A new temperature and humidity dependent surface site density approach for deposition ice nucleation

Deposition nucleation experiments with Arizona Test Dust (ATD) as a surrogate for mineral dusts were conducted at the AIDA cloud chamber at temperatures between 220 and 250 K. The influence of the aerosol size distribution and the cooling rate on the ice nucleation efficiencies 5 was investigated. Ice nucleation active surface site (INAS) densities were calculated to quantify the ice nucleation efficiency as a function of temperature, humidity and the aerosol surface area concentration. Additionally, a contact angle parameterization according to classical nucleation theory was 10 fitted to the experimental data in order to relate the ice nucleation efficiencies to contact angle distributions. From this study it can be concluded that the INAS density formulation is a very useful tool to decribe the temperature and humidity dependent ice nucleation efficiency of ATD particles. 15 Deposition nucleation on ATD particles can be described by a temperature and relative humidity dependent INAS density function ns(T,Sice) with ns(xtherm) = 1.88×10 ·exp(0.2659 ·xtherm) [m−2] (1) where the temperature and saturation dependent function xtherm is defined as xtherm =−(T −273.2)+(Sice−1)×100 (2) with the saturation ratio with respect to ice Sice > 1 and within a temperature range between 226 and 250 K. For lower temperatures, xtherm deviates from a linear behavior with temperature and relative humidity over ice. Two different approaches for describing the time depen20 dence of deposition nucleation initiated by ATD particles are proposed. Box model estimates suggest that the time dependent contribution is only relevant for small cooling rates and low number fractions of ice-active particles.

fitted to the experimental data in order to relate the ice nucleation efficiencies to contact angle distributions. From this study it can be concluded that the INAS density formulation is a very useful tool to decribe the temperature and humidity dependent ice nucleation efficiency of ATD particles. 15 Deposition nucleation on ATD particles can be described by a temperature and relative humidity dependent INAS density function n s (T,S ice ) with n s (x therm ) = 1.88 × 10 5 · exp(0.2659 · x therm ) [m −2 ] (1) where the temperature and saturation dependent function x therm is defined as with the saturation ratio with respect to ice S ice > 1 and within a temperature range between 226 and 250 K. For lower temperatures, x therm deviates from a linear behavior with temperature and relative humidity over ice. Two different approaches for describing the time depen-

