The model used for this study is the particle-resolved model PartMC-MOSAIC (Particle Monte Carlo model–Model for Simulating Aerosol Interactions and Chemistry). A comprehensive description of the model can be found in and for PartMC and in for MOSAIC. PartMC is a Lagrangian box model that tracks the evolution of particles in a fully mixed computational volume. The processes of emission, coagulation and dilution are simulated stochastically. Gas-phase chemistry and gas–aerosol partitioning are incorporated by coupling with the deterministic model MOSAIC. Specifically, MOSAIC uses the carbon-bond-based mechanism CBM-Z for gas-phase photochemical reactions , a multicomponent Taylor expansion method (MTEM) for calculating electrolyte activity coefficients in aqueous inorganic mixtures and a multicomponent equilibrium solver for aerosols (MESA) for calculating the phase states of the particles . The secondary organic aerosol (SOA) treatment follows the Secondary Organic Aerosol Model (SORGAM) . Aerosol water uptake is calculated using the Zdanovskii–Stokes–Robinson (ZSR) method based on the composition of the inorganic portion of the particles. By this method, organic species are treated as hydrophobic and do not contribute to water uptake. The impact of this assumption on optical properties was quantified by , who found that errors in single scattering albedo can be up to 6 % if neglecting the water uptake of organic compounds.

Following the strategy in and , we created a scenario library of PartMC-MOSAIC simulations for this study, with a focus on the aging of carbonaceous aerosol. To produce particle populations with a wide range of compositions and mixing states, we varied the model input parameters within the ranges shown in Table . We used Latin hypercube sampling to create input parameter combinations for a total of 100 model simulations. The simulation time for each simulation was 24 h, beginning at 06:00 local time with hourly output. This yielded a total of 2500 particle populations. All scenarios were run with 10 000 computational particles. To create aerosol initial conditions with realistic mixing states, we adopted the approach described in : we carried out a first set of simulations, starting with the aerosol initial concentrations set to zero for all simulations (the “initial runs”). We then repeated the same set of simulations but replaced the aerosol initial condition with a randomly sampled population from the initial runs (the “restart runs”). For the analysis in this paper, we only used the results from the restart runs. Within our ensemble or aerosol populations, some were found with higher species concentrations than what would be expected in the ambient atmosphere. We applied upper thresholds to eliminate those which were calculated as the sum of the 75th percentile and 1.5 times the IQR (interquartile range) for each of the aerosol species. After this procedure, 1809 out of 2500 populations were used for the error analysis presented in the remainder of the paper.

We calculated the optical properties of the particle populations using Mie calculations . These properties included the asymmetry parameter g, scattering cross section σscat and absorption cross section σabs for each particle. Particles were assumed to be spherical, and when BC was present, a core–shell configuration was assumed, with BC as the core and non-BC species as the shell. In PartMC-MOSAIC, each chemical species was assigned a refractive index, and the values were the same as , as listed in Table . The shell refractive index of the particle was the volume average of all the shell species, including aerosol water. The absorptivity of brown carbon has been of great interest in recent years ; however, this was not considered in the current work. We used the values for wavelength λ of 550 nm for our analysis.

In PartMC-MOSAIC, all particles are tracked individually in a well-mixed computational volume, and we obtained the ensemble optical property values by summing over all particles in the volume. The ensemble scattering coefficients βscat(λ), ensemble extinction coefficients βext(λ) and ensemble absorption coefficients βabs(λ) at wavelength λ are given as 1βscat(λ)=∑iNσscat,i(λ)ni,2βext(λ)=∑iNσext,i(λ)ni,3βabs(λ)=βext(λ)-βscat(λ), where i is the particle index, ni is the number concentration associated with particle i and N is the number of computational particles in the population. We determined the optical properties of all particle populations of our scenario libraries using these equations.

Two additional derived quantities of interest are the absorption enhancement and the mass absorption coefficient. The absorption enhancement of BC-containing particles due to coatings is defined as 4Eabs(λ)=βabs(λ)/βabs,BC(λ), where βabs,BC(λ) is the volume absorption coefficient when the particle coatings are removed from the BC cores.

We can also calculated the BC-specific mass absorption coefficients MACBC (m2 g-1) using 5MACBC=∑iNσabs,i(λ)ni∑iNmBC,ini, where mBC,i is the BC mass in particle i.

