To account for surfactant effects, we adopt a semi-empirical effective surface tension (EST) approach described by and implemented in PartMC-MOSAIC by . This method assumes liquid-liquid phase separation (LLPS) where an inorganic core (α) is partially or fully covered by an organic-rich shell (β). The effective surface tension σ(D) is calculated as a blend of the inorganic and organic phase tensions based on the fractional surface coverage Cβ: 1σ(D)=1-Cβσα+Cβσβ, where σα and σβ are the surface tensions of the inorganic and organic phases, respectively. The fractional surface coverage Cβ is computed individually for each particle from its simulated composition. Assuming that the entire organic volume of a particle partitions to the shell phase β under LLPS, and that a complete film requires a minimum shell thickness δmin⁡, the surface coverage of a droplet with wet diameter D is 2Cβ(D)=min⁡VβVδ(D),1,Vδ(D)=43πD23-D2-δmin⁡3, where Vβ is the organic-phase volume of the particle, and Vδ(D) is the volume of a spherical shell of thickness δmin⁡. Particles containing enough organic material to sustain a complete film are fully covered (Cβ=1, σ=σβ). Because Vβ differs from particle to particle, organic-rich particles reach full coverage (Cβ=1) while organic-poor particles are only partially covered; and because Vδ(D) grows as the droplet takes up water, Cβ and hence σ(D) also vary along each particle's Köhler curve. Following , we use σβ=30 mN m⁻¹ and δmin⁡=0.16 nm.

Within the κ-Köhler framework , the EST enters by replacing the constant water surface tension with the composition-dependent σ(D) in the Kelvin term: 3S(D)=D3-d3D3-(1-κ)d3exp⁡4σ(D)MwRTρwD. Full details on the implementation in PartMC-MOSAIC and sensitivity tests regarding σβ and δmin⁡ are provided in .

To simulate the simplification used in modal or sectional aerosol schemes, we apply a composition averaging mapping . The aerosol population is partitioned into three size regimes based on dry diameter d: Aitken mode (10–100 nm), accumulation mode (100 nm–1 µm), and coarse mode (>1 µm). Within each size range, the chemical composition of individual particles is homogenized. This mapping preserves the original dry diameter d of each particle but replaces its specific species distribution with the mode-average composition. This procedure reduces particle-to-particle compositional variability within each mode, allowing us to isolate the resulting shifts in critical supersaturation sc. The detailed mathematical derivation of this averaging procedure is described in .

We use the mixing state index χ as a diagnostic tool to quantify the distribution of chemical species across the population. χ is defined as the ratio of average per-particle species diversity (Dα) to the bulk population diversity (Dγ): 4χ=Dα-1Dγ-1. A value of χ=0 % represents a fully external mixture, while χ=100 % indicates a perfectly internal mixture. The selection of species for computing χ can depend on specific research objectives, such as elemental composition or hygroscopicity components . In this study, χ is computed based on hygroscopic and non-hygroscopic components. For each grid cell, Dα and Dγ are evaluated from the per-particle species masses in the WRF-PartMC output (Sect. ), giving one value of χ per grid cell. The index is used only as a diagnostic to interpret CCN prediction errors and does not enter the activation calculations directly.

All analyses in this study are based on the per-particle output of a single regional-scale, particle-resolved simulation using WRF-PartMC . For each simulated particle, this output provides its dry diameter, per-species mass, and number concentration, from which all quantities reported here (e.g., κ, sc, CCN concentrations, and χ) are derived. The simulation covers a California domain at 4×4 km horizontal resolution. We examine a representative near-surface snapshot at 12:00 LST, 18 June 2010, focusing on land grid cells to capture variability in anthropogenic emissions and aerosol mixing state. WRF-PartMC explicitly tracks the evolution of individual particles through emission, coagulation, transport, and multiphase partitioning of semi-volatile species, providing the heterogeneous particle populations needed for the present CCN analysis. Although this partitioning is treated dynamically within the simulation, our CCN calculations are based on the fixed snapshot and assume that dry particle composition does not change during activation. In particular, the additional co-condensation of semi-volatile and intermediate-volatility organics that can occur as relative humidity increases toward activation is not represented. Such co-condensation would add organic mass to the growing droplets, increasing both the soluble and the surface-active material at the droplet surface , and because it preferentially raises the organic fraction, its neglect most plausibly leads the surfactant-related CCN differences reported here conservative. The simulation used here should be interpreted as a process-level case study rather than a climatological evaluation. In particular, new particle formation and biogenic secondary organic aerosol are not included in the simulation. Full details on the meteorological setup, emission inventory, gas-phase chemistry and aerosol thermodynamics are provided in .

