Homogeneous freezing of supercooled liquid droplets can be described using classical nucleation theory for detailed treatments, see, e.g.,. Here, we restrict our discussion to the temperature range relevant for ice formation in the troposphere, below the melting temperature of water T<Tm=273.13K. In a supercooled liquid droplet (T<Tm), thermal molecular motion can lead to the spontaneous formation of molecular clusters that serve as ice embryos. Although the ice phase is thermodynamically favored below Tm, nucleation is impeded by a Gibbs free-energy barrier. Formation of a critical embryo is required before further molecular attachment to the crystalline lattice becomes energetically favorable. Once such a critical cluster forms, macroscopic freezing is initiated and the entire droplet freezes. The Gibbs free-energy barrier arises from (i) the change in chemical potential between the liquid and solid phases (a volume term), and (ii) the interfacial free energy associated with the liquid-ice boundary (a surface-area term). On a kinetic level, molecules continuously attach to and detach from the solid embryo, a process that can be described as a diffusive molecular flux. Both the thermodynamic and kinetic effects determine the homogeneous nucleation rate Jhom[m-3s-1]. This rate represents the number of critical ice embryos formed per unit droplet volume and unit time. In general, Jhom depends only on the droplet temperature Td, under the assumption that the droplet is in thermodynamic equilibrium with the ambient air such that Td=T and the droplet is pure, hence there are no foreign substances present that affect the freezing behaviour. Because the ice embryo forms inside the liquid, it has no interface with the ambient environment, and thus the formulation of Jhom should not depend directly on the ambient vapour pressure or saturation ratio. However, as discussed below, the saturation ratio becomes relevant in the parametrisation of homogeneous nucleation for aqueous solution droplets. The value of Jhom is sensitive to the parametrisation of the diffusion activation energy, the liquid–ice interfacial energy, and the saturation ratios . Theoretical constraints on these dependencies are limited in the temperature regime relevant for homogeneous nucleation, and they are therefore obtained through fits to nucleation rates measured in laboratory experiments. Significant uncertainties remain, both in the derivation of Jhom and in the applicability of classical nucleation theory itself . A detailed discussion of these uncertainties is beyond the scope of this study, but they should be kept in mind when evaluating the impact of the choice of Jhom parameterization. We define Pfrz as the probability of homogeneous freezing of a droplet. First, we formulate this probability as a simple temperature dependent threshold process Pfrz,thr 1Pfrz,thr(T)=1forT≤Thom0forT>Thom where the common choice for Thom=235K which corresponds to Jhom∼1015m-3s-1 (see Fig. ). A more physical approach is to parameterize homogeneous freezing as a Poisson counting process see, e.g.,. Here Pfrz is the probability that at least one ice embryo of critical size forms inside a droplet of volume Vl within the time interval Δt and thus initiates homogeneous freezing. Assuming that the process rate rproc=JhomVl remains constant over Δt, the droplet-freezing probability is 2Pfrz(Jhom,Vl,Δt)=1-exp⁡(-JhomVlΔt). Pfrz therefore depends on the droplet volume (Vl=4πrl3/3), the thermodynamic state variables that control Jhom (primarily T), and the elapsed time t. An important consequence of this time dependence is that even small droplets, or droplets in thermodynamic states with low values of Jhom, will eventually freeze if sufficient time is available. The same framework has also been applied to heterogeneous freezing, where the droplet volume is replaced by the surface area of the immersed ice-nucleating particle . In the model, freezing is treated using a Monte Carlo approach. At each time step, a uniform random number xi∼U(0,1) is generated for each super-particle i. Freezing occurs if xi<Pfrz(Vl,Jhom,Δt). Because each super-particle represents ξ real droplets, once freezing is initiated, all real droplets represented by that super-particle freeze. Statistical convergence thus depends not only on the physical time step Δt but also on the sampling of the droplet spectrum – specifically, the number of super-particles nsd and sampling method (e.g., constant multiplicity, uniform in size) used to discretise the size distribution. The formulation of the homogeneous nucleation rate Jhom is critical for representing homogeneous freezing in numerical models. The study by (hereafter KP00) is a foundational contribution in this regard. Solutes inhibit the nucleation ability of droplets, allowing aqueous solution droplets to be supercooled to lower temperatures than pure water droplets