PartMC (Particle-resolved Monte Carlo) is a stochastic, zero-dimensional aerosol model that simulates the evolution of per-particle composition of an aerosol population within a well-mixed computational volume. The particle positions within the computational volume are not tracked. The composition of individual particles evolves through emission, dilution, and coagulation, modeled using a stochastic Monte Carlo approach. Dry and wet deposition of aerosol particles are not included.

To allow for the treatment of aerosol chemistry, PartMC is coupled to the aerosol chemistry model MOSAIC (Model for Simulating Aerosol Interactions and Chemistry) . This includes the gas phase photochemical mechanism CBM-Z (Carbon-Bond Mechanism) , the Multicomponent Taylor Expansion Method (MTEM) for estimating activity coefficients of electrolytes and ions in inorganic multicomponent solutions , the Multicomponent Equilibrium Solver for Aerosols (MESA) to compute intraparticle solid–liquid partitioning and a solver for dynamic gas-particle partitioning . To simulate secondary organic aerosol (SOA) we use the Secondary Organic Aerosol Model (SORGAM) scheme . The CBM-Z gas phase mechanism includes 77 gaseous species. MOSAIC treats key aerosol species including sulfate (SO₄), nitrate (NO₃), ammonium (NH₄), chloride (Cl), carbonate (CO₃), methanesulfonic acid (MSA), sodium (Na), calcium (Ca), other inorganic mass (OIN), black carbon (BC), primary and secondary organic aerosol (POA and SOA). Species such as Cl, Na, Ca, and other mineral components are not included in this study. The simulations were conducted using PartMC version 2.6.1 and MOSAIC version 2019-01-05.

The details of the scenario library were described in . In short, the scenario library generated from previous work focuses on urban environments, and in particular on the aging process of carbonaceous aerosol by coagulation and condensation of secondary aerosols, which result in changes in per-particle chemical composition. The simulations were performed in a two-stage process: a spin-up run starting from zero aerosol initial conditions, followed by a second run using randomly selected aerosol populations from the spin-up as initial conditions. Each 48 h simulation featured a diurnal emission pattern starting at 06:00 LST (local solar time), with aerosol and gas phase emissions during the 12 h daytime period, representing a well-mixed, polluted boundary layer. Then we discontinued emissions during the 12 h nighttime period, representing the polluted air remaining in the nocturnal residual layer. We repeated the same process 100 times using different input parameters, yielding 4900 aerosol populations with diverse mixing states and chemical compositions. The input parameter space was constructed using a Latin hypercube sampling approach following the strategy of .

In this study, we used “N₂O₅-ON” as a synthetic validation library, though N₂O₅ chemistry is not the focus of the current analysis. Note that primary sea salt emissions were not included in the library. Accordingly, the validation results presented here are primarily applicable to continental anthropogenic aerosol regimes. For each population, we first selected particles with dry diameter (Dp) within narrow size ranges centered at 50, 100, 150, 200, and 250 nm (Table ). This procedure mimics HTDMA measurements, in which dry, quasi-monodisperse aerosol particles are selected using a differential mobility analyzer (DMA). We then calculated χ using two approaches. First, we calculated χPMC, the particle-resolved reference mixing state index, following the mixing-state framework of , using single-particle volume fractions from the PartMC-MOSAIC output, which represents the particle-resolved reference mixing state of each population. For the χPMC calculation, species were assigned to low- or high-hygroscopic groups. The low-hygroscopic group comprised SOA species represented in SORGAM, including aromatic-derived species (ARO1 and ARO2), alkane-derived species (ALK1), alkene-derived species (OLE1), α-pinene-derived species (API1 and API2), and limonene-derived species (LIM1 and LIM2), as well as OIN, POA, and BC, while the high-hygroscopic group included SO₄, NO₃, Cl, NH₄, MSA, CO₃, Na, and Ca. We then constructed the population-level κ-PDF based on individual particle hygroscopicity parameters κ derived from particle composition using the ZSR mixing rule, and applied the YZ method to obtain χYZ. The deviation between χPMC and χYZ quantifies the systematic bias associated with the YZ method.

