Can we define an asymptotic value for the ice active surface site density for heterogeneous ice nucleation?
Abstract
The immersion freezing behavior of droplets containing size-segregated, monodisperse feldspar particles was investigated. For all particle sizes investigated, a leveling off of the frozen droplet fraction was observed reaching a plateau within the heterogeneous freezing temperature regime (T >− 38°C). The frozen fraction in the plateau region was proportional to the particle surface area. Based on these findings, an asymptotic value for ice active surface site density ns, which we named 
, could be determined for the investigated feldspar sample. The comparison of these results with those of other studies not only elucidates the general feasibility of determining such an asymptotic value but also shows that the value of 
strongly depends on the method of the particle surface area determination. However, such an asymptotic value might be an important input parameter for atmospheric modeling applications. At least it shows that care should be taken when ns is extrapolated to lower or higher temperature.
Key Points
- Immersion freezing behavior of droplets containing feldspar particles
- Theoretical description of investigated immersion freezing process
- Determination of an asymptotic value for the ice active surface site density
1 Introduction
The formation of ice in atmospheric clouds has a substantial influence on the radiative properties of clouds as well as on the formation of precipitation. Therefore, much effort has been made to understand and quantify the major ice formation processes in clouds. Immersion freezing has been suggested to be a dominant primary ice formation process in low-level and midlevel clouds (mixed-phase cloud conditions) [e.g., Ansmann et al., 2009; de Boer et al., 2011]. This process is characterized through ice nucleation occurring heterogeneously on a foreign body which is immersed within a supercooled droplet [Cantrell and Heymsfield, 2005].
Various field measurements have been conducted [e.g., DeMott et al., 2003; Mertes et al., 2007; Pratt et al., 2009] where, e.g., ice crystal numbers in clouds have been determined as well as ice residuals have been sampled in order to determine the amount and the composition of possible ice nucleating particles (INP). It has been shown that mineral dust particles are the most abundant INP in the atmosphere and thus may play an important role for atmospheric ice nucleation [Murray et al., 2012]. Additionally, it has been suggested that biological particles like bacteria and pollen are potentially involved in atmospheric ice formation, at least on a regional scale [Murray et al., 2012].
In two recent laboratory studies, the ice nucleation ability of nonviable Pseudomonas syringae bacteria (by using Snomax®) [Hartmann et al., 2013] and birch pollen washing water [Augustin et al., 2013] was investigated. Droplets with different amounts of Snomax® and pollen washing water material, respectively, were supercooled until they froze. It was found that droplet freezing occurred at temperatures from −7°C to −10°C for droplets containing Snomax® and at temperatures from −16°C to −22°C for droplets containing birch pollen washing water. At temperatures below these, down to the homogeneous freezing temperature, the fraction of frozen droplets, i.e., the number of frozen droplets divided by the total number of frozen and unfrozen droplets, leveled off at values lower than 1. That means in both cases, not all droplets froze heterogeneously. The actual plateau value depended on the average number of ice nucleation active macromolecules (INM) per droplet, i.e., protein complexes in the case of bacteria and sugar-like macromolecules in case of pollen. In other words, in both studies not every investigated droplet contained an INM and it could be assumed that the INM were distributed among the droplet population following a Poisson distribution in each case.
The amount of primary biological aerosol particles in the atmosphere can be significant as they can add up to about 20% by number (for particle diameters larger than 0.4 μm) of the total aerosol [see Murray et al., 2012, and references therein]. But, although poorly quantified yet, the average number of biological INP is probably low compared to, e.g., the total number of atmosphericparticles. It can be therefore assumed that the average probability of cloud droplets containing at least one INM is fairly low, i.e., the assumption of Poisson distributed INM is valid for atmospheric considerations, too.
For mineral INP, as mentioned above, it is known that their average number is higher compared to the number of biological INP in the atmosphere but their number is still small compared to the total number of particles in the atmosphere. However, the question still remains whether all of the mineral particles feature ice nucleating sites, i.e., two-dimensional surface areas of finite extent which induce ice nucleation, and how these sites are distributed.