To quantify the impacts of mixing state on aerosol optical properties, we employed the strategy of “composition averaging” similar to to create sensitivity scenarios. The technique is shown conceptually in Fig. . For each population in our reference scenario library, we averaged the dry particle compositions within prescribed size bins. We chose eight size bins between 0.039 and 10 µm, consistent with the bin structure of the sectional aerosol module MOSAIC used in WRF-Chem .

The composition averaging procedure preserves the bulk mass concentration of each species, the total number concentration and the particle diameters within each bin Appendix B1; i.e., after composition averaging, each bin still contains particles of different sizes. It changes the per-particle compositions so that each bin becomes internally mixed; however the composition can vary between bins. This mimics the assumption frequently made in sectional models, namely that each size bin contains an internally mixed aerosol. PartMC-MOSAIC represents particles outside the MOSAIC bin range, especially for the lower boundary, and we used an extra bin (bin 0) to preserve the total number and mass concentrations. Since the aerosol water content plays an important role in aerosol optical properties, we further calculated water uptake for the reference populations and for the composition-averaged populations for 50 % (P2) and for 90 % (P3) relative humidity, respectively. At RH =50 %, depending on the exact composition, some particles take up water, and at RH =90 %, most particles take up water, except particles that only contain hydrophobic species, such as pure black carbon or primary organic carbon. Note that while the dry aerosol mass was conserved by the composition averaging procedure, the water content was recalculated after composition averaging and could change compared to the reference population.

Figure  illustrates the changes of two important parameters for aerosol optical properties due to composition averaging, BC mass fraction and the real part of the refractive index. In the reference case, a wide range of BC mass fractions exists within the same size bin (Fig. a). After composition averaging, all particles within a size bin have the same BC mass fraction (Fig. b). Since composition averaging preserves the particle diameters, BC and other species are redistributed so that all particles within a size bin are assigned the same mass fractions. Specifically, if a particle has lower BC mass fraction than the average level in the same size bin, BC is added to this particle from those with higher BC content. The coating species are also redistributed after composition averaging, which causes the refractive index of the coating to change (Fig. c and d). Hence, comparing optical properties before and after composition averaging in the dry population P1 isolates the impact of mixing state on aerosol optical properties. We will discuss the impact of composition averaging for dry conditions in Sect.  and the impact of water uptake in Sect. .

We quantified the optical properties error introduced by a simplified mixing state representation using the metrics developed by . These metrics include the single-particle diversity Di, the average particle species diversity Dα and bulk population species diversity Dγ. For a population with N particles, total mass μ and A species, we can calculate these metrics from the total mass of particle i, μi, total mass of species a in the population, μa, and mass of species a in particle i, μia, for i=1, …, N and a=1, …, A. The mass fraction of species a in particle i, pia, mass fraction of particle i in the population, pi, and mass fraction of species a in the population, pa, are given by 6pia=μiaμi,pi=μiμ,pa=μaμ.

The single-particle diversity Di describes the effective species number in each particle and is defined as 7Di=∏a=1A(pia)-pia. As shown in Fig. S1 in the Supplement, if a particle only contains one species, Di is 1. If the chemical species are present in equal amounts in the particle, Di equals the number of species. If the species are unevenly distributed, Di is a real number ranging between 1 and the number of species in the particle.

Based on Di, we can construct Dα and Dγ, which describes the average effective species number in each particle and bulk population, respectively: 8Dα=∏i=1N(Di)pi,9Dγ=∏i=1A(pa)-pa. Finally, the mixing state metric χ is defined as the affine ratio of Dα and Dγ: 10χ=Dα-1Dγ-1. The values of χ vary between 0 % and 100 %. Take the three particle populations in Fig. S1 as an example. All three populations have the same bulk species mass concentration. Thus, they have the same bulk effective species diversity Dγ. However, the species are distributed differently within the populations. When the particles are externally mixed, each particle only contains one species and Dα is 1, which results in χ=0%. When all particles have the same species mass fractions, Dα equals to Dγ, and we obtain χ of 100 %. The population is fully internally mixed. For the intermediately mixed population, χ ranges between 0 % and 100 %. In many applications, χ is used to describe the mixing state of chemical species, and we can therefore also refer to it as the chemical abundance metric χchem .

