Time-dependence of Heterogeneous Ice Nucleation by Ambient Aerosols: Laboratory Observations and a Formulation for Models

. The time dependence of ice-nucleating particle (INP) activity is known to exist, yet for simplicity it is often omitted in atmospheric models as an approximation. Hitherto only limited experimental work has been done to quantify this time 10 dependency, for which published data are especially scarce regarding ambient aerosol samples and longer time scales. In this study, the time dependence of INP activity is quantified experimentally for ambient environmental samples. The experimental approach includes a series of hybrid experiments with alternating constant cooling and isothermal experiments using a recently developed cold-stage setup called the Lund University Cold-Stage (LUCS). This approach of observing 15 ambient aerosol samples provides the optimum realism for representing their time dependence in any model. Six ambient aerosol samples were collected representing aerosol conditions likely influenced by these types of INPs: marine, mineral dust, continental pristine, continental polluted, combustion-related and rural continental aerosol. Active INP concentrations were seen to be augmented by about 40% to 100% (or 70% to 200%), depending on the sample, 20 over 2 (or 10) hours. This degree of time dependence observed was comparable to that seen in previous published works. Our observations show that the minority of active ice nuclei (IN) with strong time dependency on hourly time scales display only weak time dependence on short time scales of a few minutes. A general tendency was observed for the natural time scale of the freezing to dilate increasingly with time. The fractional freezing rate was observed to steadily declines exponentially with the order of magnitude (logarithm) of the time since the start of isothermal conditions. A representation of time dependence 25 for incorporation into schemes of heterogeneous ice nucleation that currently omit time dependence is proposed. 555 atmospherically relevant (Kanji et al . 2017), as used in some IN schemes (DeMott 2015; Phillips 2008, 2013). However, in the present project the samples investigated are from the ambient environment and must be assumed to contain a complex composition, where multiple INP species are abundant. Compared to opting for more well-defined artificial samples (e.g. Arizona test dust, Snowmax®) as done in some past studies (Welti et al . 2012; Budke et al . 2014), the approach of sampling aerosols from the real troposphere entails several challenges (sections 1 and 2.2). In particular, the identity of the INPs 560 dominating the ice initiation in each of our samples is uncertain. for the classification assumed above, which then in turn is added to the temperature input for the IN scheme.


Introduction
The presence of ice nucleating particles (INPs) has been shown to influence cloud formation and cloud properties, precipitation and thereby both local and global weather systems and climate (Phillips et al. 2003;Gettelmann et al. 2012;Kudzotsa 2014;30 Storelvmo 2017;Phillips and Patade 2021). Even though INPs have been studied for many decades, some aspects of their influence are still not fully understood (DeMott et al. 2011). One aspect where much uncertainty remains is the relevance of time in atmospheric ice processes.
According to the Intergovernmental Panel of Climate Change (Stocker et al. 2013) much of the uncertainty in projections of 35 climate change by current global models are associated with the effects of atmospheric particles and aerosol-cloud-radiation interactions, and recent reviews indicate that a large degree of uncertainty prevails (Bellouin et al. 2020;Sherwood et al. 2020).
The mechanisms for aerosol interaction with cold clouds have been explored with cloud models and are complex (Kudzotsa et al. 2016), as is also true globally (Storelvmo 2017).

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An emerging area of interest is ice initiation for which there are many possible pathways (Cantrell and Heymsfield 2005;Phillips et al. 2007Phillips et al. , 2020. At any given moment, only a small fraction of all condensed water in the atmosphere resides in the form of ice crystals. However, this small fraction has a disproportionately large impact on global precipitation which is mostly associated with the ice phase (Field and Heymsfield 2015). Homogeneous freezing of cloud droplets in the atmosphere normally occurs at about -37 °C (Heymsfield et al. 2005), but heterogeneous freezing can occur at much warmer temperatures 45 in the presence of rare, usually solid, aerosol particles that catalyse ice formation, termed ice nucleating particles (INPs). The presence of INPs has been shown to influence cloud formation and cloud properties (Phillips et al. 2003;Cantrell and Heymsfield 2005;Boucher et al. 2013;Kudzotsa 2014;Kudzotsa et al. 2016), precipitation (Lau and Wu 2003;Lohmann and Feichter 2005) and thereby both local and global weather systems and climate (Murray et al. 2012;Schill et al. 2020a,b;Sanchez-Marroquin et al. 2020). INPs can be influential since they initiate crystals that can grow to become snow and graupel, 50 which may melt, forming the 'ice crystal process' of precipitation production (Rogers and Yau 1989).
The first ice in any mixed phase cloud is from activation of INPs. These have variable chemical composition, concentrations and activities in nature (Knopf et al. 2021). Mineral dust particles (e.g. from deserts) and soil dust particles may efficiently act as INPs of relevance to mixed-phase clouds (e.g. Kanji et al., 2017). A range of primary biological aerosol particles (PBAPs) 55 such as e.g. bacteria, viruses, marine exudates, phytoplankton, fungal spores, pollen, lichen and plant fragments may facilitate immersion freezing potentially at relatively high temperatures (e.g. Kanji et al., 2017). It is less clear to what extent combustion emissions, for example soot particles, may play a role as INPs in mixed-phase clouds (e.g. Kanji et al., 2017). For some thin wave clouds without precipitation, the observed number concentration of ice crystals is similar to that of active 60 ice nuclei in the cloud (Eidhammer et al. 2010). Clearly, in such cold clouds, the chemistry and loading of INPs in the environment must affect the cloud properties. More generally there are many other potentially more prolific mechanisms for ice initiation (Cantrell and Heymsfield 2005;Field et al. 2017). Anyway, it is beneficial to simulate the first ice in mixed phase clouds accurately if detailed models are to represent the subsequent ice-microphysical processes adequately.