Figure  compares the hygroscopicity-based aerosol mixing state indices estimated from the YZ method (χYZ) with those computed from single-particle composition data (χPMC) for five dry diameters (50–250 nm) in the validation scenario library. The value of χPMC represents the reference mixing state by preserving the chemical composition and hygroscopicity of individual particles. Overall, the YZ method agrees well with the particle-resolved calculations, with a mean absolute error (MAE) of ∼ 2 % across all diameters. However, notable overestimations are observed, particularly at 150 nm for more externally mixed populations (χPMC < 40 %), where the maximum bias (Δχ = χYZ-χPMC) reaches 29 %. It is also worth noting that larger errors are associated with SOA-rich populations.

The retrieval framework assumes a representative hygroscopicity for the inorganic-salt component. To assess the sensitivity of the retrieval to this assumption, we repeated the analysis using κMH values of 0.60, 0.65, and 0.70 (Fig. S1 in the Supplement). The sensitivity arises because the inferred MH volume fraction in each κ bin depends directly on the assumed end-member values through Eq. (1). For a given particle hygroscopicity, decreasing κMH increases the inferred PMH and shifts particles with hygroscopicities near the MH end member closer to pure-MH composition, leading to a systematic underestimation of χ. In contrast, increasing κMH decreases the inferred PMH, causing particles that would otherwise be assigned closer to the pure-MH end member to be interpreted as mixtures of LH and MH components, resulting in a systematic overestimation of χ. Across particle diameters, the MAE increases from 1.6 %–2.4 % for the baseline κMH = 0.65 case to 7.2 %–11.8 % for κMH = 0.60 and 10.5 %–17.3 % for κMH = 0.70 (Fig. S1). Nevertheless, strong correlations are maintained across all tested values, indicating that the retrieval is robust to reasonable uncertainty in the representative hygroscopicity assigned to inorganic salts, although accurate specification of κMH is important for minimizing systematic bias.

To investigate how the presence of intermediate-hygroscopicity components (e.g., SOA) breaks down the binary assumption, we next examine the κ-PDFs.

We quantified the uncertainty associated with κ-based mixing state retrievals by computing the retrieval errors (Δχ = χYZ-χPMC) from Fig.  and summarizing its distribution in Fig. . The retrieval exhibits a systematic positive bias across all particle sizes and the uncertainty (width of the shaded area) increases with diameter. Notably, the error remains within ±10 % for χ > 70 % for all diameters, indicating a reliable retrieval regime for more internally mixed populations. These error distributions are subsequently applied to quantify uncertainties in χ retrieved from long-term HTDMA measurements.

For populations with higher SOA content, the bias in χ depends critically on how the majority of SOA is mixed with other components (Fig. ). A large overestimation of χ occurs when species are segregated into particles (Fig. a), since a substantial fraction of pure SOA particles that do not fit into either the LH or MH categories introduce the bias. In contrast, when SOA is primarily internally mixed with hygroscopic species (Fig. b), the bias becomes negligible.

To illustrate the fundamental limitation of the binary assumption in the YZ method, we constructed two simplified cases. First, consider a fully externally mixed population (Fig. c) with half the particles pure SOA (κSOA=0.1) and half pure ammonium sulfate (AS; κAS=0.65) . Under the binary assumption, the YZ method misinterprets pure SOA particles as artificial mixtures containing 15 % MH species and 85 % LH species (following Eq. (1): Pi,MH = (κSOA-κLH)/(κMH-κLH) = (0.1-0)/(0.65-0)≈15%). By assigning SOA particles a higher effective number of species than they physically contain, the method estimates a higher average per-particle diversity (Dα), thereby shifting the population toward a more internally mixed state. Alternatively, consider a monodispersed population (Fig. d) in which each particle is an internal mixture of 50 % SOA and 50 % AS. In this case, all particles have the same κ of 0.375, resulting in a narrow, unimodal κ-PDF. The YZ method correctly identifies this state (χYZ = 1), as the bulk population diversity directly reflects the average per-particle diversity (Dγ = Dα).