In the past, much of the laboratory research on ice nucleation focused on clay minerals like kaolinite and illite using various measurement techniques (see reviews by Murray et al. [2012] and Hoose and Möhler [2012]). But Atkinson et al. [2013] recently reported that “feldspar minerals dominate ice nucleation by mineral dusts under mixed-phase cloud conditions, despite feldspar being a minor component of dust emitted from arid regions” and that “clay minerals are relatively unimportant ice nuclei.” Wex et al. [2014] obtained indications that K-feldspar which may be present in some kaolinite samples (e.g., in those provided by FLUKA) dominates the ice nucleation behavior of particles generated from these samples. In this context, Augustin-Bauditz et al. [2014] further showed that the ice nucleating ability of various mineral dusts like kaolinite as well as illite and Arizona Test Dust is related to the amount of K-feldspar included in these dusts. These studies illustrate the atmospheric relevance of feldspar.
Augustin-Bauditz et al. [2014] furthermore observed a leveling off of the frozen droplet fraction at about 0.8 within the heterogeneous ice nucleation limit when investigating the immersion freezing behavior of droplets each containing a single feldspar particle with a diameter of 300 nm. To our knowledge, this is the first time that such a plateau in the frozen droplet fraction, as found for Snomax®; and birch pollen washing water, has clearly been reported for mineral INP in the immersion freezing mode.
In this study, we intensify the investigations concerning the ice nucleating ability of feldspar by focusing on size-segregated particle investigations and verifying whether such a plateau can also be found for droplets containing feldspar particles of different sizes and whether the plateau value scales with particle surface area. This would give a hint about the number and distribution of ice nucleating sites in the particle population. But note that the clarification of the nature of these ice nucleating sites is beyond the scope of this study. (Augustin-Bauditz et al. [2014] assume the lattice structure of K-feldspar to play an important role. But further chemical and/or mineralogical analyses are needed for clarification.)
The ice nucleating behavior of size-segregated, monodisperse feldspar particles of four different particle sizes is investigated utilizing the Leipzig Aerosol Cloud Interaction Simulator (LACIS) [Hartmann et al., 2011] in the immersion freezing mode. The Soccer ball model (SBM) [Niedermeier et al., 2011, 2014], which is based on classical nucleation theory (CNT), is used to parameterize the determined ice nucleation behavior. This SBM parameterization is then used to derive the ice active surface site density, ns, an ideally measurement independent ice nucleation quantity. We do this in order to compare our experimental results with those of Atkinson et al. [2013] who also investigated the ice nucleation ability of similar feldspar particles. Finally, we want to verify whether an “asymptotic value” for ns can be defined which would strongly limit the possibility to extrapolate parameterizations describing the ice nucleation process.
2 Methods
The investigated feldspar sample was provided by the Technical University Darmstadt and originated from Minas Gerais, Brazil. It consists of 76% microcline (K-feldspar) and 24% albite (Na-feldspar) based on X-ray diffraction measurements [Augustin-Bauditz et al., 2014]. The procedure of dry particle dispersion as well as particle size selection closely follows those used during previous mineral dust experiments performed with LACIS [e.g., Niedermeier et al., 2010] and will therefore only be introduced briefly.
A fluidized bed generator was applied to generate airborne feldspar particles. After charging the particles within a bipolar diffusion charger (Krypton 85), quasi-monodisperse particles were selected utilizing a Differential Mobility Analyzer (DMA-type Vienna Medium [Knutson and Whitby, 1975]). Four different mobility particle diameters Dmob were selected and independently analyzed: 200 nm, 300 nm, 400 nm, and 500 nm. In all cases care was taken to reduce the number of doubly charged particles substantially by removing large particles applying a Micro-Orifice Uniform-Deposit Impactor (MOUDI, Model 100R, MSP Corporation) and a cyclone prior to the bipolar diffusion charger. However, for the two smallest particle sizes investigated (i.e., Dmob=200 nm and 300 nm) we could observe larger particles by means of an Ultra-High Sensitivity Aerosol Spectrometer (Droplet Measurement Technologies). In case of Dmob= 200 nm, about one third of the selected particles were doubly or triply charged. In case of Dmob= 300 nm, about 8% of the particles were doubly charged. Multiple charged particles were not observed for the two other particle sizes selected. To correct for the effects of the larger, multiply charged particles a correction was performed following the procedure described in S. Hartmann et al. (Immersion freezing of kaolinite particles—Scaling with particle surface area, submitted to Journal of the Atmospheric Sciences, 2015). That means the frozen fractions presented in the scope of the study are adjusted according to this correction procedure.