before freezing becomes likely. The freezing of submicrometre-sized aqueous solution droplets plays a key role in cirrus formation. KP00 found empirically that Jhom for aqueous solution droplets depends solely on thermodynamic state variables and is independent of the type of solute. Moreover, assuming that the droplet is in thermodynamic equilibrium with the ambient environment, Jhom can be expressed by shifting the melting curve of water by an offset in water activity, Δaw=(Si-1)esat,iesat,l, where esat,i and esat,l are the saturation vapour pressures over ice and (pure) water, respectively. This formulation extends the dependence of Jhom from temperature T alone to include the saturation ratio with respect to ice Si. The KP00 rate is formulated as a third-order polynomial in Δaw. The study of (hereafter KM16) revisited the parameterisation of Jhom for micrometre-sized pure water droplets. They used new methods to constrain the relevant thermodynamic and kinetic properties controlling Jhom and used a selected dataset of nucleation-rate measurements to fit Jhom. KM16 use a sixth-order polynomial in T as a fit 3log⁡10(Jhom)|JHOM-T=∑k=06ck(T-Tm)k with the melting temperature of ice at standard pressure Tm=273.15K. The coefficients ck can be found in , Table 7. This parameterisation is valid for a temperature regime 245≤T≤227K. The label JHOM-T denotes that Jhom depends only on T.

Figure a shows three formulations of Jhom at water saturation Sw=1. Here we focus first on the parameterisation for JHOM-T (black line), which depends only on T. We note that Jhom increases from 240 to 235K by over 5 orders of magnitude highlighting the sharp increase of Jhom with decreasing temperature. Over the same temperature range, the earlier parameterisation of KP00 (blue line), which depends on Δaw and thus on the saturation ratios (Sw or Si), follows the same slope as JHOM-T for Sw=1 but shows an offset of about 1.5 orders of magnitude. The slope of the JHOM-T rate diverges from KP00 at T<234K. While it remains unclear which behaviour is correct, the temperature range below T=235K is rarely relevant for the homogeneous freezing events investigated in this study . This highlights the highly uncertain nature of Jhom. The JHOM-T parameterisation of KM16 is the latest fit to describe homogeneous freezing of pure water droplets and should therefore be used instead of the earlier KP00 parameterisation in that case. However, from a modeling viewpoint it is beneficial to have a unified, Δaw-based parameterisation that is valid for both submicrometre-sized aqueous solution droplets (in the pure-ice regime) and micrometre-sized pure water droplets (in the mixed-phase regime). The study of (hereafter SP23) provides such an option. They reformulated the T-based parameterisation of KM16 as a function of Δaw 4log⁡10(Jhom)|JHOM-DWA=∑k=03bkΔawk where the bk coefficients can be found in . The label JHOM-DWA denotes that Jhom depends on Δaw. The black line in Fig. a actually shows Jhom not only for JHOM-T but also for the JHOM-DWA formulation at Sw=1 as they coincide by construction. So, in principle, the Δaw-based rate of JHOM-DWA should be valid for both thermodynamic regimes – near water saturation (Sw≈1) for the homogeneous freezing of micrometre-sized pure droplets, and below water saturation (Sw<1) for the homogeneous freezing of submicrometre-sized aqueous solution droplets. However, at temperatures close to T=235K Δaw varies strongly with small changes in Sw. Figure b shows Jhom for JHOM-DWA at water-saturation ratios ranging from 0.95 to 1.04 (red curves) together with the Sw-independent rate of JHOM-T (black). Here we note that even for small supersaturation or subsaturation, Jhom differs strongly from the base JHOM-T rate. As previously discussed, Jhom should in general be independent of the vapour pressure in the ambient air (as the ice embryo has no interface with it), at least for the case of pure-water droplets. However, cloud droplets in the atmosphere generally contain foreign substances, e.g., the CCN responsible for droplet activation. In this model, we also assume a soluble CCN. Hence, we should expect some effect on the freezing behaviour from these substances. However, the magnitude of the effect is uncertain, especially since the concentration of foreign substances is only substantial if the droplet is unactivated and thus in the submicrometre size range. Hence, JHOM-T would be the better choice for micrometre-sized cloud droplets, even if the ambient air is super- or subsaturated with respect to water. Note that JHOM-DWA and JHOM-T coincide, as long as water saturation is maintained. In many models featuring bulk schemes, it is a basic assumption that Sw=1 for as long as cloud droplets are present using saturation adjustment, see. However, we will later see that this assumption is often violated during homogeneous freezing events.