These examples demonstrate that systematic overestimation of χ arises specifically when intermediate-hygroscopicity components are externally mixed, causing the κ-based binary mapping to misclassify chemically simple particles as artificial mixtures.

Before turning to field applications, we use the particle-resolved simulations to illustrate a more general property of aerosol populations: bulk hygroscopicity alone does not uniquely determine aerosol mixing state. Figure  illustrates how aerosol mixing state is encoded in particle-to-particle variability rather than bulk hygroscopicity alone. From the full scenario library, we identified seven populations with a dry diameter of 150 nm that share the same mean hygroscopicity (κmean = 0.31) but differ markedly in their particle-level composition distributions and mixing state indices. This example highlights that even within a narrow size range, particles can exhibit substantial variability in composition, a defining characteristic of aerosol mixing state that the particle-resolved model explicitly resolves.

Because κmean constrains only the bulk fraction of less- and more-hygroscopic material, these populations are indistinguishable based on κmean alone. In more externally mixed populations, (χpmc < 30 %), the distribution of PMH is dominated by two distinct modes near 0 % and 100 %, indicating segregation of hygroscopic components among particles. As the populations become increasingly internally mixed (χpmc → 1), these modes broaden and merge, and the distribution of PMH becomes continuous and centered at intermediate values. For nearly fully internally mixed populations, the distinction between subpopulations is effectively lost.

The distinction between aerosol populations with identical κmean but different mixing states may have important implications for aerosol–cloud interactions. Activation of a particle depends on whether its hygroscopicity is sufficient to lower its critical supersaturation below the ambient supersaturation . In an externally mixed population (lower χ), a substantial fraction of particles are dominated by less hygroscopic species, such as black carbon or primary organic aerosol, yielding high critical supersaturations that may exceed the maximum supersaturation reached in clouds. These particles are therefore unlikely to activate as cloud condensation nuclei (CCN). In contrast, a more internally mixed population (higher χ) with the same κmean distributes hygroscopic material more uniformly across all particles, lowering the critical supersaturation for a larger fraction of the population and increasing the number of CCN-active particles. As a result, aerosol populations with similar κmean but different χ values may exert different indirect radiative effects.

This result is consistent with the conceptual analysis of , and we explicitly demonstrate through particle-resolved simulations that identical mean hygroscopicity can correspond to fundamentally different aerosol mixing states. Information on particle-to-particle variability, as quantified by the mixing state index χ, is therefore required to distinguish aerosol populations with the same κmean.

An example of HTDMA-measured κ-PDFs across the four ARM sites is shown in Fig. , which can be considered as the normalized aerosol number fractions varied with κ between 0 and 0.65. These long-term averaged κ-PDFs are often broad and, in some cases, distinctly multi-modal, indicating substantial particle-to-particle hygroscopicity heterogeneity. Notably, the distributions for 50 nm particles consistently peak at lower κ values and exhibit narrower spreads compared to larger particles across all sites, likely reflecting the greater abundance of freshly emitted, low-hygroscopicity combustion particles.

While the shape and structure of these κ-PDFs provide insight into aerosol mixing state, they do not uniquely determine chemical composition. A given κ value may arise from different combinations of chemical species, and without additional speciation measurements, one cannot reliably attribute a peak in the distribution to a particular aerosol type of chemical component. Therefore, mapping from hygroscopicity distributions to aerosol mixing state is non-trivial and needs careful consideration when making compositional assumptions.

