction Simulator (LACIS [Stratmann et al., 2004]). The main part of the present configuration of LACIS is a laminar flow tube of 1 m length and a diameter of 15 mm (for the specifications see also Table 1). LACIS was developed to measure both, hygroscopic growth and CCN activation under realistic atmospheric conditions [Stratmann et al., 2004; Wex et al., 2005].
Table 1. LACIS Operating Parameters During the Experimentsa
The particle dry diameters are mobility-equivalent diameters. Therein, a shape factor of 1.08 was assumed [Kelly and McMurry, 1992] to convert from the mass equivalent diameters in the calculations.
Flow tube length, L
Flow tube radius, R
Operating pressure, p
Average inlet velocity,
0.4 m s−1
Inlet temperature, Te
Wall temperature, Tw
Inlet relative humidity/saturation, Se
Particle/droplet number concentration, Np
about 300 cm−3
Inlet particle dry diameter, dp,e
54 nm, 108 nm
Tube wall material
 In the following, a very short overview of the measurement system is given. The experimental system is described in more detail by Stratmann et al. . Figure 1 shows the experimental setup. The main part is the laminar flow tube. The system is operated by pushing air through it. All flow rates are adjusted using mass flow controllers (MKS 1179, MKS Instruments Deutschland GmbH, Munich, Germany). Polydisperse aerosol particles are generated from an NaCl solution using an atomizer (TSI 3075, TSI Incorporated, Shoreview, MN, USA). The generated particles pass through a drying unit, a neutralizer and then a differential mobility analyzer (DMA [Knutson and Whitby, 1975]). The DMA extracts a quasi-monodisperse fraction of aerosol particles. The particle number concentration is determined by means of a condensation particle counter (TSI 3010, TSI Incorporated, Shoreview, MN, USA). Afterward, the aerosol flow is saturated with water vapor (with saturator, MH-110-12S-4, Perma Pure, Toms River, New Jersey) at a desired temperature, controlled by means of a high-precision thermostat (HAAKE GmbH, Karlsruhe, Germany). A stream of particle-free sheath air is also saturated. At the inlet of LACIS, both air flows are combined isokinetically, i.e., the aerosol flow (2 mm in diameter) is surrounded by a sheath air flow (15 mm in diameter). The wall temperature of the flow tube is also controlled by the means of a high-precision thermostat (HAAKE, GmbH, Karlsruhe, Germany). The exiting droplets are measured and analyzed by means of an optical particle counter (white-light optical particle spectrometer [Kiselev et al., 2005]) using Mie scattering analysis.
 The operating conditions in LACIS (flow rates, temperatures, relative humidity, particle size and number concentration) can be adjusted over a wide range. Supersaturated conditions can be achieved using cooled walls with temperatures below the dew point of the aerosol and sheath air flows. Thermodynamical parameters used for the measurements presented in this study are listed in Table 1.
 Considering supersaturated conditions, temperature and saturation ratio vary inside the tube. Calculated saturation ratio profiles along the tube center of LACIS are shown in Figure 2a. In Figure 2b the corresponding profiles of the droplet diameter at the tube center are given. To show the evolving characteristics of the profiles, Figures 2a and 2b are depicted for a tube length of 1.5 m, whereas it was only 1.0 m in the experiments. Figure 2a shows that the maximum saturation is reached at about 0.4 m of the tube. Downstream of that point, saturation ratio drops down below unity again. At the end of the tube the saturation ratio profiles at different wall temperatures equal each other and converge to unity (Figure 2a).
 With decreasing wall temperature the maximum saturation ratio increases (Figure 3a). Under supersaturated conditions, the particles may activate and grow further (Figure 2b). In the second half of the tube, they shrink again. If supersaturation in the first part is large enough, activated particles will not shrink back to equilibrium state conditions (Figure 2b). The point of change between equilibrium state and dynamical growth regime can is called “critical wall temperature” and can be seen as a sharp edge in the slope of the droplet diameter versus wall temperature curve plotted in Figure 3b. The critical wall temperature can be related to the critical saturation ratio (Figure 3a).
