Comparative Study On Immersion Freezing Utilizing Single Droplet Levitation Methods

Immersion freezing experiments were performed utilizing two distinct single-droplet levitation methods. In the Mainz vertical wind tunnel (M-WT) supercooled droplets of 700μm diameter were freely floated in a vertical air stream at constant temperatures ranging from −5◦C to −30◦C where heterogeneous freezing takes place. These investigations under isothermal conditions allow applying the stochastic approach to analyze and interpret the results in terms of the freezing or nucleation rate. In the Mainz acoustic levitator (M-AL) 2 mm diameter drops were levitated while their temperature was contin5 uously cooling from +20◦C to −28◦C by adapting to the ambient temperature. Therefore, in this case the singular approach was used for analysis. From the experiments, the densities of ice nucleating active sites (INAS) were obtained as function of temperature. The direct comparison of the results from two different instruments indicates a shift of the mean freezing temperatures of the investigated drops towards lower values that was material dependent. As ice nucleating particles, seven materials were investigated, two representatives of biological species (fibrous and microcrystalline cellulose), four mineral 10 dusts (feldspar, illite NX, montmorillonite, and kaolinite), and natural Sahara dust. Based on detailed analysis of our results we determined a material dependent parameter for calculating the freezing temperature shift due to a change in cooling rate for each investigated particle type. The analysis allowed further classifying the investigated materials to be described by a singleor a multiple-component approach. From our experiences during the present synergetic studies, we listed a number of suggestions for future experiments regarding cooling rates, determination of the drop temperature, purity of the water used to 15 produce the drops, and characterization of the ice nucleating material. The observed freezing temperature shift is significantly important for the intercomparison of ice nucleation instruments with different cooling rates.


Introduction
Immersion freezing is considered to be the most effective freezing/nucleation process for ice particle production in mixed-phase clouds (Diehl and Grützun, 2018). The nucleation abilities of atmospheric particles have been investigated very intensively 20 for the last decades (Hoose and Möhler, 2012). Beside in-situ measurements, laboratory-based investigation techniques are widely used to discover the basic physical and chemical processes and properties of ice nucleating particles (INP). Laboratory immersion freezing experiments aim at the characterization of the temperature dependent ice nucleation ability of different types of INP under controlled conditions. The ice nucleation efficiency of INP is commonly expressed in terms of ice nucleation active sites (INAS) density n s (T ). This gives the total number of nucleating sites per unit surface area of the particles that are 25 active between 0 • C and the sub-zero temperature T (DeMott, 1995;Connolly et al., 2009;Murray et al., 2012;Hoose and Möhler, 2012). Another important parameter employed for describing the INP nucleation ability is the nucleation rate, i.e. the probability of nucleation at a certain temperature per unit time per unit surface area of the particle (Vali, 2014).
Intercomparisons of measurement techniques revealed a wide scatter of measured ice nucleation activities of particles. This is due to differences in the measuring methods employed by the different instruments and the diversity in the sample prepara-30 tion at different research sites. One essential and still not understood discrepancy arises between dry dispersion and aqueous suspension measurement techniques (Hiranuma et al., 2019). In the former, experiments employ water vapor condensation onto dry dispersed particles followed by droplet freezing (e.g., cloud chambers, continuous-flow diffusion chambers), while the latter denotes experiments starting with test samples pre-suspended in water before cooling (e.g., freezing arrays, drop levitators). Several studies have focused on identifying potential reasons of this data diversity. Recently, two major international 35 research activities were conducted and produced a large amount of new data and results, one organized around the German INUIT (Ice Nucleation Research Unit) research community, and the FIN (Fifth International Ice Nucleation Workshop). These intercomparison campaigns revealed data diversity over several orders of magnitude in n s already among aqueous suspension techniques also in case of a recommended protocol for sample treatment and preparation (Hiranuma et al., 2019;DeMott et al., 2018). As these studies concluded, a key strategy would be to rigorously examine and define the functionality, configuration 40 and limitations of the measurement techniques and instruments (DeMott et al., 2018).
The most widely employed measurement instruments for investigating the immersion freezing of aqueous suspensions are freezing arrays (Murray et al., 2011;Hader et al., 2014;Budke and Koop, 2015;Schrod et al., 2016;Reicher et al., 2018;Harrison et al., 2018). Their inexpensive and easy operation, and the large number of simultaneously measurable droplets offering good count statistics, made them standard devices for INP characterization experiments. However, the droplets are 45 continuously in contact with the supporting surface of the arrays which potentially introduces additional freezing sites and initiates freezing. Furthermore, cross-contamination and evaporation or condensation during experiments might also influence the measurement results (Budke and Koop, 2015).
To avoid the influence of the supporting surface, and to go a step further to real atmospheric conditions of cloud droplets, single droplet levitation techniques are employed. They offer experiments with natural droplet shapes and contact-free levita-50 tion, where the heat conduction of the released latent heat during freezing also better meets atmospheric conditions. The main disadvantage of the single droplet levitation techniques is the limited number of individual droplet measurements they provide.
In order to get statistically relevant numbers of data points, a series of experiments has to be conducted by an operator over a long time period, and, therefore, the long-term variation of the environmental conditions might lead to measurement uncertainties. Prominent single droplet levitation techniques used for immersion freezing are an electrodynamic balance (Rzesanke 55 et al., 2012;Hiranuma et al., 2015) in which a charged supercooled droplet of about 100 µm is levitated between electrodes; an acoustic levitator (Ettner et al., 2004;Diehl et al., 2014), and a vertical wind tunnel (von Blohn et al., 2005;Diehl et al., 2011, 2 Theoretical background of heterogeneous freezing The heterogeneous nucleation of ice, i.e. the phase transition from liquid to solid state of water (Stage 2 in Fig. 1) induced by the growth of ice embryos on nucleation sites on INP, takes place at different temperatures, depending on the properties of the particles immersed in water (Vali, 2014). The larger the particle, the higher is the possibility that some part of its surface favours ice nucleation. Hence, the probability of freezing (or nucleation) is dependent on the total surface area of the particles, 85 or more specifically on the surface density of the ice nucleation active sites (Hoose and Möhler, 2012). Nevertheless, freezing is a dynamic process in which molecules from the metastable liquid state are joining to (and detaching from) the growing ice embryo. Therefore, nucleation is also a time-dependent process and occurs under isothermal conditions, as well (i.e. when the temperature remains constant). Specifically, the parametric description of immersion freezing and the interpretation of the experimental results in the literature are based either on the stochastic (time-dependent) or on the singular (temperaturedependent) hypothesis, depending on the experimental conditions.