A dilution system downstream of the DMA was used to control the particle number concentration. Subsequent to dilution the aerosol was split into two streams with one fraction being fed into a Condensation Particle Counter (TSI 3010) to measure the particle number concentration and the other fraction being fed into LACIS.
LACIS itself is a laminar flow tube made up of seven 1 m tubes (see, e.g., Hartmann et al. [2011] for details). In the inlet of LACIS, the aerosol flow is combined with a humidified sheath air flow such that the aerosol is confined by the sheath air to a narrow beam of about 2 mm in diameter at the center axis of LACIS. The volume flow rates of both flows were chosen such that both enter LACIS in an isokinetic fashion with a velocity of about 0.4 m s−1. Supersaturated conditions were achieved by cooling the inner tube walls. The particles were activated to supercooled liquid droplets first with each droplet containing a single, size-segregated particle. Due to further cooling, these supercooled droplets may freeze (see Hartmann et al. [2011] for details). At the outlet of LACIS, the Thermally Stabilized Optical Particle Spectrometer [Clauss et al., 2013] was used to discriminate between frozen and unfrozen droplets in order to determine the frozen droplet fraction fice. For the investigations presented here, fice was measured within the temperature range of −20°C to −40°C.
3 Results and Parameterization
3.1 Immersion Freezing Behavior of Droplets Containing Feldspar Particles
In Figure 1, the experimentally determined frozen fractions are presented for droplets containing feldspar particles of various sizes. We detected frozen droplets already at approximately −23°C for all particle sizes investigated. Two features are obvious: First of all, the measured fractions of frozen droplets level off at values lower than 1 at temperatures well above the homogeneous freezing temperature with the actual ice fraction in the plateau range, 
, depending on the particle size. The larger the particle the higher is 
. In all cases, the frozen droplet fraction finally reaches 1 at T <− 38°C, i.e., at a temperature where homogeneous ice nucleation becomes the dominant ice nucleation mechanism. Second, with increasing particle diameter the frozen fraction curve shifts to higher temperatures.
This observed ice nucleating behavior indicates that not all droplets/particles feature an ice nucleating site. But the number of these sites per particle/droplet increases with increasing particle diameter. The latter observation also explains the shift of the frozen droplet fraction curves to higher temperatures because the probability of heterogeneous freezing to occur increases with increasing surface area, i.e., increasing number of ice nucleating sites [e.g., Lüönd et al., 2010; Murray et al., 2011; Welti et al., 2012].
3.2 Theoretical Considerations— Poisson Distributed Ice Nucleating Sites
(1)
denoting the ice fraction in the plateau range. For simplification particles are assumed to be spherical and 
represents the Stokes diameter-based surface area with Dp being the Stokes diameter.
The experimentally based λINS values, which have been calculated using equation 1, as well as their uncertainties are shown in Table 1 and in Figure 2 as a function of particle surface area. From Figure 1 (and Table 1 as well) it can be seen that we determined the highest uncertainty for 
for the smallest particle diameter investigated and vice versa. However, due to the strong nonlinearity of λINS as a function of 
, the absolute uncertainty for λINS is highest for the largest particle surface areas (Figure 2) because here 
is close to one and minimal changes in 
have large effects on λINS (compare uncertainties given for 
and λINS in Table 1). In other words, equation 1 yields reasonable values only for 
clearly smaller than 1.
and λINS As Well As Their Absolute Uncertainties 
and U(λINS,exp), Respectivelya
| U(λINS,exp) | |||||
|---|---|---|---|---|---|
| Dp (nm) |  |
 |
λINS,exp | Plus/Minus | λINS,SBM |
| 200 | 0.55 | 0.10 | 0.80 | +0.25/−0.20 | 0.81 |
| 300 | 0.84 | 0.04 | 1.83 | +0.29/−0.22 | 1.83 |
| 400 | 0.93 | 0.04 | 2.66 | +0.80/−0.44 | 3.24 |
| 500 | 0.98 | 0.015 | 3.91 | +1.42/−0.56 | 5.07 |
- a The calculated values of λINS,SBM used for CHESS-SBM are shown, too.
