## 1. Introduction

Atmospheric particles may modify earth's climate directly by light scattering or absorption, or indirectly by cloud modification (Houghton et al. 1996). Both effects are not fully understood, but the most serious gaps in understanding lie in the field of cloud microphysics. Cloud droplets are formed when the air becomes supersaturated with respect to water vapor, but the supersaturations found in the atmosphere are not large enough for homogeneous nucleation. Instead, the presence of suitable particles in the atmosphere allows the water to condense at significantly lower supersaturations. Methods for detecting such particles (starting with sizes of a few nanometers) exist now for more than a century (see the review on the history of condensation nucleus counters, McMurry 2000) but instrument development is continuing, using various condensable substances.

*p*

_{eq}over the surface of a droplet of pure liquid of diameter

*d*

_{K}is given by Kelvin's law where

*σ*is the surface tension,

*M*

_{υ}is the volume in the liquid state,

*R*is the universal gas constant,

*T*is the temperature, and

*p*

_{sat}is the saturation vapor pressure over a flat surface of the pure liquid. In all our calculations, we use the empirical relation of Leiterer et al. (1997) for

*p*

_{sat}of water, which was used in all the experiments and calculations presented here. Other liquids were studied in Holländer et al. (2000).

Particles of arbitrary size, shape, and physicochemical composition also have an equilibrium vapor pressure that is no longer a simple function of the size and the properties of the pure liquid alone. If the actual ambient vapor pressure *p*_{act} exceeds the equilibrium value *p*_{eq}, vapor starts to condense on the particle, and the particle is activated by heterogeneous condensation.

*S*− 1 and often given in percent. The equilibrium supersaturation of water droplets [Eq. (1)] is shown in Fig. 1. The corresponding equilibrium diameter is called Kelvin diameter

*d*

_{K}; it is widely used also for solution droplets (see section on particle activation and Kelvin diameter) or nonsoluble particles with hydrophobic surface, etc. This abstraction is useful for the simple description of the heterogeneous nucleation behavior as is the aerodynamic particle diameter for the description of gravitational settling of nonspherical particles of unknown density.

Instruments capable of measuring droplet concentrations at given supersaturations are called Kelvin spectrometers, and they are important for reliably characterizing atmospheric particles.

Understanding cloud drop nucleation is an important issue in the global change context (see, e.g., Twomey 1977; Hudson's 1993 review; Houghton et al. 1996; Rosenfeld 2000), and a variety of (continuous and discontinuous) measuring devices has been developed for this purpose. Twomey's (1959, 1963) description of a static cloud diffusion chamber and his field data triggered the development of a variety of similar devices based on the same principle of wet plates maintained at different temperatures, which is continuing up to now (Delene et al. 1998). Later on, alternating and streamwise thermal gradient flow diffusion cloud chambers (Hoppel and Wojciechowski 1981; Hudson 1989) were developed, and recently the performance of an instrument combining both features was thoroughly analyzed (Chuang et al. 2000; they also give a good survey on various types of diffusion cloud chambers).

All the above-mentioned instruments are based on the diffusion cloud chamber (DCC) principle, which allows setting up small supersaturations as they prevail in clouds. The success of DCCs stems from their simplicity and the fact that low supersaturations can be maintained, but they have shortcomings, too.

Normally, in assessing their performance, the static supersaturation profile is taken at face value and transient effects are neglected. However, DCCs exhibit pronounced transient behavior and close to the upper plate briefly a supersaturation larger than maximum equilibrium supersaturation is achieved (due to the different diffusivities of heat and water vapor), which may last long enough to activate nuclei resulting in misleading activation profiles.

Another important effect is that the stationary supersaturation is greatly reduced in the presence of sinks such as growing droplets. This is due partly to the vapor consumption and partly to the release of latent heat, which increases the saturation vapor pressure and hence decreases the saturation ratio even if the actual vapor pressure is kept constant.

Figure 2 shows the stationary profiles for two different temperature differentials Δ*T* = 4 K (left) and Δ*T* = 8 K (right) with the same average temperature *T*^{–10} kg (m^{3} s)^{–1}. In the right part, the sink strength is 7 × 10^{–10} kg (m^{3} s)^{–1}. The above sink strengths correspond to the volume of 10 droplets per cubic centimeter growing to 0.33- and 0.5-*μ*m diameter in 1 s. This means that even such small sinks considerably affect saturation ratio. On the other hand, the forced continued water flux means that activated droplets continue to grow until they settle out of the measuring volume. In the atmosphere constant supersaturation may (orographic cloud) or may not be (radiation fog) maintained and an instrument capable of measuring at quickly established, well-defined initial supersaturations in the whole measuring volume seems desirable. Such an instrument could be an expansion-type device operating with a fixed amount of water (no flux from the walls).

