Interactive comment on “ Technical Note : Hygroscopicity distribution concept for measurement data analysis and modeling of aerosol particle hygroscopicity and CCN activity ”

This study makes a relevant contribution concerning a useful way of describing aerosol mixing state with respect to hygroscopic growth and CCN activity. Presenting mixing state resolved HTDMA or CCNC measurements as cumulative distribution functions (CDF) or equivalently probability density functions (PDF) could make results reported by different research groups much better comparable than often the case in previous studies. Bearing this in mind it is important that the formalism used here is unambiguous, complete and as general and clear as possible. Below I provide some input which


Introduction
Aerosol particles serving as Cloud Condensation Nuclei (CCN) play an important role in the cloud formation process (Pruppacher and Klett, 1997).At a given water vapor supersaturation, the activation of CCN into cloud droplets is determined by particle size and composition, according to K öhler theory (K öhler, 1936).Petters and Kreidenweis (2007) proposed a κ-K öhler theory using a simple parameter, κ, as a quantitative measure of aerosol water uptake characteristics and CCN activity.The values of κ can be determined experimentally by fitting Hygroscopicity Tandem Differential Mobility Analyzer (HTDMA) and CCN measurement data.

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Full very few studies reported different hygroscopicity values among particles of the same size.It would be useful to know such differences, and the influence this would have on the cloud formation process.Internally mixed particles have the same chemical composition and hence the same hygroscopicity, so differences in κ among particles of the same size indicate how well they are mixed.If the κ distribution among particles of a given size can be measured, the (hygroscopicity related) particle mixing state can also be derived.Thus, κ distribution data complement information about aerosol mixing state obtained with other measurement techniques like the Volatility Tandem Differential Mobility Analyzer (VTDMA) (Orsini et al., 1996), the Single-Particle Soot Photometer (SP2) (Schwarz et al., 2006), Scanning and Transmission Electron Microscopes (STEM) and Single Particle Mass Spectrometers (SPMS) (McMurry et al., 1996;Buzorius et al., 2002;Krejci et al., 2004;Murphy et al., 2006).
In this paper, we introduce a concept of particle hygroscopicity distribution and we show how it can be related to hygroscopicity measurements.Model aerosols are used to explain the concept, and exemplary applications are shown with HTDMA and CCN field measurement data.

Hygroscopicity distribution
Particle size distributions are widely used in atmospheric and aerosol science.Here we introduce a similar concept, namely, particle hygroscopicity distribution.In an aerosol population, the hygroscopicity of each particle can be described by an "effective" hygroscopicity parameter, κ (Petters and Kreidenweis, 2007;Sullivan et al., 2009).Here "effective" means that the parameter accounts not only for the reduction of water activity by the solute but also for surface tension effects (Rose et al., 2008a;Gunthe et al., 2009;P öschl et al., 2009).Introduction

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For atmospheric aerosols, the range of κ typically varies from ∼0.01 for combustion aerosol particles up to ∼1 for sea-salt particles (Andreae and Rosenfeld, 2008;Niedermeier et al., 2008;Petters et al., 2009a).If we assort the particles by κ, a hygroscopicity distribution can be defined and described in analogy to the size distribution of the aerosol population.In the following, we apply the same terminology and formalisms as used by Seinfeld and Pandis (2006).
The cumulative hygroscopicity distribution function N(κ) is hence defined as, N(κ) = number of particles per cm 3 having a hygroscopicity parameter smaller than κ Note that N(κ) can be defined for the whole aerosol population, or just for particles of a specific size (bin).In this paper, N(κ) generally refers to a certain particle size (dry particle diameter, D d ).For aerosols with size-dependent particle composition, different distributions, N(κ), can be expected for different values of D d .The concept of the particle hygroscopicity distribution is not limited to the κ parameter.Similar distributions can also be defined and applied with other hygroscopicity parameters, e.g., N(ρ ion ) for the equivalent ion density as proposed by Rissler et al. (2006) and Wex et al. (2007) or N(ε AS ) for the equivalent soluble fraction as used in many earlier studies mostly referring to ammonium sulfate (or bisulfate; Gunthe et al., 2009 and references therein).Moreover, similar formalisms could also be based on the van not Hoff factor i s (McDonald, 1953) or the product of the stoichiometric dissociation number and osmotic coefficient of the solute ν s Φ s , (Robinson and Stokes, 1959;Rose et al., 2008a) averaged over all chemical components of a particle according to mixing rules (e.g., the Zdanovski-Stokes-Robinson), or more advanced models taking into account complex solute interactions and concentration dependencies (e.g., extended aerosol inorganic model, Clegg et al., 2008).Here we focus on the effective hygroscopicity parameter κ, which can be directly and efficiently linked to field measurement data as detailed below.
The normalized cumulative hygroscopicity distribution function N * (κ) is defined by the following equation, where N t is the total aerosol particle number concentration at Introduction