Stochastic approach
In experiments under isothermal conditions the number of unfrozen supercooled aqueous suspension droplets in a population decays exponentially with time, because at any point in time the number of freezing droplets is a function of the (decreasing) number of still unfrozen droplets. The underlying assumption here is that each droplet freezes with the same probability when 95 they contain identical INP. The rate R n which is used to describe this decay at a fixed temperature is determined from the number of the observed freezing events per unit time as (see Vali (2014) where n f r denotes the number of frozen droplets at time t, and N tot the total number of droplets in the population, i.e. the total number of the investigated individual droplets. After integrating Eq. (1) and assuming constant, i.e. time independent, cooling 100 rate at a fixed temperature, the well-known expression of the freezing rate follows: In the stochastic approach, the time dependence of nucleation is taken into account by introducing the heterogeneous nucleation rate coefficient J s -similarly to that for homogeneous nucleation (Pruppacher and Klett, 2010) -, which gives the rate of change of the number of ice embryos per unit surface area of the ice nucleating particle. (In case of homogeneous nucleation J hom 105 is given per unit volume of liquid drop.) If all droplets in the population contain the same amount of particle surface, and any part of the surface has an equal likelihood of containing an ice nucleating site (named single-component system after Broadley et al. (2012), and Herbert et al. (2014)), then by definition, Here S p is the total particle surface area in each aqueous suspension droplet which can be calculated as, with V d being the drop volume, c the particle mass concentration in the sample solution, and SSA the specific surface area of the particle. In case of any interparticle variability in the ice nucleating ability of the particle population Eq. 3 cannot be used.
Such a system is called multiple-component, which, however, can be divided in subpopulations of equally ice active entities.
Each subpopulation i can be treated as single-component and characterized by its number density n s,i and nucleation rate 115 coefficient J s,i (Herbert et al., 2014).

Singular approach
The concept of the singular approach is based on the observation that freezing of drops containing INP occurs at a characteristic temperature once they are subjected to cooling. This also implies that supercooled droplets remain unfrozen arbitrarily long Hence, the time dependence of ice nucleation is negligible in comparison to the particle to particle variability of the icenucleating ability (Connolly et al., 2009). Therefore, it is assumed that ice nucleation occurs on particular sites on the surface of a particle, the so-called ice-nucleating active sites (INAS), as soon as a temperature is reached which is characteristic for the INP material and its nucleating properties. Reaching this temperature by cooling, the droplet including the INP freezes instantaneously. For the singular approach, the INAS surface density n S (T ) is defined as the cumulative number of sites per 125 surface area that become active between 0 • C and T , and can be expressed as where f ice (T ) = n f r (T ) N tot is the so called frozen fraction, i.e. the cumulative fraction of droplets frozen between 0 • C and T in the population.
If all droplets were of same size and contained identical INP with homogeneous surfaces and uniform ice-nucleating sites, 130 then f ice would be a unit step function at a characteristic temperature. Variability of INP in experiments arising from diverse composition, particle size, and location of INAS on particles' surface results in a distribution of characteristic temperatures, i.e. freezing probabilities of aqueous suspension droplets, which is represented by f ice (T ) (Niedermeier et al., 2014).