As long as the last mentioned criteria is fulfilled, a linear relationship between λINS and the particle surface area can be determined as seen in Figure 2. In contrast, for Snomax® a linear relationship between λINS and the particle volume was found because there the ice nucleating material was dissolvable in the water droplets [Hartmann et al., 2013; Wex et al., 2015]. The surface area dependence for the feldspar sample is additionally strengthened when looking at Figure 3 (see 8) where the Here immersion freezing results are presented in terms of the ice active surface site density ns (introduced in 8). The ns values, determined for the differently sized particles, fall together within the range of uncertainty. Based on the 
values of the 200 nm and 300 nm particles, for which uncertainties in λINS are smallest, we determined λINS,SBM=(6.46 ± 1.18) × 1012 m−2×Sp,Stk for the investigated feldspar sample. Note that this equation also describes the 
values of the 400 nm and 500 nm particles within the respective uncertainties and, as it will be shown later in detail, the prefactor in the equation represents the asymptotic value of ice active surface site density ns (m−2). This λINS,SBM equation will be used as an input parameter for the SBM. The SBM model will be used on the one hand to parameterize our measurement results by determining the contact angle distribution. On the other hand we want to verify whether this LACIS-based contact angle distribution together with the λINS,SBM equation can be used to represent the data of Atkinson et al. [2013] who investigated the immersion freezing behavior of similar feldspar particles as well.
3.3 Theoretical Considerations—Soccer Ball Model
(2)
(3)
, i.e., the ice fraction in the plateau range. Note that Pfr ≠ Pfr,INS, i.e., while Pfr can be smaller than 1, Pfr,INS will approach unity since the latter describes droplets that include ice nucleating sites, whereas the former includes all droplets, even those without ice nucleating sites. In terms of SBM, Punfr,INS is given through [Niedermeier et al., 2014] :
(4)
3.4 Parameterization of the Feldspar-Induced Droplet Freezing
As it is possible to directly derive λINS,SBM for the feldspar sample, only two unknown CHESS-SBM parameters, μθ and σθ, are left which can be determined through fitting equation 3 to the experimental data. The calculated curves (with μθ= 1.29 rad and σθ= 0.10 rad) are shown in Figure 1 fitting the experimental LACIS data with reasonable accuracy (the root-mean-square error between the fitted curves and data points is RMSE = 0.23). For comparison, curves are presented applying a single contact angle of 1.29 rad which indicate that the assumption of identical nucleation properties among the particles cannot explain the observed freezing behavior. The slope is too steep and the effect of particle size (i.e., the frozen fraction curve moving toward higher temperature with increasing particle surface area) is less pronounced compared to the CHESS-SBM curves. The observed particles' ice nucleation ability can only be explained if an ice nucleating variability is supposed among the feldspar particles, e.g., in terms of a contact angle distribution. With increasing particle surface area, the probability for particles increases to feature ice nucleating sites with smaller contact angles than the mean μθ which shifts the frozen fraction curves to a higher temperature.
