The campaign was implemented for two periods from 23 January to 25 February and 19 July to 8 September 2019, respectively, at the meteorological station (30.58∘ N, 103.98∘ E) inside the campus of Chengdu University of Information and Technology (CUIT) located in the district of Shuangliu, southwest of the main urban area of Chengdu, China. The elevation of the observation site is approximately 500 m. Figure S1 in the Supplement shows the map of the site. It is surrounded by residential neighborhoods with no nearby sources of significant industrial pollution. Aerosol particles here are representative of the urban environments. More details can be found in .

A custom-built H-TDMA designed by was employed to obtain the hygroscopic properties for aerosol particles with dry diameters Dp(Dry) of 40, 80, 110, 150, and 200 nm. The schematic structure of the H-TDMA is shown in Fig. S2. This H-TDMA was placed in a temperature-controlled (25∘ C) container. The aerosol inlet equipped with a PM2.5 impactor was set on top of the container (approximately 3 m aboveground). It mainly consists of two differential mobility analyzers (DMA) (Model 3081A, TSI Inc., USA), a humidification system, and a condensation particle counter (CPC) (Model 3787, TSI Inc., USA). Aerosol samples with a flow of 0.6 L ⋅ min-1 are first introduced into a Nafion dryer (Model MD-700-24S-3, Perma Pure Inc., USA) by an external vacuum pump to maintain the relative humidity (RH) below 10 %. These aerosol particles flow through an advanced aerosol X-ray neutralizer (Model 3088, TSI Inc., USA) to be single-charged and are then screened by the first DMA (DMA-1) with set voltages. Subsequently, the selected quasi-monodisperse dry particles feed into a Nafion humidifier (Model PD-07018T-24MSS, Perma Pure Inc., USA) with a RH of 90 % (±0.3 %). Finally, the PNSDs of humidified particles out of the humidifier are measured by the second DMA (DMA-2) coupled with the CPC. The details of the H-TDMA refer to .

The growth factor g of the quasi-monodisperse aerosol particles g[Dp(Dry)] selected by DMA-1 equipped in the H-TDMA can be then calculated by 1g[Dp(Dry)]=Dp(RH)Dp(Dry), where Dp(RH) represents the particle diameter screened by DMA-2 at a specific relative humidity (RH), e.g., 90 % in this study. The measured distribution function of g (g-MDF) is calculated from aerosol particle number size distribution (PNSD) counted by the CPC installed downstream of DMA-2. The TDMAfit algorithm is employed to convert the g-MDF to the actual probability distribution function (g-PDF) in the campaign . For each measured dry diameter, the g-PDFs are thereafter normalized as 2∫1∞c(g)dg=1, where c(g) represents the normalized g-PDF. For each g, κ can be converted using κ-Köhler theory : 3κ=(g3-1)×[1RH×exp⁡(4σs/aMwRTρwDp(Dry)g)-1], where σs/a of 0.072 J ⋅ m-2 is the surface tension of the solution–air interface, Mw of 18.015 g ⋅ mol-1 is the molecular weight of water, ρw of 1.0 g ⋅ cm-3 is the density of water, R of 8.315 J ⋅ K-1 ⋅ mol-1 is the universal gas constant, and T of 298.15 K is the temperature in Kelvin. Then, the κ-PDF can be obtained and normalized as 4∫0∞c(κ)dκ=1, where c(κ) is the normalized κ-PDF as the examples shown in Fig. .

The particle-to-particle heterogeneity in aerosol hygroscopicity is calculated by using the H-TDMA-measured κ-PDF (e.g., the examples in Fig. ). The key assumption is that an aerosol containing N aerosol particles is a binary system consisting of the LH and MH components with respective κ of 0.01 (κLH) (typical for organics) and 0.6 (κMH) (typical for sulfate and nitrate). Each aerosol particle in the population contains one or two of the LH and MH components, and thus their κ varied between 0.01 and 0.6. κ-PDF of the aerosol can be considered the normalized aerosol number fractions varied with κ between 0.01 and 0.6 (Fig. ). The volume fraction of these two components in each aerosol particle can be calculated based on κ according to the Zdanovskii–Stokes–Robinson (ZSR) mixing rule .

By referring to information-theoretic entropy measures , three indices, including the average per-particle species diversity Dα, the bulk population species diversity Dγ, and their affine ratio χ, are calculated to together describe the heterogeneity. The details are described as follows.

Firstly, for the aerosol with a given dry diameter and a known κ-PDF with X bins, the volume fraction of the LH and MH components at bin i (i=1,2,3,…,X), i.e., Pi,LH and Pi,MH, can be calculated by the combination of 5Pi,LH+Pi,MH=1 and 6Pi,LH×κLH+Pi,MH×κMH=κi, where κi is the κ at bin i.