Another difficulty with DCC is the need for calibration with a known number concentration standard. Probably the most important shortcoming, however, is the slow establishment of supersaturation, which may result in erroneous results when at high cloud condensation nuclei (CCN) concentrations easily activable particles reduce supersaturation by their growth so that smaller CCN (which could have grown at the initial supersaturation) cannot be activated any more (Holländer and Schumann 1979).

In view of this lack of well-defined characteristics of diffusion chambers, we explored employing adiabatic expansion (see Wagner 1982, 1985) because of

- the quick establishment and the wide range of supersaturations;
- the supersaturation homogeneity over the whole chamber volume immediately after expansion;
- the easy and reliable calculation of the resulting supersaturation from first principles;
- the development of a cloud of monodisperse droplets giving rise to distinct Mie scattering and extinction signatures, which allow concentration determination without calibration just by comparing experimental signals with Mie theory. This is not an assumption but will be checked in each individual measurement;
- the easy use of operating liquids other than water facilitating a wide range of (even binary) heterogeneous nucleation studies (see Holländer, et al. 2000).

The next section briefly describes the basic features of the spectrometer design while details of the spectrometer theory and its comparison with experimental results are discussed in the modeling section and detailed aspects are deferred to the appendix.

## 2. Spectrometer design considerations

*κ*has the approximate value of 1.4 for air, but we use Richarz' precise formula (1906) for a mixture.

Let us for the moment assume that an initial volume *V*_{0} filled with an aerosol at temperature *T*_{0} = 300 K and relative humidity RH_{0} = *S*_{0} ≤ 1 is expanded to volume *V*_{1} as shown in Figs. 3a and 3d (RH is used to designate humidifier performance, while *S* is used for the saturation ratio after expansion). The corresponding changes in temperature *T*_{1} are shown in Figs. 3b and 3e. The resulting supersaturation SS developing under adiabatic conditions is shown in Figs. 3c and 3f. It is important to note that very small temperature differences suffice to produce supersaturations of the order encountered in clouds, which underlines the importance of adiabaticity. Also, nonperfect humidifier operation RH_{0} < 1 is less important for high expansion ratios than for lower ones where it may prevent *S*_{0} > 1 altogether. In conclusion, the humidifier and expansion chamber must be dimensioned carefully to allow operation at droplet concentrations and supersaturations relevant for the atmosphere. For example, the chamber diameter must be large enough to prevent (convective and molecular) heat conduction from the warm walls to reach the cool measuring volume.

The operating principle of the present device optimized for adiabatic operation at low supersations is shown schematically in Fig. 4 with a short explanation given in the caption. The aerosol flow enters the instrument through the inlet I, a valve V1, and a humidifier H and has a relative humidity RH ≲ 1 when it reaches the measuring chamber (MC). The instrument measuring cycle consists in flushing the device at ambient pressure by opening valves V1, V2, and V3 by pump P. Next, by closing V1 and V2, the buffer chamber (BC) is evacuated to a certain preset value after which V3 is also closed. Upon opening valve V2, the pressures in the MC and BC equilibrate. The rapid pressure drop in MC is accompanied by an adiabatic temperature decrease, which may eventually lead to sufficient supersaturation followed by droplet growth. Normally, the cloud droplets are monodisperse, and characteristic Mie scattering and extinction is observed and can be used to determine concentration. It can be easily recognized, if this is not the case.

There are good reasons for the validity of fundamental modeling of the performance of such an instrument.

- Shortly after adiabatic expansion the supersaturation in MC can be calculated from first principles if the relative humidity before expansion is known.
- After an extremely short initial period following activation, growth will proceed as if it were a droplet of pure liquid (even for hygroscopic nuclei because of dilution, see below). After this initial period, droplet growth speed for some time only depends on the supersaturation and the amount of condensable vapor available and should follow growth theory precisely. Later, growth slows down due to vapor depletion and becomes concentration-dependent. Both processes can be easily described by theory.
- Droplets are spherical, so their optical behavior can be quantitatively described by Mie theory. In our instrument, they are also monodisperse because they start growing at nearly the same time in the same homogeneous supersaturation (and because of an additional mechanism described in the section on droplet growth) so that their collective optical properties can be described by those of single droplets. If droplets were not really monodisperse, no Mie structures would show up in the scattering signal. These Mie structures allow the sizing of droplets at any time.
- Given the known size and measuring the cloud extinction [i.e., its optical depth (OD)] over the chamber diagonal, the number concentration can be directly determined. Applying Lambert–Beer's law seems justified as the maximum practical OD ≈ 0.04 so that multiple scattering may be safely neglected (Filipovicova et al. 1995).