Introduction
Aerosol particles interacting with clouds have a significant influence on the global climate by impacting cloud life cycles, precipitation formation and the global radiation budget. Interaction between clouds and aerosol particles may 30 occur via the initiation of ice crystal formation within clouds. There are four heterogeneous ice nucleation modes involving insoluble aerosol particles (Vali, 1985). Immersion freezing occurs if a particle is already immersed within a cloud droplet when ice nucleation is initiated, whereas condensation nucle-35 ation happens shortly after or at the time of water condensation on the particle which acts as condensation and freezing nucleus at the same time. For deposition nucleation, water vapor is directly transformed into ice at the particle's surface. Contact freezing may occur if a particle collides with 40 a supercooled droplet. Laboratory studies and field campaigns have investigated the role of mineral dusts and single mineral species as ice nuclei in the atmosphere. Mineral dust acts as ice nucleus over a wide range of temperatures and supersaturations over ice, 45 with the most active dusts nucleating ice at approximately 260 K (Welti et al., 2009;Eastwood et al., 2009;Murray et al., 2012;Yakobi-Hancock et al., 2013). Using numerical modelling to estimate the climate impact of mineral dust through ice formation requires rela-50 tions which connect aerosol properties, thermodynamic variables and resulting ice crystal concentrations. Two different approaches are typically used to find approximations to describe the measured ice formation rates, namely a nucleation rate approach based on classical nucleation theory (also 55 called "stochastic" or "time-dependent" approach), or an iceactive surface site approach assuming a deterministic, timeindependent behavior of ice nucleation ("singular hypothesis"). Both approaches are described briefly in the following paragraphs. 60 The deterministic approach implies that for heterogeneous ice nucleation the stochasticity is masked by the influence of variable aerosol properties (Vali, 2008). The observed ice formation seems to occur instantaneously upon cooling and does not explicitly depend on time. Therefore, the deterministic approach describes ice formation as a function of temperature and -for deposition nucleation -relative humidity over ice. The proposition of active sites which seemingly nucleate ice as soon as certain thermodynamic thresholds are reached motivates the ice nucleation active surface 70 site (INAS) density concept (Fletcher, 1969;Connolly et al., 2009;Niemand et al., 2012;. The INAS density concept was applied to results from cloud chamber experiments by Connolly et al. (2009) to derive INAS densities n s for various mineral dusts. The ice nucleation active surface site density for immersion freezing is described by where ∆N is the observed ice crystal concentration at a certain temperature, N s the initial number of droplets, A the aerosol surface, and T the temperature. Note that for immersion freezing, A exclusively refers to particles being immersed within droplets. Also, this relation (Eq. 3) is only valid for a certain aerosol particle size. Equation 3 has been expanded towards a formulation which can be applied to a polydisperse aerosol population, yielding an approximate form of the ice nucleation active surface site density valid for ice fractions smaller than f ice ≈ 10% (Niemand et al., 2012) with where n ice is the observed ice crystal concentration and A aer the aerosol surface area concentration. Like the INAS density approach, classical nucleation theory formulations have already been employed in several studies investigating heterogeneous ice nucleation, e.g. in the studies by Marcolli et al. (2007); Lüönd et al. (2010); Murray et al. (2011); Wheeler and Bertram (2012); Broadley et al. (2012); Rigg et al. (2013). Classical nucleation theory is based on the premise that the ice nucleation efficiency of aerosol particles can be quantified by the contact angle θ which is a measure for the reduction of the energy barrier that impedes the formation of ice germs at the surface of aerosol particles (Pruppacher and Klett, 1997). For deposition nucleation, the nucleation rate J(θ) per particle is given by following the notation used by Chen et al. (2008) with r N being the aerosol particle radius, r g the radius of the ice germ, e the water vapor pressure, m w the mass of a water molecule, k the Boltzmann constant, T the temperature in [K], n 1 the number concentration of single molecules in contact with the aerosol surface, n g the number of water molecules per ice germ and ∆g g (θ) the energy needed for forming a critical ice germ. Note that for calculating n 1 the desorption energy ∆g d is set to an average value of ∆g d = 4 × 10 −20 J (Chen et al., 2008). The formalism used by Chen et al. (2008) explicitly considers the temperature and humidity dependence of n 1 and r g with and The surface tension σ i/v is described as a temperature dependent function according to Pruppacher and Klett (1997). The activation energy ∆g g (θ) is given by where σ i/v is the surface tension at the ice/vapor interface and f (θ) = 0.25 · (2 + cos(θ))(1 − cos(θ)) 2 is the so-called form factor with θ formally being the contact angle between particle surface and the ice germ. Physically, the form factor f (θ) which reduces the activation energy can be taken as a measure for the ice nucleation efficiency. Several studies have pointed out that often a single contact angle is not sufficient to characterize the ice nucleation behavior of a non-homogeneous aerosol population (Marcolli et al., 2007;Lüönd et al., 2010;Wheeler and Bertram, 2012;Broadley et al., 2012;Rigg et al., 2013). Thus, the nucleation rate approach was extended towards including not only a single contact angle but a distribution of contact angles (Marcolli et al., 2007;Lüönd et al., 2010). For this study, the distribution p(θ) is assumed to be log-normal with with µ θ being the median contact angle and σ θ the logarith-75 mic width of the contact angle distribution. Note that some parameterizations have sought to reconcile classical nucleation theory and INAS density concept because both approaches offer frameworks for fitting and parameterizing experimental data while they treat the inherent 80 time dependence of ice nucleation differently (Vali, 1994;Niedermeier et al., 2011). However, in this study only the INAS density approach and classical nucleation theory will be compared to each other.
Besides the INAS density approach and classical nucleation theory which can be used to describe the ice nucleation efficiencies of certain well-defined aerosol species, there are also parameterizations which have been derived for either unidentified aerosols or certain subgroups of the aerosol population. Meyers et al. (1992) used laboratory data from diffusion chamber experiments to derive a saturation dependent relation for immersion freezing and deposition nucleation. The ice crystal concentration c IN [L −1 ] is described by which is valid for temperatures between 253 and 266 K and only depends on the supersaturation over ice S ice −1. The parameterization developed by Phillips et al. (2008Phillips et al. ( , 2012 links aerosol properties and ice crystal concentration in a more direct way by explicitly including the aerosol surface area and aerosol-specific freezing thresholds. The contribution of mineral dusts and metallic compounds to atmospheric ice nuclei (c IN,DM ) is given by where µ DM (D DM ,S ice ,T ) is the average number of activated 85 ice embryos per aerosol particle. µ DM (D DM ,S i ,T ) is defined in Phillips et al. (2008) as a function of aerosol diameter D DM , temperature T in [ • C] and the saturation over ice, S ice . n DM is the number mixing ratio of aerosol particles belonging to the dust/metallic compounds group, given in [kg −1 ] of 90 air. The approaches that are described in this section can all be used to develop ice nucleation parameterizations. For immersion freezing, several studies have investigated the performance of different approaches regarding the description of 95 ice nucleation efficiencies Murray et al., 2012). For deposition nucleation, only very few studies have compared different parameterizations, e.g. Wheeler and Bertram (2012). In this study deposition nucleation experiments conducted at the Aerosol Interaction and Dynamics in 100 the Atmosphere cloud chamber (AIDA, Karlsruhe Institute of Technology) are presented and accompanied by a comparison of the INAS density approach with classical nucleation theory.
The manuscript is organized as follows: the instrumenta-105 tion used at the AIDA cloud chamber and a typical deposition nucleation experiment are described in Sect. 2. In Sect. 3, the experimental results are presented, starting with ice-active fractions and ice nucleation active surface site densities. The impact of temperature, aerosol particle size and cooling rates 110 on the observed deposition nucleation efficiency was investigated.
In this work, Arizona Test Dust (ATD, Powder Technology Inc.) is used as a substitute for naturally occuring desert dusts. ATD consists of desert dust that was washed, dried 115 and milled in order to provide enough material in all size classes. Thus, the composition of individual ATD particles is probably more homogeneous than the composition of original desert dusts and also the surface properties may differ from natural dusts.

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Several sets of experimental runs were conducted, starting at approximately 250 K, 235 K or 223 K. In order to investigate the impact of time dependence and variations in the aerosol size distribution on the deposition nucleation efficiency of ATD, the experimental cooling rate was varied 125 between 0.3 and 2.9 Kmin −1 and also the aerosol size distribution was varied by either including or discarding particles larger than about 1 µm.
In Sect. 3, ice nucleation thresholds, INAS densities and contact angle distribution parameters as derived from the 130 experimental data are presented. Additionally, an average INAS density function is derived and compared to two empirical parameterizations (Meyers et al., 1992;Phillips et al., 2008) with regard to their sensitivity to temperature and relative humidity over ice.

