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 they contain identical INPs. The rate Rn 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 , for detailed discussion) 1Rn(t,T)=-1Ntot-nfrdnfrdt, where nfr denotes the number of frozen droplets at time t and Ntot the total number of droplets in the population, i.e.,  the total number of the investigated individual droplets. After integrating Eq. () and assuming a constant, i.e., time-independent, freezing rate at a fixed temperature, the well-known expression follows 2R(T)=-ln⁡1-nfrNtott In the stochastic approach, the time dependence of nucleation is taken into account by introducing the heterogeneous-nucleation-rate coefficient Js – similarly to that for homogeneous nucleation – which gives the rate of change in the number of ice embryos per unit surface area of the ice-nucleating particle. (In case of homogeneous nucleation, Jhom is given per unit volume of liquid drop.) If all droplets in the population contain the same amount of particle surface, each ice-nucleating site of the particles is equivalent, and any part of the particle surface has an equal likelihood of containing an ice-nucleating site, the system is named single-component . Then, by definition, 3Js(T)=R(T)A. Here A is the total particle surface area in each aqueous-suspension droplet, which can be calculated as 4A=Vd⋅c⋅SSA, with Vd 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. () cannot be used. Such a system is called multiple-component , which, however, can be divided into subpopulations of equally ice-active entities. Each subpopulation i can be treated as single-component and characterized by its number density ns,i and nucleation rate coefficient Js,i .

The concept of the singular approach is based on the observation that freezing of drops containing INPs occurs at a characteristic temperature once they are subjected to cooling. This also implies that supercooled droplets remain unfrozen arbitrarily long when exposed to a temperature T, even if they contain INPs which trigger freezing only at an INP-specific Tc<T. Hence, the time dependence of ice nucleation is assumed to be of secondary importance in comparison to the particle-to-particle variability in the ice-nucleating ability . In this concept, ice nucleation occurs at particular sites on the surface of a particle, the so-called ice nucleation active sites (INASs), as soon as a temperature is reached which is characteristic of the INP material and its nucleating properties. Reaching this temperature by cooling, the droplet including the INPs freezes instantaneously. For the singular approach, the INAS surface density ns(T) is defined as the cumulative number of sites per surface area that become active between 0 ∘C and T and can be expressed as 5ns(T)=-ln⁡1-fice(T)A, where fice(T)=nfr(T)Ntot is the frozen fraction, i.e., the cumulative fraction of droplets frozen between 0 ∘C and T in the population.

If all droplets were of the same size and contained identical INPs with homogeneous surfaces and uniform ice-nucleating sites, then fice would be a unit step function at a characteristic temperature. Variability in INPs in experiments arising from diverse compositions, particle sizes, and locations of INASs on particles' surfaces results in a distribution of characteristic temperatures, i.e., freezing probabilities of aqueous-suspension droplets, which is represented by fice(T).