The instrument inlet for atmospheric applications normally is a 5-*μ*m cutoff impactor designed for a 2 L m^{–1} flow rate to make sure that in clouds droplets are removed, and only interstitial aerosol is measured. Normally, the interior instrument temperature is kept floating at 5° above ambient to make sure that even in clouds without additional humidification by a humidifier no supersaturation can be achieved after expansion. The minimum temperature is normally set to 5°C to safely prevent saturator freezing. If required, however, instrument heating may be turned off, and under such conditions the saturator is allowed to freeze and growth of supercooled droplets is observed in the chamber (ice nuclei are too rare to change the overall scattering signature). The total operating temperature range covered by the instrument so far is between −9° and 35°C. With a saturator (at the chamber inlet) and optical collimator (in front of the PM tube) we use a cordierite car catalyst support with a 1-mm honeycomb structure, which carries about 0.4 g water per g cordierite; this suffices for about 3 days of continuous operation in ambient air of typical middle-European humidity. Expansion typically takes about 50 ms. The sampling rate of the transmission and scattering analog-to-digital converters (ADCs) is 200 Hz and resolution is 12 bits. Maximum droplet growth time is not limited by heat conduction from the walls (which would imply a timescale of significantly more than 10 s) rather than by internal convection currents presumably set up by small temperature gradients.

## 3. Modeling spectrometer performance

The first part of this section is devoted to theoretical modeling of the device while later performance aspects will be discussed.

*S*

_{th,1}as derived from adiabatic ideal gas behavior in the appendix [Eq. (A11)] is given by where the temperature

*T*

_{10}and the pressure

*p*

_{10}before expansion are measured, and the final values

*T*

_{11}and

*p*

_{11}follow from theory.

First, we seek to assess theoretically the relative humidity after the humidifier before expansion RH_{0} [the only unknown and not easily measurable quantity in Eq. (4)], which is required for the instrument design. Next, a variety of factors potentially affecting the actual saturation ratio during the measuring process is discussed.

### a. Saturation quenching by heat conduction from the wall

*S*

_{th,1}will have only a limited lifetime: due to its huge thermal capacity, the chamber wall always remains at the start temperature

*T*

_{10}, while immediately after expansion, the interior is at a temperature

*T*

_{11}. From then on, heat conduction will increase the interior temperature

*T*

_{i}(with

*T*

_{11}<

*T*

_{i}<

*T*

_{10}) and reduce the saturation ratio. If convection can be neglected and molecular heat conduction only is assumed, the temperature and the resulting saturation ratio profiles can be easily calculated following Carslaw and Jaeger (1980) assuming perfect saturation in the humidifier. The characteristic timescale available for droplet growth scales is where

*L*is a characteristic chamber dimension and

*a*is the heat diffusivity. Such calculations show that up to about 10 s after expansion in the center nearly the full supersaturation remains, and observation of the optical properties of the droplet cloud confirm this.

*L*is a typical chamber dimension,

*g*is the gravitational acceleration,

*β*is the expansion coefficient,

*υ*is the kinematic viscosity,

*a*is the heat diffusivity, and Δ

*T*is the temperature nonuniformity along the wall. The onset of laminar flow occurs at Ra ≈ 1100 and of turbulent flow at Ra ≈ 5 × 10

^{4}. Therefore, Δ

*T*< 9 mK is required to suppress convective flow for

*L*= 0.1 m.

For instrument design, a compromise must be found since small *L* are required for suppressing convection, while large *L* are required for obtaining large droplet growth times undisturbed by heat conduction.

### b. Humidifier performance

Knowing the exact relative humidity of the air past the humidifier is crucial to instrument performance according to (4) and Fig. 3. The method chosen here is humidifier overdesign so that RH = 1 is ensured for all operating conditions. If this is the case, *S* ≥ 1 everywhere in the chamber for the total measuring cycle. If, however, humidification were not perfect, increasing fractions of the chamber would become subsaturated and growth would become spatially nonuniform at later times.

#### 1) Humidification efficiency

_{wall}= 1, while in the diffusional loss context

*c*

_{wall}= 0. Applying, for example, Hinds's (1982) diffusion battery approximation to the humidifier scenario with where RH

_{in}, RH

_{out}are the relative humidities at the inlet and the outlet, respectively. Here

*D*

_{υ}is the water vapor diffusion coefficient in air (2.4 × 10

^{–5}m

^{2}s

^{–1}),

*L*is the channel length (4.1 cm in our case), and

*Q*is the flow in a single channel (there are more than 6200 channels). Equation (7) is applicable for

*μ*

_{υ}> 0.007, and the actual value is

*μ*

_{υ}≈ 183. The residence time in the humidifier is about 7.6 s, which may be of importance later when we estimate the deposition losses of growing hygroscopic particles.

