The PNSD of atmospheric aerosols can be roughly represented by a logarithmic normal distribution, i.e.,

(1)

where D is the diameter of aerosol particles. The aerosol number concentration, N_a, is assumed to be constant (20000 cm³), since N_a does not influence the calculation of activation ratios. Here, the geometric deviation, , is set as 1.5-2.0 ( Wu et al., 2008; Shen et al., 2011). The geometric mean diameter, D_g, which usually varies from several to hundreds of nanometers, has a significant influence on the calculated results. Here, D_g is chosen as a set of values between 50 nm and 200 nm, which covers the range of peaks for CCN-active aerosols.

The hygroscopicity of atmospheric aerosols is complicated, because aerosol particles are composed of a large number of substances and these compounds can be internally or externally mixed. Petters and Kreidenweis (2007) proposed a single parameter, κ, to represent the hygroscopicity of aerosols, which avoids a detailed description of chemical compositions and allows measurements and parameterizations of hygroscopicity from different methods to be linked. To simplify the characterization of hygroscopicity, we adapted the size-dependent κ from Liu et al. (2014) as central κ values at the given dry sizes. The SRAR at supersaturation, S, can be expressed as

(2)

where the critical dry diameter, D_c( S), is determined by the supersaturation S, the temperature T, and the size- dependent κ. The parameter C, which is zero for aerosols with homogenous chemical compositions, represents the effect of the aerosol-mixing state on hygroscopicity. C is set as D_c( S)/10, which well characterizes the aerosol activation curves according to previous measurements.

The number size distribution of particles serving as CCN (CNSD) at supersaturation S can therefore be expressed as

. (3)

The DMPS/SMPS-CCNC system measures CN and CCN number concentrations, ( N_CN( D) and N_CCN( D, S)), at different sizes and different supersaturations, and thus provides measurements of SRAR, A_m( D, S) = N_CCN( D, S)/N_CN( D). The DMA selects particles with a certain range of electrical mobility. The selected portion of particles from the DMA, however, contains not only particles with specified diameters, but also ones with the same mobility but multiply charged. Both the DMA transfer function width and the multiple charging contribute to skewing the direct measurement of activation ratios. Here, we simulate the concentrations measured in CPC and CCNC following Hagen and Alofs (1983).

The DMA selects particles with mobility Z_i (i = 1, 2, …, I) when assigned a diameter D_i, and the measured concentration

(4)

where the size parameter x = log D. The kernal function, G(i, x), is

, (5)

where is the probability that a particle of size x carries charges. ( Wiedensohler, 1988). is the transfer function of the DMA ( Knutson and Whitby, 1975), and is related to the sample to sheath flow ratio.

Equation (4) is calculated numerically at intervals much smaller than the measurement, in size range [ x_int,1, x_{int, J}]. The particle number size distribution at size x_{int, j} is expressed as a linear interpolation of those at selected sizes, x _i^*,

(6)

where i(j) satisfies .

Equation (4) is thereby written as

(7)

where

, (8)

, (9)

, (10)

, (11)

Dirac Function

. (12)

Let

, (13)

then Eq. (4) becomes

. (14)

The equation set-up can be written in matrix form as

R=TN, (15)

where T is an I × I matrix determined by the DMA hardware and settings, and R and N are vectors with length I. With known number size distribution, , the concentrations measured by CPC and CCNC can be calculated.

Figure 1 presents an example of the calculated results, assuming PNSD with a geometric mean diameter of 120 nm, a geometric deviation of 1.5, and a critical dry diameter of 70 nm (equivalently, κ = 0.2257 at a supersaturation of 0.4372% and a temperature of 20°C). In this case, the sample to sheath ratio is set as 1:4.

Figure 1a shows the PNSD and corresponding CNSD induced from the assumed SRAR. Using the charging probability of particles and the flow ratio setting in the DMA, the raw concentrations of CN and CCN measured by CPC and CCNC are calculated (Fig. 1b), which provides an estimation of SRAR taking account of the non-zero width of the DMA transfer function width and the multiply charged particles (Fig. 1c). The activation ratios are generally higher than the true values. As the activation ratio is larger for larger sizes, the raw activation ratios for a selected particle size are enhanced due to the inclusion of larger particles with the same mobilities and multiple charges. The width of the activation curve is also larger than the true value.

