Consider an internally mixed, monodisperse INP population composed of multiple species, with Np particles, each immersed in a supercooled droplet. By the Law of Large Numbers, the expected frozen fraction ffrz equals the arithmetic average of the freezing probabilities of all INPs, 11ffrz=1Np∑j=1NpPfrz,j, where Pfrz,j is the freezing probability of the jth INP as expressed in Eq. ().

Let Ns denote the number of species present in the population, and wi=Si/Sp represent the surface area ratio of the ith species, where Si is the area covered by species i and Sp is the total INP surface area.

For the externally-mixed case, the following applies. Since each particle contains only one species, the whole INP population can be conceptualized as a collection of distinct species. The unfrozen fraction of the whole INP population after time t can be written as 12funf(ext)=∑i=1Nswi⋅funf(i), where wi is the ratio of the total surface area of INPs in the ith mode relative to the total surface area across all INPs and funf(i) is the unfrozen fraction for droplets containing single-species monodisperse INPs, whose surface is 100 % covered by the ith species (for a detailed derivation, see Appendix ). The frozen fraction is therefore 13ffrz(ext)=1-∑i=1Nswi⋅funf(i)=1-∑i=1Nswi⋅(1-ffrz(i)).

In contrast, for the internally-mixed case the unfrozen fraction funf(int) can be expressed as 14funf(int)=∏i=1Ns(funf(i))wi. This form arises because a droplet containing internally mixed INPs remains unfrozen only when none of its surface components nucleates ice. Assuming independent nucleation events, the probabilities of remaining unfrozen multiply, and each term is raised to the power of its surface fraction wi to account for proportional surface coverage. Thus, this leads us to an expression that relates the frozen fraction of droplets containing internally mixed INPs to a combination of the single-species reference cases, 15ffrz(int)=1-∏i=1Ns(1-ffrz(i))wi. Here ffrz(int) represents the frozen fraction for droplets containing internally mixed INPs, whereas ffrz(i) denotes the frozen fraction that would occur if the particle surface were entirely covered by the ith INP species (for detailed derivation, see Appendix ). Equation () implies that for monodisperse INPs, the internal mixture's unfrozen fraction is a weighted geometric mean of the single-species counterparts' unfrozen fractions, weighted by the species surface ratio. A mathematical description similar to Eqs. () and () is also discussed in .

To recap, Eqs. () and () explain the differences in ice nucleation between external and internal mixtures. Mathematically, this difference corresponds to the inequality between the arithmetic and geometric means: for any set of positive values that are not all equal, the arithmetic mean exceeds the geometric mean. As a result, internal mixtures generally yield higher frozen fractions than external mixtures. Moreover, the difference between the two increases as the variability among species increases, meaning that the sensitivity of the frozen fraction to the INP mixing state becomes more pronounced when species have diverse freezing efficiencies (i.e., different funf(i)).

Despite having the same total surface area covered by a species in both the internal and the external mixture scenario, the difference in ice formation between internal and external mixtures arises because each INP freezes its droplet only once, regardless of its nucleation rate. In external mixtures (Fig. , left), the efficient freezing species (“good nucleator”) only covers half of the INPs, so only half of the INPs are highly effective. Once these INPs freeze their droplets, any further nucleation events occurring within these already frozen droplets, though predicted by the theoretical probability model, do not lead to the formation of new ice, rendering the high efficiency of the good nucleators redundant. The remaining half consists of less effective species (“bad nucleator”), leading to fewer freezing events. In contrast, internal mixtures distribute the good nucleator across all INPs, increasing the probability of new nucleation events in unfrozen droplets, producing more ice.

To illustrate the results derived so far, Fig.  compares frozen fractions for internally-mixed and externally-mixed monodisperse INP populations consisting of two species, species A and B. We assume that the species each cover 50 % of the total INP surface area. Species A nucleates ice very efficiently with the same Jhet as Fe2O3, while the Jhet ratio of species B to A is varied (shown on the x axis). All frozen fractions are calculated using Eqs. (), (), (), and ().