For a total flow of 2 L m^{–1} we obtain a theoretical worst-case (RH_{in} = 0) output humidity of 1–6 × 10^{–915}, which certainly can be called overdesign. The resulting relative humidity estimate is indistinguishable from unity, but this may be misleading because the assumption for the boundary condition assumes an isothermal wall. Actually, if dry air is entering the humidifier, its walls cool down and the boundary condition is still *S* = 1 but at a lower temperature.

In our measuring campaigns in clouds or just below cloud base, we never had reason to doubt RH = 1, but in laboratory experiments with dry or only partially humid air, we occasionally found 0.975 < RH < 1. Under such conditions RH must be derived from droplet growth dynamics as discussed later.

To cope with such situations, one can add an external prehumidifier, but in any case a trade-off between humidification efficiency and particle losses must be made.

#### 2) Particle losses in the humidifier

Here we have to discriminate between losses of nonhygroscopic particles, which do not grow when exposed to humidity, and hygroscopic ones, which may become so large that they may be subjected to gravitational settling. Therefore, diffusional losses need to be taken into account as well as sedimentation.

*P*

_{dp}through a diffusion battery is given by using the coefficients from (7). With the particle diffusion coefficient

*D*

_{p}= 2 × 10

^{–9}m

^{2}s

^{–1}for a 50-nm particle, we obtain a particle penetration of more than 0.7, and for larger particles, which form the majority of CCN, penetration is still higher.

*P*

_{sp}of particles sedimenting in laminar flow in horizontal circular channels is given by (Natanson 1964, cited by Fuchs 1989) with where

*d*

_{c}is the channel diameter and

*υ*

_{sed}is the particle settling velocity. This may to be taken into account since in our present setup, we have a horizontally oriented humidifier to prevent drops falling onto the photomultiplier (PM). For a particle of aerodynamic diameter of 0.5

*μ*m, penetration would be about 0.85, while for 1-

*μ*m particles penetration would drop to about 0.55. The situation is more complicated for hygroscopic particles because they grow during passage (see next section) but most of the difficulties can be avoided by placing the humidifier vertically so that only diffusion losses remain.

### c. Particle activation and Kelvin diameter

Soluble particles start growing earlier than droplets of pure liquid since their activity is lower due to Raoult's effect (see Fig. 5). Consequently, activated salt particles will first start as saturated solutions (*c* ≳ 0.26) with a solid nucleus, but once the latter has dissolved (at a droplet size 1.92 times the salt nucleus diameter), dilution quickly brings activity back to the level of the pure liquid so that after this short transient period growth will be that of a pure water droplet. For example, after a dilution by 1000, the solution activity has increased to 91% of that of pure water.

For droplets of finite size, the Kelvin effect also has to be taken into account. If, for instance, a 200-nm diameter NaCl nucleus is exposed to RH = 1, its equilibrium size will be 3.8 *μ*m due to the combined effects of Raoult and Kelvin (horizontal line in Fig. 6). Droplets of such size could be lost due to gravitational settling in a horizontal humidifier provided that hygroscopic growth is fast enough during residence time (discussed below). Only if *S* > 1.000 18, that is, the maximum equilibrium saturation ratio (left dot), the droplet will continue growing and, therefore, its Kelvin diameter will be 11.6 *μ*m since a pure water droplet of this size has the same equilibrium saturation ratio of 1.000 18 (right dot on upper curve for pure water).

### d. Droplet growth

Once the droplet has been activated, its growth speed depends on the absolute amount of vapor present and the supersaturation. For insoluble activated particles of a certain diameter, *p*_{sat} is that of a pure liquid droplet of the same size. Solute droplets, however, have a reduced vapor pressure that may affect activation as well as growth as discussed in the following.

#### 1) Growth of single droplets of pure liquids

*λ*

_{g}is the gas mean free path. The full single droplet growth equation (neglecting release of latent heat for the moment) expressed in Knudsen number reads (Holländer 1995) where Λ =

*λ*

_{n}/

*λ*

_{g}involves the nucleation scaling length

*λ*

_{n}For water, its numerical value is roughly 1 nm. Other variables are

*σ,*the liquid surface tension;

*ρ*

_{liq}, the liquid density;

*M,*the molecular weight;

*R,*the gas constant, and

*T,*the absolute temperature. The dimensionless droplet volume

*β*takes into account the reduction of growth rate due to vapor consumption by the growing droplets, and