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In the last part of Sect. 3, the relevance of time dependence for deposition nucleation initiated by ATD particles is investigated by using either a linear time dependent source term or a time dependent exponential function in addition to the formerly time-independent average INAS density relation. The

Experimental methods
The experiments presented in this study were conducted at the AIDA cloud chamber facility, located at the Karlsruhe Institute of Technology. With the AIDA cloud chamber, the 150 ascent of air parcels can be simulated by expanding moist air within the chamber vessel. Thus, the ice nucleation properties of various aerosol types can be investigated under atmospherically relevant conditions for mixed-phase and cirrus clouds.
155 Figure 1 shows a schematic drawing of the AIDA cloud simulation chamber: the cloud chamber itself is enclosed by a thermally isolated housing. With two pumps the chamber volume can be expanded at controllable rates corresponding to defined cooling rates. Background aerosol concentrations 160 within the cloud chamber were typically below 0.1 cm −3 .
On the left side of Fig. 1, the aerosol instrumentation is shown. A rotating brush generator (RBG 1000, Palas) is used for dry dispersion of the dust samples. Additionally, cyclone impactors are generally used to eliminate particles 165 larger than about 1 µm. Aerosol number concentrations are measured with a condensation particle counter (CPC3010, TSI) whereas the aerosol size distribution was measured by combining SMPS (Scanning Mobility Particle Sizer -TSI) and APS (Aerodynamical Particle Sizer -TSI) measurements. From this data, the total aerosol surface area concentration can be inferred by translating the size distribution into a surface distribution after converting mobility and aerodynamic diameters into equivalent sphere diameters (Möhler et al., 2008). To this surface distribution a lognormal fit is applied from which the total aerosol surface area concentration can be estimated through integrating the distribution. An exemplary aerosol surface distribution is shown in Fig. 2. Note that APS and SMPS data in combination cover the whole size range.

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Values for the relative humidity over ice (RH ice ) and over water (RH water ) are derived from tunable diode laser (TDL) absorption spectroscopy measurements. Infrared absorption is measured at a wavelength around λ = 1.37 µm and converted into water vapor concentrations with an accuracy of ±5% (Fahey et al., 2014). From these water vapor concentration values, the relative humidities RH water and RH ice are calculated using the water vapor saturation pressures over liquid water and ice (Murphy and Koop, 2005) and measurements of the gas temperature in the cloud chamber. The total wa-190 ter content is also measured by a chilled-mirror hygrometer. For the deposition nucleation experiments, however, only the TDL measurements were considered.
The AIDA cloud chamber is also equipped with several optical instruments (Wagner et al., 2009) -three of these 195 instruments (welas/welas2 and SIMONE (Scattering Intensity Measurements for the Optical Detection of Ice)) are also sketched in Fig. 1. Ice crystal concentrations are derived from the particle concentrations and size distributions measured with two optical particle counters (welas and welas2, 200 Palas GmbH). Particle sizes are calculated from the intensity of light scattered by particles crossing the beam of an internal white light source. Note that aerosol particles, droplets and ice crystals are detected alike if they are large enough to scatter sufficient light into the detector, but only ice crys-205 tals grow rapidly to sizes which eventually exceed those of the initial aerosol particles. Droplet formation is expected to be negligible during the experimental runs presented in this work because ice nucleation was only investigated at condition subsaturated with respect to liquid water and the 210 amount of soluble components is expected to be very small (Vlasenko et al., 2005). The distinction between aerosol particles and ice crystals is made by selecting a suitable size threshold. The formation of small ice crystals is also indicated by the change in depolarization detected by SIMONE 215 (Scattering Intensity Measurements for the Optical Detection of Ice) (Schnaiter et al., 2012). SIMONE is used for observing scattering signals from particles crossing the pathway of a polarized laser beam (λ = 488 nm) which horizontally traverses the cloud chamber. Besides scattering in forward (at 220 2 • ) and near-backward (at 178 • ) direction, the depolarization is measured using a Glan laser prism to separate the parallel and the perpendicular polarized components of the near-backward scattered light.
The course of a typical AIDA expansion experiment is de-225 picted in Fig. 3 and briefly described in the following paragraphs. The first panel shows the pressure p which decreases during an expansion run while the gas temperature T g within the vessel drops simultaneously. During this expansion experiment, the pressure p within the AIDA vessel is low-230 ered from ambient pressure to approximately 800 mbar. The starting temperature was 235 K whereas the minimum temperature was about 226 K. Over the course of an expansion experiment, the temperature T w at the chamber walls remains virtually unchanged. Panel b depicts the relative humidity 235 values (RH water and RH ice ) as derived from the TDL measurements. Water saturation is not reached during this experiment. Therefore, neither significant droplet activation nor immersion freezing can occur: ice crystals form almost completely by deposition nucleation. The peak relative humidity 240 over ice was about 118%. Figure 3c shows the forward-to-backward scattering ratio as derived from the SIMONE scattering signals alongside with the depolarization signal measured for the backward scattered light. The ice nucleation onset with the for-245 mation of small ice crystals is indicated by an increase in depolarization as well as a slightly delayed increase of the forward-to-backward scattering ratio. The increase in depolarization is a further indication that only deposition nucleation was observed because the formation of spherical 250 droplets leads to a clear decrease in the depolarization signal. The last panel in Fig. 3 shows the aerosol concentration (measured by CPC3010) and the ice crystal concentrations (derived from welas/welas2 data). The aerosol concentration given in [cm −3 ] is decreasing over the course of the 255 experiment due to the volume expansion. The ice crystal concentration as derived from the welas/welas2 data shows a steep onset at approximately RH ice = 103%. The maximum fraction of ice-active particles observed during this experiment was f ice = 40%. Note that for the calculation of 260 the ice nucleation active surface site densities only ice fractions f ice < 10% were considered. Initially, the growing ice particles deplete the vapor phase only negligibly and relative humidity over ice is an almost linear function of temperature. For the experiments starting at 250 K, the cooling rate was varied between 0.3 and 2.7 K min −1 . The variations of the aerosol surface area concentration during these experiments I. Steinke et al.: Describing deposition ice nucleation by an active site density approach 5 were achieved by varying the aerosol number concentration. In addition to varying the cooling rate between 0.7 and 280 2.9 K min −1 and to systematically changing the aerosol number concentration, two experiments starting at about 235 K (exps. 14 and 15) were conducted without using cyclone impactor stages which results in a shift of the aerosol size distribution towards larger particles. The ice nucleation effi-285 ciency was also investigated at colder temperatures, i.e. for expansion runs starting at approximately 223 K.