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Full the size bin in which N*(κ) is defined.After normalization, the maximum value of N * (κ) equals one.
Aerosol hygroscopicity parameters determined from HTDMA and CCN measurements are often presented in discrete forms (tables or graphs), which are difficult to generalize and use in theoretical studies.Thus, we pursued the idea of using simple mathematical functions to represent measured distributions.In atmospheric aerosol science, lognormal distributions (Aitchison and Brown, 1957) often provide a good fit to observed data and are commonly used (Seinfeld and Pandis, 2006).
For an aerosol population with lognormally distributed hygroscopicity, κ, the normalized cumulative hygroscopicity distribution function is given by where erf is the Gauss error function, and κ and σ g,κ are the two parameters describing the distribution.The parameter κ is the geometric mean value of κ.The parameter σ g,κ is the geometric standard deviation.For a homogeneously mixed, i.e., fully internally mixed aerosol population, σ g,κ =1.Increasing values of σ g,κ indicate increasing heterogeneity (external mixing) of the aerosol population.
The normalized hygroscopicity distribution function n * (κ) can be calculated from the normalized cumulative hygroscopicity distribution function as follows:

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HTDMA data analysis
In HTDMA measurements, a nearly mono-disperse dry particle fraction of the size D d is selected by the first Differential Mobility Analyzer (DMA) and afterwards equilibrated to a defined Relative Humidity (RH).Then a second DMA is used to measure the size distribution of the equilibrated wet particles, n(D w ) and to calculate the cumulative wet particle size distribution function, N(D w ).According to κ-K öhler theory (Eq.4; Petters and Kreidenweis, 2007), the equilibrium wet particle size, D w , depends on particle hygroscopicity (κ) and on RH, or the water vapor saturation ratio, respectively (s=RH/100%).
Here σ sol is the surface tension of a solution droplet (wet particle), M w is the molar mass of water, R is the universal gas constant, T is the temperature and ρ w is the density of pure water.Assuming that σ sol equals the surface tension of water (σ w =0.072 J m −2 at 298.15 K), wet particle diameters can be directly converted into effective hygroscopicity parameters: By applying Eq. ( 5), the normalized cumulative wet particle size distribution function N * (D w ) can be converted into N * (κ). Figure 1 shows an exemplary case of N * (κ) derived from HTDMA measurements in Beijing, China (Massling et al., 2009).The exemplary case in Fig. 1 is based on one-day average HTDMA measurement results from Beijing, on 12 June 2004.The dry particle diameters selected by the first DMA were 80 nm and 150 nm, respectively.Further details about the measurement campaign, instrumentation and experimental conditions are given by Massling et al. (2009).The Introduction