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There exists a paradox in observations of natural mixed-phase stratiform glaciated clouds. Westbrook and Illingworth (2013) observed that ice precipitation from such a thin layer-cloud (mixed-phase from -12 to -13 °C) persisted for many hours and was produced by the ice crystal process. They argued qualitatively that this longevity could not be explained by mixing of environmental INPs into the layer cloud as the vertical motions were very weak and the cloud top level was constant, although they did not quantify the turbulent fluxes of INP entrained from the environment. They hypothesised that time dependence of 70 activity of ice nuclei (IN) was the cause. Despite in-cloud temperatures being isothermal, the stochastic nature of IN activity meant that a weak yet persistent long-term source of primary crystals would arise from this time dependence over times of many hours. However, their interpretation of the observations with this hypothesis has been contested by Ervens and Feingold (2013), who instead suggested the alternative explanation of in-cloud vertical motions and weak turbulence continually activating INPs. The issue has not been resolved conclusively. 75 The first lab studies of time dependence of freezing began in the 1950s (reviewed by Pruppacher and Klett, 1997; 'PK97').
Two categories of models were proposed to explain the lab data, one with time dependence ('the stochastic hypothesis' that eventually became classical nucleation theory), (Bigg 1953ab) and one without time dependence ('the singular hypothesis', sometimes referred to as 'the deterministic model'), (Langham and Mason 1958). The singular hypothesis is an approximation 80 and treats ice nucleation as a process occurring on active sites that become active instantaneously at distinct conditions that vary statistically among the INPs. An ice crystal is initiated immediately when an INP's characteristic conditions of freezing temperature and humidity are reached, as if it were a digital switch. This neglect of time dependence yields a simple dependence of primary ice initiation on thermodynamic vertical structure of the environment.

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Classical or stochastic theory assumes that that embryonic ice clusters are continuously forming and disappearing (reviewed by PK97) at the interface of immersed aerosol particles. This is assumed to occur with a frequency that depends on the temperature. If an ice embryo reaches a critical size of stability, determined by the features of the surface, then ice is nucleated.
Although modern lab observations have confirmed the existence of time dependence of IN activity, nevertheless the singular hypothesis is still used in most cloud models owing to its validity as an approximation to the leading order behaviour of crystal 90 initiation. Moreover, classical stochastic theory ('classical nucleation theory') can be difficult to represent because it is complex. In reality there is a probability distribution of efficiencies of active sites among INPs of any given aerosol species, https://doi.org/10.5194/acp-2021-830 Preprint. Discussion started: 15 November 2021 c Author(s) 2021. CC BY 4.0 License. even for a population of identically sized particles (Marcolli et al. 2007). Classical nucleation theory can easily over-predict active INPs by orders of magnitude if such statistics are not properly considered.