Surface temperature of freely levitating droplets in freezing experiments
As becomes obvious from the description above, the correct representation of the drop temperature in freezing experiments is of 135 crucial importance. In freezing array experiments the droplet temperatures are assumed to be equal to the substrate temperature which is directly measured by a thermometer. Since the contact area between a droplet and the substrate is large, this is an appropriate assumption even for relatively large drops with volumes of microliters. In single levitating techniques, as in M-WT or M-AL, the droplets are subjected to continuous cooling by heat diffusion and convection. The surrounding medium is air, which is a far worse heat conductor than the substrates used in freezing array experiments. The effect of this adaptive droplet 140 cooling becomes significant for drops with volumes in the microliter range (or, equivalently, with sizes in the millimeter range), because the amount of latent heat to be dissipated increases with volume as does the surface area of the drops.
The freezing process of a single aqueous solution droplet is depicted in Fig. 1. (The description hereinafter follows the concept of Hindmarsh et al. (2003).) First, the relatively warm droplet cools down after injection (Stage 1 in Fig. 1), and its surface temperature T a approaches an equilibrium temperature T e determined by the ambient temperature T inf , the dew point, 145 and ventilation (Pruppacher and Klett (2010); and Appendix), In case of an evaporating droplet the equilibrium surface temperature is always lower than the ambient temperature due to evaporative cooling. For a droplet in a continuous air flow, the temperature difference between the drop and its environment is further enhanced by ventilation, resulting in a net temperature deviation δ (s. Appendix),

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T ∞ − T e = δ (7) Figure 1. Schematic plot of the temporal surface temperature evolution of a freezing droplet. 1) Supercooling of the liquid droplet until nucleation is initiated; 2) Adiabatic freezing stage where rapid kinetic crystal growth takes place until supercooling is exhausted. No heat exchange with the environment; 3) Diabatic freezing stage in which ice crystal growth inside the droplet is governed by heat transfer with the environmental air; and 4) Cooling stage where the ice particle cools down adapting to the ambient temperature.
The temporal evolution of the surface temperature for a droplet placed in a cold environment is described mathematically by an exponential decay function (s. Appendix for the derivation): with τ being the relaxation time, i.e. the time constant of the temperature adaptation. The main physical parameters that 155 determine τ , and therefore, the total cooling time of the droplet, are the drop size, the ventilation coefficient, as well as the ambient temperature and dew point (s. Appendix B2). Hence, for given experimental conditions, the temporal evolution of the drop's surface temperature in Stage 1 can be calculated using Eq. 8. In cold stage experiments freezing Stage 1 proceeds very quickly due to the large contact area (Harrison et al., 2018); in single levitation techniques this can take up to several minutes. Drop freezing occurs at some instant in time or at some specific temperature. As soon as nucleation is initiated inside 160 the supercooled drop rapid kinetic crystal growth takes place (Stage 2 in Fig. 1). This process is characterized by a sudden temperature increase due to the release of latent heat (which predominantly diffuses into the droplet) until the supercooling is exhausted, and the drop surface temperature rises to the ice-water equilibrium temperature (i.e. to 0 • C when the water activity of the investigated sample is ≈ 1, as it was in our experiments). (For the drop freezing experiments this characteristic temperature or time instant is to be measured, see Sections M-WT and M-AL.) Subsequently, a diabatic freezing of the whole 165 droplet takes place (Stage 3). The temporal duration of this freezing stage is determined by the heat exchange between the hydrometeor and its environment, therefore it proceeds slower than Stage 2. In the end, the frozen particle cools down to the ambient temperature (Stage 4).
In practice, n s can be determined when a population of aqueous suspension droplets is continuously or stepwise cooled down, and the number of freezing events as the function of temperature is registered. Cooling rates in freezing array (or cold 170 stage) experiments range from 1 to 10 Kmin −1 , representing also typical atmospheric rates. Employing a constant cooling rate r in the experiments, the temperature decreases with δT = rδt within a time interval δt. The number of droplets freezing in this time interval at temperature T can be calculated using the multiple-component system approach of Herbert et al. (2014) as where i denotes the components in the system, A i is the surface area of component i, and n liq,i is the number of unfrozen droplets at the beginning of the time interval. Hence, employing a higher cooling rate in the experiments results in a decrease in δt for a fixed δT , which, in turn, decreases the probability of a freezing event to occur between T and T − δT . Thereby the cumulative frozen fraction spectrum and, consequently, n s (T ) will be shifted to lower temperatures. This temperature shift for any given change in cooling rate from r 1 to r 2 can be calculated following Herbert et al. (2014) as: The parameter lambda is the gradient of the logarithm of the nucleation rate coefficient in a simple temperature-dependent formulation of J s introduced in recent immersion freezing studies (Murray et al., 2011;Broadley et al., 2012;Wright and Petters, 2013;Herbert et al., 2014): In Eq. (11) φ is the relative nucleating efficiency of the INP (see Herbert et al.,2014). It is obvious from Eq. (10) that experiments employing different cooling rates provide different INAS spectra due to the difference in their related temperature shifts ∆T f . In order to become independent of this systematic artefact, the experimental cooling rate r exp is normalized to a standard value, typically to 1 K min −1 , which results in Similarly, in isothermal experiments the relative change in cooling rate can be expressed as a relative change in residence time, i.e. the period of time for which the particles are exposed to a constant temperature (Herbert et al., 2014). In these experiments the temperature shift can be calculated as, where t res is the characteristic residence time of droplets under isothermal condition in the experiments, and t std is a standard 195 time required for an isothermal experiment to be comparable to a normalized cooling experiment (i.e. t std = 60s). Another important quantity which is used for discussing the results of the present experiments is the temperature derivative of the logarithm of the freezing rate normalized by the aerosol total surface area (Vali, 2014): Note that for a single-component system Eq.
In this study we measured the frozen fraction f ice (T ) of different INP with two different methods. From f ice (T ) the INAS densities n s (T ) and the freezing rates R(T ) were determined. Based on these quantities the temperature shifts ∆T f and ∆T iso , and the material parameters ω and λ were calculated and analyzed.