Experimental results
The deposition nucleation experiments described in Table 1 are used to derive different measures for the ice nucleation 290 efficiencies. In particular, humidity thresholds at ice nucleation onset, INAS densities and contact angle distribution parameters were analyzed.

Ice nucleation properties of ATD
3.1.1 Thermodynamic ice nucleation thresholds 295 Figure 4 shows trajectories in the T /RH ice space for all AIDA expansion experiments listed in Table 1. Also, the temperature and humidity conditions at which an ice fraction f ice = 1% was observed are represented. All trajectories in Fig. 4 start shortly after ice formation was observed and 300 end when ice crystal growth leads to a deviation from the initially linear increase in RH ice . Note that all experimental runs began at initially subsaturated conditions with respect to ice. For the experiments starting at about 250 K, ice nucleation occurs for relative humidity values between 112 and 305 125%, whereas for temperatures below 235 K ice nucleation is already observed slightly above saturation with respect to ice. From Fig. 4 it can also be observed that trajectories for experiments starting below 235 K are more similar to each other than those of the experiments at warmer temperatures.

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The relative humidity values (RH ice ), for which an ice number fraction f ice = 1% was observed, are considered as ice nucleation thresholds in this study. These ice nucleation thresholds are depicted in Fig. 4 for all experiments. For the experiments starting at 250 K, the ice nucleation thresholds 315 scatter around RH ice ≈ 120%. Note that for two experiments, the ice fraction remained below f ice = 1%. The humidity threshold values suggest that warm temperature deposition nucleation does not depend primarily on the cooling rate. At lower temperatures (T start ≈ 235 K and T start ≈ 223 K), the ice 320 nucleation thresholds mostly scatter around RH ice ≈ 104%. Only the two experiments which investigated the influence of larger particles (exps. 14 and 15) are characterized by ice nucleation starting already slightly above saturation with respect to ice. This finding agrees with other studies finding 325 that larger particles lower the observed ice nucleation thresholds (e.g Welti et al., 2009).
It should be noted that the spread of the observed humidity threshold values -considering experiments with a similar starting temperature -lies within the measurement uncer-330 tainty ∆RH ice = ±(3-5)%. Only for experiments including larger particles a shift towards lower ice nucleation thresholds is observed. Therefore, deposition nucleation seems to be only weakly time dependent over the range of variations in cooling rate and aerosol surface area concentrations 335 investigated in this study. If ice nucleation had to be described by a time dependent heterogeneous nucleation rate approach, the freezing thresholds would have been shifted towards lower relative humidities for low cooling rates. Because neither a completely singular behavior (i.e. always 340 the same ice nucleation threshold) nor a relation between cooling rate and thresholds could be deduced from our measurements, it is not possible to directly infer the influence of different cooling rates (corresponding to ice nucleation time scales) on the observed ice fraction. Therefore, the impact 345 of time dependence will be investigated in more detail in the following subsections. Figure 4 also shows that the ice nucleation thresholds are clearly divided into two groups depending on the ambient temperature, with higher humidity thresholds at T start ≈ 350 250 K and lower ice nucleation thresholds for the experiments at colder temperatures (T start ≈ 235 K and T start ≈ 223 K). Therefore, it can be also concluded that the deposition nucleation efficiency of ATD particles depends not only on relative humidity, but also on temperature. The ice nucleation efficiency can also be expressed as the INAS density averaged over the whole aerosol population for each experiment. The INAS density values (Niemand et al., 2012) are calculated from with the ice crystal concentration n ice [cm −3 ] and the total aerosol surface area concentration A aer [µm 2 cm −3 ]. Note that n s can also be interpreted as a way of normalizing ice crystal concentrations.