Now, we want to verify whether the LACIS-based contact angle distribution together with the λINS,SBM equation can be used to represent the data of Atkinson et al. [2013] (AT13 from now on) who investigated the immersion freezing behavior of similar feldspar particles (80.1% K-feldspar, 16.0% Na/Ca feldspar, and 3.9% quartz) using a cold stage cell at a cooling rate of 1 K min−1. In their study, a leveling off of the frozen droplet fraction could not be observed presumably due to the large amount of feldspar particles per droplet. AT13 investigated the freezing behavior of 14 to 16 μm sized droplets with each droplet having multiple particles included (they reported a 10 μm droplet to contain about 5 particles and a 17.5 μm droplet to contain about 27 particles). The mean diameter of the particles was given as 700 nm in their supplement. The particles surface area Sp,BET per droplet was about 1.95 × 10−11 m2 based on the Brunauer, Emmett, and Teller (BET) gas adsorption method. This surface area is supposed to be higher than the Stokes diameter derived surface area which we applied. However, AT13 expect surface areas determined for K-feldspar using BET gas adsorption method and other techniques (like the Stokes diameter derived surface area) to differ by less than a factor of 3.5.
In order to compare our results with those of AT13 we apply our surface area dependent λINS,SBM equation (as shown above as well as in Figure 2) using their given surface area of 1.95 × 10−11 m2 as an upper and 1.95 × 10−11 m2 divided by a factor of 3.5 as a lower limit. We further follow Augustin-Bauditz et al. [2014] who showed that the ice nucleating ability of mineral dust is related to the amount of K-feldspar included in them. We obtain λINS=1.95 × 10−11 m2×6.46 × 1012 m
132 (based on BET surface area) and 
38 (based on BET surface area divided by 3.5), respectively. That means we assume the average number of ice nucleating sites per droplet in AT13 to be between 38 and 132, where the spread results from uncertainties in the determination of the surface area.
The corresponding SBM curves are also shown in Figure 1. These curves are calculated based on the contact angle distribution determined from LACIS data and the cooling rate of 1 K min−1, together with a λINS of either 38 or 132, and this range is indicated by the area underlaid in green. For the highest λINS= 132, our predicted curve represents the frozen fractions determined by AT13 (RMSE = 0.11). For the other case, λINS= 38, the curve is shifted about 1.0 K to 1.5 K to lower temperature.
For both data sets, the one from LACIS and the one presented in AT13, the same contact angle distribution can be used in order to represent the slopes of the experimentally determined frozen fractions. That suggests the nature of the ice nucleating sites to be very similar in both studies. But it is obvious that the different methods of particle surface area determination have a large influence on λINS determination which could bias the intercomparison of different studies leaving the question open which method for particle surface area determination is more appropriate for ice nucleation studies. But it should be also noted that the general applicability of the combined CHESS-SBM to represent the experimentally determined frozen fractions is not impaired. The same is true when determining and applying ice active surface site densities, as will be outlined in the following.
4 Determination of the Asymptotic Value of the Ice Active Surface Site Density and Comparison With Other Studies
Besides the heterogeneous ice nucleation rate coefficient a second, instrument-independent quantity has been established in the past in order to describe and compare the ice nucleating behavior of various particle types: the temperature-dependent ice active surface site density ns [e.g., Connolly et al., 2009; Niemand et al., 2012]. Similar to jhet, for the determination of ns it is assumed that ice nucleation for insoluble particles depends on particle surface area. The main difference is that the determination of nucleation rates implies nucleation time to be an important factor for ice nucleation (“stochastic description”). For the determination of ns (“singular description”), it is suggested that the time dependence of ice nucleation is less important compared to temperature and can therefore be neglected [e.g., Hoose and Möhler, 2012; Murray et al., 2012].
4.1 Determination of the Asymptotic Value of the Ice Active Surface Site Density
In contrast to the original definition of the ice active surface site density, we use the CHESS-SBM in the following in order to determine a time-dependent (as also presented in Hoose and Möhler [2012]) formulation for ns. The reason for this approach is that the majority of experimentally determined ice nucleation data is presented in terms of ns, but we want to keep the time dependence of freezing.
Several laboratory studies quantifying the ice nucleating ability of various desert dust samples, which are summarized in Figure 11b of Hoose and Möhler [2012], suggest the existence of an asymptotic value for ns because in this figure a leveling off of ns can be observed for T >− 38°C. Based on these results and our findings concerning the leveling off of the frozen droplet fraction we will now verify whether an asymptotic value for ns can be defined.