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In recent decades, laboratory experimental work to investigate time dependency of heterogeneous ice nucleation has been scarce. This is especially true for ambient environmental aerosols. Vali (1994) reported his earlier results for a small number of isothermal experiments (4 simple isothermal and 4 with a brief warming of the sample by 0.5 K or 1.3 K before the isothermal period) with an isothermal period of 10 to15 min on samples described as "distilled water containing freezing nuclei of unknown composition", and isothermal temperatures between -16 and -21 °C. They concluded that the rate of freezing was 100 dependent both on temperature and on time (see also Vali and Stansbury 1966). Welti et al. (2012), investigated aerosolized kaolinite particles in the immersion mode with the Zürich Ice Nucleation Chamber (ZINC) and observed time dependence by altering the flowrate through the instrument. They concluded that immersion freezing is at least partly a stochastic phenomenon, and recommended that time dependence should be included in numerical calculations of the evolution of mixedphase clouds. 105 Wright and Petters (2013) did experiments with Arizona Test Dust (ATD) with a droplet freezing array for various cooling rates between 0.1 and 5 K/min. They also included data for a total of two isothermal experiments (13.3 and 15.9 hours respectively). They concluded that their results implied a limited effect from time dependence equivalent to a few degrees of error in the freezing temperature of aqueous droplets. Equally, Budke and Koop (2014) performed experiments with the 110 commercially available snow inducer, Snowmax®, which was derived from Pseudomonas syringae, in the Bielefeld Ice Nucleation ARraY (BINARY). The technology applied in the present study is similar to that of BINARY. Budke and Koop measured time dependency with experiments for cooling rates ranging between 0.1 and 10 K/min and were able to show a weak dependency on time for Snowmax®. Knopf et al. (2020) investigated time-dependent freezing of illite for up to 2 hours and confirmed that classical nucleation theory applies. 115 None of these aforementioned studies investigated environmental aerosol. No experiments studied time periods longer than a few minutes, except for Vali (1994), Wright and Petters (2013) and Knopf et al. (2020). Only a limited degree of time dependency has been observed. For example, an enhancement by about 50% in numbers of active INPs during the initial 20 s for a frozen fraction of about half was measured by Welti et al. (2012). Yet Vali (1994) observed this 50% change after 120 about 15 min. In view of such controversy about the extent of time dependence in ambient INPs, the aim of the current study is to use an experimental approach to quantify time dependency for ambient environmental samples and to suggest a way to represent it in atmospheric models.
Our rationale here is that the optimum approach is to study ambient aerosol sampled directly from the environment if the time 125 dependence of IN activity in atmospheric clouds is to be understood. The empirical parametrization by Phillips et al. (2008, https://doi.org/10.5194/acp-2021 November 2021 c Author(s) 2021. CC BY 4.0 License. 2013) follows a similar approach by treating the dependency of active INPs on chemistry, size and loading of aerosol species in terms of field observations of the background troposphere. Studying ambient aerosol samples provides the optimum representativeness of the aerosols observed, conferring realism on the cloud models that use the inferred schemes. On the other hand, there is an inevitable cost from lack of identification of the precise chemical species initiating the ice in observed samples. 130

Overview
There were three major stages to the experimental work performed in this study. Firstly, aerosol samples were collected for a period of about a year (from 2020-02-28) at the Hyltemossa research station, which is located in southern Sweden. Background aerosol data was also collected at the research station during this period. 135 Secondly, the collected data were analysed to identify candidate samples that could be assumed to be likely dominated by, or at least possibly influenced by, aerosol particles representing six broadly defined aerosol classes: • Marine dominated aerosol Thirdly these candidate samples were analysed with respect to ice nucleation activity with a combination of experiments for both continuous cooling rates and isothermal experiments for more than 10 hours.
The experimental setup enables automated control of the evolution over time of temperature of the freezing array for many hours (> 10) with minimal risk of contamination. As noted below, any sample may be exposed to repeated freezing 150 experiments with high precision (Sec. 2.3). This enables fresh questions to be addressed about time dependence of ice nucleation in natural clouds. Ambient air samples were collected at the Hyltemossa research station in southern Sweden (56°06'00"N 13°25'00"E).
Hyltemossa is located in a forested area, and it is part of the ACTRIS network. Daily air samples were collected with a continuous sequential filter sampler (model SEQ47/50-RV, Sven Leckel Ingenieurbüro GmbH, Berlin, Germany) with a PM10 inlet and a flowrate of 1 m 3 /h. The samples were collected on 47 mm polycarbonate track-etched membrane filters with 0.4 µm pore size. The sampling was set to 24-hour sampling for each filter and filter change was initiated at midnight. Because of 160 the high pressure drop over the membrane filters not all filters were able to achieve a full 24-hour sampling. This issue unfortunately limited the selection of available samples, but the sampling coverage was deemed sufficient for this study. Filters were retrieved from the field station every 1-2 weeks, placed in sterile petrislide filter cassettes and stored at a temperature of about -20°C until analysis. Field blank samples were collected and handled in a manner identical to that for the sampled filters.