The growth of droplets by water vapour deposition (condensation) and evaporation is modeled here by κ-Köhler theory . The hygroscopicity parameter κ is defined by the relation of the water activity aw to the volume ratio of the solute vs and water Vl 51aw=1+κvsvl The implementation of condensation in PySDM is described following the formulation in libcloudph++ . Following , vapour deposition on ice is implemented in a separate integration step from condensation evaporation. We obtain the mass rate for the growth and sublimation of ice crystals by deposition of water vapour as 6dmidt=4πCDv*Gv(Si-1) The capacitance C in m accounts for the non-spherical nature of the ice crystals. Here we use prolate ellipsoids as approximation for columnar ice crystals fitted as in with the (ice crystal) mass mi in kg 7C=0.015755mi0.3+0.33565mi0.43 The diffusion of vapour onto the ice surface is a gas kinetic process. For large ice crystals (>1 µm) we can treat the diffusion of vapour in the continuous regime . There the diffusion constant Dv only depends on temperature and pressure . For larger ice crystals (>100 µm) a ventilation correction must be applied, whereas for smaller ice crystals (<1 µm) a correction for the kinetic regime is required. The latter one is especially relevant in this work, as we investigate the freezing of droplets in the micrometre size range, where kinetic effects dominate. The correction factor for ventilation can be neglected for the ice crystal sizes considered in this study. Dv* is the diffusion coefficient of water vapour Dv corrected for the kinetic regime 8Dv*=Dvriri+λa+4Dvαdvm,vri with the equivalent radius of the ice crystal ri, the mean free path of air λa and the mean thermal velocity of water molecules vm,v=8RvTπ. Not every water molecule that impacts the surface of an ice crystal is integrated into the crystal structure. The ratio of colliding to integrated molecules is given by the mass accommodation coefficient αd. The value of αd is highly uncertain and might depend on the ice crystal size and temperature . Here we set αd=0.5 as in other studies that investigate homogeneous nucleation and which results are compared to this study . The phase change of vapour to ice at the surface of the ice crystals results in latent heating. Hence, the surface temperature of the ice crystals is warmer than the ambient air. This changes the local supersaturation experienced by the ice crystal. The growth rate Eq. () needs to account for temperature difference between ice crystal and ambient air and the thermal diffusion. This is modeled here by the Howell-factor Gv 9Gv=LsRvT-1LsDv*TKT*+RvTes,i-1 with the latent heat of sublimation Ls, the specific gas constant of water vapour Rv and the thermal conductivity of moist air KT. Similar as for Dv we apply a correction for the kinetic regime to KT 10KT*=KTriri+λa+4KTαTcpρvm,ari with the specific heat of dry air at constant pressure cp, and the mean thermal velocity of air vm,a=8RdTπ. The thermal accommodation coefficient αT describes the ratio of colliding molecules that attain a thermodynamic equilibrium with the ice crystals during contact. Here we set αT=1 . The change of water vapour mixing ratio due to deposition onto ice crystals and evaporation of ice crystals is expressed as 11∂qv∂t|dep=∑k=0Ndmi,kdtξkmdry where we iterate over all super particles of the ice phase with their mass rate dmi,kdt and multiplicity ξk with the mass of dry air mdry.