We first isolate the impact of composition averaging while retaining the constant surface tension (CST) assumption. The analysis shown in Fig.  is based on a representative near-surface grid cell from the WRF-PartMC simulation, selected to illustrate particle-level behavior in a polluted coastal region. This comparison separates the structural effect of homogenizing particle composition within size ranges from any surfactant-related changes in surface tension. Figure  presents a two-dimensional histogram comparing the particle-level critical supersaturation sc in the particle-resolved CST case (PR-CST, x-axis) and the composition-averaged CST case (Comp-CST, y-axis).

The dashed 1:1 line separates particles whose critical supersaturation increases after composition averaging (points above the line) from those whose critical supersaturation decreases (points below the line). To diagnose CCN activation, we impose an environmental supersaturation threshold senv of 0.3 % (red dashed lines), chosen as a representative polluted continental supersaturation and because it places both Aitken and accumulation particles near the activation threshold. Particles to the left of the vertical line activate in the particle-resolved reference case, and particles below the horizontal line activate in the composition-averaged case.

This partition defines four quadrants . Particles in quadrant A remain inactive in both cases, while particles in quadrant C activate in both cases. These particles do not contribute to bulk CCN bias despite shifts in their critical supersaturation. In contrast, quadrant B contains particles that activate in the reference case but not after composition averaging (CCN losses), and quadrant D contains particles that activate after composition averaging but not in the reference case (CCN gains). The net CCN bias under CST is therefore determined by the imbalance between particle gains (quadrant D) and losses (quadrant B), consistent with , who found that composition averaging generally leads to CCN overestimation except at very low supersaturation thresholds.

This gain-loss framework establishes that bulk CCN bias arises from threshold-crossing behavior at the particle level. We next examine how this balance is modified when surfactant effects alter critical supersaturation through changes in surface tension.

We next examine composition averaging within the effective surface tension (EST) framework. Figure  shows the corresponding sc-sc comparison between the particle-resolved EST reference (PR-EST, x-axis) and the composition-averaged EST case (Comp-EST, y-axis). As in the CST case, most particles cluster near the 1:1 line, and the same environmental supersaturation threshold partitions the diagram into gain and loss quadrants.

Because surface tension now depends on composition and organic surface coverage, changes in sc are no longer governed solely by shifts in hygroscopicity. The combined influence of κ and σ modifies the distribution of particles relative to the activation threshold. Compared to the CST case mapping, the imbalance between CCN losses (quadrant B) and gains (quadrant D) is reduced. In other words, when activation is evaluated under EST, composition averaging produces a smaller net gain-loss asymmetry than under CST. This behavior indicates that the CCN bias introduced by composition averaging depends on the assumed surface tension framework.

To clarify the origin of this redistribution, we next quantify how changes in hygroscopicity (κ) and surface tension (σ) jointly influence critical supersaturation through Δln⁡sc.

To understand why composition averaging produces a smaller net CCN bias under EST than under CST, we examine how changes in hygroscopicity and surface tension jointly influence critical supersaturation at the particle level. The κ-Köhler analytical approximation provides a convenient scaling framework for separating these contributions.

Starting from the κ-Köhler analytical approximation , 5ln⁡sc∝σ3/2d-3/2κ-1/2, which holds for small supersaturation (sc≪1), we obtain the logarithmic sensitivity 6Δln⁡sc=32Δln⁡σ-32Δln⁡d-12Δln⁡κ. This expression provides a first-order sensitivity of critical supersaturation to changes in hygroscopicity and surface tension and is used here to interpret the dominant mechanisms driving particle-level shifts in activation.

For perturbations at fixed dry diameter d, this simplifies to 7Δln⁡sc=32Δln⁡σ-12Δln⁡κ. For each particle j, composition averaging under EST modifies both hygroscopicity and effective surface tension. Relative to the particle-resolved EST reference state, we define 8Δln⁡σj=ln⁡σjcomp-ln⁡σjPR,9Δln⁡κj=ln⁡κjcomp-ln⁡κjPR, where the superscripts “PR” and “comp” denote particle-resolved and composition-averaged populations, respectively, both evaluated with EST.

Substituting these perturbations into Eq. () yields the particle-level decomposition 10Δln⁡sc,j(σ)=32ln⁡σjcomp-ln⁡σjPR,11Δln⁡sc,j(κ)=-12ln⁡κjcomp-ln⁡κjPR, so that the predicted total response becomes 12Δln⁡sc,jtotal,pred=Δln⁡sc,j(σ)+Δln⁡sc,j(κ).

This formulation explicitly separates the EST-consistent perturbation into thermodynamic (surface tension) and hygroscopic (κ) contributions.