*N*is the number concentration,

*ρ*

_{s}is the particle solid density,

*S*

_{0}is the saturation ratio at the beginning, and the mass transfer rate factor Φ (in s

^{–1}) is given by

*β*→ 0) and very small Knudsen numbers, when

*e*

^{ΛKn(t)}→ 1, the bracket in Eq. (12) reaches the limiting value

*S*

_{0}− 1 and we obtain for the droplet diameter

*d*

_{p}(

*t*) starting with

*d*

_{0}at

*t*= 0, the analytical solution in terms of the diameter Condensing water molecules release latent heat, which must be carried away by conduction. The mass transfer rate factor Φ (which is proportional to the growth speed) is strongly reduced by latent heat release to

_{eff}

*η*

*η*takes into account the actual saturation pressure at the droplet surface (which is at temperature

*T*

_{∞}+ Δ

*T*) and is defined by It is approximately given by (e.g., Mason 1971) where

*D*is the vapor diffusivity,

*H*is the latent heat of condensation, and

*W*

_{l}is the carrier gas heat conductivity. Under typical conditions,

*η*

_{appr}≈ 0.288, so the retardation due to latent heat is considerable. For precise calculations, the approximation should be avoided.

Above, we assumed that the growth does not affect the temperature of the carrier gas far away from the growing droplet, but this is certainly not true in the final stage when growth ends. The carrier gas temperature at infinity for any amount of condensed vapor can be easily calculated using the specific heat of the carrier gas if instantaneous heat transport is assumed. This is what was actually done in our numerical growth simulations.

However, there are still some minor variable unexplained discrepancies between theoretical and observed growth curves, which are probably due to the neglect of convection and heat conduction, which are difficult to account for precisely.

#### 2) Collective growth

So far, droplet growth considerations were highly simplified by assuming a monodisperse droplet cloud forming in the instrument. Figure 7 displays Eq. (12) transformed into the size domain. We see that there are two equilibrium sizes at *d*_{Kelvin} and at a larger size when vapor is depleted by growth. We also see that polydisperse clouds consisting of droplets *d*/*d*_{Kelvin} > 1.5 will become monodisperse since smaller droplets grow faster than larger ones.

_{t}

*n*

*d,*

*t*

_{d}

*G*

*d,*

*t*

*n*

*d,*

*t*

*G*[

*d,*

*t*] is the growth law in the size domain and

*n*[

*d,*

*t*] is the number concentration distribution function). The above distribution width has to be compared with the maximum width compatible with detectable Mie signatures as discussed below. Experimentally, such distribution shrinking has been observed by, for example, Szymanski and Wagner (1983).

#### 3) Supersaturation growth of electrolyte droplets

If the CCN contains soluble material that dissociates, the vapor pressure over the droplet is reduced below what would be expected according to Kelvin's equation (see Fig. 5). Therefore, initial growth of solute droplets is faster than that of pure droplets due to Raoult's effect. Very soon, however, dilution makes the difference effectively disappear so that the sizes approach each other according to Fig. 7. As an example, we show in Fig. 9 the size ratio *d*_{s}/*d*_{p} after identical growth times at *S* = 1.03 of a solute droplet containing a 20-nm NaCl nucleus and a pure water droplet both starting from a size of 91 nm.

So we find that even a cloud containing a mixture of pure liquid and solute droplets will become reasonably monodisperse for Mie structures to appear as discussed later.

#### 4) Subsaturation growth and sedimentation losses of electrolyte droplets

In Fig. 6a we saw that the final droplet size growing from of a 200-nm sodium chloride particle at RH = 1 is 3.8 *μ*m. The question is now concerning the growth during the 7.6-s residence time in the humidifier and, consequently, the sedimentation losses in a horizontal humidifier. Upon integration of (12) taking into account the solute activity, we find a size of 3 *μ*m after 7.6 s and a total settling distance of 1.6 mm, so such nuclei would not make it through the humidifier. The 50% penetration limit is approximately 115 nm for NaCl nuclei entering the humidifier, which grow to about 1.5-*μ*m droplets.

### e. Optical detection theory

#### 1) Scattering and extinction according to Mie theory

*λ*

_{l}is the light wavelength. The unpolarized Mie forward scattering intensity for water at 30°

*S*

_{30}is shown in Fig. 10 as a function of

*α*

^{2}

_{Mie}

*μ*m) at the vertical lines in Fig. 10 (and the corresponding

*α*

^{2}

_{Mie}

*Q*

_{e}:

*Q*

_{e}

*α*

^{2}

_{Mie}

*γ*and the transmission

*T*experimentally determined over the optical path

*L*for a number concentration

*N*) according to Lambert–Beer's law: This relation provides the droplet concentration without the need for calibration once DECS has been determined. This can be done as follows. The magnitude of the Mie theory ratio DECS/

*S*

_{30}does not vary strongly over large size domains, yet exhibits well-defined structures useful for sizing the droplets (Fig. 12). In particular, it has the advantage of allowing safe identification of the size at reference points 1 and 4 even, under noisy conditions, where the scattering signal alone may not be unequivocal.