Sample classification according to likely dominant composition of INPs
Many aerosol samples were collected as noted above (Sec. 2.2.1). These were classified into six basic aerosol types as follows (section 2.1). In this study, we aimed at selecting samples likely to be dominated by different INP types at least of relevance to Northern Europe, and most likely of wider spatial-temporal relevance. In Table 1 The BC concentration spans about one order of magnitude from 0.04 to 0.4 μg/m 3 between the selected samples, and the BC level correlates with the PM1 level between these samples. In southern Sweden, the main components contributing to the PM10-PM1 are typically sea salt and/or dust particles. The 'combustion dominated' sample was so labelled, due to elevated levels of BC and relatively low levels of PM10-PM1 in the ambient air. It was sampled on Dec. 7 th during the heating season over a 180 relatively short time window (local midnight to about 6:30 am). Hence, a pronounced fraction of the sampled PM was likely to originate from residential biomass combustion from local and/or Eastern European areas, as indicated by the back trajectories ( Fig. 1 The 'continental polluted' sample appears similar to the 'combustion dominated' sample in terms of the BC, PM1 and back 185 trajectories passing over Eastern Europe during the residential heating season in March. However, the PM10-PM1 level is significantly higher for the 'continental polluted' sample (7.8 versus 3.0 μg/m 3 ). The PM10-PM1 time series (not shown) peaks in the local afternoon, and we speculate that it in part is associated with local soil dust. Hence, it is not entirely clear whether (soil) dust and/or potentially combustion emissions may dominate the INP population in that sample.

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The 'marine' sample (Sec. 2.1) is characterised by relatively low levels of BC, air mass back trajectories from the Northern Atlantic, and a relatively high level of PM10-PM1 (7.1 μg/m 3 ), which may be associated with coarse sea salt particles. However, There were only limited local aerosol measurements in February, 2021, due to instrumental break down. Nevertheless, we consider the INP population in the sample from Feb. 23 rd , 2021 likely to be dominated by Saharan dust for the following 200 reasons. The online SKIRON dust forecast predicted elevated levels of Saharan dust to be present in the boundary layer in southern Sweden some days around Feb. 23 rd . Several urban measurement stations in southern Sweden showed highly elevated levels of PM10 during those days (www.dagensluft.se), with a typical maximum on Feb. 23 rd . This period was characterised by unusually warm temperatures in southern Sweden, and the transport of Saharan air masses was covered in most popular media.
Also, the modelled 120h air mass back trajectories originate in the Mediterranean Sea with Sahara as a potential source region. 205 To summarise with respect to the 'Mineral dust influenced sample', we only have indirect evidence of high levels of Saharan dust in that sample. However, we found this sample to be the most likely candidate with an INP population dominated by dust.
Nevertheless, we cannot rule out alternative European sources contributing to INPs in that sample.

210
Although all six samples could be classified approximately by their airmass type of origin, there was an inevitable lack of certainty about the chemical identity of their active INP. This limitation was a consequence of our overall approach of sampling ambient populations of INP from the real atmosphere. The mineral dust influenced and rural continental samples might be expected to have been enriched in mineral dust and PBAPs respectively, in view of the back-trajectory analysis (Sec. 2), but this is not proven by physico-chemical analysis. The rural continental sample had passed over half the length of Norway 215 and southern Scandinavia, although it may have originated previously near the Arctic Ocean.

Sample preparation
The sampled filters were cut and a half filter was placed in sterile cryogenic vials while the other half of the filter was refrozen and stored. Two mL of ultra-pure water (18.2 MΩ, <3 ppb VOC) was added to the vials and the sample was shaken at highest 220 effect for 3 minutes on a laboratory vibrating vortex shaker. All sample preparation and handling of sampled filters were done in an ultra-clean environment in a laminar airflow cabinet, and all pipetting of sample and water was done with sterile pipette tips, discarded after single use. Field blank samples were treated identically to the collected filter samples.

Experimental apparatus for measuring the ice activity of the samples 225
The freezing apparatus used to perform the experimental work in this study was designed using elements inspired from several previously described similar cold-stage setups (Wright and Petters 2013;Budke and Koop 2014) and was named the Lund University Cold-Stage (LUCS). A schematic overview of the freezing array and the LUCS system is shown in Figure 2.
In LUCS one hundred 1 µL sample drops are dispersed on siliconized hydrophobic glass slides, mounted in a freezing assembly on a 40 x 40 mm temperature-controlled stage, (here forth termed the cold-stage) and control system (model LTS120, Linkham 230 Scientific, Tadworth, United Kingdom). The cold-stage works by means of the Peltier effect, is fitted with internal temperature sensor and control system is capable to provide cooling down to -40°C (±0.1K). The device can be programmed to apply cooling or heating rates from 0.1 -10 K/min, including isothermal temperature holds for extended periods of time.
The freezing array used to hold the sample is a layered construction (Fig. 2, upper left panel I). It consists of 1: siliconized 235 hydrophobic glass slides (HR3-217, Hampton research, Aliso Viejo, US) on which the sample drops are dispersed. As the slides are hydrophobic, the sample drops do not float out on the surface, but maintains a roughly spherical form. The slides were flushed with ultra-pure water before use, and discarded after each drop population. A silicon grid (2) was used to keep the drops separated on the glass slides, and sealed each sample drop in an individual cell between the slides and a polycarbonate lid (3), minimizing interaction between the drops (i.e. by Wegener-Bergeron-Findeisen type transfer of vapour or seeding of 240 neighbouring drops by ice-splintering, surface growth or frost halos). The drops were spread out in an approximate circle on the cold-stage to avoid the corners, where the temperature may be less precise during temperature ramps. The grid was laser cut from medical grade silicone. The assembly was centred and held on the stage by a polycarbonate holder/guide. The assembly and the stage were encased in a small environmental chamber (part of the LTS120 Linkham system) and the sample was observed through a quartz glass window ( Figure 2, panel I, 4). Figure 2 (panel II) also shows the sample mounted in the 245 assembly on the cold-stage. continuously purged with a low flow of dry, clean nitrogen gas (F), and a steady flow of dry filtered air (G) is directed over the viewing window to avoid any problems with fogging that may obscure the imaging.
The camera system captured images of the cold-stage in intervals as different cooling programs were applied to the sample (as 260 detailed below in sections 2.4.1 and 2.4.2). The ice spectra for the samples were inferred from the generated images by semisupervised image analysis. An example of an image generated by LUCS, which was used as the input for automated analysis of imagery, is shown in Figure 3. This displays unfrozen and frozen sample drops. As seen in Figure 3, the reflection of the circular light source is clearly visible in the liquid phase droplets, but rapidly disappears at the onset of freezing. This transformation was used to determine the freezing temperature and time for all sample drops from the recorded images. 265

