In the DMA sizing process, only particles within a narrow electrical mobility (Zp) range can pass through the classifier exit slit and reach the humidification and size distribution measurement system downstream. The electrical mobility is defined as 1Zp=veC(Dp)3πμDp, where C(Dp) is the Cunningham slip correction, e is the elementary charge, v is the number of elementary charges on the particle, μ is the gas viscosity, and Dp is the physical diameter of the particle. From Eq. (1), we can see that the same electrical mobility can be achieved using different combinations of particle diameter and charge, and this is where multi-charge effects come from.

The range that can pass through the DMA is termed the mobility bandwidth ΔZp, defined as 2ΔZp=qaqshZp∗, where Zp∗ is the set mobility and qa and qsh are the aerosol flow rate and sheath air flow rate, respectively. This equation does not account for diffusion broadening.

We can calculate the particle charge distribution at each size using a theoretical model developed by Wiedensohler et al. (1986). The probability that a particle will pass through a DMA classifier can then be determined using the kernel function 3GDp∗,x=∑v=1∞F(x,v)Ω(x,v,Dp∗), where Dp∗ is the diameter set in the DMA, x is the scale parameter, and F(xv) is the charge distribution of the particles that exit from a neutralizer with v elementary charges at the scale parameter x. Ω(x,v,Dp∗) is the probability that particles will pass through the DMA when the set diameter is Dp∗. In this study, the maximum value of v was set to 10.

Therefore, given a particle number size distribution (PNSD) n(x), the number of particles that can pass through a DMA with a set diameter Dp∗ is 4N(Dp∗)=∫0∞GDp∗,xnxdx. If the number of particles that are carrying a specific charge is needed, the kernel function in Eq. (4) should be replaced with 5GvDp∗,x=F(x,v)Ω(x,v,Dp∗), and the corresponding number concentration of particles that can pass through the DMA is 6Nv(Dp∗)=∫0∞GvDp∗,xnxdx. The number ratio of particles carrying different charges can be calculated from Nv(Dp∗)/N(Dp∗).

Two aerosol size distributions discerned from our field measurements during a period of relatively clean air and a period with air pollution (refer to Sect. 4) are shown in Fig. 1. In each case, the ratio of particles carrying different charges was calculated from the PNSD using the abovementioned DMA electrical mobility and charging theory. The detailed calculation procedures are described in Sect. S1 of the Supplement. During the polluted period, when the total particle volume concentration is large, an obvious feature of the PNSD is that the accumulation mode > 100 nm grows very large. The growth of this mode leads to an increase in the proportion of multi-charge particles, especially in the electrical mobility diameter range 100–300 nm. For example, when we set the diameter to 100 nm in the first DMA, more than 40 % of the selected particles are multiply charged. This ratio is about 30 % and 20 % for electrical mobility diameters of 200 and 300 nm, respectively. Thus, the HTDMA-measured size-resolved hygroscopicity will also be influenced by those multiply charged large particles.

Gysel et al. (2009) reported that the center of the kernel function is systematically offset toward smaller GFs with increasing charge. Figure 2 explains the cause of this compression effect. For an electrical mobility diameter of 100 nm, doubly and triply charged particles have physical diameters of about 151 and 196 nm, respectively. When these singly, doubly, and triply charged particles have a true growth factor of 1.6, they will grow in size to 160, 242, and 314 nm, respectively. Since the charge each particle carries remains unchanged, their peak sizes in the second DMA are approximately 160, 154, and 150 nm. Therefore, the growth factors they display in the HTDMA measurements are 1.6, 1.54 and 1.5, respectively. Thus, it is clear that the growth factor is decreased or compressed with increasing charge. We call this phenomenon the compression effect on the growth factor or hygroscopicity caused by multi-charge.

When we set a diameter Dp∗ in the DMA, the electrical mobility Zp can be calculated via electrical mobility theory as 7Zp=eC(Dp∗)3πμDp∗. The physical diameter (Dpν) of particles with the same electrical mobility and a charge number of ν is then 8Dpν=veC(Dp∗)3πμZp=C(Dpν)C(Dp∗)νDp∗. We also define the f function such that 9Dpν=fDp∗,ν. The f function describes the physical diameters of multiply charged particles given an electrical mobility diameter. The properties of and detailed calculation procedures for this function can be found in Sect. S2 of the Supplement.

For a particle of size Dpν with a charge number of ν, if we assume a true growth factor of g0, the particle will grow in size to g0Dpν. The virtual growth factor depicted in the DMA is expressed as gDMA, such that 10g0Dpν=fgDMADp∗,ν=g0fDp∗,ν. Given f, Dp∗, and ν, any g0 can be substituted into Eq. (10) to get a corresponding gDMA. If ν=1, i.e., the particles carry only one elementary charge, then gDMA is equal to g0. If ν>1, i.e., the particles are multiply charged, then gDMA will be lower than g0. Figure 2b presents an example with Dp∗=100 nm and ν=2. The x axis is the assumed true growth factor g0, and the y axis is the calculated virtual growth factor depicted in the DMA (gDMA). Generally, the larger the value of g0, the greater the difference between gDMA and g0.

According to Petters and Kreidenweis (2007), the hygroscopic parameter κ for an aerosol particle can be calculated from the growth factor at a specific RH using κ-Köhler theory. As illustrated in Fig. 3a, all the data points shown in Fig. 2b generate corresponding data points in hygroscopicity space. It is clear that a decrease in the growth factor will results in a decrease in the hygroscopicity, and the compression effect increases almost linearly with the hygroscopicity of the particle. If we fit these data points with a straight line that passes through the origin, the slope of the line can be considered the compression factor. Figure 3a provides an example of this for Dp∗=100 nm and ν=2. For each combination of electrical mobility diameter Dp∗ and charge number ν, we can repeat this calculation and linear fitting process to obtain the hygroscopicity compression factor S(Dp∗v). The algorithm used to calculate the compression factor is depicted in Fig. 4, and the results obtained with it are summarized in Fig. 3b and Table 1.