 In our study, we investigated quasi-monodisperse sodium chloride particles with mobility-equivalent diameters of 54 nm and 108 nm (see Table 1). A shape factor of 1.08 was assumed [Kelly and McMurry, 1992] to convert from mass to mobility equivalent diameter. The experimental data were compared to theoretical growth curves. The conversion between mass and mobility equivalent diameter was necessary, because in the calculations the dry particles are assumed to be spherical.
2.2. Theoretical Approach
 Gas uptake by a liquid surface and droplet growth is a complex process, controlled by both mass and heat transfer. The process includes gas phase diffusion (mass diffusion of trace species molecules to the surface), transport across the surface (adsorption and desorption, solvation, phase transition processes), liquid phase diffusion, heat transfer and chemical reactions in the liquid phase. The process of gas uptake is coupled with fluid dynamics.
 To describe the droplet growth in the laminar flow diffusion chamber, the distributions of gas-vapor mixture velocity, temperature and vapor mass fractions have to be known. These distributions can be described by the momentum, heat and vapor mass transport equations of the gas-vapor mixture and by the equations for the particle dynamics, which are listed in the following. In these equations subscript v stands for vapor and subscript g for gas.
 Assuming steady state conditions, the momentum equations can be written as:
where u is the velocity vector, p the pressure, ρ the density, μ is the dynamic viscosity of the gas-vapor mixture, g is the vector of the gravitational acceleration and V are additional viscosity terms, which are not included in ∇ · (μ∇u).
 The energy equation for a binary mixture (here water vapor in air) is given by:
where q is the heat flux, h is the specific enthalpy, α = k/(ρcp) is the thermal diffusivity, k is the heat conductivity, cp is the specific heat capacity, M is the molar weight of the vapor-gas mixture, hv and hg are the specific enthalphies, Mv and Mg are the molar weights of the vapor and the carrier gas, respectively and Sh is the heat source (latent heat released by the vapor condensation/evaporation).
 The molecular and thermal diffusion together with convective transport of the vapor is described by the mass transport equation as:
where jv is the mass flux of vapor, T is the absolute temperature of the gas-vapor mixture, ωv is the vapor mass fraction, Dv is the binary vapor diffusion coefficient, αv,g is the thermal diffusion coefficient and Sv is the source/sink term due to condensation and evaporation onto the droplets.
 The particle dynamics due to transport (convection, diffusion, external forces, which are gravity and thermophoresis) and droplet growth (condensation/evaporation) processes are described using a moving monodisperse particle dynamics model. Therein, the total particle/droplet number is solved as:
where up is the particle velocity vector, the sum of gas-vapor mixture velocity vector (u) and particle external velocities (sedimentation and thermophoretic velocity).
 Furthermore, the particle/droplet mass concentration ρi* is described as:
where ∂mp,i/∂t is the single particle growth law. We used the equation according to Barrett and Clement , which is given by:
 In equation (6), Li is latent heat of evaporation of species i, R is the universal gas constant, ξv,i is the vapor mass fraction, Dv,i is gas phase diffusion coefficient, κg is the thermal conductivity of the gas mixture, Si is the gas phase saturation ratio. Si* is the saturation ratio above the droplet surface, described by the Köhler equation. Finally, terms fmass,i and fheat,i are semiempirical correction factors for the transition regime.
 In the growth law, the droplet temperature is included implicitly in the linearized mass flux equation. The equation is valid for slow condensation/evaporation processes near the saturation point, because steady state conditions are assumed. Considering high condensation rates, the calculated droplet diameter would be overestimated and αmass therefore underestimated, because under these conditions, the droplet temperature is underestimated.
 Furthermore, radiative energy transfer is neglected in equation (6), which is a good approximation for our application. The maximum error due to radiative effects was less than 10 nm in the final droplet size as estimated with the equation given by Barrett and Clement . The error is much smaller than the experimental uncertainties and radiation was therefore not considered.
 Local turbulence in the vicinity of the droplet, caused by condensational heat release was also neglected in equation (6). In this context it should be noted, that (1) droplet sizes considered in our study (<2 μm) were several orders of magnitude smaller than the Kolmogorov scale and (2) recent studies observed the onset of turbulence for particles larger than 14 μm [Korczyk et al., 2006].