(5)
(6)
(7)
determines the asymptotic value of ns, which we will name 
. Punfr,INS→0 will occur (a) if the temperature is accordingly low such that for a fixed nucleation time/cooling rate the whole contact angle distribution can contribute to droplet freezing within the heterogeneous freezing temperature range of −38°C < T < 0°C (as observed in this study) or (b) if t→∞ at a constant temperature. This last assumption has to be explained in more detail. Let us consider a droplet population at a constant temperature within the interval [−38°C,0°C] with each droplet containing a single particle and a contact angle distribution with given μθ and σθ being valid. Droplets featuring particles with lowest contact angle, i.e., highest nucleation rate jhet (or in other words smallest mean nucleation time τ = (jhetssite)−1) have the highest probability to freeze. But with increasing nucleation time, the probability of particles featuring high contact angles (or in other words larger nucleation times) to induce ice nucleation will increase resulting in the complete freezing of the presumed droplet population for t→∞ (see equation 4 for t→∞).
Note that we already calculated this asymptotic and temperature-independent value for the investigated feldspar sample due to the observed leveling off of the frozen droplet fraction: 
based on the Stokes particle surface area. This asymptotic value is a particle property which gives the overall number of ice nucleating sites existing per surface area of the examined material. This implies that the value obtained for 
is strongly affected by the particle surface area determination method (see equation 7). This influence as well as the general functionality of 
will be shown below.
4.2 Comparison of ns With Other Studies
In Figure 3, ns values are presented for the feldspar particles which are directly determined from the LACIS frozen fractions using equation 5. The corresponding CHESS-SBM-based ns curves (using equation 6 and based on the Stokes particle surface area) represent the conditions for LACIS (t= 1.6 s), for AT13 (dT/dt = 1 K min−1) and for an additional cooling rate of dT/dt= 0.02 K min−1. The ns parameterization of AT13 for feldspar particles based on Sp,BET is shown, too.
It is clearly visible that the LACIS-based ns values increase with decreasing temperature reaching 
at about −32°C. Both the observed temperature dependence of ns as well as its asymptotic value can be represented by the combined CHESS-SBM using λINS,SBM (λINS,SBM, together with Sp,Stk, determines 
but it also influences the position of the ns curve as shown below), μθ, and σθ.
We now compare our results with those of AT13. Two features become obvious. First of all, AT13 could not report a 
value because they did not observe a leveling off of in their frozen droplet fraction (see Figure 1). The reason therefore lies in the larger number of INP contained in their investigated droplets and the resulting insensitivity concerning 
. Only methods investigating droplets containing a single or a very small number of particles are sensitive in this context. Second, there is a temperature shift of about 1 K between the LACIS-based ns values and the parameterization of AT13 in the temperature range of −20°C to −25°C. Ignoring the difference of surface area determination between both studies, this shift could be explained by the differences of the considered nucleation time/cooling rate as shown by the corresponding CHESS-SBM-based ns curves representing the conditions for LACIS (t= 1.6 s) and for AT13 (dT/dt= 1 K min−1). It turns out that for the feldspar particles investigated a change in nucleation time/cooling rate by a factor of 10 corresponds to a shift in temperature of about ΔT= 0.7 K. Furthermore, the leveling off of the LACIS-based ns curve is responsible for the increasing difference between our and AT13 original curve for decreasing temperature (The ns parameterization of AT13 is only valid until −25°C). Note that, due to the relation given in equations 6 and 7, the determined asymptotic value 
is material dependent only. That means a further increase of the nucleation time only leads to a shift of the ns curve to a higher temperature with the value of 
being unaffected (see CHESS-SBM curve for dT/dt= 0.02 K min−1 in Figure 3). Our results also imply that extrapolating ns to lower or higher temperature (as we did for the AT13 curve in Figure 3 as well as for the ns parameterization of Niemand et al. [2012] in Figure 4) may lead to orders of magnitude overestimations of ns at both sides, on the one hand due to not accounting for the asymptotic behavior of ns. On the other hand, the assumption of “linear” behavior may oversimplify the actual functional form of ns.
spread of 8.40 × 1010 to 2.13 × 1012 m−2 assuming the K-feldspar amount in the desert dusts to range between 1% and 25%.