We use an adiabatic parcel model to represent an air parcel rising with a prescribed constant velocity w. The thermodynamic state of the air parcel is described by the air pressure p, air temperature T and water vapour mixing ratio qv. For the (adiabatic) change of pressure we assume hydrostatic balance 12dpdt=-gρw with the gravitational acceleration g and dry air density ρ. The temperature changes due to adiabatic expansion during vertical motion and latent heat of vaporization Lv and latent heat of sublimation Ls 13dTdt=-gcpw-Lvcp∂qv∂t|con-Lscp∂qv∂t|dep with the specific heat of dry air cp at constant p. Note that we neglect latent heat of freezing of droplets. The change in vapour mass mixing ratio qv is controlled by phase change due to condensation/evaporation and vapour deposition/sublimation 14dqvdt=∂qv∂t|con+∂qv∂t|dep The saturation ratios with respect to ice Si and with respect to water Sw are diagnostic quantities dependent on T, p and qv 15Si=pqvϵes,i(T)Sw=pqvϵes,w(T) with ϵ=RvRa and the saturation pressure es,i and es,w of ice and water, respectively. Si and Sw control the growth of frozen and liquid super-particles, respectively, and they control Jhom for the Δaw formulation.

All simulations use the simple parcel-model setup described in Sect. . Unless otherwise specified (e.g., in sensitivity simulations), we use w=1ms-1. The thermodynamic initial conditions for all simulations are temperature T0=245K, pressure p0=500hPa, and saturation ratio with respect to water Sw=0.995. Super-particles in the considered volume represent particles which are initially unactivated and liquid phase (i.e., CCN), upon their growth and freezing they become cloud droplets or ice crystals. We prescribe a monodisperse CCN size spectrum with a dry aerosol radius of 15nm and a hygroscopicity parameter κ=0.65. Note that these parameter choices affect the simulation outcome only in a minor way. The use of a monodisperse size spectrum ensures that all CCN activate, after which droplet sizes equilibrate to the ambient environment, which is determined by the adiabatic cooling rate (and thus the updraft speed w). Because the simulations start at T0=245K, all droplets have sufficient time to activate and equilibrate before thermodynamic conditions favorable for homogeneous freezing are reached. In Sect.  we will investigate polydisperse CCN size spectra using log-normal distribution with the geometric standard deviation σ as an ensemble parameter. Unless otherwise specified (e.g., in sensitivity simulations), we use a CCN number concentration of nccn=750cm-3 at STP (standard temperature and pressure), representative of an average continental air mass . Note, that the local CCN concentration used as input for the simulation is thus lower than the stated CCN concentration at STP as it has to be scaled by ρlocal/ρSTP. Condensation, homogeneous freezing, and ambient thermodynamics are active for all simulations, whereas ice growth by vapour deposition (DEP-ICE) is only enabled when explicitly stated. The model time step is linked to w such that the parcel’s vertical displacement per time step, Δz, is constant. Hence, simulations with stronger updraft speeds employ shorter time steps than those with weaker updrafts. For simulations without DEP-ICE, Δz=2m is sufficient; for simulations with DEP-ICE, we use Δz=1m. For simulations with enhanced nccn and DEP-ICE, Δz=0.5m is used. The simulation is stopped when all super-particles are frozen or evaporated. In this study we investigate the parameter space spanned by these variables: w, nccn, and nsd. For each of these parameters, ensemble simulations are performed with a choice between the two Jhom parametrisations – the T-dependent JHOM-T formulation and the Δaw-dependent JHOM-DWA formulation – and a choice between DEP-ICE inactive (Sect. ) or active (Sect. ). Additionally, we investigate the impact of choosing a lognormal CCN size distribution instead of a monodisperse distribution, with an ensemble over the width of the distribution, σ, in Sect. .