Figure a shows the frozen fractions for internal and external mixtures after being exposed to a constant temperature for 20 min, where the temperatures vary from -14 to -23 °C. As expected, the differences are more pronounced at small Jhet ratios, where the two species' freezing efficiencies differ strongly. Concurrently, at lower temperatures, the Jhet disparity has a larger impact, whereas at higher temperatures, the frozen fractions for both mixtures become nearly identical. Figure b shows the frozen fractions over durations from 1 to 20 min at -23 °C. Longer durations show more pronounced differences at small Jhet ratios, whereas shorter durations yield negligible differences. The results in both panels show a similar pattern, because lower temperatures and longer durations both lead to higher Φ values, which is the parameter that appears in Eq. (). Since temperature and duration are combined in Φ, we will use Φ as a simpler parameter to represent freezing ability, avoiding complex cooling conditions with varying temperature profiles.

In the previous section, we suggested that using the Φ value simplifies the classification of cooling condition with varying temperature and duration. For example, the condition with a brief duration at low temperatures and those with longer durations at higher temperatures can be considered equivalent if they share the same Φ value. In this section, we continue using the model from Fig.  and analyze the effects of INP size and the total surface area fraction of each species on sensitivity of frozen fraction to mixing state.

Figure  introduces a sensitivity map for monodisperse INPs with dry diameter of 1 µm, and a surface ratio of two hypothetical species A and B of 50 %. The x and y axis are analogous to Fig.  but using Φ instead of Jhet for the reasons stated above. Each data point in the sensitivity map represents the sensitivity value defined by Eq. () for a scenario involving species A and B, with exemplary Φ values denoted as ΦA and ΦB, respectively.

For reference, some examples of species combinations A and B are highlighted as horizontal dashed lines and aligned solid dots. For example, the combination of Fe2O3 and illite is represented by the green solid dot aligned with the gray dashed line. The Φ values for each species marked in Fig.  are calculated for a scenario with a constant temperature of -33 °C and a duration of 10 min, using Eq. (). The Jhet values for each species are computed using the Water Activity-Based Immersion Freezing Model , which is explained in more detail in Sect. . In Fig. , there are three regimes where sensitivities are negligible, labeled as Regimes 1 to 3, and marked with red, green, and sky blue boxes, respectively.

Regime 1 represents the scenarios where two species have freezing efficiency close to each other, which results in negligible sensitivity, as discussed in Fig. . To explain the small sensitivities in Regimes 2 and 3, we use a simple mathematical analysis. Based on Eq. (), the unfrozen fraction of monodisperse INPs covered by single species of A or B can be formulated as funf(A)=exp⁡(-SpΦA) and funf(B)=exp⁡(-SpΦB), where the Sp=πDp2 is the dry surface area of each immersed INP. Therefore, based on Eqs. () and (), the unfrozen fraction of internal mixture, denoted as funf(int), can be expressed as 16funf(int)=exp⁡-12Sp⋅ΦA+ΦB, while the unfrozen fraction of external mixture, denoted as funf(ext), can be expressed as 17funf(ext)=12⋅exp⁡(-SpΦA)+exp⁡(-SpΦB). In the Regime 2, both species A and B have a relatively small Φ value of less than 107 cm-2, where the product of Sp and Φ is less than the order of 10-1, resulting in 18funf(int)≈funf(ext)≈1-12Sp⋅ΦA+ΦB, which explains the small frozen fraction difference between internal and external mixtures in this case. In this regime, the result also has a simple physical interpretation. Multiplying Eq. () by the total number of particles Np shows that the expected number of ice particles is approximately equal to the expected number of ice-nucleation triggering events. When nucleation probabilities are small, each triggering event typically leads to the freezing of a single particle. Since the total surface area of each species is the same for internal and external mixtures, the expected number of triggering events is also the same regardless of the mixing state. As a result, the number of frozen particles is nearly identical for the two mixing states, leading to the small sensitivity observed in Regime 2. This result is consistent with Fig.  for the conditions of higher temperature and shorter duration time.

In Regime 3, both species A and B have a large Φ value, larger than 108 cm-2, with the product of Sp and Φ being larger than the order of 100, resulting in 19funf(int)≈funf(ext)≈0. This indicates a frozen fraction of nearly 100 % for both internal and external mixtures, again resulting in a small difference between the two mixing states.

Regime 4 is where the sensitivity to mixing state is significant (exceeding approximately 20 %, as indicated by the red contour in Figs. –). In this regime, the good nucleator (species A) is efficient at freezing, and a large disparity in freezing efficiency exists between the good nucleator and the bad nucleator (species B). There are several examples of species combinations that fall into Regime 4, such as Fe2O3 mixed with illite (34.1 % of sensitivity), Fe2O3 mixed with kaolinite (29.0 %), Al2O3 mixed with illite (33.7 %), and dust mixed with Pahokee Peat (12.2 %), among others.