#### 2) Effects of polydispersity

The influence of droplet polydispersity can be seen in Fig. 13. It is important to note that monodispersity must be rather good for preserving Mie structures. This fact makes a careful selection of the scattering angle crucial since the frequency of Mie structures strongly depends on it and must not be too high to cope with the remaining cloud polydispersity. On the other hand, resolution is lost if we go too far into the forward direction where structure frequency is low.

### f. Optical signals, measuring modes, and detection limit

#### 1) Scattered light signal

^{–1}at 635 nm, the wavelength of our laser diode (laser components, 2 mW, 4-mm beam diameter). The anode area is 8 mm × 24 mm leading to an estimated optical scattering volume of about 348 mm

^{3}(taking into account the oblique view of the laser beam). The PM signal is digitized in 12 bits in the 0–5-V range. The amplification is adjusted such that electrical saturation does not normally occur for ambient concentrations. The experimental scattering intensity

*J*

_{30}measured in volts by the PM is related to the theoretical response

*S*

_{30}by

*J*

_{30}

*NV*

*S*

_{30}

*N*

*S*

_{30}

*V*of the PM; it can be experimentally determined as described in the subsection on measuring modes.

#### 2) Transmitted light signal

*L*of the beam under 30° to the chamber axis through the measuring chamber of 12-cm inner diameter is about 24 cm. Therefore, the extinction measuring volume is about

*V*

_{m}= 3000 mm

^{3}, which means that statistical noise above the 10% level due to variable droplet number concentration

*N*in the optical volume is reached for droplet concentrations

*N*< 33 cm

^{–3}. The transmission signal is also digitized in 12 bits in the 0–5-V range. For our geometry, an optical density digitizing half step (the average error) ODDHS = 2.24 × 10

^{–4}results in a relative digitizing error where OD

_{act}is actual OD on which the evaluation is based. The combined relative error of these uncorrelated contributions is Assuming DECS = 1000 and

*N*= 33 cm

^{–3}, we obtain OD = 2.5 × 10

^{–4}, which corresponds to about one ODDHS in transmission signal for our present ADC. Therefore, low concentration resolution in the extinction signal is limited by the OD rather than by counting statistics. where the optical depth OD is a measure of the fraction of the light beam obscured by the droplets [

*J*(0) is the transmitted intensity before cloud formation]. Its relation to droplet concentration will be discussed in more detail later.

*U*In our case

*U*(∞) = 0, and

*U*(0) is adjusted once such that the present dynamic range of the 12-bit ADC is well used. Equations (24), (28), and (29) complete the concentration measurement without calibration.

When theoretical Mie DECS/*S*_{30} ratios are compared with experimental data OD/*J*_{30} one has to bear in mind that the experimental ratio refers to slightly different populations: while OD is measured across the whole chamber diagonal, *J*_{30} is seen by the PM in the chamber center only; this fact combined with the above effects of polydispersity somewhat reduces the modulation depth of the experimental ratio.

#### 3) Measuring modes

*J*

_{30}with the theoretical ratio DECS/

*S*

_{30}. If multiple scattering can be neglected (which is certainly the case for ambient CCN concentrations), the concentration

*N*cancels out and we obtain In this way, the size can be uniquely determined even for considerably noisy signals.

Depending on the signal quality, different measuring modes and degrees of accuracy can be achieved.

- The most reliable is the intrinsically calibrated (IC) mode where both scattering and extinction signals are used and their ratio serves to determine the droplet size. Subsequently, the extinction cross section of the droplets at the predetermined size is used with the OD to determine the droplet number concentration. This mode provides the standard operating procedure for most atmospheric measurements. It is also prerequisite for assigning a Mie scattering efficiency factor (SEF) to the PM signal. The SEF relates PM output in volts (or AD channel numbers) at known Mie scattering theory values to droplet concentrations.
- The Mie scattering (MS) mode is used when OD is too small to be useful for precise measurements. If Mie scattering from the droplets suffices, however, to uniquely identify droplet size, the SEF obtained from previous IC mode measurements can be used to derive droplet concentrations.
- Both IC and MS modes allow one to determine supersaturation by comparing theoretical growth rates with experimental ones, thus providing simple checks for chamber leaks, drying of the humdifier, etc., since any inconsistencies between observed and theoretical growth based on coupled heat and mass transfer at supersaturations based on adiabatic expansion are indicative of malfunctions.
- There are, however, cases when the droplet cloud produced in the spectrometer is too polydisperse to exhibit Mie structures. This happens, for example, when a considerable fraction of activated particles is to the left side of the growth speed maximum in Fig. 7. Under such conditions, noise statistics provide only limited means to estimate droplet concentration.