Secondly, the mixing entropy for particles at bin i (Hi) can be calculated by 7Hi=-Pi,LH×lnPi,LH-Pi,MH×lnPi,MH.

Then the average particle mixing entropy for the aerosol (Hα) is calculated by 8Hα=∑j=1NPj×Hj, where Hj is the mixing entropy for particle j (j=1,2,3,…,N) in the aerosol. Pj is the volume proportion of particle j to the total volume of the aerosol, which is calculated by 9Pj=VjVtotal=1N, because all particles have the same diameters. Considering that particles in the same κ bin have the same physical and chemical properties, they have the same mixing entropy. Equations () and () can be simplified as 10Hα=∑i=1XHi×c(κ)i×Δκ, where, c(κ)i is the probability density value of the normalized κ-PDF at bin i, and Δκ is the bin width.

Additionally, the population-level mixing entropy (Hγ) is calculated by 11Hγ=-PLH×lnPLH-PMH×lnPMH, where PLH and PMH are the respective volume fraction of the LH and MH components in the population, and they can be calculated by 12PLH=∑i=1XPi,LH×c(κ)i×Δκ and 13PMH=∑i=1XPi,MH×c(κ)i×Δκ.

Thirdly, the per-particle species diversity (Di), the average per-particle species diversity (Dα), and the population species diversity (Dγ) can be calculated as 14Di=eHi,15Dα=eHα, and 16Dγ=eHγ.

Finally, a hygroscopic heterogeneity parameter χ can be calculated as 17χ=Dα-1Dγ-1, which varied from 0 % that all particles in the population purely consist of the LH or MH component to 100 % that the LH and MH components are homogeneously distributed across all particles in the population with identical volume fractions.

A typical four-mode (i.e., a nucleation mode, an Aitken mode, an accumulation mode, and a coarse mode) PNSD with a wide size range of 3 nm–10 µm obtained by is used for the calculation and presentation of the size-resolved heterogeneity. According to the assumption that aerosol particles in a specific mode have common sources or have experienced similar aging processes, the corresponding κmean and κ-PDF of one mode should be the same. The campaign average κmean and κ-PDF for particles with diameters of 40, 80, and 200 nm measured by the H-TDMA are used to deduce the respective κmean and κ-PDF for the nucleation mode, Aitken mode, and accumulation mode of the fitted PNSDs. Although the primary chemical compositions in the coarse-mode aerosol are nearly hydrophobic, some studies showed that this aerosol is slightly hygroscopic due to the coating in atmospheric aging processes . However, κ for the coarse-mode aerosol is hard to be observed by the H-TDMA technique so far, due to the technical bottleneck. Considering that this aerosol has limited influence on radiative forcing and cloud because of its short lifetime and low number concentration, κ for this aerosol is assumed to be 0 as set. According to the contribution of each mode to the κmean of specific particle size, the size-resolved κmean for aerosol particles ranging from 3 nm–10 µm can be estimated from the known κmean of each mode (κmean,i) as 18κmean(Dp)=∑i=14κmean,i×Ni(Dp)∑i=14Ni(Dp).

The size-resolved κ-PDF c(κ, Dp) can be calculated using the similar method with Eq. (): 19c(κ,Dp)=∑i=14Ni(Dp)×c(κ)i×Δκ∑i=14Ni(Dp)×Δκ×ΔlogDp.

Then, the size-resolved heterogeneity Dα(Dp), Dγ(Dp), and χ(Dp) for the wide range of 3 nm–10 µm can be obtained by using Eqs. ()–().

From the calculation of Eqs. ()–(), Dα can represent the average of the effective number of species existing in each particle, and Dγ can represent the effective number of species in the population. Note that 1 ≤Dα≤Dγ. Dα=1 when all particles are pure, while Dα=Dγ when all aerosol particles have identical components. The value of Dγ ranges from 1 to 2 according to the volume ratio of the LH and MH components in the population. Specifically, it is 1 when the population purely consists of the LH or MH component, while it is 2 when the population has the equivalent volume of the LH and MH components. A triangular region is thus presented in Fig. , in which χ is represented by contours.

Figure a–d show an example of the evolution of the heterogeneity in aerosol hygroscopicity. The distributions of the NH and MH components vary between and within populations, causing different heterogeneities for these aerosols. Twelve red dots in Fig. e represent aerosols, each consisting of six aerosol particles with varied fractions of the LH and MH components. Their respective sketch maps and heterogeneity indices are summarized in Table , and the corresponding κ-PDFs are listed in Fig. S3.