Material and sample preparation
The experiments were carried out using seven different types of materials which are listed in Table 1. All of these materials are considered to be important constituents of atmospheric ice nucleation particles. As biological INP surrogates we investigated two cellulose types, microcrystalline and fibrous cellulose (hereafter MCC and FC, respectively). Among the investigated mineral dust materials feldspar (especially K-feldspar) exhibits the highest ability to initiate ice formation. It is a prevalent 215 component of desert dusts so that by scaling down it is representative for dust samples in dependence on their composition (Atkinson et al., 2013). Illite NX can be considered as proxy for desert dust since their mineralogical compositions are similar (Broadley et al., 2012). Montmorillonite K10 and kaolinite (SigmaAldrich) are commercially available and characterized mineral dust materials of relevance for the atmosphere, which also have been subject of several previous studies. Furthermore, we used a natural desert dust particle sample, the ice nucleation abilities of which have been investigated with different 220 measurement techniques during the INUIT09 measurement campaign (Ullrich et al., 2019).
Atmospherically relevant INP exhibit an extremely wide range in their ability to heterogeneous freezing. Furthermore, there is a large spread in the specific surface area (SSA) of the investigated materials from around 1 m 2 g −1 to 245 m 2 g −1 (s. Table 1). We chose therefore diverse mass concentrations for each of the different particle types to obtain reasonable numbers of freezing events within the temperature ranges of our measurement facilities. Furthermore, since the volume of the investigated 225 droplets in M-AL was approximately 20 times larger than in M-WT, we used reduced mass concentrations in M-WT to obtain overlapping freezing curves with the two methods (s. Eq. (5)).
Prior to each set of experiments, 20 to 40 mL aqueous suspension was prepared by mixing sample particles of known weight (measured by an analytical balance from Sartorius) with high purity water (CHROMASOLV water for HPLC, Sigma-Aldrich).
Between the measurement runs the aqueous suspension was continuously stirred at a very low rate using a magnetic stirrer to Table 1. Aerosol material and sources measured in the current study. Also given are the specific surface area ( an idle time, because in this case we could presume already homogeneous suspension due to the low particle concentration.

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Furthermore, the syringe was shaken prior to droplet injection both in M-AL and M-WT experiments to ensure homogeneous particle distribution in droplets. Otherwise no pre-treatment procedures were applied.

Experimental setups and procedures
The characteristics and deliverables of the M-WT and M-AL instruments essential for the present study are summarized in Table 2 and will be described in the following subsections. Detailed descriptions of the experimental facilities are given, e.g.,

M-WT (Mainz vertical Wind Tunnel)
The For appropriate measurement statistics, at each temperature, particle type, and INP concentration, a total number of 70 aqueous 250 suspension droplets were investigated. After injection, each droplet was floated in the M-WT until it froze, or until the experiment was terminated because of reaching a predefined time limit. The onset of freezing is characterized as a sudden significant change in floating behavior of the droplet caused by the irregular shape of the frozen particle. This changing behavior was visually observed and registered by the operator during the experiments. In this way, the total observation time, i.e. the time duration from injection until the onset of freezing was recorded (Diehl et al., 2014). When a droplet did not freeze within 255 35 seconds, it was counted as unfrozen. In our earlier immersion freezing studies we levitated the supercooled droplets for at most 30 seconds (if freezing was not initiated sooner). We extended this total observation time with 5 seconds to consider the approximate time period a drop needed to approach its equilibrium temperature in M-WT (Fig. 2). Furthermore, wind and −25°C (plotted by black, red, and blue dots, respectively). The data points were calculated using Eq. (B21). The regression line is speed, air temperature, and dew point temperature (typically in a range from −20 to −35°C) in the wind tunnel were recorded continuously at 2 Hz temporal resolution.

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The drop surface temperature was calculated using Eq. (B21) considering thermal steady state condition between the levitating drop and its surrounding air. The time needed to approach the equilibrium temperature in the M-WT experiments within 0.2 K difference (i.e. below the temperature measurement precision of the applied PT100 sensor) was calculated at distinct M-WT air temperatures and plotted in Fig. 2 for different dew points. In the calculations the starting drop temperature was set to 20°C, and the time period for reaching a temperature difference of 0.2 K (i.e. below the temperature measurement precision 265 of the applied PT100 sensor) between the drop and the equilibrium temperature was computed. One can observe a slight dependence of the adaptation time on the air temperature, but T e was typically reached within 5 to 6 seconds for air temperatures between −15 and −28°C. In the calculations, dew points of −30, −27.5, and −25°C were applied and the results are merged in the plot shown in Fig. 2. Hence, the adaptation time was found to be practically independent on the dew point for the M-WT experiments.