From a physical perspective, the differences among these regimes can be understood in terms of how ice-nucleation triggering events are distributed among particles. For a given cooling condition and fixed total surface area of each species, the expected total number of triggering events in the particle population is approximately independent of the mixing state. The mixing state instead determines how these events are distributed among particles. Because only the first triggering event on a particle causes droplet freezing, additional events occurring on the same particle are effectively wasted. Section  explains the difference between external and internal mixtures precisely from this perspective. In Regime 2, nucleation rates of both species are very small, so the expected number of triggering events per particle is much less than one. Multiple events on the same particle are therefore rare, little waste occurs, and the mixing state has negligible influence on the frozen fraction. In Regime 3, nucleation rates of both species are very large, and each particle surface experiences many triggering events. Even if the nucleation efficiencies of the two species differ by several orders of magnitude, the bad species still produces multiple triggering events on particles composed solely of that species. Consequently, the frozen fraction approaches unity (≈ 100 %) for both internal and external mixtures, because the additional events generated by the good species are effectively wasted once a particle has already frozen. Only in the Regime 4 does the mixing state become important. In this case, particles containing the good species generate more than one triggering event on average, while particles composed of the bad species generate fewer than one. Internal mixing distributes these excess events across particles and reduces waste, whereas external mixtures concentrate them on particles that already freeze easily, leading to a larger difference in frozen fraction.

Figure  presents the same sensitivity map as Fig.  but for varying INP sizes, ranging from Accumulation mode (Dp=0.1 µm) to coarse mode (Dp=10 µm). The pattern in the figure exhibits a vertical shift with changes in INP size: as the INP size decreases, the pattern shifts upward. In the case of smaller INPs (panels a and b), the immersed surface area of each INP is small, making it difficult for even efficient species to trigger multiple ice nucleation events on the same particle. As a result, Regime 2 dominates the entire sensitivity map. Conversely, for larger INPs with larger immersed surfaces, inefficient species can trigger multiple ice nucleation events, leading to the disappearance of Regime 2 and the dominance of Regimes 1, 3, and 4.

Figure  shows the same sensitivity map as Fig. , but for varying immersed surface ratios of species A and B. When the total immersed surface ratio of the efficient ice nucleating species (species A) is small (Fig. a), the sensitivity increases across the entire map, especially in Regime 4, where the maximum sensitivity reaches 90 %. In contrast, as the total immersed surface ratio of the efficient ice nucleating species increases, the sensitivity decreases accordingly. Once the surface ratio of species A is 90 %, the maximum sensitivity is reduced to only 10 %. These results indicate that when a mixed-INP population contains a small proportion of highly efficient species, the influence of the mixing state becomes significant, and failure to accurately estimate the mixing state can lead to high error in the frozen fraction calculation.

The influence of mixing state can also be interpreted in terms of changes in the INP spectrum, i.e., the temperature dependence of freezing efficiency. Figure  presents the predicted frozen fraction as a function of temperature for different mixing states under a constant cooling rate (ccr), showing that external and internal mixtures define lower and upper bounds of the INP spectrum for a given bulk surface area composition.

It is important to note that the theoretical analysis presented in this section applies to mixtures of insoluble ice-nucleating species only. When soluble species are present in mixed particles, their influence on immersion freezing arises through different mechanisms. After dissolution, soluble species do not contribute nucleating surface area but instead modify the nucleation efficiency of insoluble species through changes in water activity . In addition, soluble species can affect the activation of particles into cloud droplets through their hygroscopic properties , which is a prerequisite for immersion freezing. These processes introduce additional couplings between particle composition, nucleation efficiency, and droplet activation that are not represented in the present theoretical framework. Investigating the role of mixing state in systems containing both soluble and insoluble species therefore requires an extension of the theory developed here and is left for future work.

The particle-resolved Monte Carlo (PartMC) model is a box model for simulating the evolution of aerosol particles in the atmosphere. The volume of interest contains a large quantity of discrete computational particles. Each computational particle represents a sample of real particles with identical properties .

