This study used eight soil samples to produce laboratory-generated dust aerosols. Two were collected from the Dunhuang dust source area, Gansu Province, China (S1, S2). Three were collected from the Inner Mongolia dust source area (Ulanqab and Hohhot), China (S3–S5). Three were collected from non-dust-source areas in Qingxi Country Park, Shanghai (S6), Fudan University's Jiangwan Campus, Shanghai (S7), and Baoshi Town in Anju District, Suining, Sichuan Province (S8). All samples were passed through a 100 mesh nylon sieve to remove coarse particles and organic debris. Site coordinates and soil texture are listed in Table S1 in the Supplement.

A dust aerosol generator (SyGAVib) was used to produce laboratory-generated dust aerosols (Qu et al., 2020). The device consists of a cylindrical acrylic chamber, a loudspeaker, and an aluminum cup with an open top. Operating parameters were set following Qu et al. (2020) to generate a stable dust plume. Each experiment used 0.3 g of soil placed in the aluminum cup, with the cup fixed to the loudspeaker diaphragm. The loudspeaker was driven at 100 Hz with medium output power to loosen particles via collisions. Three bottom inlet ports supplied dried and filtered air at a total flow rate of 8 L min⁻¹. The two lower side ports each supplied 3.5 L min⁻¹ of sheath air to provide upward momentum, and the port above the cup delivered air of 1 L min⁻¹ into the cup to simulate turbulence and loft the dust aerosols. To avoid the effect of humidity on Coulomb forces between aerosols (Ma et al., 2023), the plume relative humidity was maintained at 25 %–35 %. A temperature-humidity probe was placed at the top of the chamber. The setup is shown in Fig. S1 in the Supplement. It should be noted that these aerosols are more representative of freshly generated dust aerosols under near-source conditions than of aged atmospheric dust after long-range transport.

Size distributions of dust aerosols were measured with an Aerodynamic Particle Sizer (APS, TSI Inc. Inlet flow rate: 1 L min⁻¹) over 0.5–20 µm, whereas aerosols <0.5 µm were measured with a Scanning Mobility Particle Sizer (SMPS, TSI Inc. Sampling flow rate: 0.3 L min⁻¹). For consistency, all aerosol sizes were converted to electrical mobility diameter, which more accurately approximates the geometric diameter of non-spherical aerosols. The detailed conversion formula is provided in Eq. (3). Electrical mobility distributions were measured with an SMPS without a neutralizer, covering 1.56 × 10⁻⁹–1.07 × 10⁻⁶ m² V⁻¹ s⁻¹. The schematic is in Fig. S1.

The experimental setup is shown in Fig. 1. The laboratory-generated dust aerosols were then directed into the Differential Mobility Analyzer (DMA, Models 3080 and 3082, TSI Inc., inlet flow rate: 1 L min⁻¹, sheath flow rate: 10 L min⁻¹) at prescribed electrical mobilities Zp (Table S2), and the aerodynamic diameter of the classified aerosols was measured with an APS. The distribution of the number of charges carried by individual dust aerosol particles was retrieved through joint analysis of the prescribed electrical mobility and the electrical mobility diameter (Dp).

Electrical mobility equation: 1Zp=neC3πμDp Zp is the electrical mobility (m² V⁻¹ s⁻¹), n is the number of charges carried by a particle, e is the elementary charge (1.6 × 10⁻¹⁹ C), π is the circular constant, and μ is the dynamic viscosity of air (1.85 × 10⁻⁵ Pa s at 25 °C), Dp is the electrical mobility diameter (m).

Here, C is the Cunningham slip correction factor and is calculated as (Seinfeld and Pandis, 2016): 2C=1+2λD1.257+0.4exp-1.1D2λ D is the aerosol diameter (m), λ is the molecular mean free path (68×10-9 m).

