1. Introduction

Low vapor pressures are difficult to measure in general, and it is especially challenging to perform measurements at low temperatures typical for aerosol particles under upper tropospheric conditions.

Amongst the many different substances present in atmospheric aerosols, in this paper we focus on oxygenated organic compounds (e.g. dicarboxylic acids) of very low vapor pressures. They originate from direct emissions as well as from photochemical reactions of biogenic and anthropogenic precursors. There is considerable interest in understanding the formation but also the partitioning between gas and particle phase of these compounds because of their influence on atmospheric chemistry and the radiative properties of the aerosol [1]. In order to predict the partitioning it is important to know the vapor pressure of the compounds and its dependence on temperature and composition of the aerosol [2]. If the compounds are water soluble like the dicarboxylic acids, they will form an aqueous solution in the condensed aerosol phase and the concentration of the organic acid and its vapor pressure will depend on relative humidity.

A possible approach to determine vapor pressures is to measure evaporation rates of aerosol particles of known composition at fixed temperatures and relative humidities. One way for measuring evaporation rates of aerosol particles is by using the Tandem Differential Mobility Analyzer (TDMA) [3].

In the case of very low vapor pressures, this technique is limited by the finite residence time (some seconds) of the aerosol particles in the laminar flow tube. Alternatively, the size of single, levitated particles of known composition and at fixed temperatures and relative humidities may be monitored for prolonged times (of the order of days) to deduce evaporation rates.

There are several methods to measure the size of a levitated particle, e.g. direct imaging [4] or mass change monitoring by measuring the change in DC voltage to compensate gravitational force in an electrodynamic balance [5]. However, analyzing light scattering data using Lorenz-Mie theory is the method of choice to size dielectric microspheres. Measuring and analyzing the change in the angular scattering pattern (phase function) can be used to determine size and vapor pressure as demonstrated by Ray et al. [6].

Extremely high precision is achieved if one uses light scattering to measure cavity resonances in spherical droplets, a method called Mie resonance spectroscopy [7],[8].

Besides elastic light scattering these resonances have been detected also in fluorescence [9], Raman [10], [11] and infrared absorption spectroscopy [12].

For elastic light scattering typically a fixed wavelength laser source is used to measure light scattering intensity as a micron sized evaporating droplet passes through resonances [13]. This method of recording resonance spectra is working if the change in radius is monotonic, meaning in practice: sufficiently fast. It is not well suited for sizing low vapor pressure aqueous aerosol particles, because fluctuations in relative humidity and temperature lead to size changes, and hence to light scattering intensity fluctuations. For instance this will cause a particle to pass back and forward through a particular resonance several times, while its size slightly shrinks and rises because of the fluctuating relative humidity, a situation which makes the analysis impossible. Therefore, instead of a fixed wavelength source, inelastic scattering may be used [15], or a tunable light source may be employed to scan the resonance spectrum, as in the pioneering work of Ashkin and Diziedzic [7]. Refined procedures for measurement and analysis permit the determination of size and refractive index with a precision of 3×10^-5 [14].

Here we report an attractive and technically simple alternative approach, namely using a “white” light source with high spatial coherence to illuminate a levitated microdroplet and excite its resonance spectrum which is recorded by an optical multichannel analyzer.

2. Principles of the method

The extinction and scattering spectra of dielectric spheres with small absorption exhibit a ripple structure, i.e. irregular spaced sharp peaks as a function of the size parameter x = 2πr/λ, with r being the radius of the sphere, and λ the wavelength [16]. These peaks are often called Morphology Dependent Resonances (MDRs) and can be interpreted as electromagnetic waves that, striking the surface of the sphere, are strongly confined within the particle by a series of almost total internal reflections and then focused on the surface in phase.

In Mie theory, the electromagnetic normal modes are resonant when the denominators of the scattering coefficients a_n and b_n in the infinite series expansion [16] are minima. For each index n there is a sequence of values of x for which the mode associated to a_n or b_n is excited. Following the convention introduced by Chýlek [17], we label the peaks with the type of mode (electric a_n or magnetic b_n ), the index n, and the sequential order number l of x. The identification of an arbitrary mode of resonance in a measured spectrum, i.e. the unique determination of l and n and hence the correspondent size parameter and refractive index is possible but difficult because of the periodic recurrence of almost identical sets of resonance peaks [18].