Each computational particle is assumed to be a sphere, and is represented by a vector that contains the masses of the species comprising that particle, i.e., the jth particle is described as μj=(μ1j,μ2j,…,μNsj), where μij is the mass of the ith species in the jth particle (i∈[1,Ns]), and Ns is the total number of species. Other characteristics of a particle such as the volume of each species, the total mass and the wet diameter (the diameter of the whole particle, including water and dissolved and insoluble aerosol material) and dry diameter (the diameter excluding water) can be derived from its mass vector.

PartMC encompasses multiple physical aerosol processes, including emission of primary aerosols, dilution with background air, coagulation , and condensation of water vapor to simulate cloud formation . Gas-particle partitioning is included when PartMC is coupled with MOSAIC . During these processes, the evolution of the mass of each species in a particle is tracked. This feature makes PartMC a valuable tool for investigating the impact of mixing state on the aerosol population and on climate-relevant properties of the aerosol. The PartMC has been used in several particle-resolved studies, e.g., used PartMC-MOSAIC to simulate the heterogeneous oxidation process on soot particles and derived the half-lives parameter of surface-bound PAHs for large-scale model. used it to investigate the chemical evolution of aerosols in a ship plume and its impact on CCN properties, coupled PartMC-MOSAIC with WRF model to study the vertical distribution of aerosol mixing state during turbulent diffusion and dry deposition within the planetary boundary layer, used the mixing state simulation result from PartMC to train the machine learning model for the global estimation of aerosol mixing state distribution, provided a systematic quantification of mixing state impact on aerosol optical properties based on the PartMC model simulation.

So far, PartMC has not yet been applied to questions related to heterogeneous ice nucleation. In this study, we extended PartMC by including the process of immersion freezing based on classical nucleation theory. Several parameterizations can be used to calculate the heterogeneous nucleation rate coefficient Jhet . Following , we will use the activity-based immersion freezing model (ABIFM) proposed by .

The water activity-based immersion freezing model (ABIFM), proposed by , is a useful tool for the estimation of heterogeneous ice nucleation rate Jhet in immersion freezing simulations. Following their study, when an INP is immersed in an aqueous solution, the rate coefficient Jhet can be determined by the difference of solution water activity aw and the water activity in equilibrium with ice awice. For an aerosol particle in equilibrium with ambient relative humidity, the numerical value of water activity equals the 100 % of humidity . The parameterization of Jhet is 20log⁡10Jhet=m⋅(aw-awice)+c, where, m and c are the ABIFM parameters for a specific aerosol species. derived the value of m and c for ten different species fitted from experimental data, such as Fe2O3, Al2O3, fungal spores, dust, 1-Nonadecanol, kaolinite, illite, Leonardite, and Pahokee Peat.

Combining Eqs. (), (), and (), the immersion freezing probability of a supercooled droplet containing one INP with multiple species can be formulated as 21Pfrz=1-exp⁡-∑i=1Ns(wi⋅Jhet(i))⋅Sp⋅Δt;log⁡10Jhet=mi⋅(aw-awice)+ci;aw=RH≈1;awice=esice(T)es(T);wi=SiSp, and the descriptions of each variable used in the Eq. () are listed in the Table .

To use Eq. () to calculate the immersion freezing probability of a single droplet with a multi-species insoluble INP, the immersed surface area that each species contributes must be known. However, in PartMC each particle is characterized by the masses of each species, and a detailed description of the particle's morphology is not tracked, and therefore appropriate assumptions need to be made. Here, we assume that the volume ratio of the species equals their surface area ratio, and that all surfaces on the insoluble INP are immersed in water.

The ice nucleation rate coefficient, Jhet(i), is determined by the difference between aw and awice, along with the parameters mi and ci. Assuming the droplet remains in equilibrium with the ambient conditions and the INP it contains is insoluble, hence there is no solute present, the value of aw is equal to 1. The value of awice is determined by the ratio of the saturated vapor pressure with respect to ice to the saturated vapor pressure with respect to water. As temperature is the only variable that affects these terms, awice is thus determined by temperature. The values of mi and ci for a specific species are obtained from .

This section introduces the immersion freezing simulation algorithms that we developed and implemented for the PartMC model. Simulating immersion freezing of a computational particle at a given temperature involves the following five steps (also shown in Algorithm ): #1:

Check whether the particle has an insoluble component.