In practice, fice can be determined when a population of aqueous-suspension droplets is cooled down continuously or stepwise, and the number of freezing events as a function of time or temperature is registered. Cooling rates in, e.g., freezing-array (or cold-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 frozen droplets per unit temperature interval in a single-component system is then calculated using the stochastic approach by rearranging Eqs. () and () to (e.g., ) 61Ntot-nfrdnfr=-A⋅Js(T)rdT. This equation indicates that the number of frozen droplets at a given supercooling T decreases with an increasing cooling rate. Observations revealed that the nucleation rate coefficient is an exponential function of the temperature : 7ln⁡Js(T)=-λ⋅T+ϕ. The gradient of the logarithm of the nucleation rate coefficient, λ, is a material-dependent parameter, while ϕ is the relative nucleating efficiency of the INPs. The conventional unit of λ is K-1, reflecting its empirical definition by neglecting the units of ln⁡Js. Integration of Eq. () then yields to 8-ln⁡1-nfrNtot=Aλ⋅rexp⁡-λ⋅T+ϕ. This equation shows that the same number of frozen droplets of a population occurs at different temperatures when using different cooling rates. The temperature difference calculated from Eq. () for cooling rates r1 and r2 is 9ΔTf(r1,r2)=1λln⁡r1r2. The shift in the mean freezing temperatures ΔTf was analyzed by for a set of experimental data from the literature. For discussing the data, the temperature derivative of the logarithm of the experimentally determined freezing rate normalized by the aerosol total surface area was utilized: 10ω=-dln⁡(R/A)dT. Following this empirical definition, the unit of ω is K-1, similarly to λ (cf. Eq. ). When λ-1=ω-1, then the single-component stochastic approach leading to Eq. () holds and can be applied for calculating the temperature shift caused by different cooling rates. It was found that for kaolinite and volcanic-ash samples shown in , this approach was applicable. For the majority of the revised data Vali found λ-1>ω-1; thus, the observed temperature shifts were smaller than predicted by the stochastic model. This deviation might be the result of ice-nucleating sites of different effectiveness in INP samples. showed that applying a multiple-component stochastic model can indeed describe this behavior. For single-component systems Eq. () can be applied (i.e., R/A=Js), and therefore, the approach of resulted in ω=λ, while for a multiple-component system ω≠λ. In this approach, λ is calculated from the temperature adjustment, which brings two data sets into agreement. The two data sets in were ns(T) spectra determined either by isothermal experiments utilizing two different residence times or by cold-stage experiments using two different cooling rates. For the former case, used a temperature shift analog to Eq. (): 11ΔTiso(t1,t2)=1λln⁡λ⋅t1t2, where t1 and t2 are the periods of time for which the particles are exposed to a constant temperature. For the cooling rate experiments, ns(T) was determined by applying the singular approach. Although the singular approach excludes any temperature shift due to a change in cooling rate, there is experimental evidence contradicting this prediction (e.g., ). Such observations resulted in the so-called modified singular description , which offsets the ns(T) spectrum to lower temperatures when higher cooling rates are applied. In accordance with this empirical description, shifted the ns(T) spectra to 12ns(T)→nsT-ln⁡|r|-λ. From their analysis revealed that kaolinite and volcanic-ash samples can be described by the single-component stochastic approach, whereas for K-feldspar and mineral dust a multiple-component approach has to be applied.

For comparative analysis of the ice-nucleating ability of particles investigated by different experimental approaches, λ is a crucial parameter. Large values of λ indicate effective INPs and, therefore, weak time dependence, while less-effective INPs possess small λ values. determined λ for a set of ice-nucleating materials and compared it to several literature data. The large variability in λ on the material of the INPs necessitates further quantification of λ for other atmospherically relevant INP species. The lack of laboratory data of λ and ω in the literature was also highlighted by .

In this study we measured the frozen fraction fice(T) of seven INP materials with two different methods, one utilizing isothermal conditions (M-WT) and the other demonstrating a continuously decreasing temperature (M-AL). From the isothermal M-WT experiments the freezing rate and its gradient ω were determined. Furthermore, from the measured fice(T) we calculated the INAS density ns using Eq. (). From the non-isothermal M-AL measurements ns(T) was obtained by applying the singular description. Following the approach of , the temperature adjustment was determined, which brings the two ns spectra into agreement. In this way the material-dependent parameter λ was calculated. We analyzed the λ and ω values of different INP materials and tested whether a single- or a multiple-component description can be applied to model their ice nucleation behavior. For the temperature shifts we normalized the cooling rate in Eq. () using a standard value of 1 K min-1, which results in 13ΔTf=1λln⁡1|rexp|. In isothermal experiments the temperature shift was normalized by applying a standard time of 60 s (corresponding to a cooling rate of 1 Kmin-1) as 14ΔTiso=1λln⁡λ⋅tres60s, where tres is the characteristic residence time of droplets in the experiments.

The isothermal M-WT and cooling rate M-AL measurements were related to each other in terms of frozen fraction. The total observation time in the M-WT experiment was calculated to reach the same fice as a cooling rate experiment using the equation given by : 15tiso=1λ⋅r. Applying the standard cooling rate of 1 Kmin-1 and a typical λ value of 2, a total observation time of 30 s is obtained.