For aerosols with particles purely consisting of the LH component or MH component (e.g., populations 1–4 in Fig. e and Table ), Dα equals 1 and χ equals 0, although Dγ varies from 1 to 2 due to the change of the bulk volume ratio of the LH and MH components. Their κ-PDFs show that κ distributes only at 0.01 for the LH component or 0.6 for the MH component due to the pure component in each particle.

For aerosols with equal bulk volume fractions of the LH and MH components (e.g., populations 4–7 and populations 10–12 in Fig. e and Table ), Dγ have constant values of 2 and 1.5, respectively. However, the different distributions of the LH and MH components in each particle of an aerosol contribute to different Dα and χ. Specifically, the heterogeneous distribution (e.g., populations 4–6 and populations 11–12 in Fig. e and Table ) leads to relatively small values of Dα and χ and the dispersed κ-PDF. On the contrary, the homogeneous distribution (e.g., populations 7 and 10 in Fig. e and Table ) generates a narrow κ-PDF and thus relatively high Dα and χ.

For aerosols consisting of particles with identical volume fractions of the LH and MH components (e.g., populations 7–10 in Fig. e and Table ), equal values of Dα and Dγ for each population result in χ of 1, although the bulk ratio of the LH to MH components changes between populations. Accordingly, their κ-PDFs are concentrated at a single value of κmean.

From the populations, it can be concluded that the particle-to-particle heterogeneity in hygroscopicity resulting from the distribution discrepancy of the LH and MH components within particles can be quantified according to the κ-PDF using the proposed algorithm of this work, which is of importance to the further analysis of the existing κ-PDF and can help the heterogeneity to be considered in the calculation of aerosol climate-relevant properties.

In this section, we apply the proposed algorithm to the in situ measurement of the H-TDMA that was implemented for two periods in 2019, located in Chengdu, China. In the following, the expression of the κ-PDF is equivalent to the g-PDF due to their one-to-one corresponding relationship. In order to show the variation characteristics and the significance of Dα, Dγ, and χ in the real atmosphere, especially for the condition that the total hygroscopicity of aerosol remained constant, two episodes with a tiny change of κmean during the observation periods are presented. The 110 nm aerosol is chosen as an example to show the typical evolution processes for aerosol heterogeneity, because it has a large number concentration and lies in the transition zone between the diameters that show the processes of gas-to-particle (<100 nm) and aging (>100 nm) in the atmosphere. The variation of the heterogeneity in a 110 nm aerosol can better reflect the evolution of aerosol particles in the real atmosphere and may have a great impact on the direct radiative forcing. The first episode appeared from 22:00 (LST) 23 January 2019 to 08:00 (LST) 24 January 2019, and the other one occurred from 00:00 27 January 2019 to 12:00 27 January 2019. The variations of Dα, Dγ, χ, and κmean during these two episodes are shown in Fig. . The tiny change of κmean indicates the slight variation of the bulk fraction of the LH and MH components in the aerosol. The decreasing trends of Dα and χ reveal that the existing aged aerosols with uniformly distributed LH and MH components among particles at night, indicated by the unimodal distributions of κ-PDF and relatively high χ (0.831 for episode 1 at 22:00 and 0.832 for episode 2 at 00:00), were intruded upon by the fresh emissions during midnight as reflected by the bimodal and wider distribution of κ-PDF . This process led to component discrepancies among particles in populations and thus the decrease of Dα and χ during midnight.

The statistics of Dα, Dγ, and χ for aerosols with five measured diameters are shown in Fig. . The patterns of the distributions for Dα and Dγ move to larger values with increasing diameter, indicating that the effective number of species increases in aerosols with larger diameters due to the longer aging time in the atmosphere. This trend is more obvious in the summer campaign. Taking the aerosol of 110 nm for example, Dα varies between 1.386 and 1.850, and Dγ varies between 1.470 and 2.000 during the winter field measurements. The ranges of Dα and Dγ in summer are 1.280–1.928 and 1.371–2.000, respectively, contributing to a wider distribution of χ. This indicates that the fraction of the LH and MH components varied more pronouncedly due to the more complex aging processes in summer compared to that in winter. χ ranging from 0.6 to 0.9 reveals the large variation of heterogeneity in aerosol hygroscopicity in the real atmosphere.

The details in the variation characteristics of the heterogeneity and the difference among the aerosols of five measured diameters will be further discussed in our future studies. It can be concluded from the discussion above that aerosols with a tiny variation of κmean can have ever-changing heterogeneity in aerosol hygroscopicity in the atmosphere. The algorithm proposed in this work can efficiently characterize this heterogeneity with time and provide detailed information about the evolution of aerosols in the air. More importantly, this algorithm offers an unexplored understanding of the H-TDMA measurement.