To outline the influence of the different methods of particle surface area determination on ns, a second curve for ns is presented in Figure 3 which is based on AT13 curve but multiplied with the above mentioned factor of 3.5, i.e., indicating the uncertainty in the surface area as given in AT13. This multiplication leads to a shift of this curve of about 1.2 K to higher temperature. This shift in temperature corresponds to a shift in cooling rate from 1 K min−1 to about 0.02 K min−1 (dash-dotted line in Figure 3). In fact, we also could apply the surface area factor of 3.5 to the corresponding CHESS-SBM-based ns curves. In this case, the surface area Sp,Stk would change to Sp,Stk=3.5Sp,BET which leads to 
. That means the resulting ns curve (representing the conditions for LACIS, i.e., t= 1.6 s, not shown in Figure 3) would be shifted by 1.2 K to lower temperature and would level off at the lower 
value compared to the ns curve which is based on Sp,Stk.
The observed influence of different particle surface areas on ns and 
shows that care has to be taken when comparing ns values which are based on different methods for its surface determination. Furthermore, the
question is raised which method (BET gas absorption, surface area based on Stokes diameter, etc.) is more appropriate for ice nucleation studies. But this influence of different particle surface areas on ns and 
does not impair the principle applicability of the combined CHESS-SBM to derive such values.
Laboratory studies quantifying the ice nucleation ability of various desert dusts in the immersion as well as deposition/condensation mode [Connolly et al., 2009; DeMott et al., 2011; Kanji et al., 2011; Koehler et al., 2010; Niemand et al., 2012; Pinti et al., 2012] further support the existence and applicability of 
. In some of these studies, which are also summarized in Figure 11b of Hoose and Möhler [2012], a leveling off of ns curves can be observed at values of about 6 × 1010–6 × 1011 m−2 in the heterogeneous temperature regime. Our determined 
value for feldspar is larger by about an order of magnitude for reason given below.
In Figure 4, the ns values for the different desert dust samples are shown. In addition to Hoose and Möhler [2012], we included the ns parameterization of Niemand et al. [2012] which was developed for desert dusts in the temperature range of −12°C to −36°C. Furthermore, the CHESS-SBM-based curve is shown (applying equation 6) based on the parameters which we determined for our feldspar sample, i.e., μθ=1.29 rad and σθ=0.10 rad. This curve further includes a range for the nucleation times/cooling rates which are commonly used in laboratory studies. We set 0.1 K min−1 as an upper and 10 K min−1 as a lower limit noting that the ice nucleation probability decreases with increasing cooling rate. We further constrain 
to range between 8.40 × 1010m−2 and 2.13 × 1012m−2 based on the following reasoning. As mentioned in the beginning, AT13 showed that feldspar minerals dominate the ice nucleation by mineral dusts under mixed-phase cloud conditions. They made a comparison based on their feldspar parameterization which was scaled by assuming the feldspar content of various desert dusts—the ones presented in Connolly et al. [2009] andNiemand et al. [2012]—to be between 1% and 25%. They used this range because Nickovic et al. [2012] showed that the feldspar content in Saharan soil can range between about 2% and more than 16%. The highest feldspar content was found in Asian soil being up to about 28% by mass. It has further been reported that the K-feldspar concentration in airborne mineral dust can range between a few percent [Glaccum and Prospero, 1980] and about 25% [Kandler et al., 2011]. For our study we take further note of findings by Augustin-Bauditz et al. [2014] who showed that the ice nucleating ability of various mineral dusts is related to the amount of K-feldspar included in these dusts. The feldspar sample investigated in this study included about 76% of K-feldspar (microcline). To determine the range for 
to be used for the CHESS-SBM parameterization, we assume the K-feldspar content in the desert dusts to range between 1% and 25%. Therefore, we set 
as a lower and 
as an upper limit. The different desert dusts can be very well represented by this CHESS-SBM-based curve (see Figure 4). In contrast to AT13 who were able to represent the different desert dusts for a temperature range from about −15°C down to −25°C, we obtain good agreement even for temperatures below −25°C because we could determine 
. These findings on the one hand confirm that the ice nucleating ability of mineral dusts is dominated by the included K-feldspar amount. On the other hand they strengthen the applicability of 
.