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The INAS densities are depicted in Fig. 5 with respect to RH ice (left) or with respect to a function x therm (right) which is defined as Note that in Eq. (13), T represents the numerical value of the average temperature within the cloud chamber in [K] and is therefore dimensionless. S ice corresponds to the ice saturation ratio. Eq. (13) can be understood as a very generic and simple way to describe the combined dependence of deposition nucleation on temperature and relative humidity over ice within a certain range of thermodynamic conditions. More general formulations of Eq. (13) would read or (15) with α(T ) and β(T ) being temperature dependent weighting coefficients. However, the improvement of fits relying on Eqs. (14) or (15)  suggest that different x therm or orther approaches might be needed within different temperature regimes. Also, deposition nucleation close to water saturation may coincide with pore condensation freezing (Marcolli et al., 2014). In Fig. 5 (left) the two groups of experiments starting ei-375 ther at 235 K/223 K or 250 K are clearly separated. Thus, in agreement with the behavior of the ice nucleation thresholds, Fig. 5 (left) confirms that within the temperature range between 223 and 250 K, deposition nucleation as a process does not only depend on RH ice but it is also strongly con-380 trolled by temperature. Also, experiments including larger particles (dashed lines in Fig. 5) are characterized by similar INAS densities as the experiments targeting a narrow particle size distribution. Therefore, within this experimental setup, aerosol particle size does not impact the observed INAS den-385 sity values much. This finding supports the concept of a surface area related density of ice nucleation sites. By representing the INAS densities as a function of relative humidity and temperature (Fig. 5, right), the INAS trajectories representing warm temperature deposition nucle-390 ation fall much closer together which means that deposition nucleation can be described by the change in x therm as defined by Eq. (13). Note that the length of each n s trajectory generally corresponds to a time period of ∆t ≤ 25 s starting at the first observation of ice nucleation. For experiments 395 during which the growth of ice crystals led to an early deviation from the linear increase of relative humidity over ice, this time interval ∆t was chosen to be shorter than 25 s in order to minimize systematic errors of the measured ice crystal concentrations caused by settling of ice crystals. The time 400 interval ∆t was defined with regard to excluding reductions of the observed ice crystal concentration by sedimentation, assuming that the largest ice crystals grow to approximately 100 µm. These large crystals determine the sedimentation time scale and sediment with terminal velocities between 0.1 405 and 10 cms −1 (Westbrook, 2008). This corresponds to sedimentation times between 35 and 3500 s for an average fall distance of 3.5 m (half of the cloud chamber height). Thus, as a conservative estimate the time scale was chosen to be ∆t = 25 s since the maximum dimensions of the observed 410 ice crystals were not measured directly.
The n s trajectories as shown in Fig. 5 are afflicted with two sources of uncertainty of which the n s values themselves are the first source. The measurement uncertainty of n s is determined by the uncertainties of n ice and A aer with 415 ∆n ice /n ice ≈ 25% and ∆A aer /A aer ≈ 25% which results in ∆n s /n s ≈ 35%. Secondly, the position of each trajectory within the T /RH ice space is affected by the uncertainties ∆T = ±0.3 K and ∆RH ice up to 5%. These uncertainties then translate into an uncertainty of the thermodynamic vari-420 able x therm with ∆x therm ≈ 5. Figure 5 shows that the experiments at higher relative humidities over ice (corresponding to warmer starting temperatures at about 250 K) are characterized by a much larger variation in the slopes of INAS density trajectories than the experiments at lower relative humidities 425 over ice (corresponding to colder temperatures).

Ice nucleation active surface site density approach and comparison to other parameterizations
In this section, we will first present an overall INAS density fit to all measurements above 226 K. This means that 430 the two measurements starting at 223 K will be excluded. The average INAS density function is then compared to the dust-adapted parameterization by Phillips et al. (2012) and the parameterization by Meyers et al. (1992) which does not distinguish between different aerosol species. Complement-435 ing the INAS density approach, also results from fitting nucleation rates according to classical nucleation theory to the measured ice fractions are presented. Additionally, in the last subsection, the time dependence of deposition nucleation initiated by ATD particles is expressed either as a linear source 440 term or a time dependent exponential function. Figure 5 shows that the n s values observed for temperatures above 226 K do not diverge by more than one order of magnitude which suggests that the INAS density values may be described by an average n s function. According to least-square fitting, all measurements above 226 K can be described by the fit function n s (x therm ) = 1.88 × 10 5 · exp(0.2659 · x therm ) [m −2 ] (16)

General ice nucleation active surface density approach
The measurements together with the fit (r 2 = 0.49) are depicted in Fig. 6. Note that the quality of the fit only slightly improves by defining x therm as  Table 1. Isolines with constant INAS density values indicate the increase of the fit function n s (x therm ) with supercooling and relative humidity over ice. The measurement uncertainties are given by ∆T ± 0.3 K, ∆RH ice up to 5% and ∆n s /n s ≈ 35%.
For comparison, n s values from other experimental studies (see references) investigating the ice nucleation properties of ATD in the deposition nucleation mode are shown. Note that the experimental setups which were used to derive 455 the INAS densities differ among these studies. INAS densities calculated for previous AIDA cloud chamber experiments with ATD agree well with n s (x therm ) from Eq. (16) (Möhler et al., 2006).
INAS densities were also derived from ice fractions f ice observed in studies investigating the deposition nucleation mode properties of monodisperse ATD particles (Koehler et al., 2010;Sullivan et al., 2010;Welti et al., 2009) with where d is the diameter of the size selected ATD particles.