The aerodynamic diameter was converted to the electrical mobility diameter using the size conversion relationship (Seinfeld and Pandis, 2016): 3Dp=Daχ3ρ0C(Dp)2C(Da)ρpC(Dve)3 is derived from the following two equations: 4DpC(Dp)=DveχC(Dve)5Da=DveρpC(Dve)χρ0C(Da) χ is the dynamic shape factor under the experimental conditions (set to 1.0 assuming spherical dust aerosols), Da is the aerodynamic diameter (m), Dve is the volume equivalent diameter (m), ρ0 is the reference density (1 g cm⁻³), and ρp is the density of dust aerosol particles, taken as 2.65 g cm⁻³ (Haywood et al., 2003). Missing charge bin concentrations were obtained by linear interpolation along the charge axis.

Under each prescribed electrical mobility condition Zp, the APS provided the PSD (dN/dlogDp) of the DMA-classified dust aerosols for different aerodynamic diameter bins. The aerodynamic diameter Da measured by the APS was converted to the electrical mobility diameter Dp using Eq. (3). For each logarithmic particle-diameter bin (log⁡Dpbin), the corresponding number of charges per particle n was then calculated from Eq. (1) using the prescribed Zp and the converted Dp. Therefore, each prescribed electrical mobility condition produced one aerosol number concentration line in the two-dimensional size-charge space.

For simplicity, the concentration was not jointly normalized over the particle-size and charge dimensions. It was expressed as dN/dlogDp, rather than as a combined two-dimensional concentration form such as dN/(dlogDpdn) or dN/(dlogDpdlogn). Each dN/dlogDp value was assigned to the corresponding calculated charge number n. The full two-dimensional size-charge distribution was reconstructed from multiple measured lines obtained under different prescribed electrical mobility conditions, and the bins not directly covered by these measured lines were filled by linear interpolation between adjacent lines.

The interpolation uncertainty is mainly associated with the high-charge tail, where aerosol number concentrations are inherently low and therefore more sensitive to data-processing methods. However, the prescribed electrical mobility range in this study is relatively broad and still includes measured points in the high-charge tail. Therefore, the uncertainty associated with the high-charge tail is expected to have a limited influence on the reconstructed size–charge distribution.

It should also be noted that a dynamic shape factor of χ=1 was assumed in the conversion from aerodynamic diameter to electrical mobility diameter as a simplifying assumption. Since irregular mineral dust aerosols typically have χ>1, this assumption may shift the particle size range associated with highly charged aerosols, while this assumption affects the converted particle size, but does not change the measured charge characteristics. This uncertainty does not affect the qualitative comparison between the two cases, because both cases start from the same initial PSD and therefore share the same underlying Brownian coagulation pathway, with the only difference being whether electrostatic interactions are included.

In addition, the finite DMA mobility selection range may introduce a small bias toward higher inferred charge states for large aerosols. This is because aerosols with large diameters and only a few charges may fall below the lower DMA mobility selection limit and therefore cannot be fully selected. According to the electrical mobility definition equation, the lower DMA mobility selection limit used in this study, 4.68×10-9 m² V⁻¹ s⁻¹, corresponds to a minimum required charge of only ∼ 5 e for aerosols with diameters of ∼ 1 µm, ∼ 10 e for ∼ 2 µm, and ∼ 35 e even near the upper end of the measured size range (∼ 7 µm). Importantly, ∼ 59 % of the total dust aerosol number concentration is distributed within the 1–2 µm size range, where the lower DMA mobility boundary corresponds to only ∼ 5–10 e. However, the mean charge level of dust aerosols in this dominant 1–2 µm size range is still on the order of 10² e, much higher than the lower-boundary charge values.

The number fraction of dust aerosols outside the lower DMA mobility limit was estimated on a particle size bin basis. For each particle size bin, the charge value corresponding to the lower DMA mobility boundary was first calculated based on the electrical mobility definition equation. The number concentration below this boundary was then estimated from the fitted charge distribution for that particle size bin. Specifically, the charge distribution in each particle size bin was empirically fitted using a Gaussian-shaped function in ln⁡(n) space: 6NDp(n)=ADpexp⁡-(ln⁡n-μDp)22σDp2 where n is the number of charges per particle, Dp is the particle diameter of a given size bin, NDp(n) is the number concentration at charge state n for particles with diameter Dp, and ADp, μDp, and σDp are empirical fitting parameters obtained separately for each particle size bin. Based on this particle size bin based estimation, only ∼ 0.6 % of the dust aerosols in this study are estimated to fall below the lower DMA mobility limit. Therefore, under the present operating conditions, the dust aerosols considered in this study would generally remain above the lower DMA mobility selection limit.