As becomes obvious from the description above, the correct representation of the drop temperature in freezing experiments is of 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 on the order of microliters. In single-droplet levitation techniques, as in the 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 cooling becomes significant for drops with volumes in the microliter range (equivalent to 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.  following the concept of . After injection, the relatively warm droplet cools down (stage 1 in Fig. ), and its surface temperature Ta approaches an equilibrium temperature Te determined by the ambient temperature T∞, the dew point, and ventilation (; and Appendix ): 16Ta(t→∞)=Te. 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 airflow, the temperature difference between the drop and its environment is further enhanced by ventilation, resulting in a net temperature deviation δ (see Appendix ): 17T∞-Te=δ. The temporal evolution of the surface temperature for a droplet placed in a cold environment is described mathematically by an exponential-decay function (see Appendix  for the derivation): 18Te-Ta(t)=T∞-Ta(t)-δ=T∞-Ta(t=0)-δexp⁡-t/τ, with τ being the relaxation time, i.e., the time constant of the temperature adaptation. The main physical parameters that determine τ, and therefore the total cooling time of the droplet, are the drop size, the ventilation coefficient, and the ambient temperature and dew point (see Appendix ). Hence, for given experimental conditions, the temporal evolution of the drop's surface temperature in stage 1 can be calculated using Eq. (). In cold-stage experiments, freezing-stage 1 proceeds very quickly due to the large contact area ; in single-droplet 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 the supercooled drop, rapid kinetic crystal growth takes place (stage 2 in Fig. ). 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 Sect. ). Subsequently, a diabatic freezing of the whole droplet takes place (stage 3). The temporal duration of this freezing stage is determined by the heat exchange between the particle and its environment; therefore it proceeds slower than stage 2. In the end, the frozen particle cools down to the ambient temperature (stage 4).

The experiments were carried out using seven different types of materials, which are listed in Table . All of these materials are considered to be important constituents of atmospheric ice nucleation particles. We investigated two cellulose types as biological INP surrogates: 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 component of desert dusts so that, by scaling down, it is representative of dust samples in dependence on their composition . Illite NX can be considered to be a proxy for desert dust since their mineralogical compositions are similar . Montmorillonite K10 and kaolinite (Sigma-Aldrich) are commercially available and characterized mineral dust materials of relevance for the atmosphere, which also have been the 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 measurement techniques during the INUIT09 measurement campaign (Ullrich et al., 2019).

Atmospherically relevant INPs exhibit an extremely wide range in their heterogeneous-freezing ability. Furthermore, there is a large spread in the specific surface area (SSA) of the investigated materials from around 1 to 245 m2g-1 (see Table ). We therefore chose 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 droplets in the M-AL was approximately 20 times larger than in the M-WT, we used reduced mass concentrations in the M-WT to obtain overlapping freezing curves with the two methods (see Eq. ).

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 high-precision liquid chromatography, HPLC, Sigma-Aldrich). Between the measurement runs the aqueous suspension was continuously stirred at a very low rate using a magnetic stirrer to avoid coagulation and sedimentation of the particles in the suspension. A hypodermic syringe was used to inject suspension droplets into the measuring instruments. For the M-AL measurements, the syringe was filled with aqueous suspension after an idle time of about 30 min without stirring (following the sample preparation protocol of ) so that at the uppermost part of the solution a homogeneous suspension was generated. For the M-WT measurements we abandoned an idle time because in this case we could presume already-homogeneous suspension due to the low particle concentration. Furthermore, the syringe was shaken prior to droplet injection in both M-AL and M-WT experiments to homogenize the particle distribution in droplets. Otherwise no pre-treatment procedures were applied.

Although efforts were made to unify and standardize the sample generation, we cannot rule out INP surface area variation among the investigated droplets. There are several sources of uncertainty in total surface area inside droplets like inhomogeneous distribution of particles among injected droplets, externally or internally mixed particles, aggregation due to sedimentation, and internal circulation. The most appropriate way to determine the actual INP surface area would be the continuous measurement of the surface area inside each droplet under investigation, but that seems not feasible currently. Another possibility is the measurement using size-selected particles, as in the study of . Neglecting the variability in composition and surface area of INPs may introduce significant error in calculated ns(T) and Js(T) . Furthermore, the assumption of identical INP surface area in each droplet imposes a cooling rate and surface area dependence on Js(T) . In our analysis we considered the error sources in concentration determination, in SSA, and in droplet size for determining the propagated error for the calculated parameters.