In Step 4, if a computational particle is frozen, we assume that all real particles it represents are also frozen. Regarding Step 5, we assume that when an ice particle collides with a supercooled droplet and coalesces into a single particle, the supercooled water will undergo instantaneous freezing. Algorithm  outlines the process for determining the freezing state of an individual particle. The function randUnif() creates a random number that is uniformly distributed between 0 and 1, and the function state(p) returns “frozen” or “unfrozen”, showing the current phase state of particle p. To apply this in a particle-resolved model, it is equally important to develop a particle-looping algorithm that applies Algorithm  to each particle for the entire population, which we will address next.

We define Π as the set of particles that contain an insoluble core and sufficient water to satisfy the conditions necessary for immersion freezing. In the numerical implementation, this condition is defined as particles whose water mass fraction exceeds 2 %, which is used as a practical criterion to identify particles activated to droplets. In the simulations presented in this study, all particles are initialized with sufficient water to satisfy this condition, so the specific threshold does not influence the results.

A straightforward algorithm to simulate immersion freezing in each time step of the PartMC model consists of iterating over all particles in Π and determining the freezing state for each particle one-by-one. In this paper, we refer to this method as “Naive Algorithm” (shown in Algorithm ). In the initial stage of PartMC simulation, the state of the particles needs to be set to “unfrozen”. Then, in each time step, the PartMC model runs the Algorithm  to update the phase state of each particle.

proved the accuracy of this type of algorithm. However, the naive algorithm's efficiency is low when the average freezing probability Pfrz is much less than 1. This is because it iterates through each particle, irrespective of how many actually freeze. In such scenarios, the model expends most of its time examining “unfrozen” particles one by one, only to find that the majority remains unfrozen. This process results in an inefficient use of computational resources and time.

To optimize the use of computational resources and time, we apply the Binned Tau-Leaping Algorithm . This approach offers both efficiency and accuracy for stochastic particle-resolved simulations. In this approach, particles are categorized into several bins based on their size. Within each bin, the tau-leaping method is employed to enhance computational efficiency. Additionally, a secondary selection process is implemented to ensure that the algorithm accurately reflects the freezing probability associated with each particle. In this study, we have integrated the Binned Tau-Leaping Algorithm into the PartMC model to update the phase-state of particles in each timestep, detailed in Algorithm .

We still use Π to represent the set of particles that meets the prerequisite of immersion freezing. If the number of bins in PartMC is Nbin, define Πi(i∈[1,Nbin]) as the ith bin set, and Π=Π1∪Π2∪…∪ΠNbin, the details of the algorithm are as follows.

The function randGeom(Pimax) returns a random number that follows the geometric distribution with a parameter of Pimax. The Πi refers the number of elements in Πi. Lines 2–9 constitute the tau-leaping process among the particles in the ith bin. Pimax signifies the upper bound of the freezing probability in the ith bin, a limit that the freezing probabilities of all particles within this bin should not exceed. Practically, in this algorithm, Pimax is calculated by considering the largest particle in the ith bin and assuming it to consist of the most efficient ice nucleation species.

Instead of checking all particles, the algorithm selects a subset of particles and bypasses the rest to enhance computational efficiency. The “jump length,” which is the number of particles skipped between two selected particles, follows a geometric distribution with a parameter Pimax. In this sampling method, each particle in Πi has a Pimax probability of being selected. Lines 13–16 detail the secondary selection process, in which each chosen unfrozen particle has a probability of Pfrz,kΠi/Pimax to transition into a frozen state. After Tau-Leaping and secondary selection, the final probability for each unfrozen particle to become ice, Pkfinal, is: 22Pkfinal=Pimax⋅Pfrz,kΠiPimax=Pfrz,kΠi which matches its inherent freezing probability. This attribute guarantees the precision of the Binned Tau-Leaping Algorithm, thus, at every timestep, the probability of an unfrozen particle being selected by the algorithm to change its phase state is always equal to its theoretical freezing probability as computed by the ABIFM method. A detailed proof of the algorithm's efficiency and exactness is provided by . Our runtime tests demonstrate that the accelerated algorithm reduces the runtime by 87.26 % on average, corresponding to a 7.85× speedup, as shown in Fig. . The simulated frozen fractions returned by both algorithms are consistent, with RMSE less than 0.1 % and R2>0.99, as presented in Fig. . For detailed discussion on the efficiency and exactness of algorithms in PartMC run instances, please see Appendix .