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Interactive Discussion lower end of the exemplary distribution curves in Fig. 1 reflects the first HTDMA size channel larger than the nominally selected diameter, indicating that 10-20% of the mono-disperse particles have even smaller κ values close to zero.These particles are most likely externally mixed soot particles freshly emitted from strong local sources (e.g., Garland et al., 2008;Rose et al., 2008b;Cheng et al., 2009;Garland et al., 2009;Massling et al., 2009;Wehner et al., 2009).The steep increase around kappa ∼0.5 indicates a lognormal mode of particles, which consists likely of varying amounts of sulfate, organics and aged soot (Massling et al., 2009;Wiedensohler et al., 2009).Equation ( 5) has already been used in earlier HTDMA studies (Petters and Kreidenweis, 2007;Petters et al., 2009b;Wex et al., 2009), and its application for deriving hygroscopicity distributions will not be discussed any further in this paper.HTDMA data can easily be further analyzed and plotted as illustrated below for CCN data (Sects.3 and 4).

CCN data analysis
The size-resolved CCN measurement usually provides two size-dependent parameters, N CN and N CCN .The parameter N CN is the total number concentration in one size bin.The parameter N CCN is the number concentration of particles activated at a given water vapor supersaturation, S, and according to κ-K öhler theory, particles activated at S have a hygroscopicity parameter κ larger than a critical value κ cri : particles (D d,cri ) and the "critical" supersaturation (S cri ) above which particles will be activated: 27κ(ln(S/100% + 1)) 2 ( 8) Since N(κ cri ) means the number concentration of particles with hygroscopicity smaller than κ cri , the sum of N(κ cri ) and N CCN equals N CN .Equations ( 10) and ( 11) show the expression of N(κ cri ) and its normalized form N * (κ cri ) by CCN measurement results.
According to κ-K öhler theory, κ cri depends on the size of particles and the supersaturation, S, as described by Eq. ( 7).For CCN measurements at a fixed dry particle size D d , a given value of S corresponds to a specific value of κ cri (as in Eq. 7) and the measured values of N CCN and N CN yield a value of N * (κ cri ) (as in Eq. 11).To obtain the complete distribution function N(κ) for a given particle size D d , S can be varied so that κ cri covers the whole relevant range of κ.By applying the same procedure for particles at other D d , the complete representation of N * (κ) in the D d −κ plane can be obtained.This approach (called method I or "S scan"), of first keeping D d constant and varying S, then choosing another D d and doing the same procedure, has been adopted in several studies (e.g., Dusek et al., 2006;Frank et al., 2006).Alternatively, S can be kept constant while varying D d .Then another S is selected and the same procedure Introduction

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Full In principle, method I and II are mathematically equivalent with regard to probing the surface of the N * (κ) distributions in the D d −κ plane.Method I is easier to interpret and understand, but because most of the recently reported CCN measurement studies have applied method II, our subsequent discussions will be focused on method II (sizeresolved CCN measurements, varying D d at a constant S).As mentioned above, N(κ) can be defined for the whole aerosol population, or just for particles of a specific size (bin).However, only size-resolved CCN measurement data can be used to determine N(κ), because of the dependence of κ cri on D d .Therefore N(κ) derived from sizeresolved CCN measurements always refers to a specific D d .

Uncertainties
The uncertainty of N * (κ) derived from size-resolved CCN measurements is determined by the uncertainty of the N CCN /N CN measurement data, which depends on various factors like instrument calibration, counting statistics, and various correction factors (counting efficiency, electric charge, DMA transfer function, particle shape, etc.; (Rose et al., 2008a).Moreover, κ values determined by HTDMA measurement can be different from κ values determined by CCN measurements (Petters and Kreidenweis, 2007), because of the general dependence of κ, i s and Φ s on solute concentrations (Mikhailov et al., 2004;Rose et al., 2008a;Reutter et al., 2009) and potential solubility effects (Petters and Kreidenweis, 2008).Thus, the uncertainty, applicability, and extrapolation of hygroscopicity distributions determined by HTDMA or CCN measurements depend on the ambient and experimental conditions and on the quality of the measurement data (e.g., Rissler et al., 2006;Svenningson et al., 2006;Vestin et al., 2007;Gunthe et al., 2009;Petters et al., 2009a;Wex et al., 2009).If the chemical composition and properties of a particle population are known or can be estimated, it would be possible to calculate hygroscopicity distributions as a function of relative 1013 Introduction

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Full Screen / Esc Printer-friendly Version Interactive Discussion humidity.As indicated above, simple mixing rules or advanced models could be used for such purposes, and surface tension effects could also be taken into account.A detailed discussion of such aspects, however, would go beyond the scope of the present manuscript, which is aimed at introducing and illustrating the basic concept.

