3.1.1 DMA operation and electrical mobility

DMAs used in atmospheric science include commercially available instruments from Grimm Aerosol Technik, TSI Incorporated, and MSP Corporation. They have also been custom-built by a number of research groups (Mei et al., 2011; Barmpounis et al., 2016; Jokinen and Makela, 1997; Seol et al., 2000). All models allow for the selection of particles through electrical mobility, the ability of a particle to move through a medium (such as air) while acted upon by an electrical field. The DMA size-selects near-monodisperse aerosol from a polydisperse aerosol source, as shown in Fig. 2 (modeled after the Vienna-type long DMA from Grimm Technologies). The electrical mobility Z_p of a particle with mobility diameter d_m can be calculated according to

where n is the number of charges on the particle (assumed to be one in this study), e is the elementary unit of charge, η is the gas dynamic viscosity, and C_C(d_m) is the Cunningham slip correction factor:

where λ is the mean free path (DeCarlo et al., 2004). For the Vienna-type long DMA from Grimm Technologies, Inc. considered here, α_CC=1.246, β_CC=0.42, and γ_CC=0.86 (Grimm Aerosol Technik, 2009).

Figure 2Simplified flow diagram of a DMA with an inner electrode radius r₁, outer electrode radius r₂, distance between aerosol inlet and sample outlet L, clean sheath air flow Q_sh, aerosol flow Q_a, excess air flow Q_e, and sample air flow Q_s.

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Particle-laden flow enters the DMA through the aerosol inlet (flow Q_a) and travels down the DMA column (inner radius r₁, outer radius r₂) with the clean air sheath flow Q_sh. Positively charged particles are attracted by the negatively charged inner electrode, to which voltage V₀ has been applied. Ideally, selection of a voltage allows only particles of a specific mobility diameter to exit the DMA through the sample flow Q_s. All particles with a larger diameter (lower Z_p) or smaller diameter (higher Z_p) will exit the DMA through the excess flow Q_e. In other words, Q_s would ideally consist only of aerosols with diameters equal to, or very nearly equal to, the selected diameter.

In reality, the aerosol flow that leaves the DMA through Q_s is polydisperse with a mobility distribution determined by instrumental parameters. A triangular approximation has been chosen as a model for this distribution, as particle inertia is negligible for the diameters considered in this study (Stratmann et al., 1997; Mamakos et al., 2007). The probability that a particle at the aerosol inlet will exit with the sampling flow is defined by transfer function $f\left(Z_{\mathrm{p}},Z_{\mathrm{p},\mathrm{mid}}\right)$:

where Z_p, mid is the midpoint mobility of the transfer function, and α_TF and β_TF are flow-derived constants, defined as

and

The midpoint and half-width of the transfer function are respectively calculated according to Knutson and Whitby (1975):

and

where L is the distance between the DMA inlet and outlet.

3.1.2 κ_app artifacts arising from DMA flow ratios

Next we assess the ramifications of the DMA transfer function for the derived κ_app. A lognormal theoretical aerosol number distribution was used to represent a polydisperse ambient aerosol population (Fig. 3a). This distribution was converted to an electrical mobility distribution using Eqs. (7) and (8), assuming that the aerosols in the distribution were spherical and singly charged. From the distribution, a series of single aerosol sizes were selected (25, 50, 100, and 200 nm diameter). For each aerosol size, the resulting DMA transfer functions were calculated for seven cases using Eq. (9) and the various parameters for DMA sheath, excess, aerosol, and sample flow listed in Table 1. These seven cases were chosen to represent possible measurement scenarios that may be encountered in a CCN experiment. The aerosol ∕ sheath ratio is varied in cases 1–4 in order to study the effects of chosen experimental parameters. Sheath flow is predetermined in some DMAs (for example, the Grimm Vienna DMA considered in this study), but can be varied in other instruments. The aerosol flow rate may also be selected in an experiment. Cases 5–7 vary the excess ∕ sheath ratio in order to take proper instrument operation into account. The excess and sheath flow should be identical, but small discrepancies may occur.

Table 1Theoretical DMA flow test cases.

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Figure 3(a) A theoretical aerosol distribution generated using a lognormal function centered at 50 nm. (b) The transfer function calculated using Eq. (7). (c) Multiplying the distribution by the transfer function gives the downstream aerosol concentration (cm⁻³).