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

Figure  shows the INAS densities computed using Eq. () from fice spectra obtained from M-WT measurements of kaolinite with concentrations of 0.1 and 1.0 gL-1, marked with light- and dark-blue symbols, respectively. The number of data points is limited to five, which is the issue of the M-WT experiments being very laborious for collecting statistically relevant numbers of measurements for each temperature. Comparing Fig.  with the INAS densities of other investigated materials presented in Appendix  reveals that kaolinite is a good atmospheric INP, exhibiting large ns values that, nevertheless, vary steeply over 1 order of magnitude within the investigated temperature range of only 4 K, i.e., here from 252 to 256 K.

For computing ns for Fig.  using Eq. (), the fraction of frozen droplets fice was determined by employing the singular approach, i.e., by counting the number of droplets frozen in an experimental run, disregarding the time from injection until freezing. A droplet remaining liquid for up to 35 s (i.e., the end of the experimental run) was classified as unfrozen.

From the time-resolved measurement data from the M-WT, the time dependence of the freezing process was analyzed. For that, Eq. () was rearranged to 21ln⁡nliq(T,t)Ntot=-R(T)⋅t, where nliq(T,t)=1-nfr(T,t) is the number of droplets remaining liquid after time t at temperature T. Figure  depicts the time dependence of liquid ratio from the M-WT measurements at five different temperatures and using the two distinct concentrations of kaolinite as for Fig. . The times needed for the injected droplets to reach their equilibrium temperatures (i.e., 6 s; see Fig. ) were subtracted from the recorded time interval between injection and freezing. At lower temperatures and with higher particle surface areas per drop (i.e., higher INP concentration), the curves get steeper, indicating that freezing proceeds faster. Figure a clearly shows the expected exponential decay of liquid drops predicted by the stochastic approach. The temperature dependence of the normalized freezing rate according to Eq. () as shown in Fig. b was determined by computing the slopes of the curves in Fig. a and dividing them by the total surface areas of INPs immersed in the examined water droplets. Figure b reveals the expected linear dependency of R/A in agreement with Eq. (). Hence, ω, which is the slope of R/A (see Eq. ), can readily be determined for the investigated kaolinite sample from Fig. b by linear regression as ω=0.49±0.03. (Here the error in ω is the standard error in the linear fit.) Note that if the INPs can be considered to be single-component, then Js=R/A. In our experiments, the total surface area A was estimated from the concentration of the aqueous solution and from the specific surface area. Accurately measuring the actual total surface area of INPs inside the droplets, which should be taken into account for calculating ω and Js, is currently not feasible. Therefore, the error in A might be significantly higher than estimated, which would result in a false classification of the INPs as single-component.

Frozen fractions of kaolinite suspension with 1.265 gL-1 concentration as a function of temperature are shown in Fig. a. Error bars are associated with the temperature bin interval (±0.5 K), and the uncertainty in the determination of fice stems from the counting statistics and the experimental temperature uncertainty. The active-site density ns calculated from Eq. () using fice in Fig. a is plotted in Fig. b. Here the error bars originate from Gaussian error propagation when using the measured data in Fig. a. From the calculation we excluded the data points for which fice was above 90 % or below 10 %. This cutoff was introduced because in these cases the uncertainty in fice was very large due to the poor counting statistics when freezing or unfreezing events occur very rarely.

Another criterion for using fice for further evaluations was that it should significantly exceed the background nucleation caused by impurities in the water used for generating the aqueous suspension. To determine this background spectrum, we investigated pure water droplets before each experimental run in the M-AL, similarly to the M-WT measurements. However, in contrast to the findings for the M-WT, some of the HPLC water droplets froze in the M-AL. This indicates that the abundance of impurities in the HPLC water was high enough in the relatively large (∼4 µL) drops in the M-AL to initiate freezing. Therefore, fice spectra for pure water samples were measured in a similar way as for the INPs (Fig. ). Although the number of freezing droplets was 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.