Experimental design
The ice nucleation activity of the samples was measured on five drop populations from each sample consisting of 100 drops, each with a volume of 1 µL. There were six samples of ambient environmental aerosol material in total (Sec. 2.1), collected and classified as noted above (Sec. 2.2). For each of these drop populations at least three 2-hour isothermal experiments and 270 one longer (11-16 hours) isothermal experiment were performed. The number of 2-hour isothermal experiments on each drop population was dictated by practical reasons, and the longer isothermal experiments were performed overnight. Measurements with a constant cooling rate of 2K/min were performed before and after the isothermal experiments, and also between some of the isothermal experiments to assure that the freezing spectra remained unchanged during the experimental time for each drop population. The cooling programs used are defined as follows. 275

Experiments with constant cooling rate
The cooling program used for the experiments at constant rate of cooling is illustrated in Figure 4 (left panel). The sample was dispersed on the cold-stage and initially cooled with a fast cooling rate (-8K/min) from room temperature to -5°C. The sample was then held at -5°C for one minute to assure thermal stability before a constant cooling rate of -2K/min was applied to the 280 sample until it was fully frozen.

Isothermal experiments
The cooling program used for the isothermal experiments is illustrated in Figure 4 (right panel). The sample was initially cooled with a fast cooling rate (-8K/min) from room temperature to -5 °C. The sample was then held at -5°C for one minute 285 to assure thermal stability before a constant cooling rate of -2 K/min was applied to the sample until 1K warmer than the target isothermal temperature, where the cooling rate was decreased to -1K/min to avoid "overshooting" the target temperature. When the target temperature was reached, the sample was held at this temperature for a determined period of time, ranging from 2 hours to over 10 hours. When the isothermal phase had elapsed, a constant cooling rate of -2K/min was then applied to the sample until it was fully frozen. 290 The target temperature was chosen based on the initial constant cooling rate experiments for each sample to correspond to a temperature where about 20-30% of the sample was frozen at the onset of the isothermal phase. This resulted in an isothermal temperature of -16°C for the 'continental polluted' sample and -14°C for all other samples in this study. 295

Quality control
The experiments with constant cooling rates performed before, between and after the isothermal experiments (Sec. 2.4.2) were primarily included to ensure that the freezing spectra of the samples remained consistent during the experimental time. This was effectively a check on both the consistency of performance of the LUCS apparatus and the stability of samples with respect 300 to repeated measurements.
It was seen that the constant cooling experiments remained consistent for all samples during the experimental time, although statistical variations were observed between individual drop populations. In addition, measurements were also carried out with field blank samples to assure that no significant freezing could be observed to rise from either the measurement apparatus, the 305 polycarbonate filters or the handling procedures at temperatures overlapping with the samples. Specifically, isothermal experiments were also done with both ultra-clean water and field blank samples, and no freezing events were observed during 2-hour isothermal experiments at the target temperatures used in this study (-14°C and -16°C).    (Vali 2008). 360 In Figure 7 the normal freezing temperatures (± 1 standard deviation) are displayed for those drops that were also observed to freeze during the isothermal experiments. As seen in Figure 7, a majority of these drops that froze during the isothermal experiments had a normal freezing temperature lower than the isothermal temperature for the experiments (marked in the figure by the dotted cyan line). The difference for most drops displayed is about 1-2 K. The practically sigmoidal-like 365 distribution of normal freezing temperatures (Fig. 7) arises because the probability of any drop freezing during any isothermal experiment decreases with decreasing normal freezing temperature below the isothermal temperature.
In summary, the effect from time-dependence of freezing over 2 hours is to raise the freezing temperature by mostly about 1-2 K relative to the 'normal' value in cooling-only experiments. 370 https://doi.org/10.5194/acp-2021-830 Preprint. Discussion started: 15 November 2021 c Author(s) 2021. CC BY 4.0 License.

