In this section we present a parameterization for the water coverage on iron silicate particles under mesospheric conditions. Traditionally, the amount of water vapour adsorbed on a surface is described using adsorption isotherms where the water coverage is plotted as a function of the saturation ratio (e.g. Venables et al., 1984). The water coverage Θ is defined as the number of adsorbed water monolayers and is calculated using the wet and dry particle radius as Θ=(rwet-rdry)/dML. For dML we use the average distance of water molecules in the condensed state dML=(mH2O/ρ)1/3 with molecular mass mH2O and the density of ice (ρ=930 kg m-3). For hydrophilic materials such as iron oxides and silica it is known that multilayer adsorption occurs (Mazeina and Navrotsky, 2007; Navrotsky et al., 2008; Sneh et al., 1996).

We measured the mass of adsorbed water vapour in the equilibrium regime for iron oxide, silica, and mixed iron silicate particles (rdry=1.05–3.05 nm) covering the temperature range between 128 and 152 K in a total of 192 experiments. We converted the measured adsorbed mass to H2O coverages and plot the results in Fig. 2 as a function of SASW. The coverage on iron oxide particles is typically higher compared to iron silicate and silica particles which is in line with measurements showing that iron oxide exhibits a higher desorption energy than silica (e.g. Mazeina and Navrotsky, 2007; Sneh et al., 1996). We observe, however, that the data do not show the typical multilayer behaviour where coverage increases with saturation. This is a result of the curvature effect, which strongly influences the equilibrium coverage for nanometre-sized particles. Instead, we find that the adsorption model that we used recently (Nachbar et al., 2018a) to describe the water affinity of iron silicate particles describes the water coverage more adequately. In this work we modify this model to account for the influence of the curvature on the equilibrium coverage, which has not been done before.

The model was originally used to describe the surface concentration of adsorbed monomers at sub-monolayer coverage on a planar surface (Pruppacher and Klett, 2004). It assumes a homogeneous particle surface in which all surface binding sites for adsorbent molecules are characterized by the same surface desorption energy Edes0. The equilibrium between desorbing water molecule flux density jdes and adsorbing flux density jads yields n⋅vth4︸jads=c⋅f⋅exp⁡-Edes0RT⋅SK(rdry)︸jdes. Here, n and vth are the number density and the mean thermal velocity of gas phase water molecules, respectively. The number of adsorbed water molecules is described by c=mads/(mH2O⋅Adry) with the surface area of the dry particle Adry=4πrdry2. The other parameters are the vibrational frequency f=1013 Hz for H2O (Pruppacher and Klett, 2004), the universal gas constant R, and particle temperature T. Compared to our previous work (Nachbar et al., 2018a), we added a Kelvin-effect-like term, SK=exp⁡2σMRTρrdry, to the flux density of desorbing molecules in order to take the increased desorption due to the curvature effect into account. Here, M is the molar mass of water, ρ is the density of the adsorbed water film which we assume is similar to the density of ice, and σ is the interfacial tension between the water film and air. We do not take into account the influence of the particle charge and of the collision radius of water molecules on the equilibrium saturation. Both effects are small compared to the Kelvin term and would make the parameterization unnecessarily complicated. Additionally, they would render Eq. (3) analytically unsolvable. By neglecting these effects we can re-arrange the equation to yield the following: RT⋅ln4cfnvth︸≡Edes=Edes0-2σMρ⋅1rdry. In Fig. 3 we plot the curvature-dependent desorption energy Edes as defined by the left-hand side of Eq. (5) versus the inverse of the dry particle radius using measured data for the adsorbed water vapour mass mads. From the intercept the curvature-independent desorption energy can be obtained while the slope is directly proportional to the surface tension of the water ad-layer–air interface. The figure shows that the three particle materials exhibit different desorption energies and that the water molecules are on average less strongly bound to the particle surface on smaller particles due to the curvature effect.