5 Summary and Implications
We investigated the immersion freezing behavior of droplets containing differently sized feldspar particles. We observed a leveling off of the frozen droplet fraction reaching a plateau at temperatures well above −38°C which is proportional to the particle surface area. Based on these findings, we developed a combined CHESS-SBM which can be applied to parameterize and describe the experimentally determined frozen droplet fractions. Subsequently, an asymptotic value for ns, namely, 
, could be determined for the investigated feldspar sample using this model.
It should be noted that the value of 
strongly depends on the particle surface area used for its deduction. Small differences in the particle size determination could cause noticeable (linear) shifts in 
and could shift the ns curve to higher or lower temperature. In order to enable the determination and intercomparison of reasonable ice active surface site densities, the surface area of the investigated particles has to be known as precisely as possible or at least similar assumptions concerning particle size should be made. However, the question remains which method (gas absorption, surface area based on Stokes diameter, etc.) for particle surface area determination is more appropriate in terms of heterogeneous ice nucleation. But this observed influence of the method for particle surface area determination on the value of 
does not impair the principle feasibility to derive and apply such a value.
In particular, we could show that the existence of such a 
value is strengthened by other experimental studies for various desert dust samples, summarized in Hoose and Möhler [2012], which show a leveling off of ns in the heterogeneous temperature regime. The contact angle distribution which we derived for our feldspar sample, together with a lowering of the 
value which represented K-feldspar fractions of 1% to 25%, lead to results which compared well with the data for desert dust samples collected in Hoose and Möhler [2012]. Based on this we can say that the comparison of the obtained ns / 
values for our feldspar sample and for these desert dusts confirm earlier studies [Atkinson et al., 2013; Augustin-Bauditz et al., 2014] which showed that the ice nucleating ability of mineral dusts is dominated by the feldspar or more precisely by the K-feldspar amount included in them. Our results also suggest that care has to be taken when extrapolating ns curves for mineral dusts to lower or higher temperature. It might lead to orders of magnitude overestimations of ns at both sides.
Our findings also lead to another interesting conclusion: If it is possible to determine a 
value for a given particulate material, it is possible to determine the (average) number of ice nucleating sites per particle surface area. In other words, for each particle size the specific (average) number of ice nucleating sites can be calculated. This finding could be used in atmospheric modeling applications in order to define size-class specific numbers for ice nucleating sites for various materials/substances.
Last but not least, the nucleation time/cooling rate should be reported as carefully as possible in order to compare results utilizing different measurement techniques. For feldspar particle samples, an influence of nucleation time was observed although being of minor importance compared to the effect of temperature and particle surface area. However, if a droplet population including such types of particles with the determined contact angle distribution would be set to a fixed temperature below 0°C for a long period of time (like in an arctic stratocumulus cloud as investigated by Westbrook and Illingworth [2013] which was well-mixed and bounded above and below by stable layers over a long time scale), then residence time will affect the freezing probability of this droplet population. In this case, with increasing residence time the probability increases for larger contact angles from the distribution to contribute to freezing of the droplet population approaching a limited value which we have determined in this study. Therefore, the combined CHESS-SBM model could be a useful tool for cloud resolving models which include time influence on freezing since it is able to represent both the time dependence on freezing as well as a possible asymptotic behavior of the frozen droplet fraction.
Acknowledgments
This work is partly funded by the Federal Ministry of Education and Research (BMBF—project CLOUD 12) and by the German Research Foundation (DFG project WE 4722/1-1, part of the research unit INUIT, FOR 1525). D. Niedermeier acknowledges financial support from the Alexander von Humboldt-Foundation. Please contact D. Niedermeier ([email protected]) or S. Augustin-Bauditz ([email protected]) for raw and processed data.
as test substance