The time series of χYZ is shown in Fig. . For each dry diameter, the shaded uncertainty range is obtained by applying the particle-size-dependent retrieval-error distributions derived from the PartMC-MOSAIC evaluation (Sect. ; Fig. ) to the corresponding HTDMA-derived χYZ values. Across all sites and seasons, χYZ typically falls between 70 % and 90 %, a regime where our model evaluation indicates that the κ-based retrieval is generally reliable. Comparable ranges have been reported in Chengdu measurements (60 %–90 %) using the same method . The relatively narrow range of retrieved χYZ values is notable given the diversity of the four ARM sites and the variability in colocated aerosol and meteorological measurements. It suggests that, for the HTDMA-selected submicron particles considered here, site-specific source and transport effects modulate particle-to-particle hygroscopicity variability within a predominantly intermediate-to-high mixing regime, rather than producing persistently strongly externally mixed populations. This interpretation is consistent with previous studies suggesting substantial atmospheric aging and mixing of submicron aerosol populations , although definitions of χ vary among studies and are not necessarily based on LH/MH species.

To place the HTDMA-derived κ distributions and χYZ retrievals in environmental context, we examined colocated ARM measurements of aerosol size distributions, trace gases, meteorology, wind direction, and bulk submicron aerosol composition derived from the aerosol chemical speciation monitor (ACSM) (Figs. S2–S6). These measurements show that the four sites differ substantially in aerosol loading, bulk non-refractory composition, transport patterns, and meteorological conditions. The particle number concentrations in different size range provide context for differences in the CCN-relevant aerosol population (Fig. S2), while trace gases and wind direction provide information on source influence and transport (Figs. S3–S4). The ACSM measurements further show variability in the relative contributions of organic and inorganic aerosol components (Fig. S5). These colocated observations support a qualitative interpretation of site-to-site and seasonal differences in χYZ, but they are not used as direct inputs to the retrieval because they do not provide particle-resolved composition for the same HTDMA-selected particles.

The site-to-site differences are most apparent at HOU and EPC. The HOU site shows generally lower χYZ values compared to the other continental sites, consistent with a heterogeneous aerosol population in which fresh urban/industrial emissions coexist with more aged regional aerosol. This coexistence can maintain particle-to-particle differences in hygroscopicity, even if atmospheric processing rapidly ages part of the population . This interpretation is supported by elevated SO₂ concentrations and variability in trace-gas conditions (Fig. S3). The ACSM measurements further show enhanced sulfate and reduced organic during summer (Fig. S5), consistent with seasonal changes in anthropogenic emission influences. reported a broader range of mixing-state index values (5 %–95 %) for soot-containing particles at HOU. These values are not directly comparable to χYZ because they are based on a different particle population and a different mixing-state definition. However, both studies point to substantial particle-to-particle heterogeneity at HOU.

At EPC, lower wintertime χYZ values for larger particles are consistent with seasonal changes in source influence and transport. This interpretation is supported by wintertime enhancements in CO and SO₂ (Fig. S3), indicating stronger anthropogenic influence than during summer, together with shifts in prevailing wind direction (Fig. S4) and seasonal changes in the relative abundance of different particle-size ranges (Fig. S2). Similar to HOU, the enhanced anthropogenic influence likely increases the diversity of aerosol sources and particle compositions, thereby maintaining greater particle-to-particle hygroscopicity variability and reducing χYZ. The ACSM measurements also show enhanced organic fractions during winter and enhanced sulfate fractions during summer (Fig. S5), further indicating seasonal changes in the dominant aerosol sources. At the other sites, the auxiliary measurements also show variability in aerosol loading, meteorology, and bulk composition, but these bulk or population-level measurements do not uniquely determine the particle-level hygroscopicity distributions that control χYZ.

The ACSM measurements provide useful information on bulk non-refractory submicron composition, including organic and inorganic aerosol fractions (Fig. S5), but they do not uniquely identify the composition or mixing state of the HTDMA-selected particles. In particular, ACSM does not directly constrain refractory components such as sea salt and black carbon, and the organic aerosol signal does not distinguish POA from SOA without additional source-apportionment analysis. This limitation is particularly relevant at the coastal or coastal-influenced sites, where marine aerosol may contribute during some periods. We therefore interpret the coastal-site χYZ retrievals with this additional caveat rather than attributing specific κ modes to sea salt.