 The Köhler equation to describe Si* in equation (6) is given by:
where aw is water activity and represents the Raoult term, σ is the droplet surface tension, ρ is droplet density and Mw = 18.0148 g/mol is the molecular weight of water. In this study, dynamical growth of activated NaCl particles is studied. These droplets can assumed to represent highly diluted solutions. Therefore the surface tension of water was used in the calculations (see Table 2). The Raoult term was described as:
where ν is the total number of ions the salt molecule dissociates into (νNaCl = 2), Ms is the molecular weight of the salt, ms and mw are the masses of salt and water in the solution, respectively and ϕs is the osmotic coefficient, which was calculated following Pruppacher and Klett .
Table 2. Physicochemical Properties of the Water-Air Systema
where α is the accommodation coefficient and A = 1, B1 = 0.377 and B2 = 4/3 are empirical constants. In the equation, Kn = 2λ/dp is the Knudsen number, where λ is the mean free path of the vapor molecules and dp is the droplet diameter.
 The theoretical description of the droplet growth in our study is similar to recent studies [e.g., Winkler et al., 2006] and makes our results comparable to those of other investigations and also applicable for larger scale of modeling applications. Furthermore, the model has been already used in a very similar form in the work by Stratmann et al. .
 The equations have been solved using the Computational Fluid Dynamics (CFD) code FLUENT6 (Fluent Inc., Lebanon, NH, USA) combined with the Fine Particle Model (FPM, particle dynamics GmbH, Leipzig, Germany). The FPM is an Eulerian model to describe the particle dynamical processes. The FPM model is integrated into FLUENT, which solves for the fluid dynamical processes.
 In the calculations, laminar flow conditions were assumed.Additionally, coagulation processes were neglected, which is a good approximation for the low particle number concentrations used inthis study. The number concentration was set to 300 #/cm3. A monodisperse particle size fraction was assumed. A sensitivity analysis applying a polydisperse particle population (with ageometric standard deviation of σ = 1.1 and 1.2) was made,but no significant differences in the results could be observed.
 Calculations were performed for dry NaCl particles with mobility sizes of 50 nm and 100 nm. Because the FPM assumes spherical particles, these diameters correspond to the chosen experimental diameters of 54 nm and 108 nm, assuming a shape factor of 1.08 (according to Kelly and McMurry ).
 The mass accommodation coefficient αmass was varied in the calculations. Values between 0.01 and 1.0 were used. As suggested in previous studies [e.g., Winkler et al., 2006], the thermal accommodation coefficient was set to both 1.0 and 0.85. With our experimental setup, we are not able to determine both, mass (αheat) and thermal (αheat) accommodation coefficient simultaneously, because the influences of both coefficients are of similar importance under the considered thermodynamical conditions (inlet temperature of the laminar flow tube: 293.15 K, pressure: 1000 hPa). The results of a sensitivity study are depicted in Figure 4. Figure 4 shows the calculated droplet diameter at the outlet of a tube of 1.0 m length dependent on the wall temperature using different values for the mass and heat accommodation coefficient. One of them was set to unity while the other was varied. It can be seen, that the influences of both coefficients cannot be separated. For example, assuming αmass = 0.1 and αheat = 1.0 yields similar results to αmass = 1.0 and αheat = 0.1. Furthermore it becomes obvious that actual values of αmass and αheat do not affect the location of the edge in the dp versus Twall curves. In other words the accommodation coefficients do not influence the critical supersaturation needed for activation.
2.3. Data Analysis
 Particle diameters have been measured at the outlet of the laminar flow tube (LACIS) using a Mie scattering analysis (white-light optical particle spectrometer [Kiselev et al., 2005]). Experiments were performed for at least three times for each set of parameters, respectively. Measurements have been carried out on different days over several weeks.
 The measured data were tested statistically. In lack of a large data set, tests like χ2 could not be used. Instead we used both, a Shapiro-Wilk Test [Shapiro and Wilk, 1965] and an Anderson-Darling Test [Stephens, 1974], which can be applied for small data sets. According to these tests, all measured data have normal distributions with a statistical significance level of 5%.
 Thus mean value and standard deviation were calculated for each data point. The experimental uncertainty was assumed as the standard deviation of the measurements. Therein, variations, e.g., in system pressure, temperature, DMA-Voltage are included.