In order to determine the surface tension of the water ad-layer we performed independent linear fits on the three data sets which are shown as dashed lines in Fig. 3. The determined surface tensions are σ (H2O-on-Fe2O3) = (87±5) mN m-1, σ (H2O-on-Fe0.65Si0.35O3) = (99±13) mN m-1, and σ (H2O-on-SiO2) = (53±18) mN m-1. To keep the intended parameterization for the water coverage as a function of the particle material as simple as possible, we refrained from using a material-dependent surface tension of the water ad-layer. We performed a combined fit with a single value for the surface tension for all three materials. The resulting fits yielded σ=(94±11) mN m-1, which compares very well to extrapolated values of the surface tension of supercooled liquid water (SLW) at these temperatures (σSLW (T=155 K) = 92 mN m-1 and σSLW (T=128 K) = 96 mN m-1) and which we recently proposed to use for lack of available surface tension data of ASW (Nachbar et al., 2018c). The combined fit is shown as solid lines in Fig. 3. From the intercepts of the linear fits we determined the curvature-independent desorption energies Edes0 (Fe2O3) = (42.5±0.3) kJ mol-1, Edes0 (Fe0.65Si0.35O3) = (42.1±0.2) kJ mol-1, and Edes0 (SiO2) = (40.8±0.4) kJ mol-1. We also re-analysed the adsorption data for mixed iron silicate particles published in Nachbar et al. (2018a) using Eq. (5) and σ=94 mN m-1. The resulting curvature-independent desorption energies are shown in Fig. 4. Here, the three labelled data points are the intercepts from Fig. 3 while all other data points represent single measurements analysed using Eq. (5).

The average curvature-independent desorption energies follow the following linear relation: Edes0[kJmol-1]=40.8±0.1+(1.899±0.103)⋅X, where X=Fe/(Fe+Si) represents the relative iron content. Inserting this in Eq. (3), we arrive at a parameterization for the water coverage for iron silicate particles at mesopause conditions, mads=AdrymH2On⋅vth4f⋅expEdes0RT-2σMRTρrdry, which can be used to predict the mass of adsorbed water for iron silicate particles as a function of the dry particle radius, temperature and water vapour concentration. The wet-particle radius rwet and the water coverage Θ can be calculated from mads using rwet3=rdry3+34πmadsρ;Θ=rwet-rdrydML.

In Fig. 5 we show the ratio of calculated to measured water coverages for all three materials. The average calculated water coverage deviates by about 7 % from the measurements (solid line) with a mean square deviation of about 22 % (dashed lines). The comparison shows that the measured water coverage is well represented by our water adsorption parameterization for particles between r=1–3 nm and temperatures between 128 and 152 K. Applied water vapour partial pressures range between about 2×10-7 and 10-4 Pa.

We have measured the critical saturation for the activation of ice growth on small iron silicate nanoparticles (rdry=1.05–2.8 nm) in the temperature interval between 128 and 147 K for three different particle compositions in a total of 53 independent experiments. Figure 6a shows the measured critical saturations as a function of the initial dry particle radius for pure iron oxide (Fe2O3) particles at four different temperatures. Note that here we plot the saturation ratio with respect to the vapour pressure of hexagonal ice to facilitate comparison with other studies. Figure 6b compares the results at 139.6 K with measurements on silica and one mixed iron silicate with an elemental ratio Fe/(Fe+Si)=0.65. Note that in all measurements ice growth is activated below Sh=50, which compares to observations of saturation ratios exceeding Sh=100 when temperatures drop below 140 K on a regular basis during NLC season (Lübken et al., 2009). The solid lines in Fig. 6 are the results of an adsorption-activation model which we will present below.

We have shown previously that ASW is the initial form of condensed water that deposits on hydrophilic iron silicate nanoparticles at temperatures below 160 K (Nachbar et al., 2018c). It is well known that, depending on growth conditions, microporous ASW may be formed and that background gas molecules may be trapped during deposition. However, ASW samples produced above 100 K in the absence of a background gas were found to be compact or exhibited the same surface area as crystalline ice reference samples indicating the formation of compact ASW (Mayer and Pletzer, 1986; Stevenson et al., 1999; Westley et al., 1998). We thus argue that at the experimental conditions employed in our study (surface temperature of T≥128 K, maximum deposition rate 3Å s-1, or ∼1 ML s-1, Helium as a background gas) compact ASW is created on the nanoparticles with at maximum a marginal degree of porosity. Here, we will rely on a simple approach to describe ice formation in the mesopause by assuming that the nuclei are spherical and perfectly wettable (contact parameter close to 1, i.e. the activation barrier to form ASW is low). Under such conditions, and taking into account that the ice particle is charged, ice formation is likely to occur near the equilibrium saturation which is given by the Kelvin–Thomson equation for the particle radius including the adsorbed water rwet: lnSKT=MRTρ2σrwet-Q232π2ε0rwet41-1εr, where Q is the particle charge, and ε0 and εr are the permittivity of vacuum and the relative permittivity of water, respectively. Above we showed that the MSP analogues are already covered with more than one monolayer of water at saturations below the ice-activation threshold. This water film is well described using bulk properties for the surface tension of ASW. For the surface tension of ASW we use σASW = (114.8–0.144⋅T) mN m-1 (where T is measured in kelvins, K), a parameterization which is based on a low-temperature extrapolation of measured data for supercooled water (Vinš et al., 2017). Our previous study on the vapour pressure of compact ASW indicates that this parameterization is consistent with the properties of ASW to within 10 % at the investigated temperatures between 133 and 147 K (Nachbar et al., 2018c). For the density of compact ASW we use a constant value of ρASW=0.93 g cm-3 (Brown et al., 1996; Loerting et al., 2011).