The particle-resolved simulations demonstrate that the YZ method provides a useful framework for inferring aerosol mixing state from κ-PDFs, particularly for continental aerosol populations in which the prescribed less- and more-hygroscopic end members provide a reasonable representation of the dominant hygroscopicity range. However, the retrieval is sensitive to how these end members are defined. For typical continental inorganic salts, varying the assumed more-hygroscopic end member within a plausible range (κMH = 0.60–0.70) changes the magnitude and sign of the bias but preserves the overall correlation between χYZ and χPMC (Fig. S1). This indicates that accurate end-member selection is important for minimizing systematic bias, but that modest uncertainty in κMH does not by itself invalidate the retrieval.

A more fundamental limitation arises when the aerosol population cannot be represented well by a binary less-/more-hygroscopic system. In such cases, κ no longer maps uniquely onto a two-component composition. This ambiguity is most evident when chemically distinct particle types have intermediate hygroscopicities or when additional highly hygroscopic components are present. For example, externally mixed SOA-rich particles can be misinterpreted as artificial LH–MH mixtures, leading to overestimation of χ. Similarly, in marine-influenced environments, submicron sea-salt particles have been observed and may extend the hygroscopicity range beyond that of typical continental inorganic salts . Because sea salt is not included in the present PartMC validation library, the results presented here should not be interpreted as a full validation of the YZ method for marine aerosol populations.

To illustrate the sensitivity of the retrieval to this type of end-member mismatch, we repeated the YZ retrieval using a sea-salt-like more-hygroscopic end member, κMH = 1.28, while retaining the same PartMC aerosol populations as the reference (Fig. S7). This test is not intended to represent a realistic marine aerosol simulation. Instead, it shows how the binary retrieval behaves when the assumed upper hygroscopicity end member is shifted beyond the hygroscopicity of common non-sea-salt inorganic species. Under this assumption, sulfate-, nitrate-, and ammonium-rich particles become intermediate relative to the prescribed LH–MH end members and can be interpreted as mixtures, leading to systematic overestimation of χYZ relative to χPMC.

These examples highlight the broader non-uniqueness of κ-based mixing-state retrievals. Retrieval uncertainty increases when multiple chemically distinct particle types can produce similar κ values, or when important aerosol components fall outside the assumed end-member range. Applications in coastal, marine-influenced, or SOA-rich environments should therefore be interpreted cautiously, especially when externally mixed intermediate- or highly hygroscopic components are expected. Colocated bulk composition measurements, such as ACSM, provide useful context for the non-refractory submicron aerosol composition, but they do not directly constrain refractory sea-salt mass, particle-resolved composition, or the mixing state of the HTDMA-selected particles.

One possible extension would be to represent aerosol populations using more than two hygroscopicity classes, for example by adding an intermediate-hygroscopicity class for organic aerosol or a highly hygroscopic class for sea salt. Such an extension could reduce some of the biases identified here, because intermediate-hygroscopicity particles would no longer be forced into artificial LH–MH mixtures. However, this would also remove a key advantage of the original YZ framework: in the binary case, the measured κ value constrains the relative LH and MH fractions for each κ bin. Adding additional classes introduces additional unknown component fractions, so the same measured κ value can generally be produced by multiple combinations of those fractions. A multi-class retrieval would therefore require additional observational constraints, such as independent size-resolved composition or source-apportionment information. Developing and evaluating such an extension is beyond the scope of the present work.

Despite these limitations, the YZ method remains useful for aerosol populations in which the dominant hygroscopicity range is reasonably represented by the prescribed LH and MH endmembers and intermediate-hygroscopicity components are not predominantly externally mixed. These conditions are expected to be most common for aged continental and anthropogenic aerosol populations, where condensation, coagulation, and multiphase processing tend to reduce particle-to-particle compositional contrasts. In such cases, κ-PDFs provide a physically meaningful constraint on particle-to-particle hygroscopicity variability, and the YZ framework enables statistically consistent mixing-state inference from long-term HTDMA observations. This is an important advantage because comparable time coverage is difficult to obtain from particle-resolved composition measurements.