However, the approximate determination of the particle size is possible without identification of the mode if the refractive index is known and ∆x is measured; ∆x being the size parameter spacing between consecutive resonance peaks in index number n, ∆x = $x_{n+1}^l$ - $x_n^l$ , with the same order number l. Under the assumption x ≫ 1, x~n and mx ~ n, mx < n, where m is the index of refraction, which holds when the broader resonance peaks are considered and the particle is larger than a few micrometers, the size parameter spacing becomes as in [17]:

The accuracy of this approximation has been evaluated to be better than 1% for |x - n| < 4 [19].

The wavelengths ${\lambda }_{n+1}^l$ and ${\lambda }_n^l$ associated with two consecutive modes of resonance can be determined directly from a resonance spectrum. The spacing ∆x is derived from Eq. (1), where the refractive index can be interpolated from literature data if the composition of the particle is known [20]. This yields for the radius:

If the radius and radius change of a particle is known, the vapor pressure can be calculated applying the evaporation equation, here in the form as in [21], rearranged from the original formulation given by Maxwell in 1877 [22]:

where r is the radius of the particle, D the gas diffusivity of the constituent species with respect to ambient air, M is the molecular weight, ρ the density, R the gas constant, p and p _∞, are the vapor pressures at the surface of the particle and at infinite distance.

The diffusivity D is calculated through:

where n_o is the number density concentration of the ambient air, and d_c the collision diameter between a molecule of ambient air and a molecule evaporated from the particle [21].

We assume that the gas system is dilute enough to neglect the vapor pressure of the species at a large distance from the particle, hence p _∞ = 0. Also, the evaporation rate is very slow, because of low vapor pressure, that no detectable cooling due to evaporation at the particle surface occurs, so that T _∞ = T. For particles bigger than 1μm, the correction term Φ, calculated for instance by Fuchs [23] is Φ ≅ 1. These considerations simplify Eq. (3), and the vapor pressure can be expressed as follows:

which implies that the quantity $\frac{rdr}{dt}=\frac{1}{2}\frac{d{r}^2}{dt}$ stays constant during the evaporation of a particle at constant temperature.

Thus, Eq. (5) can be conveniently reformulated as:

The size change of an evaporating particle can be determined without identifying the exact mode by considering that it will be reflected in a shift of the resonance spectrum. If one tracks only the position of one arbitrary distinct mode of resonance during its shift, the index number n and order number l will remain of course constant, meaning that the resonance size parameter of the mode, x_mode , remains constant as well [15]. This property allows the calculation of the radius at every time through the mere definition of the size parameter:

where the size parameter is calculated at the beginning of the analysis knowing the location of the chosen peak in the spectrum, λ_p , and the radius from Eq. (2). The vapor pressure follows now from the linear time evolution of r ² and Eq. (6).

3. Experimental setup

The basic experimental setup has been described previously [24], [25]. Briefly, an electrically charged particle (typically 5–15 μm in radius) is levitated in an electrodynamic balance [26], see a schematic of the setup in Fig. 1. The balance is hosted within a three wall glass chamber with a cooling agent flowing between the inner walls and an insulation vacuum between the outer walls. A constant flow (typically 30 sccm) of a N₂/H₂O mixture with a controlled H₂O partial pressure is pumped continuously through the chamber at a constant total pressure adjustable between 200 and 1000 mbar. The temperature can be varied between 330 K and 160 K with a stability better than 0.1K and an accuracy of ±0.5 K. Relative humidity (RH) in the chamber is set by adjusting the N₂/H₂O ratio, using automatic mass flow controllers.

During an experiment, the temperature and the relative humidity are kept constant while measuring a particular evaporation rate. The relative humidity is registered by a capacitive thin film sensor that is mounted in close vicinity of the levitated particle (< 10 mm). The sensor was calibrated directly in the electrodynamic balance using the deliquescence relative humidity of different salts. Its accuracy is ±1.5% RH between 10% and 90% RH.

A single-particle generator (Hewlett-Packard 51633A ink jet cartridge) is used to inject a liquid particle from solutions prepared by mass percent with MilliQ water using an analytical balance and analytical grade reagents with purities of 99% or higher. Two collinear laser beams illuminate the particle from below (HeNe @ 633 nm, Ar⁺ @ 488 nm).

To size the particle three different, independent methods are employed. First, we use the video image of the particle on CCD detector 1 and an automatic feedback loop to adjust the DC-voltage for compensating the gravitational force [27]. A change in DC voltage is therefore a direct measure of the mass change, allowing to calculate a radius change when the density of the particle is known or can be estimated.