Isothermal time series and relaxation time
All examined samples showed the same general pattern during the isothermal phase (Sec. 2.4.2), with more abundant freezing events occurring during the first minutes of the experiment and then a decreasing frequency of observed freezing events as 375 times increase (Fig. 8, blue dots). There was a significant variability between individual experiments and drop populations from the same sample, but this should be expected both from the natural diversity of possible INPs in environmental samples and the stochastic nature of ice activation.
Additionally, the data in the time series of freezing fraction were binned in logarithmically spaced time intervals and the 380 average freezing fraction was plotted for each bin (Fig. 8, yellow points). Figure 9 shows the relative enhancement of IN activity during the entire 10-hour period of the isothermal phase (from final and initial averages of freezing fraction). The corresponding enhancement over the first 2 hours is shown in Table 4. The two samples with the most enhancement are the mineral dust influenced and rural continental samples, whereas the two with the least enhancement are continental pristine and combustion dominated samples (Fig. 9). 385 Figure 10 shows the same averaged freezing fractions in terms of the fractional rate of freezing, / .
The fractional freezing rate has the same order of magnitude among all samples for any given time. The fractional rate decreases almost exponentially with time since the start of the isothermal phase. This fractional rate must vary inversely with the natural time scale of the freezing. This is explicable in terms of the more active INPs nucleating faster on shorter time 390 scales such that progressively less active and slower IN remain as the isothermal experiment progresses.
The limited literature of observations show that the unfrozen fraction is often seen to decay exponentially with time (Bigg 1953ab, Vali 1994Knopf et al. 2020). Consequently, from our isothermal measurements the time dependency effects were inferred by fitting the observations with this empirical model: 395 In this model is the frozen fraction observed after time since the start of the isothermal phase. Also, , is the average initial ice fraction for the sample at the beginning of the isothermal phase, while Δ , is the eventual increase of 400 ice fraction during the entire period of the isothermal phase. Here τ is a relaxation time. A novel feature of our model (Eq (1)) is that the relaxation time-scale is allowed to evolve somehow over time (τ =τ (t)).
Numerically, τ(t) can be determined from our empirical data by re-arranging Eq (1): The fitting of Eq (1) to the measurements was done as follows. First, , and Δ , were estimated from the initial and final averages of freezing fraction during the isothermal phase (Fig. 8, initial and final yellow points). During the isothermal period, from each average of the measured freezing fraction ( ( ); yellow dots in Fig. 8) the relaxation time was inferred 410 using Eq (2), as shown in Figure 11 (blue dots).
These inferred values of were seen to conform to a power law, as shown in Figure 11 (red lines): Note that both in the data and in the fits, as → 0 always ̂ decreases (Fig. 11). Also, ̂ increases monotonically with time throughout each isothermal period. This dilation of the relaxation time-scale with the age of exposure to constant conditions of temperature and humidity is explicable in terms of a statistical distribution of active sites among all the INPs. The most active sites nucleate ice on shorter time-scales and are then 'lost', so the less active sites remain and they activate on longer 420 time-scales, as time progresses.
The values for the fit parameters of Eq (3) are given in Table 2. With these, ( ) was reconstructed by applying the empirically fitted relaxation time, ( ) from Eq (3) for in Eq (1). Figure 8 (red lines) confirms that this empirical model agrees with the experimental data used in its design. 425 In summary, the observed isothermal dependence on time of freezing conforms with a simple law of a freezing fraction increasing exponentially initially and then approaching an asymptotic value after an extended time of a few hours. The relaxation time increasingly dilates as time progresses throughout the isothermal phase, as expected from the most active INPs being steadily depleted and leaving the less active INPs with longer characteristic times for activation. 430