460
The particle size selection in the aforementioned studies was achieved by using differential mobility analyzers (DMA). Note that in Fig. 7 all studies, the ATD sample was dispersed by either using a rotating brush generator or a fluidized bed generator. The INAS density values derived from the aforementioned studies are much lower than the INAS densities derived within this experimental study. These deviations might be partially 480 explained by differences in the temperature and humidity profiles compared to the AIDA experiments. Figure 8 shows a comparison between the INAS densities from the n s (x therm ) parameterization (Eq. 16), the ice for-485 mation as parameterized by Phillips et al. (2008Phillips et al. ( , 2012 and the ice crystal concentration derived by using the purely humidity dependent parameterization proposed by Meyers et al. (1992). For our calculations we assume that the ice was formed on a generic aerosol population with an aerosol sur-490 face area concentration of A aer = 2 × 10 −6 m 2 m −3 as pro-posed in Phillips et al. (2012). The grey dashed line in Fig. 8 indicates the upper limit of observed ice nucleation active surface site densities in this study (f ice < 10%). The INAS density lines as shown in Fig. 8 are also restricted by depo-495 sition nucleation to occur only below water saturation. Note that for this comparison not the absolute INAS density values are considered to be most relevant but the slopes of the n s curves, because the absolute values also depend on the assumed aerosol surface area concentration A aer . Neverthe-500 less, in a recent immersion freezing study ice crystal concentrations derived from an INAS density parameterization based on cloud chamber experiments with desert dusts were observed to differ by more than one order of magnitude from estimates made with the Phillips parameterization for immer-505 sion freezing at temperatures above 255 K (Niemand et al., 2012). For deposition nucleation, the parameterizations by Phillips et al. (2012) and Meyers et al. (1992) predict INAS densities with significantly smaller slopes (i.e. humidity de-510 pendence) compared to the results from our ATD measurements. Additionally, the temperature dependence of the parameterization by Phillips et al. (2012) is weaker whereas the parameterization by Meyers et al. (1992) is a priori not considering any changes in supercooling. Applied in climate 515 models, paramaterizations describing deposition nucleation without considering the temperature dependence will predict largely deviating ice crystal concentrations in comparison to calculations based on our parameterization.

520
Classical nucleation theory can be used to fit results from deposition nucleation experiments with ATD particles. For each experimental run, the observed ice nucleation efficiency can be expressed by an apparent median contact angle µ θ and an apparent contact angle distribution width σ θ . These 525 parameters µ θ and σ θ can be derived from using Eqs. (5), (8) and (9) to fit the observed ice fractions.
For most experiments, the aerosol size distribution was assumed to be lognormal with the median diameter µ d = 0.23 µm and the geometric size distribution width σ d = 1.56.

530
Only for the experiments without cyclone impactors (i.e. larger particle being present) the aerosol size distribution parameters were chosen to be µ d = 0.35 µm and σ d = 1.73.
For the experiments starting at about 250 K, the median contact angles µ θ vary between 17 • and 48 • (excluding one 535 outlier), whereas for experimental runs starting at about 235 or 223 K, the median contact angles µ θ were found to scatter between 25 • and 39 • and between 8 • and 13 • .
The contact angle parameters derived from the ATD experiments presented in this study vary considerably between different experimental runs and also slightly depend on the thermodynamic conditions (i.e. temperature and relative humidity over ice). The nucleation rate approach with the assumption of a lognormally distributed range of contact angles did not result in a consistent set of fit parameters for the available data set.
Note that even though both T and S ice enter the classical 555 nucleation theory formulation of the nucleation rate J het , the dependence on S ice is much stronger than the dependence on T . This can be seen e.g. in Fig. A1 of  by the near-horizontal isolines of J het . The experimentally observed T and S ice dependence in this study, how-560 ever, is markedly different from the CNT prediction. More experimental studies in a wider range of temperature, aerosol surface area and cooling rate may provide a better basis for constraining the results from nucleation rate fits to measured ice formation rates. Time dependent ice nucleation may be described by where x therm is defined by Eq. (13) and t [s] is the time starting from the first observation of ice crystals, neglecting ice formation below the detection limit. For deriving the coefficients in Eq. (19) only the first 25 s after ice formation was observed are considered. The first part of Eq. (19) expressed 580 asñ s describes the formation of ice crystals caused by the "best" ice nuclei among the dust particles. Upon reaching certain thermodynamic thresholds (i.e. x therm values) these particles initiate ice nucleation immediately within the temporal resolution of our experimental setup. The linear source 585 term then describes the formation of ice by the less efficient ice nuclei components which (at comparable x therm conditions) have lower freezing probabilities and are only activated after a certain period of time. Therefore, this linear contribution will become apparent especially at low cool-590 ing rates. The coefficients in Eq. (19) are determined from least-square fitting as a 1 = 1.9 × 10 3 [m −2 ], a 2 = 0.363 and a 3 = 3.7 × 10 6 [m −2 s −1 ] (r 2 = 0.74).
A second time dependence parameterization assumes that there is a certain ice nucleation active surface site densityñ s (x therm ) towards which the measured INAS densities would converge eventually at a certain x therm value. This time dependent behavior is then described by Again, the coefficients are derived from the measurements for ice fractions smaller than f ice < 10%. The coefficients in 595 Eq. (21) are determined as b 1 = 6.1 · 10 5 [m −2 ], b 2 = 0.254 and b 3 = 0.065 [s −1 ] (r 2 = 0.70).
Note, however, that Eqs. (19) and (21) need to be viewed as very simplistic approaches. Nevertheless, these equations could be used to evaluate the time dependence of ice nucle-600 ation initiated by other particle species.