Second, the two-dimensional angular scattering pattern is recorded with CCD sensor 2 by measuring the elastically scattered light from both lasers over observation angles ranging from 78° to 101°. If the particle is liquid, and therefore of spherical shape the scattering pattern is regular, with the mean distance between fringes being a good measure of the radius of the particle, almost independent of its refractive index [28].

Third, we have added a ball lens type point source LED (EPIGAP GmbH, Germany) as a “white light” source with high spatial coherence (50 μm source diameter, peak wavelength ≃ 589 nm, spectral bandwidth at 50% ≃ 16 nm, radiant power ≃ 150 μW) using a bestform lens (f = 32 mm, f# = 2.0) to focus the light on the levitated particle and a pierced mirror to collect the Mie resonance spectra in a backscattering geometry (as shown in Fig. 1, collection angle 180°±4°).

Although resonance spectra may be measured in other scattering geometries, backscattering gives the advantage of using approximate solutions for calculating the resonance positions [18]. Two holographic notch filters are used to reject the elastically scattered light of both lasers and an optical fiber is employed to deliver the backscattered light from the particle to a 150 mm spectrograph with a slow scan backilluminated CCD array detector as an optical multichannel analyzer (OMA). The resolution of the spectrograph-OMA combination is about 0.5 nm; this excludes the observation of the narrowest resonances in the backscatter spectrum.

 

Fig. 1. Schematic of the apparatus. A three-wall glass chamber hosts four metal rings which supply the electric field needed for particle levitation. The particle is illuminated by two laser beams from below and by the LED from aside. The scattered light is collected in the near and far field view by two CCD cameras, and feeded to a spectrograph via an optical fiber.

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Figure 2 illustrates how a resonance spectrum is obtained with this setup. The red curve in panel (a) shows the intensity measured when the particle is removed from the illuminated region by applying an additional DC voltage to the endcaps of the electrodynamic balance. This is the emission spectrum of the LED originating from reflections from the walls of the chamber, which serves as reference in the following. The black line in panel (a) shows the backscattered intensity when the particle is moved back into the center of the balance. Dividing this spectrum by the reference leads to the spectrum of panel (b), which is further processed with a smoothing FFT filter employing the bandwidth of the spectrograph-OMA combination. The resulting Mie resonance spectrum is plotted in panel (c) of Fig. 2, showing the typical double-peak structure.

To compare the proposed method for measuring evaporation rates with a fixed wavelength approach [13], we recorded the glare spot intensities measured with CCD 1 at a wavelength of 633 nm with a 1 Hz temporal resolution. A critical comparison of the two methods will be found in the next section.

4. Results and discussion

The setup described above has been used to determine the vapor pressure and the heat of vaporization of aqueous malonic acid. As an illustration, the temporal evolution of the experiment is depicted in Fig. 3. The uppermost panel displays the relative humidity and the balancing DC voltage proportional to the mass of the particle. The relative humidity, after an initial transient, is kept constant at 52%, then lowered to reach a constant value of 42% at 60 ks, and then decreased further to 33% at 122 ks, so that the particle experiences three long periods of constant relative humidity separated by fast changes during which it shrinks while adjusting to thermo-dynamic equilibrium. We focus our attention on the three periods with constant RH, where the shrinking is due to evaporation of malonic acid to the gas phase, and stress the fact that they correspond to three different concentrations of the aqueous solution particle, from a more dilute one at the beginning to a more concentrated one at the end. The concentrations, corresponding to the relative humidities mentioned above, were determined from literature data based on EDB measurements [29], and are 80%, 86% and 92% in weight respectively. These values are used for the calculation of the refractive index [20]. Together with spectra as that shown in Fig. 2, these values allow to deduce size parameter spacing and finally the radius through Eq. (2) and Eq. (7).

 

Fig. 2. Example of the experimental dataset processing. Panel (a): The LED reference spectrum (red curve) and the original intensity of light scattered by the particle (aqueous malonic acid particle, radius ca. 8.1 μm, refractive index ca. 1.41, black curve). The latter is plotted in panel (b) after background correction and dividing by LED reference. The signal is smoothed by application of an FFT filter and shown in panel (c).

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In panel (b) it is shown how the resonance spectra evolved during an experiment. A shift of the spectra in the regions of constant RH is evident. More, it is also clear from the slope that the shift is faster when the RH is lower, and hence the concentration of the particle is higher, meaning that a more concentrated particle is evaporating faster than a more dilute one.

A maximum located at 600 nm in the first spectrum is chosen and followed during its shift (panel (c) of Fig. 3). When its location exits the wavelength domain of our detection device at 100 ks a new peak is chosen at 580 nm and the tracking is continued.