Figure 1

	Figure Option ViewDownloadNew Window
	Figure 1 An example of particle/cloud condensation nuclei (CCN) number (a) size distributions, (b) the measured concentrations, and (c) activation ratios.

An activation critical diameter, D_50,raw, can be calculated as the diameter where the raw activation ratio is 0.5. In this case, D_50,raw is 60.8 nm, while the true value is 70 nm. If the SRAR are not corrected for multiple charging and the DMA transfer function, the derived hygroscopicity is overestimated as κ = 0.3455 at a diameter of 60.8 nm.

The numerical calculation of the concentrations measured by CPC and CCNC is performed for PNSD with different geometric mean diameters and geometric deviations at different supersaturations. The critical diameters and hygroscopicity are calculated for the SRAR without data inversion (Fig. 2a). Here only the results for the geometric deviation of 1.5 are shown, since the results for other geometric deviations are similar.

The true values of hygroscopicity, κ, are between 0.22 and 0.24 at diameters of 40-120 nm; they slightly increase with diameter, and reach 0.33 at a diameter of 270 nm. Without data inversion, the measured hygroscopicity for PNSDs with a small geometric mean diameter is overestimated at small critical sizes (large supersaturations) and is underestimated at large critical dry sizes (small supersaturations). For PNSDs with large geometric mean diameters, the hygroscopicity is generally overestimated, especially at small critical sizes (large supersaturations). Super-100 nm particles have a high probability of being doubly charged, leading to a large deviation of the calculated hygroscopicity from the true values for PNSDs with large geometric mean diameters.

To quantify the contribution of multiply charged particles to the biases of hygroscopicity, we also perform a calculation considering only the singly charged particles. The thin lines in Fig. 1b present the concentration of singly charged particles measured by CPC and CCNC. The measured number concentrations for particles smaller than 200 nm contain a fraction of multiply charged particles, since super-100 nm particles have a large probability of being multiply charged. The resulting SRARs (Fig. 1c) are not an exact fit but are close to the true values compared to the SRARs with multiply charged particles. The hygroscopicity derived from singly charged particles (Fig. 2b) decreases with diameter between 40 nm and 120 nm, while the true value increases with diameter. The measured hygroscopicity for a singly charged particle is higher for PNSDs with larger geometric mean diameters. However, the measured hygroscopicity for singly charged particles are only slightly different from the true values, less than 0.04, indicating that multiply charged particles account for most of the deviation.

Figure 2

Figure Option ViewDownloadNew Window

Figure 2 True values of the hygroscopicity parameter, κ, and those from measurements without data inversion with a differential mobility analyzers (DMA) sample to sheath flow ratio of 1:4. The black solid line represents the true hygroscopicity, while the others are the measurements of aerosols with different geometric mean diameters: (a) particles with all charges; (b) only singly charged particles.

The width of the DMA transfer function also influences the derived hygroscopicity. The width of the DMA transfer function, which is determined mainly by the DMA sample to sheath flow ratio, F_sa/ F_sh, is usually characterized in the range of mobility by a triangle with a height of one-and-a-half times the width of Z _i F_as/ F_sh centered at the prescribed mobility, Z_i. The hygroscopicity parameters are recalculated using a narrower DMA transfer function at an F_sa/ F_sh of 1:10. The derived hygroscopicity parameters are also highly biased when multiply charged particles are included (Fig. 3a), although they are closer to the true values than those using an F_sa/ F_sh of 1:4. The derived hygroscopicity parameters without multiply charged particles are very close to the true values (Fig. 3b), with the largest deviation less than 0.01. A narrower DMA transfer function results in a smaller bias, because the quasi-mono-dispersed particles selected by the DMA have a smaller size range.

Figure 3

	Figure Option ViewDownloadNew Window
	Figure 3 True values of the hygroscopicity parameter, κ, and those from measurements without data inversion with a DMA sample to sheath flow ratio of 1:10. The black solid line represents the true hygroscopicity, while the others are the measurements of aerosols with different geometric mean diameters: (a) particles with all charges; (b) only singly charged particles.