It is crucial to recognize that the method of grouping particles into bins and the approach used to calculate the Pimax for each bin can significantly influence the algorithm's efficiency. If there is a large variance in the selection probability of particles within the same bin – meaning the upper bound value, Pimax, is substantially higher than the selection probabilities for the majority of particles in that bin, the algorithm's efficiency is reduced. This scenario occurs when factors other than particle size affect their selection probability. In the context of immersion freezing simulations for single-species or internally-mixed INP populations, variations in freezing probabilities are solely due to size differences, as all INPs have the same composition and are exposed to the same environmental conditions. However, for INP populations with differences in composition within a size range, Pimax is calculated based on the assumption that the largest particle in the ith bin is covered by the most efficient species. This can lead to a large overestimation for INPs only covered by less efficient species within that bin. A potential solution to this issue is to group INPs by both size and ice nucleation efficiency. This bin structure would ensure that Pimax does not excessively exceed the probabilities of most particles in the bin, potentially enhancing the algorithm's overall efficiency. However, maintaining a two-dimensional bin structure would come with additional computational cost.

In Algorithm , we omit the handling of the case where an unfrozen particle coagulates with a frozen particle (i.e., Lines 24–28 in Algorithm ). In the actual implementation of the PartMC model, this case is handled in the coagulation subroutine rather than in the immersion freezing subroutine.

The study in this section serves as an illustration of the application of the Binned Tau-Leaping Algorithm to accelerate particle-resolved aerosol simulations. For simulations of other physical processes that involve iteratively looping through particles and stochastically selecting some particles based on a size-related probability, the Binned Tau-Leaping Algorithm offers a promising approach for significantly reducing computational time.

PartMC, equipped with the multi-species ABIFM algorithm, is able to simulate the changes in ice number concentration Ñfrz, and frozen fraction ffrz over time given a prescribed temperature profile. In this study, we conducted eight immersion freezing simulation experiments: four populations of droplets were subjected to isothermal freezing conditions, and four to a constant cooling rate (ccr) freezing conditions, respectively, listed in Table .

The INP populations are shown schematically in the left panel of Fig. . Group 1 consists solely of illite, while group 2 is exclusively made up of Fe2O3. In group 3, half of the particles are composed entirely of illite, and the other half entirely of Fe2O3, representing an external mixture. Group 4 features each particle with an even split of illite and Fe2O3 on its surface, constituting an internal mixture. Note that in both the external and internal mixtures, the total surface area ratio between the two species is 50:50. All four groups of INPs share the same log-normal size distribution, with a geometric mean dry diameter of 1 µm and a log-geometric standard deviation of 0.5, shown in the right panel of Fig. . Specifically, for the external mixture case (Group 3), we assigned identical size distributions for Fe2O3 and illite particles, that is, within each size bin, the particle population comprises a 50:50 ratio of pure Fe2O3 to pure illite particles by number concentration. It is important to note that the simulations are performed for polydisperse particles, whereas the theoretical framework in Sect.  is derived for monodisperse INP populations with identical particle surface areas. The implications and conditional applicability of the theory in this context are discussed in Sect. .

The initial condition of each simulation consists of an INP number concentration of 100 cm-3. All INPs were already immersed in water (one INP per droplet) and none of the INPs was frozen. We set the initial relative humidity with respect to water at 100 % to prevent evaporation of the droplets. We used 10 000 computational particles to represent the total number concentration of aerosols, with each computational particle representing 0.01 cm-3 aerosols sharing the same size and chemical composition.

In the isothermal experiments, the temperature was maintained at -20°C for a duration of 10 min. Under the ccr freezing conditions, the temperature was reduced from -10 to -30°C at a constant rate of 2 °Cmin-1 over a period of 10 min, i.e., T(t)=-10°C-2°Cmin-1⋅t. The time interval Δt is set to 1 s. Within each interval from t to t+Δt, the temperature is assumed to remain constant at T(t). Furthermore, the simulations were limited to immersion freezing only; emissions, dilution, and coagulation were excluded from the model.

To summarize, the assumptions in the PartMC immersion freezing simulations are: (1) All INPs are spheres, the surface area is computed by the dry diameter of INP as Sp=πDp2. (2) Each droplet contains only one INP. (3) The volume ratio of each species of mixed particles equals their surface ratio.

