We describe how a time series of optical resonance spectra of an evaporating, non-spherical, irregular aerosol particle levitated in an electrodynamic balance exhibits patterns which are related to its evaporation kinetics. Simulated spectra of an evaporating, model aerosol particle show comparable features. If these patterns are used to deduce the particle size change with time, the resulting vapor pressures and enthalpies of vaporization compare favorably with literature data for both crystalline ammonium nitrate and succinic acid particles.
© 2009 Optical Society of America
CorrectionsAlessandro A. Zardini and Ulrich K. Krieger, "Evaporation kinetics of a non-spherical, levitated aerosol particle using optical resonance spectroscopy for precision sizing: Errata," Opt. Express 18, 10760-10761 (2010)
In atmospheric and climate science there is considerable interest in understanding the partitioning between gas and particle phase of chemical species. In particular, for semi-volatile substances like ammonium nitrate or certain organic species, the partitioning will strongly influence the particulate matter burden in the troposphere, the radiative properties of the aerosol, the cloud processing and the heterogeneous chemistry . In order to predict this partitioning, it is crucial to know the vapor pressure of the compounds under ambient conditions, whereas most established methods rely on high temperatures to achieve detectable vapor pressures. Very low vapor pressures have been measured either with thermal desorption coupled to mass spectrometry (e.g. ), or by precisely sizing the evaporating particles. This can be achieved by monitoring a flow of particles with Tandem Differential Mobility Analyzers , or a single levitated particle with direct imaging , mass change monitoring  or light scattering techniques. Liquid, spherical droplets can be sized with extremely high precision using Mie resonance spectroscopy (e.g. [6, 7]). Since generalized Lorenz-Mie theory allows only the treatment of the interaction of electromagnetic fields with regular particles, up to now light scattering has not been applied for precision measurements of evaporation rates of non-spherical, irregular particles. In the present work we use optical resonance spectroscopy  to size solid, non-spherical particles during evaporation with a precision superior to direct imaging and mass change monitoring.
2. Experimental setup
The experimental setup used in this study has been described previously in detail . A micrometer-sized aerosol particle is levitated in an electrodynamic balance and its evaporation is monitored. The 2-dimensional angular scattering pattern is recorded with a CCD camera to distinguish liquid (spherical) particles from solid (non-spherical) particles . A point source LED is used as a broadband light source with high spatial coherence (peak wavelength ≃ 589 nm, spectral bandwidth at 50% ≃ 16 nm). The optical resonance spectra are collected in a backscattering geometry (collection angle 180°±4°) through a pierced mirror using a spectrograph and an optical multichannel analyzer. Zardini et al.  explain in detail how a resonance spectrum is obtained with this setup. Briefly, the emission spectrum of the LED originating from reflections from the walls of the cell serves as a reference. A spectrum with a particle in the center of the cell is divided by the reference and processed with a smoothing FFT filter employing the bandwidth of the spectrograph/optical multichannel analyzer combination.
3. Results and discussion
Figure 1 shows three spectra of a solid, non-spherical ammonium nitrate (AN) particle taken at different times during an evaporation experiment. The spectra are characterized by a complex structure of weak optical resonances and by pronounced differences in their mean backscattered intensity. Consistently, the 2-dimensional angular scattering pattern  has the same basic features (not shown here), namely a complex structure similar to what has been observed with natural aerosol particles , and a significant change in intensity over time periods of seconds. As a consequence, a time series of raw spectra (like in Fig. 2a) does not allow easily to discern a shift in the resonance position, which is related to a size change, as in the case of an evaporating liquid, hence spherical particle .
To make the size change of an evaporating, non-spherical particle visible in its resonance spectra we proceed as described in Fig. 2. Panel (a) shows the times series of raw spectra (color coded intensity, max=red, min=blue). Each single spectrum is separately normalized to its own maximum and minimum and the result is plotted in panel (b). The most prominent features here are intensity extrema at certain wavelength (roughly regularly spaced) which are not time dependent and originate from etaloning (reflections between the parallel front and back surfaces of the CCD that cause them to act as partial etalons ). Therefore, further normalization is performed in panel (c) by dividing each spectrum of panel (b) by the mean spectrum of the complete times series. The non time dependent features are suppressed and optical resonances shifting with time become visible, although not nearly as distinct as in the case of an evaporating liquid, i.e. spherical particle .
The evaporation experiment was repeated at a higher temperature and the same analysis as described above was performed to produce the results in panels (e) to (g) of Fig. 2. The shift rate clearly depends on temperature, as can be seen by comparing panels (c) and (g), consistently with a particle evaporating faster at a higher temperature.