3. Results and Discussion
 Measurement results are shown in Figure 5. There, the measured droplet diameter at the tube outlet is plotted against the tube wall temperature, which is a measure for the (maximum) saturation ratio (as shown in Figure 3a).
 With decreasing wall temperature, i.e., increasing maximum saturation ratio, the droplet size also increases. In the curves, a sharp edge is noticeable. The location of this edge represents the “critical” wall temperature, described above. Considering a wall temperature below this critical value, the droplets are activated and in the dynamic growth regime. Above the critical wall temperature, the droplets are in equilibrium.
 As described above, calculations were performed for sodium chloride particles with 50 nm and 100 nm in diameter using various mass accommodation coefficients (αmass). The results are shown in Figure 6. Here, calculated and measured droplet diameters are plotted against the wall temperature. Calculations were performed for mass accommodation coefficients of 0.01, 0.04, 0.1, 0.25, 0.5 and 1.0. The calculated curves were slightly shifted (approximately by 0.1 K) to match the critical wall temperatures observed experimentally. The slope of the theoretical curves varied dependent on the value used for αmass. Lower values of αmass yielded a reduced droplet growth and smaller droplets at the end of the flow tube. The shapes of the theoretical curves are clearly more sensitive for small αmass and become almost insensitive for values of αmass greater than 0.5.
 A value for αmass was determined by comparison of experimental data (mean values and experimental uncertainties) and theoretical growth curves. Figure 6 indicates, that the mass accommodation coefficient must be in the upper range of the chosen values.
 For a more quantitative analysis the modeling results were fitted to the experimental data by adjustment of αmass. Figure 7 shows the calculated droplet diameter versus αmass exemplarily for one wall temperature. The theoretically determined dependence of droplet diameter on αmass was parameterized by a series of exponential functions. Comparing measured (mean values and mean standard deviation) and parameterized droplet diameters as shown in Figure 7 yields the theoretical αmass and its corresponding uncertainties, which describes the experimental data most accurately.
 The obtained accommodation coefficients are depicted in Figure 8. To distinguish between the data, the results are shown as function of Twall in LACIS. Considering wall temperatures below and above the critical value, the droplet size is no or only a weak function of αmass (see Figure 6). Therefore only temperatures below 276.25 K are considered in Figure 8a (276.0 K in Figure 8b).
 Considering the temperature range below the critical wall temperatures and assuming a thermal accommodation coefficient of 1.0 (or 0.85, according to Winkler et al. ), the mass accommodation coefficients are close to unity, but because of the experimental uncertainty, the lower limit of αmass is about 0.3.
 The relative large error bars in Figure 8 are caused by the reduced sensitivity of our experimental setup against variations in the upper range of αmass (see Figure 4). The determined lower limit is raised slightly from 0.27 to 0.31 (100 nm, 295.95 K) assuming a thermal accommodation coefficient of unity instead of 0.85.
 Because of the small considered range of Twall compared to the large temperature gradient in the tube (about 20 K), we are not able to give any statements about a temperature dependence of αmass.
 The mass accommodation coefficient of water vapor on highly diluted sodium chloride solution droplet has been determined under realistic lower atmospheric by the comparison of measured and calculated droplet growth data. Following Winkler et al. , calculations have been performed for two specific thermal accommodation coefficients of 0.85 and 1.00.
 Experimental data presented in this study result in values of αmass close to unity. Because of the experimental uncertainty, the lower limit of αmass was about 0.3. Thus our results agree well with recent studies by Winkler et al.  and Li et al. . The experimental uncertainty was slightly larger than in the study by Winkler et al. . However, in our study we considered saturation ratios and droplet growth times, which are closer to real lower atmospheric conditions than in the study by Winkler et al. . Therefore we cannot support the suggestion by Davidovits et al.  that the value for αmass given by Winkler et al.  is maybe overestimated because of the rapid droplet growth rate at the large supersaturation ratios considered in their experiments. In conclusion, our results support the suggestion by several recent studies [e.g., Kulmala and Wagner, 2001; Winkler et al., 2006] that a value of αmass = 1.0 should be used in cloud modeling.