 

Fig. 3. Experimental raw data from a vapor pressure measurement of an aqueous malonic acid particle at 291 K exposed to different relative humidities. The uppermost panel (a) shows the DC voltage (proportional to the mass) necessary to compensate the gravitational force (black line). The red line is the relative humidity. Panel (b) shows the corresponding resonance spectra, with color-coded intensity. The values in the palette represent the intensity of the scattered light divided by the reference LED signal, as explained in the caption of Fig.2. In panel (c) the location of the resonance peak at 600 nm at t=0 is tracked during its shift (gray line) up to t=100 ks, where it exits the domain of our spectrograph. Thus, a new peak at 592 nm and t=100 ks is chosen and tracked up to 190 ks. The corresponding r ² is calculated through Eq. (7) and displayed in panel (d) together with the linear fits in the regions of constant RH (red green and orange overlapped lines). The results of the three fits are: dr ²/dt = -3.81×10^-5 μm ²/s, -4.14×10^-5 μm ²/s, and -4.82×10^-5 μm ²/s for RH=51%, 42% and 32%.

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Fig. 4. Panel (a) shows 90’000 s (roughly 24 hours) of the same raw data already presented in Fig.3; malonic acid particle at 291 K and 32 % relative humidity, size ca. 7.2 μm. Panel (b) shows for comparison the scattered intensity at the fixed wavelength of the HeNe laser (633 nm at 90° scattering angle, sampled with 1 Hz). Panel (c) and (d) show the same kind of data - also for a time span of 90’000 s - for a malonic acid particle at 273.5 K and 21% relative humidity, size ca. 6.3 μm.

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Panel (d) shows the evolution of r ² and its linear fits in the regions of constant relative humidity (colored lines) which yields the quantity dr ²/dt necessary to obtain the vapor pressure from Eq. (6).

Let us briefly discuss the advantages of the proposed method for measuring evaporation rates by comparing it to the more conventional one using a fixed wavelength laser source to measure the temporal evolution of scattered light intensity as illustrated in Fig. 4. Panel (a) and (b) of Fig. 4 show raw data for a particle under conditions leading to a relatively high evaporation rate (4.82×10^-5 μm²/s). It is obvious that both, the spectral shift of a resonance peak shown in panel (a) and the scattered light intensity with time as shown in panel (b) could be used to retrieve the evaporation rate here. Panels (c) and (d) show the data of a particle evaporating at a considerably lower rate (5.09×10^-6 μm²/s). While under these conditions the resonance peak shift can be clearly evaluated this is no longer possible for the fixed wavelength data simply because there are not enough resonance peaks appearing in the data because of the small overall radius change within 24 hours. To make the method work the particle has to shrink at least 3 size parameter units. Also, note that it becomes increasingly more difficult to interpret the type of data shown in panels (b) and (d) if there are fluctuations in relative humidity during the measurements leading to a non monotonous radius change. This could lead to passing through the same resonance back and forward several times at a fixed wavelength with no simple means to correct for this effect. The spectral shift data of a resonance peak in panel (a) and (c) become more noisy under such circumstances but still allowing to deduce rates if the distortion is not too severe. We conclude that the proposed method is about two orders of magnitude more sensitive in measuring low vapor pressures and considerable less sensitive to fluctuations in relative humidity compared to a fixed wavelength measuring approach.

 

Fig. 5. Vapor pressure of aqueous malonic acid measured at different temperatures as a function of concentration in mole fraction. All experimental points are fitted simultaneously to Eq. (8) (except for the datapoint at x = 0.12 at T = 291 K), leading to the corresponding colored lines. The shaded area represents the region of unreliability of our measurements (see text for a detailed description).

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To obtain the vapor pressure the diffusivity is needed. Since we are not aware of any experimental measurements of the diffusivity of malonic acid in nitrogen, it has been evaluated through Eq. (4) taking the value of the collision diameter given in Bilde et al. [30]. For the calculation of the diffusivity Bilde et al. followed Bird et al. [31] instead of using Eq. (4). The agreement between diffusivities using Eq. (4) and those calculated in [30] is between 5% and 10% depending on the temperature.

The measurements are performed at constant RH, which means constant aqueous solution concentration. Therefore the evaporation of malonic acid is accompanied by an equal (in terms of molecules) evaporation of water to the gas phase. Thus, the vapor pressure calculated through Eq. (6), which holds for a one component system, has to be corrected by multiplying it by the molar fraction of the malonic acid concentration.

Figure 5 summarizes the results obtained from vapor pressure measurements of malonic acid at the investigated temperatures and humidity conditions.