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For example, the resulting DMA transfer functions for 100 nm aerosol conditions constrained by cases 1–4 are shown in Fig. 3b, in which an increase in Q_a ∕ Q_sh from 0.1 (black line) to 0.3 (green line) tripled the width of the number distribution, and decreasing Q_a ∕ Q_sh to 0.05 (blue line) from 0.10 halved the width of the number distribution. The result of applying the transfer functions shown in Fig. 3b to the distribution in Fig. 3a is shown in Fig. 3c.

All downstream distributions for all seven DMA cases and all aerosol sizes are shown in Fig. S1 in the Supplement. DMA cases 1–4 represent experimental conditions in which the sheath and excess air flows are equal and the aerosol ∕ sheath flow ratio is varied. As Q_a ∕ Q_sh increases, the width of the number distribution measured downstream of the DMA increases, while the midpoint diameter remains constant. It was found that doubling the aerosol-to-sheath ratio doubled the width of the downstream number distribution for 25, 50, 100, and 200 nm particles. For example, when selecting 200 nm particles, increasing Q_a ∕ Q_sh from 0.10 to 0.20 increased the downstream diameter range from 181–222 nm (a spread of 41 nm) to 167–250 nm (a spread of 87 nm). The particle diameter ranges that would be observed downstream of the DMA are summarized in Table 2.

Table 2Predicted downstream particle diameter range for each DMA case.

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To assess the variations in CCN properties resulting from DMA uncertainties, the critical percent supersaturation was calculated for representative atmospheric aerosols. The value of SS_crit was calculated for each particle diameter using Eq. (3a), using literature values for apparent hygroscopicity of 0.61 for ammonium sulfate and 1.28 for sodium chloride (Clegg et al., 1998). It should be noted that this analysis considers two homogeneous aerosol distributions of hygroscopic salts. Real aerosol distributions tend to be mixtures of many species, and the shape of the number distribution can vary among species.

To test how uncertainties in DMA diameter translate to uncertainties in κ_app, the activation of particles downstream of the DMA was assessed. First, for each case and diameter (25, 50, 100, and 200 nm) the critical saturation ratio s_crit was calculated for each particle diameter range downstream from the DMA using Eq. (3a). These critical saturation ratios were converted to critical percent supersaturation SS_crit and used to calculate the activated fraction (AF) for the aerosol particles downstream from the DMA for percent supersaturations $\mathrm{0.01}<\text{SS}<\mathrm{1.5}$, using the equation

where the standard deviation σ was equal to 1∕100 of SS_crit. The small σ ∕ SS_crit ratio was chosen in order to generate accurate activated fraction curves for each particle diameter.

Figure 4(a) Critical supersaturation of ammonium sulfate and sodium chloride particles calculated for DMA cases 1–7 for sodium chloride (triangles) and ammonium sulfate (circles). Ammonium sulfate and sodium chloride curves from κ-Köhler theory are shown for comparison. (b) Apparent hygroscopicity κ_app for DMA cases 1–7. (c) DMA-flow-derived artifacts in ammonium sulfate κ_app are shown for each DMA case. (d) DMA-flow-derived artifacts in sodium chloride κ_app are shown for each DMA case.

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The activated fraction curve for each selected diameter (25, 50, 100, and 200 nm) was then calculated as the sum of the number-weighted activated fractions of each particle diameter downstream from the DMA. For example, for a selected diameter of 25 nm, the downstream diameters ranged from 23 to 27 nm for DMA case 1 and from 20 to 36 nm in DMA case 4. The equation used for this calculation is

where AF_i is the activated fraction calculated using Eq. (12) and $\frac{n_i}{n_{\mathrm{total}}}$ is the fraction of particle downstream from the DMA of diameter i.

This calculation was repeated for each selected diameter (25, 50, 100, and 200 nm), each DMA case (1–7), and percent supersaturation (0.01–1.5) in order to construct activation curves for each selected diameter and DMA case. As an example, in Fig. S2, the shape and position of each activated fraction curve vary with the DMA flow ratios. As the aerosol ∕ sheath ratio increases, the activated fraction curve flattens out (DMA case 4). The critical percent supersaturation SS_crit was then determined for each activation curve as the percent supersaturation, for which AF =0.50. These results are shown in Fig. 4a for ammonium sulfate and sodium chloride. Equation (4) was then used to calculate κ_{app, theory} for each DMA case and selected diameter, as shown in Fig. 4b. Discrepancies between κ_{app, theory} calculated in this study and literature values (hereafter referred to as “κ_app artifacts”) are shown for both compounds in Fig. 4c–d.