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Typical droplet diameters/volumes were approximately 700 µm/0.18 µL. The size of each investigated droplet was determined from its terminal velocity (Beard, 1976), i.e. from the vertical air speed needed for freely suspending it, which can be measured with high accuracy in the M-WT (Diehl et al., 2014).
Since the M-WT experiments represent isothermal measurement conditions, the stochastic approach was applied first for data analysis. Thus, the rate constant R and the nucleation rate coefficient J het were calculated from Eqs. 2 and 3, respectively, Furthermore, from the number of freezing events over the whole observation time period the frozen fraction f ice (T ), and the INAS density n s (T ) (Eq. 5) were determined employing the singular approach.

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Background measurements were carried out before each experimental run by floating at least 10 HPLC (high purity liquid chromatography) water droplets for 35 seconds in the tunnel. We have not observed any freezing event during these test measurements, which indicate the absence of impurities (i.e. background active INP) both in the HPLC water droplets and in the wind tunnel.

M-AL (Mainz Acoustic Levitator)
285 The main component of the M-AL measurement facility is an acoustic levitator (APOS BA 10, tec5 GmbH), in which contactfree single droplet levitation is maintained by a standing ultrasonic wave (Diehl et al., 2014). The M-AL is placed inside a walk-in cold room where the ambient temperature was set to be −30 • C for the freezing experiments. In order to prevent any disturbing air motion, which might cause unsteady temperature condition and unstable levitation, or carry ice nucleating particles onto the levitating drop surface, the M-AL was surrounded by a protective acrylic housing. Using this setup, the air temperature. The number of frozen droplets was measured and binned in 1 K intervals to calculate f ice , and thereof n s applying the singular approach (Eq. 5).

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The actual cooling rate at a time instant during temperature adaptation is defined as r(t) = −dT /dt, that can be calculated after rearranging Eq. (8) to After some manipulation the actual cooling rate can be written as, The r(T a ) curve for τ = 11.3s is shown in Fig. 4. It is apparent from the figure that at high temperatures the cooling rate is substantially high and gets moderate values only at low temperatures close to the equilibrium temperature. For such large cooling rates in M-AL measurements Eq. 12 predicts significant temperature shift in drop freezing temperature.

Results and Discussion
In this section we present the results of M-WT and M-AL experiments on immersion freezing using the clay mineral kaolinite.

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The data for other materials listed in Table 1 are presented in the Supplement Material.    From the time resolved measurement data from M-WT, the time dependence of the freezing process was analysed. For that, Eq.

INAS
(2) was rearranged to where n liq (T, t) = 1 − n f r (T, t) is the number of droplets remaining liquid after time t at temperature T . in agreement with Eq. 14. Hence, ω, which is the slope of R/A (see Eq. (14)

M-AL experimental results
Frozen fractions of kaolinite suspension with 1.265 g L −1 concentration as function of temperature are shown in Fig. 7a. Error bars are associated to the temperature bin interval (±0.5 K), and the uncertainty in the determination of f ice stems from the counting statistics and the experimental temperature uncertainty. The active site density n s calculated from Eq. 5 using f ice 355 in Fig. 7a is plotted in Fig. 7b. Here the error bars originate from Gaussian error propagation when using the measured data in Fig. 7a. From the calculation we excluded the data points for which f ice was above 90% or below 10%. This cut-off was introduced because in these cases the uncertainty of f ice was very large due to the poor counting statistics when freezing or unfreezing events occur very rarely.
Another criterion to use f ice for further evaluations was that it should significantly exceed the background caused by im-360 purities in the water used for generating the aqueous suspension. To determine this background spectrum, we investigated pure water droplets before each experimental run in M-AL, similarly to the M-WT measurements. However, in contrast to the findings for M-WT, some of the HPLC water droplets froze in M-AL. This indicates that the abundance of impurities in the HPLC water was high enough in the relatively large (∼ 4 µL) drops in M-AL to initiate freezing. Therefore, cumulative nucleus spectra for pure water samples were calculated similarly as for the INP (Fig. 8). Although the number of freezing droplets was 365 relatively small at temperatures higher than 248 K, in some cases (e.g., when low concentrations were used) the INP nucleus spectra had to be corrected by considering the water background spectrum as described below. ( 2015)), we applied a background correction method described below using the f ice spectrum for pure water drops collected prior and during each set of experiments.