As an example, Fig.  presents the results of Case 6 (all INPs consist of Fe2O3 and are exposed to a ccr of 2 °Cmin-1), displaying the simulated frozen fraction, which is confirmed by the analytical solution using Eq. () in Appendix . As the temperature drops from -10 to -30 °C, more and more INPs become activated and the frozen fraction increases (green dots).

The size distributions of activated INPs at 2, 5, and 9 min are depicted in blue in panels (a), (b), and (c) of Fig. , respectively. Ice crystal formation begins in droplets containing larger INPs, which are capable of activating and freezing at higher temperatures and after shorter times. Conversely, smaller INPs require more time and lower temperatures to initiate freezing, resulting in some remaining unfrozen droplets until the end. The number concentration density function for activated INPs, analytically derived from Eq. () and depicted by the black solid line in subplots (a), (b), and (c), closely aligns with the histogram representing activated INPs as predicted by the PartMC model.

Figure  shows the simulation results for the frozen fraction for all eight cases listed in Table . Panel (a) illustrates Cases 1–4 conducted at a constant temperature of -20 °C, while panel (b) shows Cases 5–8, where the temperature decreased steadily from -10 to -30 °C.

In the constant-temperature cases, the ice nucleation rate for an INP remains constant, so the frozen fraction increases over time because more time allows for more nucleation events. However, the rate of this increase slows down as fewer unfrozen droplets remain available to freeze. In scenarios with a ccr, both the frozen fraction and its rate of increase increase over time. This is because the ice nucleation rate rises as the temperature continues to drop, accelerating the freezing process.

Regardless of the temperature profile, INPs composed entirely of Fe2O3 consistently exhibit the highest frozen fraction. Conversely, INPs made solely of illite display the lowest frozen fraction, with internal and external mixtures of these two species falling in between.

Fe2O3 is a highly effective freezing agent, with an ice nucleation rate coefficient of 3.5×104 cm-2s-1 at -20 °C, as per the ABIFM method shown in Eq. (). In contrast, illite exhibits a significantly lower coefficient of only 0.098 cm-2s-1 at the same temperature. This means that Fe2O3's ice nucleation rate coefficient is orders of magnitude larger than that of illite. The substantial disparity in the freezing behavior of different species causes the large variations in their respective frozen fractions.

Our model results also highlight the distinctions in frozen fraction between internally and externally mixed INPs. Internally mixed INPs, comprising 50 % Fe2O3 and 50 % illite by surface area, exhibit a frozen fraction of 41 % in the constant temperature scenario at the end of the simulation, and 61 % in the constant cooling scenario. In contrast, externally mixed INPs, with an equal number of pure Fe2O3 and pure illite INPs, show a lower frozen fraction of 26 % in the constant temperature simulation, and 36 % in the simulation with decreasing temperature. Even though the total surface area of the INPs is the same for both species, and the cooling condition is consistent across both groups with different mixing states, a significant disparity in the frozen fraction is observed between internal and external mixtures. These results are consistent with the main conclusion in Sect.  and further illustrate, in a particle-resolved modeling framework, how mixing state influences the quantity of ice formation.

Figure  presents another view of the comparison of frozen fraction results between analytical calculations and PartMC simulations for all eight cases listed in Table . Each point represents the frozen fraction at one simulation time step, so results from all time steps are included. Generally, the model's results closely align with the theoretical values, with an RMSE of less than 1 % and a correlation coefficient exceeding 0.9999, except for Case 1. The discrepancy in Case 1 arises from the discreteness of computational particles in the PartMC simulation. Because the frozen fraction in this case is extremely low, on the order of 10-3 %, accurately resolving it with only 10 000 computational particles becomes challenging. An order-of-magnitude estimate of the computational particle number required to suppress sampling noise is provided in Appendix . That estimate suggests that resolving frozen fractions at the level of Case 1 with a relative uncertainty of about 10 % would require on the order of 107–108 computational particles. The results in this section demonstrate the implementation of the immersion freezing algorithm for multi-species INPs in PartMC, verified against the analytical solution.

According to Theorem 1, for monodisperse particles, the frozen fractions of internally-mixed and externally-mixed INPs define the upper and lower bounds, and any intermediate mixing state will result in frozen fractions in between these two extremes. In this section, we will use PartMC to confirm this conclusion quantitatively, and discuss its applicability to polydisperse particles.