CCN efficiency spectra calculated from model hygroscopicity distributions
The concept of N(κ) helps in interpreting the CCN measurement data.Previous studies tended to calculate a single κ value from one CCN efficiency spectrum, e.g., only the κ cri corresponding to the diameter D d where 50% of the particles are activated.In fact, each N CCN /N CN at size D d represents one N * (κ) value at κ cri corresponding to D d .In this section, we show how N * (κ) are reflected in the CCN efficiency spectra (activation curves) of size-resolved CCN measurements.Three distributions of N * (κ) (in Table 1) and the corresponding activation curves by D d scans (method II) are presented.

Internally mixed aerosols
In Case A, the aerosol is assumed to be 100% internally mixed.All particles have the same composition and hygroscopicity corresponding to a mono-disperse distribution N * (κ) with σ g,κ =1 and κ=0.2.If we made a CCN measurement of this aerosol at S=0.86%, 0.26% and 0.067%, the ideally obtained CCN efficiency spectra would be perfect step functions as shown in Fig. 3.All particles are activated at D d ≥D d,cri and none are activated at D d <D d,cri (D d,cri can be calculated by Eq. 8).Note that such step functions could be observed only under assumed ideal measurement conditions.In practice, experimental uncertainties will result in some dispersion even for pure calibration aerosols (Rose et al., 2008a).Introduction

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Full In Case B, we assume an aerosol with a size-independent κ distribution consisting of a single lognormal mode, i.e., N . The values of κ and σ g,κ were set to 0.2 and 1.583, respectively, and these values are the same over all D d .Figure 4 shows the n * (κ) of Case B aerosols.If we made CCN measurements of such aerosol at S=0.86%, 0.26% and 0.067% (tilted lines in Fig. 4), the obtained CCN efficiency spectra would be cumulative lognormal distribution functions as shown in Fig. 3. Similar CCN efficiency spectra have been observed for well-mixed and aged atmospheric aerosols in several field measurements (Rose et al., 2008b).

Complex externally mixed aerosols (two-mode lognormal κ distribution)
In Case C, we assumed an aerosol with two lognormally distributed modes of N * (κ).In one mode (mode 1), κ is size-independent with κ 1 =0.05 and σ g,κ,1 =1.1.In the other mode (mode 2), κ is size-dependent with κ 2 (D d )=0.2(D d /20 nm) 0.4 and σ g,κ,2 =1.587, as illustrated in Fig. 5.If we made a CCN measurement of this aerosol at S=0.86%, 0.26% and 0.067% (tilted lines in Fig. 5), the obtained CCN efficiency spectra would show the existence of the two modes as illustrated in Fig. 3. Despite knowing the size dependence of κ in mode 2, one cannot easily see it from the CCN efficiency spectra in Fig. 3.That is one of the reasons that we recommend using retrieved N * (κ) as an analysis tool.Compared with the CCN efficiency spectra, the form of N * (κ) distributions presents the particle hygroscopic properties more explicitly.Both the shape and size dependence of particle N * (κ) can be seen from Fig. 6.In Case A and Case B, the N * (κ) calculated from CCN efficiency spectra are consistent with our assumptions, showing a lognormal distribution of κ=0.2 and σ g,κ =1 or 1.587, respectively.In Case C, the two-mode lognormal distributions are clearly seen.The first mode has a κ 1 =0.05 and the second mode shows an increasing trend in κ 2 for measurements at increasing D d (because in CCN measurements a lower S corresponds to larger D d for the same κ cri ).