First, we investigate the setup where only ambient thermodynamics, condensation, and homogeneous freezing are active. Here, frozen super-particles act as passive tracers, since without vapour deposition on ice (DEP-ICE) or collisions, they do not interact with other super-particles or the ambient air. Without DEP-ICE, the ambient air never becomes subsaturated with respect to water (Sw<1). As a consequence, all droplets eventually freeze, as there is no removal process. Thus, for the bulk quantities - droplet number concentration nc, CCN number concentration nccn, and ice number concentration ni – the following relationship holds: nccn=nc(t<tf)=ni(tend), where tf is the time of the first freezing event, and tend is the end of the simulation when all droplets are frozen. Number concentrations are expressed as number per kg of dry air. Figure  shows a sample air parcel ascent for the JHOM-T rate (Fig. a–d) and JHOM-DWA rate (Fig. e–h) with nsd=1000 and w=2.5ms-1. We show only a single simulation here, as nsd=1000 is sufficient to resolve the stochasticity for this setup, and we do not observe significant changes between runs (see Sect. ). The dashed line represents the first occurrence of ice (qi>10-7kgkg-1). Since w is constant, there is a constant adiabatic cooling rate applied. Hence, an earlier time step is associated with warmer temperatures. Panels (a) and (e) show the evolution of ambient temperature T in black, as well as the saturation ratios with respect to water (Sw, red) and ice (Si, blue). T governs Jhom for JHOM-T, while the ratio of Sw and Si governs Jhom for JHOM-DWA. Before homogeneous freezing sets in (dashed line), both simulations show the same evolution of T, Sw, and Si. After activation of the (monodisperse) CCN population at the beginning, it takes several seconds until the droplets equilibrate with the ambient water vapour and water saturation (Sw=1) is reached. T decreases continuously, mainly driven by adiabatic cooling. This leads to a decrease in esat,i, which in turn increases Si from 1.3 to 1.5 over the course of the simulation. Panels (b) and (f) show the microphysical bulk quantities for both simulations. The mixing ratio of liquid water (qw, red) increases steadily due to the constant supply of supersaturation, while the vapour mixing ratio (qv, black) decreases. Only about Δt≈75s after the first freezing event does qw begin to decrease significantly as droplets freeze and contribute to the ice mixing ratio (qi, blue). At this point, the air parcel becomes supersaturated with respect to water (Sw>1), because too few droplets remain to maintain water saturation through condensation (and DEP-ICE is disabled). Panels c and g show a cumulative histogram of freezing temperatures Tfrz. Here the cumulative frequency of Tfrz is the same as the frozen fraction of super-particles. Since all super-particles freeze in this example, the frozen fraction reaches 1 by the end of the simulations. These histograms demonstrate that the stochastic freezing approach used here leads to a gradual freezing event over a range of Tfrz from ∼239 to 235K. The first freezing event, indicated by the dashed line, occurs close to 238K for both simulations. However, note that the first Tfrz can vary slightly between simulation runs due to the Monte-Carlo approach. A significant frozen fraction (>0.01) only occurs at temperatures lower than 237K. The frozen fraction for JHOM-DWA increases more steeply than for JHOM-T, completing the freezing event at temperatures slightly above 236K, whereas JHOM-T completes freezing closer to the “classical” homogeneous-freezing limit of 235K. As discussed in Sect. , Jhom for JHOM-DWA and JHOM-T should be of the same order of magnitude for Sw=1, a condition that is maintained for most of the freezing events. Panels (d) and (h) show Jhom for JHOM-T in blue and for JHOM-DWA in red. The line indicates the active parameterisation, while the dashed line shows the other parameterisation under the same thermodynamic conditions for comparison. As freezing begins, the two parameterisations agree closely in order of magnitude. However, plotting the relative error (with JHOM-T considered the reference value) already reveals a divergence during the early onset of freezing. Although such deviations might be considered a “close match” in studies focusing on Jhom see, e.g.,, they have a significant impact on the stochastic freezing approach employed here. Since the JHOM-DWA rate is slightly higher than JHOM-T even for Sw=1, this leads to a shift of the histogram toward higher Tfrz. Once Sw=1 is no longer maintained, the difference between JHOM-DWA and JHOM-T becomes much larger. This is a consequence of the Δaw-dependent approach and would also occur even if JHOM-DWA and JHOM-T matched exactly at Sw=1.