We use the χ index based on surface area as a metric to quantify the mixing state of INPs. Originally proposed by , the χ index was designed to characterize the mixing state of an aerosol population, ranging from internally to externally mixed, based on species' mass fractions. In our study, we modify this definition by replacing mass fraction with surface area fraction for each species to better account for the ice nucleation properties of INPs (see details in Appendix ). Figure  illustrates the aerosol mixing states as characterized by the χ index (with respect to surface), where χ=0 % signifies an external mixture and χ=100 % indicates an internal mixture. Values of χ ranging between 0 % and 100 % denote intermediate mixtures, representing a continuum between external and internal mixing states. This conceptual framework is essential for comprehending the diverse nature of atmospheric aerosols and their varying properties.

Although Theorem 1 is formally derived for a monodisperse INP population, it can be reasonably extended to polydisperse populations, provided that the surface area ratio of species is specified within each size bin. When this condition is met, the polydisperse population can be treated as a collection of monodisperse subpopulations, allowing Theorem 1 to be applied to each bin individually. Under these assumptions, the internal mixture remains the most efficient one – since every particle within each bin exhibits the highest available ice nucleation efficiency – while the external mixture is the least efficient, as fewer particles possess the highly active nucleating surface. Figure  illustrates an applicability example of Theorem 1. The polydisperse INP populations shown in panels (a)–(c) share the same size distribution and an identical total surface area ratio of 50:50 between species A (gray) and B (green). Panel (a) shows an internal mixture, while panels (b) and (c) represent external mixtures. Theorem 1 is applicable to the efficiency comparison between panels (a) and (b), since each bin corresponding to the same INP size has the same species surface area ratio (50:50) in both cases. However, Theorem 1 is not applicable to the comparison between panels (a) and (c), because the species surface area ratio within individual bins differs. For example, while the first bin in panel (a) has a 50:50 ratio, it is 100:0 in panel (c). Similarly, other bins exhibit inconsistent ratios between panels (a) and (c), despite having the same total surface area ratio (50:50) when aggregated across all bins. Following PartMC simulation results illustrate how the frozen fraction varies with the mixing state index (χ) for a polydisperse INP population, while maintaining a 50:50 surface area ratio of Fe2O3 and illite particles in each bin.

We initialized the model with a series of INP populations characterized by varying intermediate mixing states, with the χ value for INPs in each bin were set to 10 %, 20 %, 30 %, 40 %, 50 %, 60 %, 70 %, 80 %, and 90 % for each respective simulation case respectively. All populations shared the same size distribution, as illustrated in the right panel of Fig. , and had the same number concentration of 100 cm-3. Each INP population was divided into 100 size bins, within which the total surface ratio is composed of 50 % Fe2O3 and 50 % illite. The method we used to construct an intermediate mixing states with a specific χ in each bin is detailed in Appendix .

Two types of simulation were conducted for each of the nine INP populations with intermediate mixing states. One is the isothermal scenario, with a constant temperature of -20 °C for 10 min, the same as Cases 1–4; the other involves a ccr of -2 °Cmin-1 over 10 min, the same as Cases 5–8. All model assumptions are the same as those in Sect. , so that the isothermal simulations for intermediate mixing state INPs are comparable to Cases 3 (external mixture) and 4 (internal mixture), and ccr simulations are comparable to Cases 7 (external mixture) and 8 (internal mixture).

The time series of simulated frozen fraction for all intermediate mixing state cases are shown in Fig. , with the upper panel representing the isothermal experiments and the lower panel representing the constant cooling experiments. Cases 3 and 4 are also shown in the upper panel as references for external and internal mixtures in the constant temperature scenario, marked as χ=0 % and χ=100 %, respectively. Similarly, results from Cases 7 and 8 are shown in the lower panel as references for external and internal mixtures in the constant cooling scenario. Regardless of whether the scenario involves constant temperature or ccr, the frozen fractions of intermediate-mixed INPs fall between those of internally and externally mixed INPs, ordered according to the χ value.

The frozen fractions of the internal and external mixtures set the upper and lower bounds for the frozen fraction of the intermediate mixing state, respectively. This confirms the argument made in Theorem 1. In cases where the mixing state is unknown, assuming the INP population is in an internal mixture during simulations would lead to the largest error in the frozen fraction estimation if the actual state is an external mixture. All other intermediate mixing states would result in smaller overprediction. The uncertainty mentioned here can be quantified by the sensitivity defined by Eq. (), which describes the maximum range of the “mixing state effect” on the frozen fraction.