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The background spectra were also corrected by shifting the freezing temperature following Vali (2014) with Although Vali proposed the factor 0.66 for the temperature correction of pure water, this parameter depends most probably on the type of impurities in the water. This is also suggested by Fig. 8 since the frozen fraction spectra are significantly different for different water samples, purity grades, and water age. Nevertheless, the temperature correction of Eq. 18 barely shifts the  where fewer freezing measurements were carried out. At surface temperatures below 258 K the difference is about 0.2 K only. Furthermore, the number of kaolinite particles in a 0.1 g L −1 aqueous suspension drop of 2 mm diameter, for instance, is approximately 300000. Thus, numerous particles will occur in the coldest region of the drop. Since a single particle is sufficient to initiate nucleation, the warmer temperature in the drop interior plays a minor role in initiating the freezing.
To correct the measurement data for a temperature shift due to interparticle variability of ice nucleation efficiency, we follow 405 the approach of Herbert et al. (2014) as described in Section 2. We present here only the case of kaolinite as an example; the approach was applied to reconcile the data for all examined materials. Those results are presented in Appendix A. The procedure for correcting for the raw data set in M-AL and M-WT is depicted as a flow diagram in Fig. 10.

Determination of λ
The optimal parameter λ for the temperature shift was determined assuming that with the correct λ value the ln(n s ) data from  Fig. 11a) and the RMSE (root mean square error) between the data and the linear fit were calculated for each set of modified 415 experimental data. The RMSE for the set of λ values for the kaolinite experiments is depicted in Fig. 11b. The optimal λ value, 1.7 in the present case, is corresponding to the minimum of the RMSE curve.This optimal value provides the best linear fit among the tested λs.  To determine the error of λ originating from the measurement error, the following procedure was used. We generated random data points around each of the actually measured data points, but within the bounds of the measurement error (assuming the 420 error bar of the measurement corresponds to 95% confidence interval). Hence, the number of data points for the λ-analysis did not change but each data point was shifted both in temperature and in n s . Then, the optimal λ value for this modified data set was determined by fitting a linear regression curve to the log-linear graph, and subsequently calculating RMSE. We repeated this procedure 1000 times by generating new random data points, and from the statistical analysis of the obtained λ values, ∆λ was calculated. As an example for the procedure, two randomly generated data sets (plotted in red and green colors) and the 425 corresponding RMSE values as function of lambda are shown in Fig. 11b.
The optimal λ value 1.7 was used to correct the temperature shift caused by the residence-time dependency of the freezing process in M-WT and by the cooling-rate dependency in M-AL. The corrected data points together with the fitted regression line are plotted in Fig. 9b. When comparing Fig. 9b to Fig. 9a, the agreement between the corrected data points from the two distinct experimental methods is apparent. This is also supported by the high R 2 value of the regression line.

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The temperature gradient of the normalized freezing rate, ω, was already determined in Section 4.1 from the time dependency of the frozen fraction measured in M-WT. The data points corrected by the ω-based temperature shift (i.e. if ω = λ) are plotted in Fig. 9c together with the best fit line. Again, an obvious agreement can be seen between the two distinct experimental methods.
For a single-component INP, ω is equal to λ, which was found in Herbert et al. (2014) for their kaolinite sample from the Clay

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Mineral Society. In our study the ω = 0.49 and λ = 1.7 values for our kaolinite sample from SigmaAldrich are differing. The deviation in the temperature correction based on λ and ω is further emphasized in Fig. 9c where the regression lines obtained by employing the optimal λ values and ω are plotted by dash-dotted and dashed lines, respectively. This plot suggests that the kaolinite sample investigated in our study is a multi-component system, and the determined λ-value should be employed for correcting the measured freezing temperatures.