Assuming a Clausius-Clapeyron relationship between vapor pressure and temperature, we can write the following equation:

where p₀ is the vapor pressure of the pure substance under standard conditions, γ is the activity coefficient at each concentration calculated using the UNIFAC parametrization by Peng et al. [29], x is the concentration in mole fraction, ∆H ^⊖ is the standard heat of vaporization.

In order to estimate the errors of the vapor pressure measurements let us consider first the pure substance under study (malonic acid). The resolution of the spectrograph is 0.5 nm at 585 nm, the size of the aerosol particle and its residence time in the electrodynamic balance are typically 8 μm and 100 ks. This leads to an uncertainty in the evaporation rate of (dr ²/dt)^min = -1 × 10^-6 μm²/s, which corresponds through Eq. (5) to a minimum detectable vapor pressure and a constant error in vapor pressure of p^min = 1.25 × 10^-6 Pa. Moreover the radius of the particle calculated from Eq. (2) with an estimated maximal error of 5% propagates to a relative error in the vapor pressure of 30% through Eq. (5). These are the error bars shown in Fig. 5.

The extension to the mixed system malonic acid/water includes an additional contribution due to a possible drift in relative humidity. This will result in a particle radius change which is superimposed on the change due to evaporation of malonic acid to the gas phase. A mere 0.5% drift at 90% RH (molar fraction concentration x = 0.18) will result in a 0.15 μm size change on a 8 μm particle. During an experiment running over 100 ks, this radius change corresponds to an apparent evaporation rate of dr ² /dt = -5.9× 10^-5 μm²/s, and an apparent vapor pressure of 1.1 × 10-5 Pa. The same calculation repeated for different relative humidities leads to the upper contour of the shaded area in Fig. 5, where we cannot measure the vapor pressure reliably with our present relative humidity control. Also datapoints close to this area are to a small degree affected, but it is not taken into account in the error bars shown in Fig. 5.

To see whether the estimation of the shaded area is correct we performed also experiments with highly diluted aqueous sulfuric acid particles at high relative humidities. Since aqueous sulfuric acid at low concentrations and room temperature has a sulfuric acid vapor pressure smaller than 10^-13 Pa an apparent drift in the resonance spectra must be due to a drift in relative humidity. The drifts observed were consistent with the shaded area shown in Fig.5.

A fit to Eq. (8) leads to a vaporization enthalpy ∆H ^⊖ of 100 ± 17 kJ/mol, a vapor pressure p₀ of (3.2 ± 1.2) × 10^-4 Pa at standard temperature T ^⊖ = 298.15 K for malonic acid as a supercooled melt, and a vapor pressure of the saturated aqueous solution of malonic acid (at x = 0.22) p_sat = (4.1 ± 1.6) × 10^-5 Pa at T ^⊖ = 298.15 K, corresponding to the vapor pressure of the crystalline solid.

These values compare with a ⊖H ^⊖ of 92 ± 15 kJ/mol and a vapor pressure of 6.0 × 10^-4 Pa at 298.15 K measured for pure malonic acid with the TDMA technique [32]. The enthalpy agrees within the experimental errors and the vapor pressure comparison suggest that in the TDMA measurements the malonic acid was also present as a supercooled melt. Considering the relative error of about 35% of both vapor pressure measurements [32], the agreement is reasonable.

5. Conclusions

A new and simple experimental approach to Mie spectroscopy using a ball-lens LED as “white” light source and a simplified treatment of the Morphological Dependent Resonances have been presented and tested on an aqueous malonic acid micron sized particle, a dicarboxylic acid of interest for atmospheric science.

The size and the evaporation rates of the particle levitated in an electrodynamic balance have been measured for different ambient conditions of temperature and relative humidity. This method of recording resonance spectra works even if the change in radius is not monotonic, making it especially suited to study binary aqueous solution aerosol particles at elevated relative humidities. For example, it might be of use when determining binary activity coefficients [33].

As a useful application we calculated the enthalpy of vaporization and the vapor pressure of the malonic acid depending on concentration and temperature: ∆H ^⊖ = 100 ± 17 kJ/mol for the enthalpy of vaporization, p₀ = (3.2 ± 1.2) × 10^-4 Pa for the supercooled melt at a standard temperature of T ^⊖ = 298.15 K, and p_sat = (4.1 ± 1.6) × 10^-5 Pa for the vapor pressure of the saturated malonic acid solution particle.

We think that this alternative approach has potential to become a widespread simple and relatively cheap method for sizing spherical homogeneous levitated particles.