The largest κ_app artifact was found in DMA case 4 (in which the aerosol ∕ sheath ratio was the highest) for both ammonium sulfate and sodium chloride aerosols. The artifact for 25 nm ammonium sulfate aerosol in DMA case 4 was 0.08, or ∼13 % of the literature value used for ${\mathit{\kappa }}_{\mathrm{app}}^{{\left({\mathrm{NH}}_{\mathrm{4}}\right)}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}$, while the artifact for 25 nm sodium chloride in DMA case 4 was 0.16, or ∼13 % of the literature value used for ${\mathit{\kappa }}_{\mathrm{app}}^{\mathrm{NaCl}}$. Artifacts were also high for DMA case 6 ($-\mathrm{0.041}\le {\mathit{\kappa }}_{\mathrm{app},\mathrm{artifact}}^{{\left({\mathrm{NH}}_{\mathrm{4}}\right)}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}\le -\mathrm{0.048}$) and DMA case 7 ($\mathrm{0.014}\le {\mathit{\kappa }}_{\mathrm{app},\mathrm{artifact}}^{{\left({\mathrm{NH}}_{\mathrm{4}}\right)}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}\le \mathrm{0.024}$), in which sheath and excess flow were unequal. This result demonstrates that artifacts may still occur when low aerosol ∕ sheath flow ratios are chosen (0.15 and 0.08 for DMA cases 6 and 7, respectively) due to small differences between sheath and excess flow rates (5 % and 2 % for DMA cases 6 and 7, respectively).

κ_app artifacts were larger for sodium chloride ($-\mathrm{0.10}\le {\mathit{\kappa }}_{\mathrm{app},\mathrm{artifact}}^{\mathrm{NaCl}}\le \mathrm{0.16}$) than for ammonium sulfate ($-\mathrm{0.05}\le {\mathit{\kappa }}_{\mathrm{app},\mathrm{artifact}}^{{\left({\mathrm{NH}}_{\mathrm{4}}\right)}_{\mathrm{2}}{\mathrm{SO}}_{\mathrm{4}}}\le \mathrm{0.08}$) across the DMA cases. As our results show, when two or more compounds are compared, the more hygroscopic compound will have larger κ_app artifacts.

This analysis was also applied to the range of apparent hygroscopicity values Svenningsson et al. (2006) reported for ammonium nitrate $\mathrm{0.577}\le {\mathit{\kappa }}_{\mathrm{app}}\le \mathrm{0.753}$, with a mean value of 0.670. If 0.670 is assumed to be the true κ_app for ammonium nitrate, then the sample ∕ sheath ratio used to determine κ_app (1.2–2.0 L min⁻¹) could lead to an experimental kappa as low as 0.665 or as high as 0.674, which would not fully explain the actual experimental range. This assessment ignores possibility of under- or overcounting, which could introduce additional errors.

In addition to the errors discussed above, accuracy in CCN measurements depends on the accuracy of the instrument calibration. Specifically, accurate determination of the percent supersaturation set points within the CCN instrument is dependent on accurate sizing of aerosols entering the CCN, and therefore is dependent on the DMA sizing during CCN calibration. CCN calibrations were performed using two standard compounds, ammonium sulfate and sodium chloride, as described in detail in Rose et al. (2008). Fortunately, if the calibration procedure described by Rose is followed and an optimal DMA aerosol-to-sheath ratio is employed, the uncertainties will be minimal. Specifically, this analysis shows that an aerosol ∕ sheath ratio of 1 : 10 or 1 : 20 (case 1 or 2, respectively) is recommended for all CCN calibrations. This will result in κ_app uncertainties of less than 1 % for all dry sizes (25–200 nm). However, if CCN calibrations are performed using a DMA operated with less-than-ideal aerosol-to-sheath ratios, substantial errors will be introduced. Analysis of the impact of DMA uncertainties on CCN calibrations is discussed in detail in the Supplement. In the worst case scenario amongst the cases evaluated here (case 4), the resulting uncertainty in κ_app is 15 %.