In earlier experiments in the M-AL (e.g., ), in-house-produced Milli-Q water was used as a solvent for the aqueous solutions. Therefore, we also analyzed the Milli-Q water in our present experiments. The results are plotted by black symbols in Fig. . Apparently, using HPLC water from a freshly opened chemical bottle (red symbols) reduces the background fice. Nevertheless, the fice spectrum of HPLC water changed with time and increased significantly after about a year (magenta symbols), indicating an aging effect. This behavior of different water types is in accord with the finding of . Since it is difficult to eliminate the contribution of INPs still present in high-purity water (see Fig. , and ), we applied a background correction method described below using the fice spectrum for pure water drops collected prior and during each set of experiments.

The background spectra were also corrected by shifting the freezing temperature following with 22β=0.66lg(r).

Although Vali proposed the factor of 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.  since the frozen-fraction spectra are significantly different for different water samples, purity grades, and water age. Nevertheless, the temperature correction of Eq. () barely shifts the background spectra: β=0.46 for 252 K, where the cooling rate in the M-AL is approximately 252 Kmin-1. Such a temperature shift would increase the background frozen fraction by less than 0.05. Therefore, no background subtraction (as, e.g., in ) was applied, but a cutoff temperature was defined where the difference between the background and the INP spectra was less than 0.05. This correction method was only necessary for FC and MCC in our experiments, while the other materials initiated freezing at higher temperatures for the investigated concentrations.

The parameter λ for the temperature shift was determined assuming that with the correct λ value, the ln⁡(ns) data from the two M-WT and M-AL experiments converge onto one single curve. Therefore, the temperatures of the unmodified data were shifted by applying Eq. () to the isothermal experimental data from the M-WT and Eq. () to the data obtained using the continuous-cooling approach of the M-AL. For each investigated INP species a set of λ values varying from 0.1 to 8.0 in 0.1 steps applied for the modification in Eqs. () and (). A linear fit to the derived ln⁡(ns) (solid black line and black data points in Fig. a) and the RMSE (root mean square error) between the data and the linear fit were calculated for each set of modified experimental data. The RMSE for the set of λ values for the kaolinite experiments is depicted in Fig. b. The optimal λ value, 1.7 in the present case, corresponds to the minimum of the RMSE curve. This optimal value provides the best linear fit among the tested λ values.

To determine the error in λ originating from the measurement error, the following procedure was used. We generated random data around each of the actually calculated ns data points but within the bounds of the measurement error (assuming the error bar of the measurement corresponding to 1σ). Hence, the number of data points for the λ analysis did not change, but each data point was shifted in both temperature and ns. Here, the distribution of ns and T was not considered; random values within the error bounds were taken. Then, the optimal λ value for this modified data set was determined. We repeated this procedure 1000 times by generating new random data points, and from the statistical analysis of the obtained λ values, Δλ was calculated. Choosing random values within the 1σ bounds around the mean data and neglecting values outside this bound might result in overestimating Δλ. As an example for the procedure, two randomly generated data sets (plotted in red and green colors) and the corresponding RMSE values as a function of λ are shown in Fig. a and b, respectively.

The optimal λ value of 1.7 was used to apply the temperature shift caused by the residence time dependency of the freezing process in the M-WT and by the cooling rate dependency in the M-AL. The modified data points together with the fitted regression line are plotted in Fig. b. When comparing Fig. b to Fig. a, the agreement between the modified data points from the two distinct experimental methods is apparent. This is also supported by the high R2 value of the regression line.

The temperature gradient of the normalized freezing rate, ω, is determined in Sect.  from the time dependency of the frozen fraction measured in the M-WT. The data points modified by the temperature shift Eq. () presuming ω=λ are plotted in Fig. c 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 for their kaolinite sample from the Clay Mineral Society. In our study the ω=0.49 and λ=1.7 values for our kaolinite sample from Sigma-Aldrich differ. The deviation in the temperature correction based on λ and ω is further emphasized in Fig. c, 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 has to be treated as a multi-component system, and the determined λ value should be employed for modifying the measured freezing temperatures.

The ω and λ values for the investigated materials are listed in Table . After the definition of and , all materials exhibit multiple-component behavior since ω<λ in all cases. Nevertheless, for some materials, e.g., illite NX, despite different λ and ω values the deviation between the data sets modified using λ or ω was not obvious (see Appendix ). To obtain further insights into this feature, we performed statistical-significance tests as follows.