For this work, our focus is the optical properties of the particles. Differing from χchem, we grouped the aerosol species by absorbing and non-absorbing species, i.e., BC and non-BC species, and defined a new index accordingly, χBC. It still ranges between 0 % and 100 % and signifies the degree to which BC and non-BC species are mixed. Since we only consider two (surrogate) aerosol species, the maximum value of Di, Dα and Dγ is 2. The same metric was chosen by to characterize the mixing state of BC-containing aerosol in Beijing and by to understand the role of mixing state in aerosol light absorption enhancement. For the remainder of the paper, unless otherwise noted, we will refer to the χBC simply as χ.

Figure  shows the range of bulk chemical species concentrations, the mixing state metric and optical properties within the selected scenario library. The simulated aerosol bulk species mass concentration in the library covered a wide range of urban conditions (Fig. a), and the values were comparable to the measurements in different locations . Both mixing state metrics χchem and χBC were larger than 30 %, with a median value of 85 %. The fact that mixing state metric values smaller than 30 % did not occur in our scenario library is consistent with the notion that aerosol species rarely exist in a completely external mixture but rather form some degree of internal mixtures already at the time of emission. Additionally, in urban environments, particles age quickly, forming internal mixtures with secondary species .

Our range of χBC values encompasses the range observed in field measurements at Taizhou, China, where χBC ranged between 68 % and 79 % for a period in May–June , and at Beijing, China, where χBC ranged between 55 % and 70 % in winter and between 60 % and 75 % in summer . The range of χchem is also consistent with the daily range of 37 %–73 % of particle samples collected in Paris, France .

Figure c shows that the single scattering albedo (SSA) was larger than 0.4 for all populations, with a median value of 0.88. While SSA values lower than 0.5 are considered extremely low (4 %), most populations (72 %) had a SSA larger than 0.85, which is consistent with fine-mode SSA observations from AERONET . The distribution of simulated total number concentration (Fig. d) is consistent with the observed number concentration of particles in the accumulation-mode size range . Note that the simulations presented here do not include the process of new particle formation. As a result, the simulated particle populations are more representative of accumulation-mode particles in a range of different environments.

Absorption was overestimated universally after composition averaging, and, as expected, the error was higher for more externally mixed populations (low χ values), with ϵ(βabs) reaching up to +70% for χ of 30 % (Fig. a). Each dot in Fig. a represents a particle population from the scenario library. As shown in the box plot inset, the mean overestimation was 18 %, and the maximum reached over 80 %. The figure further contains information of BC bulk mass concentration and relative average BC core size changes, which are the two main factors in determining absorptivity , as represented by marker size and color, respectively. The relative average BC core size change for a population is defined as 12ΔDcore=∑i=1NniDicore′-∑i=1NniDicore∑i=1NniDicore, where i is the particle index, ni and Dicore are the associated number concentration and BC core diameter in the reference scenario, and Dicore′ is the BC core diameter in the sensitivity scenario. The number concentration ni is always greater than 0, and if there is no core for particle i, Dicore is 0. It is interesting to note that ΔDcore is always positive; that is, the average core diameter after composition averaging is larger than the average core diameter before composition averaging. This is a result of particle mass being a convex function of particle diameter (assuming spherical particles). Calculating the new core diameters after composition averaging will therefore always lead to larger core diameters on average than averaging the core diameters before composition averaging, as shown in Fig. S2.

The decreasing error with increasing χ can be explained by the magnitude of ΔDcore. Evidently, composition averaging caused larger changes of BC core sizes when the populations were more externally mixed. For example, for χ=30 %, the change in core sizes was as large than +25 %, while for χ=95 %, the change in core sizes was less than 5 %. We also noticed a range of errors for populations with χ between 60 % and 70 %, i.e., partially internally mixed populations. In fact, the highest overestimation of 82 % was reached at χ=63 %. As indicated by the circle size, these populations contained very little BC (0.01 µgm-3), and even small changes in core sizes can lead to large relative errors in the volume absorption coefficient.

Given the constraint that composition averaging preserves the particle number concentration and sizes, it follows that, for some particles, this operation increases the sizes of BC cores (while at the same time decreasing the coating thickness), whereas for other particles it decreases the BC cores sizes (while increasing the coating thickness). It is therefore not immediately clear that composition averaging consistently causes overestimation of aerosol absorption coefficients.

At a per-particle scale, for particles of the same diameter, σabs increases with increasing BC core, even though the coating thickness (and hence the absorption enhancement) decreases (Fig. S3). However, ϵ(βabs) is determined by the entire population. The internal mixture in each size bin is reached by moving species from a group of particles to another group of particles. As the BC mass fraction distribution in Fig.  shows, there are two major groups of particles in the population: group 1 are particles with higher BC mass fraction, and group 2 are particles with lower BC mass. Particles in group 1 experience decreased absorbing ability because they are losing BC and vice versa for particles in group 2.