Relevance of the time dependent source term
The box model ACPIM (aerosol-cloud precipitation interaction model) which has been developed at the University of Manchester (Connolly et al., 2009) was used to calcu-605 late the ice formation within an ascending air parcel, using a prescribed ice nucleation parameterization. The ice nucleation parameterizations as described by Eqs. (16), (19) and (21) were analyzed for different updraft velocities and aerosol parameters as described in Table 2. Each parcel 610 run is initialized at cirrus cloud conditions with T = 235 K, p = 550 mbar and RH water = 68%. The parcel is then allowed to develop for t = 600 s resp. for t = 1200 s at the lowest updraft velocity. Figure 9 shows the decrease in temperature (Fig. 9a), the 615 development of relative humidity over ice (Fig. 9b), and the change in the temperature and saturation dependent function x therm as defined in Eq. (13) (Fig. 9c). The ice fractions predicted by Eq. (16) (without time dependence) for different updraft velocities are depicted as a function of time (Fig. 9d) 620 and in relation to the temperature and saturation dependent function x therm (Fig. 9e) and of temperature (Fig. 9f). For each updraft velocity value the trajectories were calculated for all aerosol parameters as described in Table 2.
-For the lowest updraft velocity (w 1 = 0.05ms −1 ), the 625 reduction in temperature is less than 1 K over the whole simulated time period. Likewise, the increase in relative humidity over ice is less than 5%. Thus, only a small supersaturation is reached. The temperature and saturation dependent function x therm increases from 630 x therm = 37 to x therm = 42. After ∆t = 1200 s, the observed ice fractions remain below 2%.
-For intermediate updraft velocities (w 2 = 0.5ms −1 ), the parcels are cooled to 232 K and reach peak relative humidity values of RH ice = 110% at high aerosol con-635 centrations and RH ice = 120% at low aerosol concentrations. The increase in x therm is strongly driven by the increase in relative humidity and thus x therm can reach peak values of x therm = 50 and x therm = 65. The observed ice fractions are strongly influenced by the 640 aerosol concentrations and vary between 2 and 70%.
-At very large updraft velocities (w 3 = 5.0ms −1 ), temperatures as low as 206 K are reached within 600 s. However, the determining factor for these simulations is the peak relative humidity which is related to the pre-645 scribed aerosol concentration. At low aerosol concentrations, all aerosol particles are activated within less than 100 s. After the ice activation process is completed, the relative humidity value increases further to values larger than RH ice = 200%. For high aerosol con-650 centrations, the conversion of all aerosol particles into ice crystals is only achieved at the end of the parcel run since the peak relative humidity (RH ice = 120%) is already reached within the first 100 s of the simulation while ice formation slows down after having reached 655 peak relative humidity.
The graphs in Fig. 9g-j show simulations similar to those depicted in Fig. 9d, e,f. However, for the simulations presented in Fig. 9g-j the ice nucleation process was assumed to be time-dependent according to Eqs. (19) and (21). Note 660 that the temperature and relative humidity trajectories are very similar to the runs without time-dependent ice nucleation ( Fig. 9a and b). Likewise, the evolution of x therm is also similar.
Comparing the predicted ice fractions at the end of the up-665 draft periods, the first time-dependent ice nucleation parameterization (Eq. 19) does not produce results deviating much from those based on Eq. (16). Only the initial increase of the observed ice fractions is steeper than for purely humidity and temperature dependent ice formation. The second time-670 dependent ice nucleation parameterization (Eq. 21) generally predicts ice-active fractions being higher than the purely x therm dependent parameterization by a factor of 2 which is largely due to the coefficient b 1 in Eq. (21). Note that the time-dependent ice nucleation parameterization described by 675 Eq. (21) predicts rapid ice nucleation at low ice-active particle fractions. The measurements shown in Fig. 5 at least partially corroborate this result. From this simple case study it can be concluded that the effect of time dependence is generally small and may only be 680 relevant at low to moderate updraft velocities and for small ice-active particle fractions.