In the derivation of the Kelvin–Thomson equation, the vapour in equilibrium with the condensed phase is assumed to be an ideal gas. This also includes the assumption that the gas phase molecules are point-like. This assumption breaks down when the size of the particle becomes comparable to the size of the water molecules. When taking the size of water molecules into account using a hard sphere collision model the equilibrium saturation changes to the following: SKT∗=SKT⋅1+rH2Orwet-2. The second term on the right-hand side represents the correction due to the finite size of the water molecules for which we use rH2O=1.5 Å (Bickes et al., 1975). The right-hand side of Eq. (10) reduces to the Kelvin–Thomson term for rH2O/r≪1. In our adsorption-activation model the onset conditions for ice growth are reached when the saturation in the environment of the particle surpasses the equilibrium saturation given by Eq. (10) (SASW≥SKT∗). The adsorption-activation model is illustrated in Fig. 7 for rdry=2 nm particles at T=140 K. Here, the Kelvin–Thomson radius rwet=f-1(SKT∗) represents the boundary between equilibrium and the ice growth regime (dashed–dotted line). For comparison, we also plotted the Kelvin–Thomson radius where we neglected the H2O collision radius (dashed line). Solid lines represent the wet-particle radius in the equilibrium regime according to Eqs. (7) and (8) for Fe2O3 in red and SiO2 in black. The onset conditions are defined in our model by the saturation ratio SASW for which the particle radius in the equilibrium regime intersects the Kelvin–Thomson radius.

We determined the critical saturation ratios using this method and plotted the results as solid lines in Fig. 6 for all temperatures and particle compositions shown. The model curves agree well with measured data, confirming the method as a good predictor for ice growth onset at mesopause conditions.

In Fig. 8a we plot the results of the adsorption-activation model (solid red curve) for iron oxide particles at 146.8 K. The blue dashed line was calculated using the adsorption-activation model but assuming that no water adsorbs prior to activation. The comparison of both model calculations shows that the equilibrium saturation is substantially reduced due to the adsorption of several monolayers of water near the activation threshold. Also shown as a green dashed–dotted line is the activation model assuming the formation of hexagonal ice and assuming that no water adsorbs prior to activation. These calculations represent the lowest critical saturations currently assumed in mesospheric models (e.g. Berger and Lübken, 2015; Schmidt et al., 2018). However, it is expected that the majority of MSPs in the mesopause is smaller than r=1.2 nm (Bardeen et al., 2008; Megner et al., 2008a, b; Plane et al., 2014). Ice formation on such small particles has to occur in order to explain observed ice particle concentrations in PMCs. According to our activation model, critical saturations are substantially smaller than currently assumed for particles with r<1.2 nm.

Figure 8b compares the critical saturation predicted by the activation model for neutral particles (solid curve) with singly charged particles (dashed–dotted curve). We note that the presence of several monolayers of water substantially increases the particle radius at the activation threshold which effectively decreases the influence of the particle charge. In our ice-activation model, the particle charge becomes significant only below rdry=0.6 nm. Note that such small iron oxide particles only consist of about 90 atoms. In Fig. 8b we also plot the critical saturation using modified versions of the Kelvin–Thomson equation according to Yu (2005) and Lapshin et al. (2002), which are in disagreement with our measurements below rdry=1.2 nm.