To further illustrate the effects at the population level, we show the effects of composition averaging on the volume absorption coefficient for a simplified case of five monodisperse populations of different sizes, starting out with completely externally mixed populations consisting of BC and ammonium bisulfate (Fig. ). Absorption coefficients are normalized by the absorption coefficient for fBC=1 (pure BC). The black line shows the normalized volume absorption coefficient for populations when all particles are externally mixed for bulk BC mass fractions fBC varying between 0 % and 100 %. For external mixtures, absorption increases linearly with increasing BC mass fraction (black line). The linear relationship applies for all five externally mixed populations with different diameters, so we can only see one black line in the figure.

The colored lines represent the internally mixed monodisperse populations (i.e., after composition averaging) for different diameters. These populations all have higher absorption coefficients compared to the corresponding externally mixed populations. The effect is more pronounced for larger particles and intermediate BC mass fractions because the maximum Δcore is reached. As the table (Fig. ) shows, for a 300 nm population, the normalized absorption is 0.76 when the particles are internally mixed, higher than an external mixed population (0.5). Although this example is an idealized case since our populations lie between external and internal mixtures before composition averaging and are polydisperse, this illustrates that assuming internal mixture will lead to absorption overestimation.

The coating redistribution after composition averaging also changes the absorption enhancement. As shown in Fig. S4, the median Eabs is 1.88 for the reference populations with BC mass fraction less than 10 %, while it is 1.98 for the corresponding populations of the sensitivity library. The absorption enhancement decreases as the bulk BC mass fraction decreases. These values are within the range of previous studies . Similar to the error in volume absorption coefficient ϵ(βabs), the errors are larger for the populations for lower mixing state metric (Fig. b). Lastly, since composition averaging conserves the bulk species mass concentrations, the denominator in Eq. () (total BC mass concentration) remains unchanged, and the errors for MACBC are the same as for βabs.

Considering the volume scattering coefficient, composition averaging resulted in a negative relative error (Fig. a). Similar to what we found for ϵ(βabs), the magnitudes of ϵ(βscat) decreased with increasing χ but were overall smaller, with the largest underestimation of -32% for a population with χ=40% and a median of -1.2%.

Two factors affect the particle scattering ability by composition averaging, the change of the BC core size (and the corresponding change in coating thickness) and the change in the refractive index of the coating. As Fig.  shows, adding a BC core decreases the scattering ability for particles with diameters less than 1200 nm, which is the typical size range considered in our study. This explains the larger scattering underestimation with higher BC mass concentration in Fig. a.

To further explore the effects of coating volume changes, Fig. b shows the size-resolved scattering coefficients before and after composition averaging for the aerosol populations from scenario 77 at t=2 h, which produced the largest scattering coefficients underestimation (-32%). There is a significant decrease of σscat in the size range of 400–800 nm in the sensitivity populations, and the core mass ratio increment in bin 4 is responsible for this decrease (Fig. S5).

The blue lines in Fig.  show the scattering cross sections for two different real refractive indices. For particles with diameters between 800 and 1200 nm, a lower refractive index leads to a larger scattering cross section, although the difference is smaller than the change caused by adding a BC core. Similar to the BC core size change ΔDcore in Fig. , we defined a volume-weighted refractive index change, Δmreal, to help understand the changes in scattering. The index change is defined as 13Δmreal=∑i=1NVimreal′-∑i=1NVimreal∑i=1NVimreal, where i is the particle index, Vi is the particle volume, mreal is the real part of the coating refractive index of the particles in the reference library and mreal′ is for particles in the sensitivity library. Here we applied the total particle volume Vi in the equation to focus on the relation between the changes in scattering and changes in the refractive index. As shown in Fig. , aerosol populations with small errors in scattering tend to be associated with small Δmreal. For more externally mixed populations (with lower χ), Δmreal tended to be larger.

For the effects of composition averaging for particle scattering, we conclude that at a given value of χ, the magnitude of ϵ(βscat) was determined by the change in core/coating volumes and by changes in the coating refractive index. The increase of BC core sizes after composition averaging is the major factor for the decrease of the scattering coefficients. Populations with large underestimation are those with higher BC mass concentrations and large refractive index changes. It is worth emphasizing that we did not consider the absorption of organic carbon that might be present in the coating .