First, we computed the arithmetic-mean curve of the two best-fit lines corresponding to λ and ω, respectively, and calculated their mean deviation d‾ from that 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 sω and sλ were calculated as 23sω=∑i=1NΔω,i-Δω‾2ΔTi2∑i=1N1ΔTi224sλ=∑i=1NΔλ,i-Δλ‾2ΔTi2∑i=1N1ΔTi2, where ΔTi is the temperature measurement error, Δω,i and Δλ,i are the deviations of the corrected data points from the corresponding best-fit curves, and Δω‾ and Δλ‾ are the mean values of these deviations. For the significance test we applied a two-sided Student t test to a significance level of 99.9 % and calculated 25ts,ω=|d‾-μ0|sω⋅N26ts,λ=|d‾-μ0|sλ⋅N, where N is the number of data points. The null hypothesis was that the two linear curves do not significantly differ; thus, μ0=0 for their deviation from the arithmetic-mean curve. In Table  we listed the calculated ts,ω and ts,λ, the number of data points, and the tabulated tsig (β=99.9 %) values for the Student t test for each material. If ts,ω or ts,λ is greater than tsig (β=99.9 %), then the null hypothesis is rejected. That means that the two best-fit lines differ significantly with respect to data scatter and measurement error, and consequently, the material is treated as multiple-component on a 99.9 % confidence level. Otherwise we consider the material to be single-component, although 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 , according to our statistical test, kaolinite, feldspar, montmorillonite, and Sahara dust are multiple-component, while illite NX, FC, and MCC are single-component INPs. This implies that the definition of to distinguish between single- and multiple-component samples on the basis of λ and ω values cannot directly be adapted to our M-AL and M-WT experiments. This is the consequence of the adaptive cooling of the drops in the M-AL, which results in a temperature dependence on the λ-based correction. Thus, the same λ value caused a higher temperature correction at higher temperatures (see Fig.  in Appendix ). Therefore, our analysis indicates that statistic tests have to be performed considering both data scatter and measurement error to compare the λ and ω values. This procedure improves the classification of the materials as single- or multiple-component.

The statistical tests supported the finding that the kaolinite that we analyzed is multiple-component. That contradicts the finding of , who showed their kaolinite sample (KGa-1b from the Clay Mineral Society) to be single-component with λ=ω=1.12. This indicates that these two kaolinite samples are different, and thus the result outputs cannot directly be compared since the ice nucleation activity of materials depends on their specific chemical composition, which is known to be very variable for kaolinite. For example, the λ value for the kaolinite used in the cooling experiments of was 1.7, which is equal to our result. In contrast, the Fluka kaolinite sample measured by , which is known to contain particles of very-ice-active feldspar, had a λ value of 2.2 (see Table 2 in ). In general, we found slightly higher λ values for biological aerosols (FC and MCC) than for mineral dusts. This results in smaller ΔT in Eq. (), and hence, biological INPs show a weaker time dependence, in agreement with the findings of and . For the investigated samples in our experiments, the temperature correction ranged from ≈0.5 K up to several kelvins, depending on the material's λ value (see also Fig. ).

The composite plot of the INAS densities for all investigated materials obtained by M-WT and M-AL measurements is shown in Fig. . In accord with the literature (e.g., ), feldspar is by far the most efficient ice-nucleating particle type among the investigated dust materials. Besides feldspar, kaolinite also has a high ice nucleation efficiency, in particular at higher temperatures. The biological particles (FC, MCC) and the clay minerals illite NX and Sahara dust have similar temperature-dependent ns values. The one exception is montmorillonite, which was found to be the least efficient within the investigated temperature range from 248 to 266 K. Also shown in Fig.  are parameterizations for feldspar and for desert dust . Our temperature-corrected feldspar data fit very well to the parameterization of , which was based on cold-stage experiments, i.e.,  using aqueous suspensions of INP material. In contrast, the desert dust parameterization of is based on dry-deposition experiments and predicts higher INAS densities as measured in the M-WT and M-AL. This is in accord with the literature as for example revealed different INAS densities when dry-deposition or aqueous-suspension techniques were utilized.