A custom charged particle remover (CPR) operated at 3.5 kV was used to determine the neutral fraction of dust aerosols (Zhang et al., 2023). The CPR consists of two coaxially aligned stainless-steel tubes that create a radial electric field, which removes charged aerosols and allows only neutral aerosols to pass through unaffected (design and operation details in Fig. S2).

In the coagulation model, aerosol number concentrations are represented on a unified matrix in size and charge. Dust aerosol model inputs were derived from three Inner Mongolia soil samples. Given that Mongolia accounted for more than 42 % of dust concentrations in northern China during March–April 2023 (Chen et al., 2023) and shares a dust source belt and similar emission and transport pathways with Inner Mongolia (Mu and Fiedler, 2025), dust aerosols generated from Inner Mongolia soils are representative. The dataset was discretized on a two-dimensional size-charge matrix, defined by electrical mobility diameter Dp (Table S5) and charge number z∈{-1000,-999,…,999,1000}, where z represents the number of elementary charges carried by a particle. The size matrix comprised 131 logarithmically spaced bins from 14.1 nm to 7 µm. This yielded three number concentration matrices Nk (Dp,z) for k=1, 2, 3. Each Nk was normalized to a joint probability density matrix: 7fk(Dp,z)=Nk(Dp,z)∑Dp,zNk(Dp,z) and the representative joint probability density function was obtained by taking the element-wise arithmetic mean: 8f‾(Dp,z)=13∑k=13fk(Dp,z) Following Chen et al. (2023), the dust aerosol number concentration of ∼ 300 # cm⁻³ (∼ 3 mg m⁻³) observed in Beijing during transport from Mongolia was mapped onto the matrix according to the above representative joint probability density matrix, yielding a representative number concentration matrix of dust aerosols. This matrix was combined with the ambient sub-500 nm aerosol matrix, or used alone, to form the initial mixed field that served as the model input. For sensitivity analysis under dust-only conditions, additional simulations were performed with initial dust aerosol number concentrations of 50, 100, 200, 300, 500, and 1000 # cm⁻³ (Fig. 5). This range of concentrations was used to evaluate model stability under varying initial conditions and to examine the concentration dependence of electrostatic coagulation.

Under Brownian coagulation, the generation and loss terms are computed on the size matrix for each size bin, following Zebel (1958) and Oron and Seinfeld (1989): 9dN(Dp)dt=12∑Dp′∑Dp′′βB(Dp′,Dp′′)N(Dp′)N(Dp′′)-N(Dp)∑Dp∗βB(Dp,Dp∗)N(Dp∗) N is the aerosol number concentration in a size bin (# cm⁻³), Dp′ and Dp′′ are the aerosol diameters (m), Dp∗ denotes the representative size of aerosols colliding with Dp (m), and βB is the Brownian coagulation kernel (m³ s⁻¹), calculated using the Fuchs (1964) formulation at 25 °C.

The Brownian coagulation kernel is given by: 10βB=2π(d1+d2)(Dp1+Dp2)×(Dp1+Dp2Dp1+Dp2+2(δ12+δ22)1/2+8(D1+D2)(c‾12+c‾22)1/2(Dp1+Dp2))-1 where: 11c‾i=8kTπmi1/212λi=8Diπc‾i13δi=13Dpiλi(Dpi+λi)3-(Dpi2+λi2)3/2-Dpi14di=kTCc3πμDpi Dpi is the aerosol diameter (m), k is the Boltzmann constant (1.380649 × 10⁻²³ J K⁻¹), T is the temperature (298.15 K), and mi is the aerosol mass (kg).