#### 4) Concentration and supersaturation detection limits

Which number concentration detection limit can be expected for our device at which supersaturation? Measurements in the IC mode become impossible when concentrations are so high that the amount of condensable vapor is not sufficient to grow particles into the Mie structure size range: this happens at ambient temperatures and supersaturations of several percent at droplet concentrations exceeding about 50 000 cm^{–3}.

The low concentration limit is much more important. Under the assumption of growth, undisturbed by heat conduction from the walls or by convection for 10 s, we can calculate the growth adjusting the concentration such that the final OD after 10 s is 0.001, which seems a reasonable detection limit after the planned instrument upgrading with high-resolution ADC for the extinction measurement. The results are shown in Fig. 14. It is clear that under these conditions nonadiabatic effects limit droplet growth and, hence, the concentration detection capability.

## 4. Practical examples

The purpose of this section is to compare the theoretical expectations obtained above with a few (unpublished) experimental results obtained at the Atmospheric Sciences Research Center 2000 Cloud Condensation Nucleus Counter Intercomparison Workshop in Albany, New York.

### a. Concentrations and growth curves in the IC mode

Figure 15 shows the result for monomobile ammonium sulfate particles (mobility of singly charged 100-nm particles). The number concentration as determined by a TSI 3020 condensation nucleus counter (CNC) ranged between 150 and 170 cm^{–3}, and the present device gave *N* = 163 cm^{–3} at SS = 3.55%. In general, we found good agreement between our IC mode results and the concentrations determined by CNC total counts, when losses due to growth of hygroscopic nuclei were absent. On the other hand, an example with 200-nm ammonium sulfate particles showed that—in agreement with the estimates presented earlier—losses in a horizontal humidifier were greater than 60%. For the humidifier in short-term experimental vertical position the losses in the humidifier were negligible.

### b. Supersaturation confidence interval in the IC mode

The question concerning the accuracy of the supersaturation is more difficult to answer. If we knew the humidifier efficiency exactly, the uncertainty of the saturation ratio would be uniquely defined by the total (analog and digitization) uncertainty of the sensors. The pressure differential in the reservoir volume is measured by a 120 000 Pa full-scale sensor with 12-bit resolution; that is, the resolution is limited to 120 000 Pa/2^{12} = 30 Pa (apart from any systematic analog errors), so the pressure sensor is not the limiting factor.

Experimental determination of RH seems impossible unless by observing the droplet growth. Therefore, we do not assume any a priori knowledge of the expected saturation ratio RH = 1 and treat it as an unknown parameter for fitting the observed scattering and OD data. The result of such a procedure applied to average-quality signals is shown in Fig. 16. We conclude that typically the supersaturation can be determined to ±20% relative to the optimum of the growth model. Under normal operating conditions, however, RH = 1 after the humidifier is a reasonable and well-justified assumption.

### c. Concentration estimate from noise statistics

Mie structures are not always present. A rather interesting and rare example of this kind is shown in Fig. 17, where there is a long delay before growth actually takes place. This indicates particles left to the growth speed maximum in Fig. 7 and very close to the Kelvin diameter. Obviously, another approach has to be taken in this case.

*V*

_{m}, and, hence, uncorrelated sampling from the droplet population, the noise caused by the stochastic droplet motions

*F*

_{stat}may be used as an independent method for determining the concentration

*N*This technique we call the statistical noise (SN) mode. For a detailed description of the statistical approach see Kyle (1979). Figure 18 shows the application of the above formula to the data of Fig. 17. Figure 19 was obtained by evaluating measurements with low concentrations but clear Mie structures so that number concentrations could be measured reliably. Shown is the minimum relative standard deviation

*F*

_{stat}(with Mie structures removed) of the ten 1-s measuring intervals for measurements with good enough quality to allow for precise independent measurement of concentration

*N*(see below). The large scatter indicates that perfect exchange (mainly due to convection) during the 1-s intervals was not warranted in all cases.

Obviously, the SN method can be even applied if there is no useful extinction signal. It extends the measuring range down to very low concentrations, but results may only be accurate within a factor of about 2. Since growth theory cannot be applied, one has to know the humidifier efficiency from previous experiments in the IC mode.