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The ω and λ values for the investigated materials are listed in Table 3. After definition of Herbert et al. (2014), all materials exhibit multiple-component behaviour since ω < λ in all cases. Nevertheless, for some materials, e.g., illite NX, despite dif- First, we computed the arithmetic mean curve of the two best fit lines and calculated their mean deviationd from that 445 mean curve. The ultimate question of our statistics tests was, whether the mean deviation is significant with respect to the measurement error and data scatter. Hence, as the next step, the error weighted standard deviations of the residuals s ω and s λ were calculated as where ∆T i is the temperature measurement error, ∆ ω,i and ∆ λ,i are the deviations of the data points from the corresponding best fit curves, while ∆ ω and ∆ λ are the mean values of these deviations. For the significance test we applied a two-sided Student t-test on a significance level of 99.9%, and calculated where N is the number of data points. The null hypothesis was that the two linear curves do not significantly differ, thus, µ 0 = 0 for their deviation from the arithmetic mean curve. In Table 3 we listed the calculated t s,ω and t s,λ , the number of data points, and the tabulated t sig (β = 99.9%) values for the Student t-test for each material. If t s,ω or t s,λ is greater than t sig (β = 99.9%), then the null hypothesis is rejected. That means that the two best fit lines are differing significantly with respect to data scatter and measurement error, and consequently, the material is treated as multiple component on a 99.9% confidence 460 level. Otherwise we consider the material as single component albeit the statistical test does not prove the null hypothesis.
Hence, we classify the material as single or multiple component within our measurement error and data scatter.
As listed in Table 3 which is equal to our result. In contrast, the Fluka kaolinite sample measured by Welti et al. (2012), which is known to contain particles of very ice active feldspar, had a λ value of 2.2 (see Table 2 in Herbert et al. (2014)). In general, we found slightly higher λ values for biological aerosols (FC and MCC) than for mineral dusts. This results in smaller ∆T in Eq. (12) The composite plot of the INAS densities for all investigated materials obtained by M-WT and M-AL measurements is shown in Fig. 12. In accord with the literature (e.g., (Atkinson et al., 2013)), feldspar is far the most efficient ice nucleating particle type among the investigated materials. Besides feldspar, kaolinite has also a high IN efficiency, in particular at higher temperatures. The biological particles (FC, MCC) and the clay minerals illite NX and Sahara dust have similar temperature 485 dependent n s values. The one exception is montmorillonite, which was found to be the least efficient within the investigated temperature range from 248 K to 266 K. In Fig. 12 also shown are parameterizations for feldspar ((Atkinson et al., 2013)) and for desert dust ( (Ullrich et al., 2017)). Our temperature corrected feldspar data fits very well to the parameterization of Atkinson   (Hiranuma et al., 2019). Therefore, we had already corrected the data published in those papers for the freezing temperature shift. Following the procedure depicted in Fig. 10, we were able to 500 bring the INAS densities obtained from the two different methods in line. We have also reconciled our earlier experiments on illite NX (Diehl et al., 2014) and ascertained that those data were burdened with a temperature shift as well. A modification of the data in Diehl et al. (2014) according to our new findings improves further the agreement of the data from M-WT and M-AL (green symbols in Fig. 12).
Taking advantage of having two independent single droplet levitation methods located in our laboratory we determined the 505 material dependent temperature correction factor for the investigated aerosol types based on the analysis method suggested by Herbert et al. (2014). Furthermore, we classified the aerosol materials investigated in this study as single-or multiple- component.
An Based on the experiences collected during the presented synergetic study we suggest the following points for future immer-515 sion freezing studies: -If the instrument used for the measurements utilizes a continuously varying cooling rate, then its temperature adaptation decay has first to be characterized in terms of equilibrium temperature and decay constant and the corresponding uncertainties. Furthermore, the drop temperature has to be measured directly, because it can significantly deviate from the ambient temperature.

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-When comparing the IN efficiencies measured by different instruments utilizing distinct cooling rates, the comparison has to be carried out very carefully and critically. We suggest to use the same cooling rate in the different instruments in such intercomparison studies.
-We note from Fig. 8 that freezing behavior and, consequently, the necessity of background correction depends on the purity grade and age of water used for producing aqueous suspension samples. Therefore, it has to be carefully charac-525 terized for all experiments, as well.
-In case the INAS densities are measured applying a non-standard cooling rate (i.e. CR = 1 K min −1 ), the freezing temperatures have to be corrected following the procedures of Herbert et al. (2014) and the one described in this study.
It has to be taken into account that the temperature correction is material-dependent, and most probably temperature dependent for most of the INP.

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-By the characterization of the aerosol material in terms of temperature correction, statistical significance tests should be carried out taking both the data scatter and the measurement error into account. Of course, by increasing the measurement sensitivity (i.e. decreasing the measurement error) or by decreasing the data scatter (either by improving measurement accuracy, or due to less natural variability of the sample material), the prediction whether the material is single or multiple component will be more accurate. Nevertheless, the classification can only be obtained within the measurement error 535 and accuracy of the applied experimental method.
-In cloud models the cooling rate has to be considered and the freezing temperatures of materials have to be corrected taking the material dependent λ-values into account.
-The λ-based temperature shifts of some aerosol species determined by Herbert et al. (2014) and in the present study serve rather for orientation. We suggest the determination of such temperature shifts for the specific material samples 540 under investigation in each future experimental study on immersion freezing of aerosol particles.
Data availability. The measurement data will be provided upon request.
In this section we provide the experimental results for the determination of the temperature dependent freezing rate (as in Fig. 6), as well as the composite INAS density spectra from M-WT and M-AL using λ and ω (as in Fig. 9) for the materials listed in Table 1.
A1 Fibrous cellulose (FC) Figure A1.       and its surrounding. In order to describe the temperature adaptation process, diffusional and convective heat and mass transfers for water vapor are considered. We will follow the concept of Pruppacher and Klett (2010) in the forthcoming derivation.
Hence, first, the heat and mass transfer of a motionless droplet will be described, and after that the effect of air ventilation will be introduced. The symbols used here are listed in the List of Symbols.