Coagulation satisfies volume conservation: 15Dp=(Dp′3+Dp′′3)1/3 At each time step a pairwise collision rate matrix is formed. Columns correspond to the size bins being depleted and rows correspond to the pairing size bins that collide with them. The total coagulation rate for a given size bin is obtained by summing all elements in its column. The remaining number concentration in the size bin Dp after one time step is: 16Nrem(Dp)=N(Dp)-ΔtN(Dp)∑Dp∗βB(Dp,Dp∗)N(Dp∗) The time step Δt was 1 s for 0–100 s, 10 s for 100–1000 s, and 100 s for 1000–10 000 s.

For the formation term, the upper-triangular part of the rate matrix was used to avoid double counting. For each pair in the upper triangle of the pairwise collision rate matrix, the equivalent product size was computed from volume conservation and mapped to the nearest discrete size bin. The number of newly formed aerosols was then added to the corresponding product bin. After traversing the upper triangle, the formation term for each size bin was obtained. The field N(Dp) was updated by adding the formation matrix to the remaining concentration matrix.

Electrostatic coagulation was modeled using the Brownian framework with a Coulomb correction applied to the coagulation kernel: 17dN(Dp,z)dt=12∑Dp′∑Dp′′∑z′=-1000zβE(Dp′,z-z′),(Dp′′,z′)⋅N(Dp′,z-z′)N(Dp′′,z′)-N(Dp,z)⋅∑Dp∗∑z∗βE(Dp,z),(Dp∗,z∗)N(Dp∗,z∗) βE is the electrostatic coagulation kernel (m³ s⁻¹), derived from the Brownian kernel with a Coulomb interaction correction. In this study, all parameters were set to 25 °C. z′ denotes particles with charge number smaller than z, and z* denotes the representative charge of aerosols colliding with z.

The electrostatic coagulation kernel is given by (Brownian kernel with Coulomb correction): 18βE=βBWc19Wc=eκ-1κ20κ=z1z2e24πε0εRp1+Rp2kT zi is the aerosol charge number, e is the elementary charge (1.60217662 × 10⁻¹⁹ C), ε0ε is the dielectric permittivity of air (8.854737 × 10⁻¹² F m⁻¹), and Rpi is the aerosol effective radius (m).

Simulating electrostatic coagulation requires accounting for atmospheric ions. Dust aerosols are typically transported at altitudes of ∼ 2–5 km, where they are influenced by cosmic-ray ionization (Kok et al., 2021; Xie et al., 2022). The initial ion number concentration is 440 # cm⁻³ and the ion production rate is 2 # cm⁻³ s⁻¹ (Tammet et al., 2006; Israel, 1970, 1973; Hoppel et al., 1986a). At each time step, the ion balance and aerosol charging are updated first and coagulation is then applied.

The pairwise collision rate matrix in electrostatic coagulation is indexed by paired size and charge (Dp,z). As in Brownian coagulation, the loss for any source cell is obtained by summing the column of the pairwise collision rate matrix corresponding to that cell. The formation term differs in that the product is indexed by both size and charge. The size of newly formed aerosols is computed from volume conservation and mapped to the nearest discrete size bin. The charge of newly formed aerosols is computed from charge conservation and assigned to the corresponding integer charge bin. The resulting size and charge indices are then used to add the newly formed aerosols to the corresponding matrix cell.

It should be noted that certain features of the coagulation model may introduce slight biases. The calculation of the size of the product aerosol follows volume conservation and the computed size may deviate from the true volume-equivalent diameter because aerosols are not ideal spheres. Moreover, although a relatively fine size discretization is adopted, binning still introduces errors, which are more apparent when the size difference between dust aerosols and ambient sub-500 nm aerosols is large. Coagulation products are often binned near the size of the larger precursor aerosol. This underestimates the contribution of small aerosols, causes a numerical loss of total aerosol volume, and subtly shifts the PSD toward smaller sizes.