### d. A Kelvin spectrum

When a Kelvin spectrometer is challenged by a monodisperse aerosol of known concentration, it should correctly reproduce the concentration at supersaturations high enough to activate all particles. Monodisperse polyvinyl alcohol particles with nominal values of *N*_{0} = 325 cm^{–3} and *d* = 150 nm were used as determined by a TSI differential mobility analyzer and a TSI ultrafine particle counter; the results are shown in Fig. 20. At a saturation ratio of *S* = 1.0308, corresponding to a Kelvin diameter of 69 nm, the measured concentration (IC mode) was 239 cm^{–3}. For the same aerosol at *S* = 1.0234 (corresponding to a Kelvin diameter of 91 nm), the concentration determined by the MS mode was 242 cm^{–3}, while at *S* ≤ 1.015 (i.e., a Kelvin diameter ≥ 141 nm) no activation could be observed at all. Unfortunately, for intermediate saturation ratios no data are available so that the resolution of our device could not be precisely determined. The data are presented in Fig. 20. It is obvious that the critical Kelvin diameter for water activation is smaller than the geometric diameters, indicating nonhygroscopic or even hydrophobic condensation behavior.

## 5. Conclusions

We have presented an analysis of an expansion-type Kelvin spectrometer and could show that the experimental performance of the Kelvin spectrometer closely agrees with theoretical expectations. Also, the measuring principle of the device makes external calibration unnecessary.

The presently used 12-bit resolution is not adequate for precisely determining droplet concentration from the optical thickness; the next version will therefore be equipped with a higher resolution ADC. A fundamental difficulty of the present single expansion device is the limited amount of condensable water available for droplet growth, which makes low concentration measurements at low supersaturations (i.e., SS < 0.5%) very difficult because the growth time is limited by heat conduction from the walls.

In the intrinsically calibrated mode, the precision of the concentration measurement is estimated to ±15%, and the precision of the supersaturation to ±20%. The lower number concentration detection limit, as measured by extinction, depends on the supersaturation as shown in Fig. 14. In the Mie scattering mode it may be as low as 30 cm^{–3}. The upper concentration limit of about 10^{5} cm^{–3} is determined by the disappearance of Mie structures. Another advantage of the device is that contrary to continuous flow devices, change of operating liquid does not pose any problem (Holländer et al. 2000).

## Acknowledgments

The instrument described above was developed during the project Physical and Chemical Characterization of the Interstitial Aerosol funded by the German Ministry of Research and Technology under Grant 07AF101/0, which is gratefully acknowledged.

The instrument participated in the ASRC Albany CCN intercomparison workshop July/August 2000, and we gratefully acknowledge the extremely helpful support provided by Drs. Lee Harrison and Gar Lala by supplying the challenge aerosols. There, we learned our lessons concerning the humidifier performance.

Also, in November 2000, another laboratory test was performed at Paul-Scherrer Institute, Villigen, Switzerland, and we thank Dr. Werner Graber for the excellent cooperation.

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## APPENDIX

### Temperature and Saturation Ratio after Expansion

*T*

_{10}and ambient pressure

*p*

_{10}, the measuring chamber of volume

*V*

_{10}is filled with air. The buffer chamber has the volume

*V*

_{20}and also starts at ambient pressure

*p*

_{10}. Next, it will be quickly (and, hence adiabatically) evacuated to the measured pressure

*p*

_{200}resulting in a (calculated) temperature

*T*

_{200}. The adiabatic relations (3) yield with the adiabatic exponent

*κ*(all symbolic and numerical calculations were done with

*Mathematica*4.0, available online at http://www.wolfram.com): Obviously,

*T*

_{200}<

*T*

_{10}but heat conduction makes

*T*

_{200}reach

*T*

_{10}after some time leading to an isochoric pressure increase from

*p*

_{200}to

*p*

_{20}in the closed buffer volume

*V*

_{20}so that with the isochoric relation we obtain

*p*

_{11}in both chambers. Immediately after pressure equilibration, the air mass originally contained in

*V*

_{10}of the measuring chamber at

*T*

_{10}is adiabatically cooled to

*T*

_{11}by expansion to

*V*

_{11}, while the air originally contained in

*V*

_{20}of the buffer chamber at

*T*

_{10}is adiabatically heated to

*T*

_{21}by compression to

*V*

_{21}where

*V*

_{10}

*V*

_{20}

*V*

_{11}

*V*

_{21}

*p*

_{act,0}

_{0}

*p*

_{sat}

*T*

_{10}

*S*

_{th,1}in the measuring chamber after expansion is Inserting Eqs. (A7) and (A8) into (A11) we can calculate the theoretical saturation ratio after expansion from measured quantities.