Multi-charge correction is commonly performed when a DMA is used to scan aerosol particle sizes, especially in PNSD measurements. The shape of the PNSD after multi-charge correction can be significantly different from that of the raw measured PNSD. Therefore, it is necessary to evaluate the effect of multi-charge correction on the size-resolved hygroscopicity obtained by HTDMA. In this study, we developed an algorithm to perform the multi-charge correction of measured values, based on the work of Deng et al. (2011) and Zhao et al. (2019).

Our correction is based on the assumption that, for each electrical mobility diameter set at DMA1, all particles that pass through DMA1 contribute to the measured mean hygroscopicity. Each of the contributing particles has the mean hygroscopicity for its physical size. In reality, when a particle with a large dry diameter Dpν and a charge number passes through the DMA1 and contributes to the MDF, it is hard to elucidate its actual hygroscopicity because there is a probability distribution function for hygroscopicity. However, in our algorithm, this particle is assumed to have the mean hygroscopicity for the particle size Dpν. This assumption is statistically correct and feasible, but may not hold at the single-particle scale.

When the scan diameter in the first DMA is set to Dp∗, the mean hygroscopicity K∗ observed by the HTDMA can be expressed as 11K∗(Dp∗)=1N(Dp∗)∫0∞GsDp∗,xKxnxdx, where Kx is the true mean κ for the scale parameter x, n(x) is the true aerosol number size distribution, and N(Dp∗) is the total number concentration of particles that pass through the first DMA. GsDp∗,x is the transformed kernel function GDp∗,x of DMA1, which includes the compression effect S(Dp∗,v): 12GsDp∗,x=∑v=1∞F(x,v)Ω(x,v,Dp∗)S(Dp∗,v)13N(Dp∗)=∫0∞GDp∗,xnxdx. Equation (5) can then be simplified to the following: 14K∗(Dp∗)=∫0∞HDp∗,xKx or 15K∗=HK, where Kx or K is the true distribution of κ we want to obtain, K∗(Dp∗) or K∗ is the measured κ distribution, and 16HDp∗,x=1NDp∗nxG∗Dp∗,xdx. HDp∗,x (the H matrix) is the forward function and can be calculated from the information available. Given a true κ distribution, we should be able to calculate the measured κ distribution influenced by the multi-charge effect. The H matrix accounts for the DMA transfer function, the particle charge distribution, compression factors, and the number concentration of particles over each size parameter. The detailed steps performed to solve this matrix inverse problem can be found in Zhao et al. (2019).

A hypothetical uncorrected κ distribution K∗(Dp∗) is shown along with two examples of a corresponding multi-charge-corrected Kx distribution in Fig. 5a. This κ distribution represents a common case in the ambient environment: relatively low hygroscopicity for ultrafine particles, high hygroscopicity at accumulation-mode sizes, and coarse-mode particles that are nearly hydrophobic. Multi-charge correction was performed using two different PNSDs representing clean and polluted conditions, leading to two corrected Kx distributions. It can be seen that when κ varies greatly with size, especially among singly, doubly, and triply charged particle sizes, large differences will arise between the pre- and postcorrected κ distributions. For example, the difference between the measured and corrected κ values is largest for sizes of 150 and 350 nm. For a singly charged particle with an electrical mobility size of 150 nm, the corresponding doubly and triply charged particle sizes are around 235 and 314 nm. These three sizes are located in the area of the distribution where κ increases steeply. Similarly, for an electrical mobility size of 350 nm, the corresponding doubly and triply charged particle sizes are about 605 and 852 nm. These three sizes are located in the area where κ drops greatly. Figure 5a also shows that for regions of the curve where κ is increasing, the measured κ will be an overestimate, and for regions of the curve where κ is decreasing, the measured κ will be an underestimate.

Aside from the size-resolved mean hygroscopicity, other key information that can be obtained from a HTDMA includes the mixing state and the detailed shape of GF-PDF or κ-PDF. The correction of GF-PDF or κ-PDF involves the inversion of two-dimensional vectors, which is too complicated for this study. However, the mixing state can be simply represented by the particle number fraction in different GF ranges. Below, we use the number fraction of less hygroscopic particles as an example.

The correction of the mixing state is similar in general to the correction of the mean hygroscopicity, but it differs in some minor aspects. When the scan diameter in the DMA is set to Dp∗, the number fraction of less hygroscopic (LH) particles observed by the HTDMA can be expressed as 17M∗(Dp∗)=1N(Dp∗)∫0∞GsDp∗,xMxnxdx, where M∗(Dp∗) is the measured number fraction of LH particles at the selected diameter Dp∗ in the DMA, and Mx represents the true number fraction at scale parameter x. Also, 18GsDp∗,x=∑v=1∞F(x,v)Ω(x,v,Dp∗)SM(Dp∗,v)19N(Dp∗)=∫0∞GDp∗,xnxdx, where SM(Dp∗v) represents the correction factor caused by the compression effect. This factor varies with the GF probability distribution function (GF-PDF) and cannot be simplified to a constant. If we assume that the compression effect on the LH number ratio can be neglected, then this parameter is 1, and the equation can be simplified to 20M∗(Dp∗)=∫0∞HDp∗,xMx or 21M∗=HM, where 22HDp∗,x=1NDp∗nxG(Dp∗,x)dx. HDp∗,x (the H matrix) can also be calculated from the available information. A hypothetical measured distribution M∗(Dp∗) is shown along with two examples of a corresponding multi-charge-corrected distribution Mx in Fig. 5b.