Time dependent temperature shift for use in empirical parametrization of IN activity
Here, , , is the active INP concentration (number per unit mass of air) as a function of time and of the ambient temperature , the saturation of vapor with regard to ice, and the available surface area , Ω , of aerosols of the -th IN species. Also , * is the corresponding concentration from the original scheme without any time-dependence. Eq (4) here is based on our empirical parameterization (EP) of heterogeneous ice nucleation by multiple species of aerosol (Phillips et al. 2008(Phillips et al. , 2013. 445 Yet the same method is generally applicable to any other IN scheme that neglects time-dependency. Figure 12 (blue dots) shows the temperature shift inferred for every measurement of freezing fraction (Fig. 8, blue dots) for all samples during the isothermal phase. This was done by averaging the freezing fraction over the first 10 s and matching this with the prediction of the scheme by adjusting the constant of a proportionality between freezing fraction and , * for = 450 0. So, by definition, ∆ = 0 initially. Then at all times subsequently ∆ is numerically solved to satisfy Eq (4).
The shift values for the ambient temperature input to the IN scheme (EP) were seen to conform to a power law, as shown in All samples show a temperature shift of about 3 K and 5 K over the initial 2 and 10 hours respectively. Here is the time since the start of isothermal conditions. Such an ambient temperature shift (downward) may be viewed as equivalent to a corresponding opposite shift (upward), of comparable absolute magnitude, in characteristic freezing temperatures of most 460 INPs. The eventual impact from time-dependence of freezing after 2 hours is evinced by the maximum difference in freezing 485 temperature, between the red and blue lines, being about 1.6, 1.4, 0.7, 1.1 ,1.0 and 1.2K for the marine dominated, mineral dust influenced, continental pristine, continental polluted, combustion dominated and rural continental samples respectively. This is consistent with Fig. 9, which shows the total fractional increase in IN activity after 2 hrs of exposure to constant conditions. Thus, Figures 9 and 13 are consistent about which samples are the most (e.g. mineral dust influenced) or least (e.g. continental pristine) time dependent. 490

Constant cooling rates with and without an isothermal phase
Even after the end of the isothermal phase for some experiments, a deviation persists between the red and blue curves in Fig.   13. Some of the INPs, which normally would have activated at temperatures a few K colder than the isothermal temperature in the cooling-only experiment, actually activated at this temperature during the prolonged isothermal phase of the hybrid experiment. Generally, as the temperature cools after the end of the isothermal phase, the hybrid experiment shows a freezing 495 fraction that becomes increasingly similar to the cooling only experiment, as expected from the strong dependency on temperature of IN activity.
In summary, the observed time-dependence of freezing involves an upward shift in freezing temperature that increases with time, by about 1-2 K after 2 hours for most drops. This further justifies the approach taken above for incorporating the effect

Discussion
The present study attempts to fill a gap in knowledge about the role of time in real-world atmospheric ice processes, and how 505 its influence should best be represented. Several previous studies have aimed to provide both a theoretical understanding and empirical data (Vali 1994;Welti et al. 2012;Wright and Petters 2013;Budke and Koop 2014;Knopf et al. 2020) so as to represent time dependence more accurately in ice nucleation. However, much of the previous published work studied the effect on idealized systems, and on relatively short time scales. Such studies are invaluable for understanding the basic physics of time dependence in ice nucleation, but may be challenging to apply for practical use in atmospheric modelling. 510 Pioneering aspects of the present study include the fact that the effects of time dependence on ambient environmental samples are measured. This is done for far longer time scales (many hours) than observed hitherto in other studies (many mins) and provides more realistic data which can be applied directly to modify IN schemes and cloud models. It is, to the best of our knowledge, the first study to date investigating time dependence on real-world ambient aerosol samples, although real 515 precipitation samples were observed previously (e.g., Wright, 2014).
Generally, the results from the cooling-only experiments (e.g., Figure 6) show that the instrumental setup provides consistent measurements on each individual drop population for all samples, agreeing well before and during, as well as after, the repeated isothermal experiments (Sec. 3.1.1). Together they comprise a total experimental time of up to 24 hours on each drop 520 population. This shows that the measurements are robust, and that the samples do not change significantly during the instrumental time.
The data derived from the isothermal experiments (Figures 8 and 12) show much variability, which may be expected both because of the stochastic nature of time dependence and from statistical variations in the composition of IN among different 525 drop populations. However, when all freezing fraction data were averaged for a given sample and then fitted by Eq (1), the assumed fit was found to conform to the data (e.g., compare red curve and yellow points in Figure 8). Thus the approach with several repeated isothermal measurements on several drop populations from each sample is likely to give a realistic, albeit approximate, estimate of the effect of time dependence for the different samples. In short, the measurement datasets are sufficient for adequate statistics to describe each sample. Curiously, as seen in Figure 7 for the rural continental sample, it indeed seems as if the overall active INP concentration is 545 low, and that there may be multiple IN types in the same sample. One IN type activates close to the isothermal temperature; another IN type is less prevalent and activates about 5 K lower than the isothermal temperature. This is consistent with the observation that this sample, overall low in IN, may show a mode IN activating at warmer temperatures (e.g., possibly PBAPs or mineral dust). It should also be noted that in nature, even though a cloud drop generally originates from a single cloud condensation nucleus (CCN), it is not unlikely that after some time several aerosol particles are present in cloud drops as a 550 result of coagulation, particle scavenging and other processes. Thus the samples in this study may be representative of the aerosols studied in that sense.