We now investigate the setup in which vapour deposition on ice (DEP-ICE) is active in addition to ambient thermodynamics, condensation, and homogeneous freezing. Frozen droplets are no longer passive tracers; instead, they interact with the ambient air through vapour deposition. As vapour is depleted, the air parcel becomes increasingly subsaturated with respect to water (Sw<1). Consequently, once a sufficient number of droplets have frozen, the remaining liquid super-particles begin to evaporate. This marks the onset of the Wegener-Bergeron-Findeisen process. The air parcel is strongly supersaturated with respect to ice (Si∼1.5) in the temperature range of homogeneous freezing in this study (239 to 234K). Thus, the thermodynamic driving force of the Wegener-Bergeron-Findeisen process is exceptionally large, owing to the large difference between Sw and Si (see also Fig. ). In this setup, the resulting ice number concentration ni at the end of the simulation is of particular interest, because not all droplets will freeze: nccn=nc(t<tf)≠ni(tend).

So far, we have used only a monodisperse mode to discretise the CCN size distribution. In the atmosphere, however, CCN size spectra are more diverse and often follow log-normal distributions or exhibit multiple modes. In this section, we use a log-normal distribution with a geometric mean radius of r0=0.05 µm and geometric standard deviations in the range σ∈[1.5,2]. These values of σ are typical for a variety of aerosol species, including several types of mineral dust, sea salt, sulfate, and ammonium nitrate . A more comprehensive approach would involve varying not only σ, but also r0 and the hygroscopicity parameter κ to represent CCN spectra more realistically. However, for simplicity, we again vary only a single parameter at a time. Moreover, in this model setup – with a single CCN mode and idealised parcel physics – the exact values of r0 and κ (chosen within a reasonable range) do not significantly affect the simulation outcome. This is because, after activation, droplet sizes increases to a radius determined by the ambient water saturation ratio Sw. The use of a polydisperse CCN size spectrum has several important consequences. First, the multiplicity ξ is no longer uniformly distributed across all super-particles. As a result, different super-particles have varying impacts on the ambient thermodynamics, since condensation, latent heating, and vapour deposition on ice are weighted by ξ. Second, super-particles exhibit different droplet radii r because they activate at different times during the ascent, with the largest CCN activating first, and have more time to grow. This affects their freezing probabilities as it is size-dependent (see Sect. ) as well as their growth rates by deposition of vapour. In this section, we exclusively employ the JHOM-T parameterization for the homogeneous freezing rate Jhom, which we consider the more accurate representation. We present simulations both with and without vapour deposition on ice. If not otherwise specified, all simulations in this section discretise the CCN size spectrum as a lognormal distribution with r0=0.05 µm and σ=2, using nsd=10000. A sample ascent for these initial conditions is shown in the Supplement (see Fig. S2). The CCN size spectrum spans radii from 0.0025 to 1 µm, with ξ varying by more than three orders of magnitude between the maximum and the tail end of the distribution. This leads to a smoother onset of droplet activation (in terms of saturation adjustment for Sw) and a wider, more gradual histogram of Tfrz. As before, we choose w=1ms-1 and nccn=750cm-3 for the ensemble simulations.

Our ensemble simulations confirm the fundamental characteristics of homogeneous freezing events. The shift of freezing temperatures Tfrz towards lower temperatures for stronger updrafts reflects the time dependency of the stochastic freezing process, as droplets have less time to freeze before reaching lower temperatures. Parameters affecting the CCN size spectrum- the width determined by σ and the multiplicity determined by nccn – control the droplet size, with larger droplets shifting Tfrz towards higher temperatures. Repeating the simulations yields different Tfrz histograms, highlighting the probabilistic formulation of homogeneous freezing. To obtain smooth Tfrz histograms, nsd=1000 super-particles were sufficient when DEP-ICE was inactive, whereas nsd=10000 were required when DEP-ICE was active. However, bulk properties such as the ice number concentration ni were already robust for nsd=1000.