The same bulk volume fraction of the LH and MH components indicates the same κmean for populations (e.g., populations 4–7 and populations 10–12). This reveals the identical diameter for all particles in each population after hygroscopic growth at high RH. Actually, due to the particle-to-particle heterogeneity in one population, different ratios of the LH and MH components in each particle result in different κ (e.g., particles in each of populations 4–7), which can lead to different ambient number size distributions, especially for the condition with high RH. Figure  depicts a sample of the hygroscopic size distribution, where RH is 90 %, for three aerosols with the same κmean of 0.305 but different χ of 0.653, 0.884, and 0.999, respectively. Each of these three populations contains 1000 aerosol particles with dry diameters of 100 nm. Significant discrepancies can be seen in the number size distribution that the width decreases, while the peak value increases with increasing χ after hygroscopic growth. Thus, the heterogeneity in aerosol hygroscopicity plays a significant role in the evolution of aerosol ambient number size distributions, which is a key factor for evaluating aerosol radiative forcing.

The heterogeneity for only one size can contribute to dramatically different ambient size distributions, especially for the condition under high RH. Thus, it is urgently needed to figure out how the heterogeneity varies with aerosol size in the real atmosphere. This section discusses the characteristic of the size-resolved heterogeneity in aerosol hygroscopicity.

The PNSD within the range of 3 nm–10 µm is commonly observed by commercial instruments, including the scanning mobility particle sizer (SMPS) and the Wide-range Particle Spectrometer (WPS) around the world . However, the widely used H-TDMA technic can only observe the growth factor of aerosols with limited sizes (generally smaller than 350 nm in the dry condition), e.g., 40, 80, 110, 150, and 200 nm in this study. Recently, extended the H-TDMA measurement of aerosol hygroscopic properties to 600 nm in the urban environment. Although κ for larger particles (>1 µm) can be derived from the size-resolved chemical composition , the size-resolved κ with high size and time resolution is scarce and is needed to be observed or reversed by more advanced technology and new algorithms. Fortunately, provided an approach to derive the size-resolved κ for aerosols within the range of 3 nm–10 µm based on the measured PNSDs and H-TDMA-determined κ. They used the κmean for each measured size by neglecting aerosol heterogeneity. Here, the heterogeneity in aerosol hygroscopicity is considered based on the method provided by .

The typical four-mode PNSD mentioned above is used and shown in Fig. a and d. The κmean and the corresponding κ-PDFs for the nucleation mode, Aitken mode, and accumulation mode are represented by those measured at sizes of 40, 80, and 200 nm, respectively, during the winter (Fig. b) and summer (Fig. e) campaigns. The calculated size-resolved κmean are shown in Fig. c (winter) and Fig. f (summer), in which κ-PDF for each size contributed by the heterogeneity in aerosol hygroscopicity is represented by the contour. The variation of both the size-resolved κmean and the corresponding κ-PDF is influenced by the contribution of each mode to the total number concentration. The corresponding χ for the size range of 3 nm–10 µm can therefore be calculated by using the framework proposed by this work.

As shown in Fig. , the consideration of the heterogeneity provides the size-resolved κ-PDF, which can further show the region where κ is mostly distributed for aerosols of any size and how aerosol particles evolve with size in the population. For example, although the κmean for aerosols smaller than 40 nm is 0.131 during the winter campaign, the κ-PDF shows most κ locate at a narrow area lower than 0.05. The continuous evolution of the κ-PDF with sizes smaller than 1 µm indicates that on the one hand, a part of the aerosol particles in the population has increasing hygroscopicity during the aging process of growing up, which is the main reason for the increase of κmean, and on the other hand, NH-dominated particles exist across all sizes. This corresponds to the slight decreasing trend of χ accompanied by a wider area of κ ranging from 0.01 to larger than 0.4, which reflects the heterogeneous distribution of the LH and MH components between aerosol particles within a population during the aging progress.

Additionally, the phenomenon that a sharp decrease of χ close to 0.1 for aerosols larger than 1 µm and the κ-PDFs for this size range shows a very narrow band lying at κ of 0 responds to the assumption that the coarse-mode particle is mainly composed of the NH component.

Overall, this newly developed aerosol heterogeneity provides more insight into the evolution of aerosols during the aging processes. The hygroscopicity of an aerosol is complicated and diverse due to the particle-to-particle heterogeneity in aerosol hygroscopicity; if only the κmean is applied without considering aerosol heterogeneity, significant uncertainty will occur.