3.1.3 Effect of double and triple charges on particles

During normal operation, the Grimm DMA employs a bipolar charger (also known as a neutralizer) to charge aerosol particles through the capture of gaseous ions. The analysis in Sect. 3.1.2 assumes that each particle carries a single (+1) charge. In reality, the methods used to charge particles prior to entering a DMA may impart two, three, or more charges to individual particles (Fuchs, 1963). The charge distribution resulting from a bipolar charger is roughly approximated using the Boltzmann law (Keefe et al., 1959). However, the Boltzmann law assumes symmetric aerosol particle charging (equal concentrations of negatively and positively charged particles). Deviation from symmetric charging is observed in regions of high ionizations, and this deviation becomes more pronounced as particle size increases (Hoppel and Frick, 1990). A more accurate estimation of stationary charge distribution has been calculated using an approximation formula for the charge distribution produced by a bipolar charger:

where f(k) is the fraction of particles carrying k charges, a_i(k) is the approximation coefficients determined using a least-squares regression analysis, and D_nm is the particle diameter in nanometers (Wiedensohler, 1988). The approximation coefficients only apply to particles with 0, ±1, and ±2 charges. In a separate study, Maricq (2008) determined approximation coefficients for poly (α-olefin) oligomer oil droplets with ±1, ±2, and ±3 charges. The approximation coefficients reported by these two studies were in excellent agreement for particles with a ±1 charge and in weak agreement for ±2 charges (+2 and −2 charging efficiencies were overestimated by 50 % and 100 %, respectively). Therefore, this analysis will use the approximation coefficients from Wiedensolher (1988) for particles with +1 and +2 charges and the approximation coefficient for particles with a +3 charge from Maricq (2008).

Figure 5Theoretical raw (green) and adjusted (blue) activated fraction curves for (a) 25 nm (+1), 50 nm (+2), and 75 nm (+3) particles; (b) 50 nm (+1), 100 nm (+2), and 150 nm (+3) particles; (c) 100 nm (+1), 200 nm (+2), and 300 nm (+3) particles; and (d) 200 nm (+1), 400 nm (+2), and 600 nm (+3) particles. All particles are pure sodium chloride.

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In order to assess the impact of multiple charges on κ_app, Eq. (14) and the approximation coefficients from Wiedensohler (1988) and Maricq (2008) were used to calculate the charge distribution of the representative aerosol population shown in Fig. 3a. The resulting charge distribution is shown in Fig. S6a. An increase in multiple charging is observed as particle diameter increases, though this is offset somewhat by the decrease in concentration with particle size above 50 nm.

Figure 6(a) Critical percent supersaturation of sodium chloride particles determined from activated fraction curves shown in Fig. 5. A κ-Köhler curve for sodium chloride is shown for comparison. (b) Theoretical κ_app for each particle diameter (gray dashed line indicates literature value for ${\mathit{\kappa }}_{\mathrm{app}}^{\mathrm{NaCl}}$). (c) Artifacts in κ_app resulting from multiple particle charges.

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It follows that aerosols incorrectly sized due to double and triple charging will be passed from the DMA to the CCN counter and result in an additional uncertainty in the CCN measurements. To illustrate this, activated fraction curves were generated for 25, 50, 100, and 200 nm sodium chloride particle selection by the DMA (Fig. 5). The activation of sodium chloride is represented by sigmoid curves, for which the midpoint of each activation curve is the κ-Köhler-derived critical supersaturation of sodium chloride, and the standard deviation of each curve is 1∕10 of this value (consistent with the standard deviation ∕ midpoint ratio observed from our instrument's ammonium sulfate CCN calibration data). For each particle diameter, D, the observed activated fraction, AF${}_{D,\mathrm{weighted}}^{\mathrm{SS}}$, for each percent supersaturation SS was determined by weighting the activated fraction AF${}_{D,i}^{\mathrm{SS}}$ of each particle diameter and charge at that percent supersaturation by the fraction of particles of that diameter:

The raw data shown in Fig. 5 (green curves) can be corrected for multiple charging by determining the fraction of particles with $>+\mathrm{1}$ charge from the lower plateau in each plot (dashed lines). The adjusted activated fraction for each percent supersaturation, AF_adjusted, is calculated using the equation

where AF_raw is the raw activated fraction at that percent supersaturation, and AF_plateau is the activated fraction corresponding to the lower plateau (Rose, 2008). The adjusted activated fraction curves are shown in Fig. 5 (blue curves). These are in good agreement with the theoretical κ-Köhler-derived activation curves for sodium chloride (not shown).