Implementation of Time dependence Results in Cloud Models
In nature, there are many types of INPs that can be classified as belonging to broader aerosol groups frequently referred to as 555 atmospherically relevant (Kanji et al. 2017), as used in some IN schemes (DeMott 2015;Phillips 2008Phillips , 2013. However, in the present project the samples investigated are from the ambient environment and must be assumed to contain a complex composition, where multiple INP species are abundant. Compared to opting for more well-defined artificial samples (e.g. Arizona test dust, Snowmax®) as done in some past studies (Welti et al. 2012;Budke et al. 2014), the approach of sampling aerosols from the real troposphere entails several challenges (sections 1 and 2.2). In particular, the identity of the INPs 560 dominating the ice initiation in each of our samples is uncertain. Eq (5) can be applied in any cloud model to modify such IN schemes for inclusion of time dependence (Table 3) may be inferred from the local predicted value of . This is then used in Eq(5) to get the temperature shift ∆ ( ) for species X, for the classification assumed above, which then in turn is added to the temperature input for the IN scheme.

585
Finally, if very long-lived clouds are being simulated, we recommend applying Eq (5) beyond = 10 hours, providing ∆ < 10K and thresholding at 10K otherwise. In view of the experimental limitations of our data, we make a simplification assumption that the temperature shift is independent of temperature but has a different functional form for each INP species.
In summary, our lab observations with Eqs (4)- (6)  In the present study we present empirical data about the time dependence of heterogeneous ice nucleation for six ambient environmental aerosol samples. Ambient environmental samples, representing a variety of aerosol types expected to be dominated by certain INP species, were investigated. As they were natural samples, they must be assumed to contain a complex 595 composition, where multiple INP species may be active. Although this approach involves less certainty about the chemical identity of the active INPs observed, it yields results with maximum realism.
The conclusions were as follows: 1. Clear effects from time dependency were observed on a level comparable to previous published works, with a 600 percentage enhancement over 10 mins and 2 hours of about 20-40% and 40-100% respectively (Vali 1994;Welti et al. 2012;Wright and Petters 2013;Budke and Koop 2014). There was variation seen among the various samples.
Time dependency effects were observed to be strongest for the rural continental and mineral dust influenced samples, and weakest for the continental pristine and combustion dominated samples. 5. Comparison of cooling-only and hybrid cooling-isothermal experiments reveals that exposure to constant conditions for long times causes an upward shift in freezing temperature that increases with time, by about 1-2 K after 2 hours 630 for most drops. There is a wide variation in the extent of this shift among the individual drops in any drop population, following a sigmoidal-like statistical distribution. 6. A technique for representation of time dependence is proposed for incorporation into schemes of heterogeneous ice nucleation that currently omit time dependence (e.g., Phillips et al. 2008Phillips et al. , 2013DeMott et al. 2015;Patade et al. 635 2019Patade et al. 635 , 2021, as are commonly used in cloud models (Eqs (4)-(6), Table 3, Sections 3.2.2 and 5). This involves a shift that depends on time for the ambient temperature input for these schemes when determining the active number of IN.
Our observations reveal a simple power law dependence of this ambient temperature shift with time, reaching about 3 K and 5 K of cooling over the initial 2 and 10 hours respectively.

Author Contributions
JJ carried out the experimental work with support from TBK. JJ did most of the data analysis with support from VTJP and minor contributions from TBK. All authors were involved in the experimental design and the interpretation of results. JJ and 660 VTJP drafted the first manuscript version together, and all authors contributed to the manuscript writing.

Competing Interests
The authors declare that they have no conflict of interest.         cooling rate experiments (blue lines) and isothermal experiments (red lines) are also shown. The offset seen for the isothermal experiments at the isothermal temperature is the effect of the 2-hour isothermal phase of the experiment. As seen in the figure, the samples behave almost identically until the isothermal phase is reached, and the isothermal data approaches the constant cooling rate data after the isothermal phase has elapsed, and constant cooling is resumed.  Table 1. Aerosol particle properties for the six studied samples during the respective sampling periods. The black carbon (BC) concentrations were inferred from the aethalometer at Hyltemossa. The concentrations of particulate matter (PM) were measured at the nearby Hallahus station. Data from February 2021 were not available due to instrument malfunctioning.    (5), ∆T(t) = −A t , with 95% confidence bounds for the fitting parameters.
975 Table 4: The averaged initial ice fractions for the samples at t = 0, and the observed average increase in ice fraction after 2-980 and 10-hours isothermal time. It should be noted that the experimental data for the isothermal time between 2 and 10 hours of exposure to constant conditions is much more limited than the data for the first 2 hours. This can also be seen in Figure 8. χ is the fractional increase in freezing fraction (FF) after 10 hours, also shown in Figure 9.