Conclusions and discussion
Deposition nucleation on Arizona Test Dust (ATD) particles was investigated with AIDA cloud chamber experiments, fol-685 lowing expansion trajectories starting from ice-subsaturated conditions at about 250 K, 235 K or 223 K. The aerosol surface area concentrations and cooling rates were varied for all expansion experiments, because one of the goals of this experimental study was to determine the relevance of time 690 scales to the observed ice nucleation efficiencies.
The ice nucleation efficiency observed for each experimental run was quantified by the measured ice nucleation thresholds at f ice = 1%, by deriving the ice nucleation active surface site (INAS) densities and by fitting the contact angle 695 distribution parameters using nucleation rate formulations.
Ice nucleation onsets (f ice = 1%) were observed at relative humidities over ice between 118 and 121% at warmer temperatures (T start ≈ 250 K) whereas ice activation of 1% of all ATD particles occured between 101 and 107% at colder 700 temperatures (T start below 235 K). No direct relation between ice nucleation thresholds and cooling rates could be deduced from the experimental data. The time dependence of deposition nucleation was presumably small and could not be quantified from the ice nucleation thresholds. It should be noted 705 that the observed freezing thresholds could also be partly explained by a freezing mechanism other than deposition nucleation, namely pore condensation freezing. Pore condensation freezing was proposed by Marcolli et al. (2014) as an explanation for freezing below water saturation. Note, however, 710 that in our experimental setup we cannot clearly distinguish between these freezing mechanisms and thus make the assumption that ice nucleation is mostly caused by deposition nucleation.
INAS densities were derived for all experiments and were found to depend both on temperature T and the ice saturation ratio S ice with n s (x therm ) = 1.88 × 10 5 · exp(0.2659 · x therm ) [m −2 ] (23) where the temperature and saturation dependent function x therm is defined by The INAS density approach was found to be independent 715 of shifts in the particle size distribution, i.e. from shifting the median diameter from d med ≈ 0.23 µm to d med ≈ 0.35 µm. As a parameterization for numerical models, the INAS density relation is only strictly valid for temperatures between 226 and 250 K and for humidities with 1 < S ice < 1.2. Especially 720 at temperatures below 220 K, x therm may be better described by a relation different from Eq. (24). Note that an extrapolation to lower temperatures relying on Eq. (24) would also predict very high INAS densities already at S ice close to 1. To describe deposition nucleation even more precisely, x therm 725 could be parameterized as a higher-order function of temperature and relative humidity over ice in order to achieve a better match with observations, both at low temperatures above ice saturation, and at higher temperatures close to water saturation. Deposition nucleation at higher temperatures 730 should be investigated for a wider range of thermodynamic conditions in order to better characterize the dependence of x therm on temperature and relative humidity and also for natural mineral dusts which are typically less ice-active than ATD particles (Möhler et al., 2006). Ice crystal concentrations predicted by n s (x therm ) match the observed ice crystal concentrations for most experiments of this study within one order of magnitude regardless of the cooling rate or the aerosol surface area concentration.
In comparison to INAS density values derived from other 740 empirical parameterizations or laboratory studies, the ice nucleation efficiency of ATD in deposition nucleation mode as derived from AIDA cloud chamber measurements is larger by at least one order of magnitude. Note that in contrast to the parameterization derived from our measurements, the parameterizations by Phillips et al. (2012) and Meyers et al. (1992) suggest a much weaker or no temperature dependence of deposition nucleation. Applying classical nucleation theory to the observed ice fractions yields average contact angle distribution parame-750 ters. For high temperature deposition nucleation (T start ≈ 250 K) the contact angle distribution parameters which best described all experimental runs (r 2 = 0.48) were µ θ = 22.1 • and σ θ = 0.095. For deposition nucleation at lower temperatures, the contact angle parameters were found to be 755 µ θ = 36.2 • and σ θ = 0.520 (r 2 = 0.52) for experiments at T start ≈ 235 K and µ θ = 16.9 • and σ θ = 0.540 (r 2 = 0.89) for T start ≈ 220 K. The large variability of the contact angle distribution parameters suggests that the application of classical nucleation theory to deposition nucleation by certain aerosol 760 species such as mineral dust would require a detailed investigation of the contact angle distribution parameters for different thermodynamic conditions. Additionally, the contribution of pore condensation freezing to heterogeneous nucleation observed close to water saturation might lead to diffi-765 culties with applying classical nucleation theory directly.
The time dependence of deposition nucleation initiated by ATD particles was investigated by assuming that time dependence might be either represented by a linear source term a 3 · t or a factor describing the a delayed activation of ice 770 nucleation active surface sites according to 1 − exp(−b 3 · t). Note that for t → ∞, n s is limited by two factors: first, the activation of all aerosol particles and, secondly, by the size of an active site which is assumed to cover A site = 10 nm 2 (Marcolli et al., 2007) with the surface area covered by ac-775 tive sites not exceeding the available aerosol surface area.
For evaluating the potential role of time-dependent ice nucleation in the atmosphere, the box model ACPIM was used to simulate the ascent of air parcels. For these case studies, ice nucleation was described by a purely thermo-780 dynamically driven INAS density function and two parameterizations with additional time-dependent terms. The timedependent terms are potentially important at low to moderate updraft velocities and for small ice fractions. However, the results obtained from the three different parameterizations 785 did not differ much from each other under the prescribed experimental conditions. It should be noted, however, that the modelling case studies in this work are based on ice nucleation results for ATD obtained under certain thermodynamic conditions.

790
The ATD experiments and modeling studies presented in this work are supposed to be a first step in rigorously investigating deposition nucleation over a wide temperature and saturation range in order to gain a better understanding of the factors which are relevant for deposition nucleation. This 795 knowledge was then used to develop a metric which can be easily employed for the comparative analysis of other heterogeneous nucleation studies. Further investigations of atmospherically relevant dust samples are needed in order to better inform future parameterizations describing deposition 800 ice nucleation.      Fig. 4. Trajectories of ice nucleation experiments with ice nucleation thresholds: trajectories are shown from the point on when ice crystal concentrations first exceed background concentrations with only the part being shown for which RHice increases almost linearly with decreasing temperature; relative humidity over ice corresponding to an ice-active particle fraction fice = 1% is indicated by • symbols for standard experiments using cyclone impactors for defining an aerosol size cutoff, and for experiments including larger particles.  Table 1) -the arrow indicates decreasing temperature and increasing RHice during expansion experiments (see Fig. 4); colored lines correspond to isolines of the fitted INAS density (log 10 (ns)) from Fig. 6 symbols indicate ice nucleation active surface site densities derived from experimental studies by other authors (same color code as for isolines).