Most cloud models employ a threshold formulation for homogeneous nucleation of pure water droplets, in which all droplets freeze once the ambient temperature drops below a prescribed threshold Thom. A commonly used value is Thom=235K (or -38 °C). In contrast, we find that most super-particles freeze at temperatures higher than Thom, with the highest freezing temperatures Tfrz close to 240K. Only for strong updrafts (w>5ms-1), high CCN number concentrations (nccn>1250cm-3), or lognormal CCN size spectra do some super-particles freeze below Thom. With vapour deposition on ice (DEP-ICE) active, the thermodynamic forcing of the Wegener-Bergeron-Findeisen process is particularly strong in this temperature range. As a result, most droplets (> 90 %) evaporate before freezing following the first few freezing events, which typically occur at Tfrz>237K. In this sense, our results are consistent with the classical homogeneous nucleation threshold Thom=235K, in that by this temperature most droplets have indeed been removed or are frozen. However, two important caveats must be considered. First, when ice is already present (e.g. due to immersion freezing), liquid droplets may evaporate before reaching Thom. In a more realistic stochastic freezing framework, these droplets would still have a significant probability of freezing. In this case, a threshold formulation underestimates the resulting ice number concentration ni. Second, in the absence of pre-existing ice or ice-nucleating particles, the threshold formulation overestimates ni, since in reality early freezing events trigger the evaporation of a large fraction of the remaining droplets via the Wegener-Bergeron-Findeisen process.

The choice of homogeneous nucleation rate parameterisation influences both the distribution of Tfrz and the resulting ni after the freezing event. Although the JHOM-T and JHOM-DWA formulations have similar slopes and are often considered to agree closely at water saturation (Sw=1), even small absolute differences between the two can lead to different onsets of freezing. Their behaviour diverges further when water saturation is no longer maintained. This divergence arises from the explicit dependence of the JHOM-DWA formulation on water activity, which makes the nucleation rate sensitive to small deviations of Sw. In our simulations, use of the JHOM-DWA formulation is associated with broader Tfrz distributions and generally lower ni compared to JHOM-T, particularly when vapour deposition on ice (DEP-ICE) is active. While JHOM-DWA in principle provides a unified framework applicable to both cloud droplets and aqueous solution droplets, our results suggest that its application to micrometre-sized cloud droplets in mixed-phase conditions requires care, especially in models that explicitly predict Sw (e.g., without saturation adjustment). However, we should not neglect the JHOM-DWA approach for cloud droplets entirely. While these can mostly be considered pure droplets, as the amount of solute from the CCN is low in this size range, these foreign substances should still have some effect on the freezing behaviour. However, constraining the magnitude of this effect is beyond the scope of this study.

In this study, we employ a highly idealised model setup. The simple air-parcel framework does not account for turbulent mixing or interactions with the surrounding flow. In addition, sedimentation is neglected, both the sedimentation of particles into the air parcel and their removal from it. Including sedimentation in a parcel framework is inherently problematic, as it depends on the arbitrary choice of parcel volume and the assumed surrounding cloud environment. Droplet coalescence is also not considered. However, this process is not expected to be relevant for most setups investigated here, as the commonly used geometric collision kernel is not applicable to the monodisperse CCN spectra employed in this study. While coalescence would lead to larger droplets that freeze at higher temperatures, it would also modify the multiplicity distribution ξ, thereby affecting how super-particle–environment interactions such as condensation, latent heating, and DEP-ICE are weighted. The Wegener-Bergeron-Findeisen process plays a key role in stochastic homogeneous freezing events, as early freezing of a small number of “lucky” super-particles can trigger the evaporation of the remaining liquid droplets. In our simulations, typically more than 90% of droplets evaporate before freezing. However, several studies suggest that glaciation driven by the Wegener-Bergeron-Findeisen process may be overestimated in cloud models, and that mixed-phase conditions can be maintained for longer periods . Improving the representation of this process is therefore critical for accurately modelling homogeneous freezing events. Finally, all simulations were performed without ice-nucleating particles (INPs) or pre-existing ice. Including INPs would introduce additional degrees of freedom, such as the freezing temperature and surface area of INPs. Here, we focus on the fundamental behaviour of homogeneous freezing. We further argue that the INP-free conditions considered here are relevant in specific atmospheric situations, such as at rising cloud tops with high droplet concentrations or in regions where pre-existing ice has already sedimented out of the air parcel.