Critical supersaturation was determined for each diameter by calculating the percent supersaturation at which the raw AF${}_{D,\mathrm{weighted}}^{\mathrm{SS}}=\mathrm{0.5}$. These critical supersaturations are shown in Fig. 6a, and the theoretical critical supersaturations calculated from κ-Kohler theory are shown for comparison. Equation (4) was used to calculate apparent hygroscopicity for each particle diameter, shown in Fig. 6b. A dashed line in Fig. 6b indicates the literature value for ${\mathit{\kappa }}_{\mathrm{app}}^{\mathrm{NaCl}}$. It is apparent that failing to account for multiply charged particles in the activated fraction curves shown in Fig. 5 leads to an overestimation of κ_app. Artifacts in κ_app are shown in Fig. 6c.

For the theoretical aerosol distribution used in this analysis (Fig. 3a), small, positive deviations from κ-Köhler theory and the literature value for ${\mathit{\kappa }}_{\mathrm{app}}^{\mathrm{NaCl}}$ were observed ($\mathrm{1}\le {\mathit{\kappa }}_{\mathrm{app},\mathrm{artifact}}^{\mathrm{NaCl}}\le \mathrm{0.04}$, 1 %–3 % of ${\mathit{\kappa }}_{\mathrm{app}}^{\mathrm{NaCl}}$). As shown in the figure, κ_app artifacts resulting from unaccounted-for multiple charges decrease with particle diameter for this theoretical aerosol population. Greater κ_app artifacts would be expected for aerosol populations with more prevalent accumulation modes.

The aerosol ∕ sheath ratio within the DMA also modulates the effect of multiple charges on κ_app. As the aerosol ∕ sheath ratio increases, the transfer function broadens, allowing particles that are both larger and smaller than the selected diameter to exit the DMA. This in turn broadens the CCN activated fraction curve (Rose et al., 2008). The larger particles will activate as CCN at lower supersaturations than particles with the selected diameter, resulting in an increase in the activated fraction plateau due to multiply charged particles and a further decrease in the determined SS_crit. Petters et al. (2007) showed that CCN activated fraction curves are significantly skewed by multiply charged particles when the mode diameter of the aerosol population upstream of the DMA exceeds the critical diameter of the size-selected particles. In an example CCN activated fraction curve, Rose et al. (2008) demonstrated that a 1 : 6 ratio of doubly to singly charged particles resulted in an underestimation of the critical activation diameter by 2 %. Zhao-Ze and Liang (2014) also showed that multiply charged particles can introduce significant uncertainty into hygroscopicity calculations.

3.1.4 Additional artifacts resulting from DMA measurements

Several additional factors that may impact experimental κ_app are beyond the scope of this study, but are worth mentioning as they represent additional potential sources of error in some cases. First, volatile aerosols may partially evaporate inside the DMA, resulting in a decrease in particle size exiting the DMA. DMA sizing error due to aerosol volatility (defined as the ratio of sampled diameter to the selected diameter) increases with volatility, though sizing error can be decreased by increasing the sheath flow rate in the DMA. Conversely, hygroscopic aerosols may grow inside the DMA, resulting in larger particles existing in the DMA. Operationally, errors in DMA sizing due to hygroscopic growth can be mitigated if aerosols entering the DMA inlet are in wet metastable states (higher aerosol relative humidity at DMA inlet) and if DMA sheath flow rates are kept low (Khlystov, 2014).

Voltage shifts within the DMA (differences between the selected voltage and the actual voltage inside the DMA) can lead to discrepancies between selected and sampled particle diameters. Voltage shifts may result from a space-charge field generated by the motion of charges within the DMA. Particles charged by the bipolar neutralizer will either be attracted towards or repelled away from the inner column of the DMA, depending on whether they are positively or negatively charged. This charge separation creates a space-charge field that shifts the actual voltage within the DMA from the selected voltage. The impact of the space-charge field on the midpoint and spread of the DMA transfer function increases as particle mobility increases (as particle size decreases) and as particle concentration increases (Alonso and Kousaka, 1996; Alonso et al., 2000, 2001).