Figure  visualizes this competition by plotting Δln⁡sc(κ) on the x-axis and Δln⁡sc(σ) on the y-axis for all particles, as well as separately for Aitken and accumulation modes. The origin represents no change in activation properties. Positive values correspond to an increase in critical supersaturation and thus suppressed activation, whereas negative values decrease supersaturation and thus enhance activation.

The four quadrants correspond to distinct physical regimes:

Quadrant I (reinforcing suppression): both κ and σ contributions increase sc and suppress activation.

When separated by size mode, the dominant compensating configuration differs. Aitken particles are more frequently found in Quadrant IV, whereas accumulation particles preferentially occupy Quadrant II. This mode-dependent partitioning reflects differences in the internal composition variability within each mode and how composition averaging redistributes organic and inorganic material across particles.

Section – established the particle-level mechanisms that govern CCN biases. Composition averaging redistributes particles across the activation threshold (Sect. ), while surfactant-induced surface tension changes introduce an additional response that competes with hygroscopicity (Sect. ). The κ-σ decomposition further showed that these contributions frequently act in opposition at the particle level (Sect. ). We now examine how these mechanisms manifest in bulk CCN biases across the WRF-PartMC domain.

The error metrics reported here refer to CCN concentrations (εCCN), not directly to cloud droplet number concentration (CDNC). CCN represents the particles that can activate at a specified supersaturation, whereas CDNC represents the number of droplets that actually form in a cloud parcel. In rising cloud parcels the peak environmental supersaturation adjusts to aerosol loading and composition, which partially buffers CDNC against CCN uncertainties. For example, showed that a twofold uncertainty in CCN typically produces only about a 15 % change in CDNC. The CCN errors quantified below should therefore be regarded as an upper bound on potential CDNC impacts, while still providing process-level evidence of how surfactants may bias simulated aerosol-cloud interactions.

Before quantifying regional CCN biases, we first summarize the spatial context of the WRF-PartMC case. Figure  shows the near-surface total particle number concentration and CCN activation fraction for the land grid cells analyzed in this study. These fields highlight the pronounced contrast between polluted coastal regions and cleaner inland areas, which provides an important background for interpreting the regional error patterns associated with composition averaging and surfactant effects.

The preceding sections showed that CCN biases arise from particle-level κ-σ competition and modal error compensation. We now examine how these effects depend on aerosol mixing state. This diagnostic links the structural simplifications examined above to a widely used metric of aerosol population heterogeneity .

Figure  shows the spatial distribution of χ across the near-surface land grid cells. The lowest values of χ, corresponding to the most externally mixed aerosol populations (with respect to hygroscopic and non-hygroscopic species), occur primarily in polluted coastal regions, whereas inland regions tend to exhibit higher χ and thus more internally mixed aerosol. This pattern is physically plausible: near source regions, freshly emitted particles from different sources remain more compositionally distinct, whereas farther inland atmospheric aging and condensational processing drive the population toward a more internally mixed state. This spatial structure helps explain why the strongest CCN sensitivities to composition averaging and surfactant effects are concentrated near source regions.

To improve sampling across mixing states, this analysis combines data from two representative time steps (daytime: 12:00 LST on 18 June and nighttime: 00:00 LST on 19 June 2010). Figure  shows all four bias metrics as a function of χ for the total aerosol population. For the three component differences (εσPR, εCSTcomp, and εESTcomp), the magnitudes decrease monotonically with increasing χ. Both surfactant effects and composition averaging produce the largest CCN biases under strongly externally mixed conditions, and these biases diminish progressively as the populations approach a more internally mixed state. This is physically consistent with the expectation that composition averaging introduces less structural error as particle-to-particle compositional variability decreases. However, their asymptotic behavior differs: the composition-averaging errors (εCSTcomp and εESTcomp) approach zero as χ→100 %, whereas the surfactant-related difference (εσPR) levels off at a finite negative value. This behavior has direct implications for large-scale aerosol models. As composition-averaged representations effectively assume internally mixed populations, the error associated with composition averaging becomes small, whereas the bias introduced by neglecting surfactant-driven surface tension reduction remains finite and may therefore represent a more persistent source of CCN bias.

By contrast, the model-relevant bias εrep remains approximately constant and negative across the full range of χ, with binned median values near -20 %. This behavior shows that the overall modal-like bias is not primarily controlled by mixing state in this case study. Rather, as χ increases, and the composition-averaging contribution weakens, the persistent negative bias from omitted surfactant effects becomes relatively more important. This finding has an important implication. Improving the representation of mixing state alone is unlikely to remove the dominant structural CCN bias if surfactant effects are still neglected.