Hygroscopicity distributions calculated from measured CCN efficiency spectra
In the CAREBEIJING 2006 campaign size-resolved CCN measurements at different S were carried out from 12 August to 8 September (Garland et al., 2009;Wiedensohler et al., 2009).The CCN data were recorded and processed in the same way as described in detail by Rose et al. (2008b) and Gunthe et al. (2009).
Figure 7 shows the measured CCN efficiency spectra averaged over the campaign, and Fig. 8 shows the corresponding distributions N * (κ), which are comparable to the ones obtained for the model aerosol Case C (Fig. 5).Over 95% of the ∼1536 individual distributions N * (κ) recorded during the campaign can be fitted by cumulative lognormal distribution functions with coefficients of determination R 2 >0.8, supporting the idea that lognormal distribution functions are well suited not only for describing the size distribution but also the hygrocopicity distribution of atmospheric particles.
Figure 9 shows the measurement-derived values of n Introduction

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Similar size dependencies of κ have also been observed in recent field studies of atmospheric aerosols (Rose et al., 2008b;Gunthe et al., 2009) and chamber experiments with combustion aerosols (Petters et al., 2009a).As illustrated in Fig. 8, the measurement data indicate a significant particle fraction (∼10%) with κ values <0.1.These are most likely externally mixed soot particles freshly emitted from strong local and regional sources (Garland et al., 2008;Rose et al., 2008b;Cheng et al., 2009;Garland et al., 2009;Wehner et al., 2009).They would likely show up as a low hygroscopicity mode at the bottom of Fig. 9 (in analogy to Fig. 5) if the available data would cover this area.The CCN measurement data available for Beijing, however, provide only an upper limit value for the effective hygroscopicity parameter of these particles and do not allow a full characterization of the low hygroscopicity mode (insufficient data at high S and high D d , respectively).

Figure 2
Figure 2 illustrates the two different methods used in measuring N * (κ) where each line represents one measurement cycle.Method I is shown by the vertical dashed lines (constant D d ).Method II is represented by the tilted solid lines (constant S).In principle, method I and II are mathematically equivalent with regard to probing the surface of the N * (κ) distributions in the D d −κ plane.Method I is easier to interpret externally mixed aerosols (single-mode lognormal κ distribution)

3. 4
Retrieval of N * (κ) from CCN efficiency spectra To obtain N * (κ) from an observed CCN efficiency spectrum (activation curve, by D d scan), each data pair of N CCN /N CN vs. D d in the CCN efficiency spectrum is converted into a corresponding data pair of N * (κ cri ) vs. κ cri by Eqs.(11) and (7).The N * (κ) obtained from the CCN efficiency spectra in Fig. 3 are shown in Fig. 6

Fig. 1 .
Fig. 1.Normalized cumulative particle hygroscopicity distributions N * (κ) calculated from exemplary HTDMA measurements performed in Beijing (one-day average for 12 June 2004).The dry particle diameters selected by the first DMA were 80 nm and 150 nm, further details can be found in the work of Massling et al. (2009).

Fig. 2 .
Fig. 2. Two methods of probing particle hygroscopicity (κ) by CCN measurements: (1) method I or "S scan" is represented by vertical dashed lines, in which the dry particle diameter D d is first kept constant and the water vapor supersaturation S is varied, then choosing another D d and doing the same procedure; (2) method II or "D d scan" is represented by the tilted solid lines, in which S is first kept constant and D d is varied, then choosing another S and doing the same procedure.
and laboratory studies.With the help of single particle analysis techniques, it should become possible to predict aerosol hygroscopicity distributions from chemical composition data.Accordingly, we propose and intend to use hygroscopicity distributions for testing the influence of observed and predicted aerosol mixing state on the formation of cloud droplets in atmospheric model studies.Introduction
d /20 nm) 0.4 1.587The parameter a represents the relative proportion of particles in mode 1.The erf function is the Gauss error function.