B1 Diffusional heat and mass transfer of a motionless drop in equilibrium 560
When computing the simple case of the diffusional heat transfer of a motionless water droplet in air latent heat from condensation or evaporation is not considered. The rate of heat is calculated by integrating the heat flux density over the entire droplet surface. The heat flux density can be derived from Fourier's law which in spherical coordinates reads as, where r is the distance from the drop center. Thus, the rate of heat transfer of a motionless drop considering pure diffusional 565 heat transfer is The temperature is determined by solving the heat conduction equation which has its form in spherical coordinates for a motionless drop under steady state thermal condition: This partial differential equation is solved using the boundary conditions, where T ∞ is the temperature in the free air, i.e. far away from the drop; and T a is the drop surface temperature, while a is the drop radius. The solution for the temperature as function of r is, Hence, Similarly to the heat transfer, the mass transfer rate of an motionless droplet in equilibrium with its surrounding air is calculated where D v is the water vapor diffusion coefficient, and ρ v is the water vapor density in the surrounding air around the water droplet. The water vapor density can be found by solving the convective diffusion equation, This differential equation simplifies for a motionless drop in steady state to Laplace's equation in the form: ∇ 2 ρ v = 0 (cf.

585
B3).The boundary conditions for the problem are where ρ v,∞ and ρ v,a are the water vapor densities in the air far away from the drop and at the drop surface, respectively. The solution of the governing differential equation is similar to that for the heat transfer. Thus, the rate of change of the mass of a 590 motionless droplet due to diffusion of water vapor under steady state conditions is given by,

B2 Heat and mass transfer of an evaporating drop in air flow
We now consider the more realistic and atmospherically relevant case involving also the effect of air motion around the droplet The total rate at which a drop falling in air gains heat is the sum of the convective heat flux from the air to the drop and the heat 595 loss of the drop by releasing latent heat due to evaporation: where the so called ventilation coefficients f h and f v are introduced accounting for the enhanced heat and mass transfer, respectively, due to ventilation. Thus, for a drop in an air flow f h > 1 and f v > 1, while a motionless evaporating drop can be where c w is the specific heat capacity of water. By equating Eqs. (B13) and (B14) we get the governing equation for the temperature adaptation of the droplet as 605 4π 3 After integration, we obtain the following solution of this differential equation: which gives the steady temperature difference between the equilibrium temperature (T e = T a (t → ∞)) of a ventilated evaporating drop and its surrounding air at a relative humidity of r v . For simplicity we did not indicate but the physical quantities 615 are represented by their averages over the integration interval. Furthermore, we assumed that f h = f v (Pruppacher and Klett, 2014).
After some manipulation, and using ρ v,sat (T e ) = ρ v,a (T e ) one can get two other forms for δ: or, by applying the ideal gas law where e sat (T e ) is the saturation water vapor pressure at temperature T e , and e ∞ is the water vapor pressure in the air. Hence, the equilibrium drop temperature is given as, In M-AL the levitating drop may gain heat from the absorbed acoustic energy at a certain constant rate: where e ac is the acoustic energy density flux and V is the drop volume. e ac varies with time at a very high frequency (≈ 20 kHz) therefore it can be considered as time independent when discussing the slow process of heat transfer. Since this term is independent on temperature and time, it does not affect τ and will not appear in Eq. (B17), which describes the time dependence of the temperature adaptation process. Nevertheless, the absorbed acoustic energy heats up the drop and increases 630 δ with a constant temperature value. This temperature difference between the theoretically calculated equilibrium temperature and the environmental temperature was also observed in M-AL, and calculated to be ≈ 4.5K.

B3 Drop surface temperature in M-AL
Although there is seemingly no air flow around a drop levitating in M-AL, the pressure distribution caused by the acoustic waves does create convection about it (Lierke, 1995). This has to be considered in the temporal evolution calculation in 635 Eq. (B17). In Fig. B1 an example of the measured surface temperature evolution of a 2-mm diameter drop placed into M-AL is plotted by black line. Neglecting ventilation around the drop (i.e. f v = 1.0 in Eq. (B17)) the cooling would be much slower (blue line) than in reality. Setting the ventilation coefficient to f v = 5.2 -which value is close to f v = 3 determined by Lierke (1995) -the temperature evolution follows accurately the measured curve (green line).  The temperature variation of three layers together with the volume averaged drop temperature change relative to the surface 655 temperature are further investigated in Fig. B3. While the temperature at drop's center (red curve) deviates from the surface temperature by up to 2.5 • C, the second outmost layer representing 40% of the entire drop volume (green curve) follows the surface temperature within 0.5 • C at subzero temperatures. At temperatures below −5 • C, which are relevant for immersion freezing experiments, the temperature difference is less than 0.3 • C, thus, within the measurement uncertainties. The volume averaged drop temperature (magenta curve in Fig. B3) is also within the measurement uncertainty of 0.5 • C for this temperature 660 range.
The simulation was carried out without considering any internal circulation, which would further and faster unify the temperature distribution inside the liquid. Considering the large number of ice nucleating particles ( 300.000) immersed in each of the drops, one can conclude, that the surface temperature measured by the pyrometer can be used as characteristic drop temperature.

665
The temperature difference between the surface and the volume average temperature of the drop is compared to the temperature shift calculated for three different λ values in Fig. B4. The figure reveals that the calculated temperature difference inside the drop (magenta curve) is a factor of 6 to 20 smaller than the temperature shift caused by the high cooling rate in M-AL for different λ values (red, green, and blue lines). Therefore, this effect cannot explain the observed freezing temperature shift in M-AL.