To deduce quantitative information about the radius change with time, we associate one of the optical resonances with a specific size parameter x 0 = 2πr 0/λ 0. If we know the initial radius, r 0, we may follow its temporal evolution by measuring the wavelength, λ(t), of the time shifting resonance through r(t) = x 0 λ(t)/2π . By assigning a radius to the optical resonances the question arises: what is meant by the characteristic radius of a non-spherical particle. Later in the paper we will use the one of the minimum enclosing ball, but others, like area mean radius could also be used. Our results, see below, suggest that evaporation and resonance spectra relate to the same characteristic radii, indicating that our particles are close to spherical.
For retrieval of the particle radius from the resonance spectra shift of Fig. 2(c) we use visual inspection, i.e. a prominent resonance feature is tracked down as indicated by the red crosses. If it leaves the wavelength domain or becomes less distinct with time, we switch to another resonance feature as illustrated in Fig. 2(c) at t ≈ 40000 s. As alternative to visual tracking a resonance line in the patterns of Fig. 2(c) and Fig. 2(g), automatic tracking was applied using upsampled cross correlation between consecutive spectra to deduce the average shift (an example of the radius deduced with this technique is shown in Fig. 6).
The initial radius may be obtained either by sizing the particle using its microscopic image or by fast Fourier transforming a single spectrum of Fig. 2(c), assuming the resulting wavenumber to correspond to the scattering wavenumber of an equivalent sphere . In case the particle undergoes a transition from liquid to solid, the initial radius can directly be obtained from the spectra in the liquid state using Mie theory (see example in Fig. 6).
The results of this analysis yields an evaporation rate for the particles shown in Fig. 2 of dr/dt = −1.9 × 10−6 μm/s at T = 283 K(panel(d)) and dr/dt = −8.6 × 10−6 μm/s at T = 293 K (panel (h)).
It is difficult to assess the validity of assigning a specific size parameter to the time shifting patterns of Fig. 2 based on light scattering theory. For axially symmetric perturbations from spherical shape explicit algebraic expressions have been derived by Lai et al. (1990) , but we do not know the exact morphology of our effloresced, non-spherical particle, and even if we knew, an exact treatment would be computationally expensive because of the combination of complex morphology and particle size. However, there is indication that effloresced particles are best described as perturbed spheres .
Therefore, we test whether the basic features of Fig. 2 can be reproduced by a model particle by using the T matrix code of Mackowski and Mishchenko  to calculate the random-orientation backscattering spectra of several close packed asymmetric aggregates of spheres. The first example shown in Fig. 3 is made of 7 spheres randomly drawn out of a lognormal distribution centered at 0.5 μm radius with a standard deviation of 0.05 μm and close packing is realized using a “drop and roll” algorithm . We let the whole aggregate shrink at a constant rate, realized by 100 discrete steps and calculate the corresponding resonance spectra. The initial and final state of the aggregate are drawn in panel (a) (see also animation in Fig. 3, (Media 1). The color coded, simulated spectra in panel (b) (see also Fig. 3, (Media 2) exhibit a pattern similar to the experimental ones (panels (c) and (g) of Fig. 2) but without intensity fluctuation as the algorithm calculates the true random-orientational averaged spectra. In addition, the backscatter intensity decreases while the aggregate shrinks, which can not be seen in our experimental data due to rescaling and to a smaller relative change in radius compared to the modeled one.
If we apply to the simulated spectra the cross correlation analysis between consecutive spectra to deduce the average shift in resonance wavelength and calculate the radius as we did for the experimental data, we obtain the results shown Fig. 3(c). The initial radius was taken as the one of the minimum enclosing ball of the aggregate at the first step of the simulated evaporation. The agreement between the retrieved radius from the spectra and the one of the enclosing ball calculated for each time step is excellent.
Since the experimental patterns of Fig. 2(c) and Fig. 2(g) display a richer structure than the simulated ones, we did a treatment of the more complex shrinking aggregate described in Fig. 4 to investigate the effect of restructuring with evaporation leading to a change in particle morphology. The aggregate of 9 spheres is randomly drawn out of a broader lognormal distribution centered at 0.5 μm radius with a standard deviation of 0.13 μm. Here, we let the spheres shrink with a rate inversely proportional to their radius and rearrange the aggregate after each time step (56 in total), assigning again an equivalent size to the entire aggregate by calculating the radius of the minimum enclosing sphere. The initial and final state of the aggregate are drawn in Fig. 4(a) (see also animation in Fig. 4, Media 3)). The rearrangement is made by subsequently moving each sphere (starting with the one closest to the center) towards the center until it touches one of the other spheres. The same “drop and roll” algorithm is then used to move the sphere until it is in contact with at least 3 other spheres. The spectral pattern in Fig. 4(b) and Fig. 4 (Media 4) is now more complex due to the rearrangement reflected by changes in the spectral shape during evaporation, but it still allows to retrieve the radius change (Fig. 4(c)) using the cross correlation method.
The deviation between the rate of radius change deduced from the simulated spectra and the one in the model of the shrinking aggregate is less than 10% for all the different aggregates we investigated (7 or 9 spheres with different widths of the lognormal distribution) and whether or not rearrangement during shrinking was allowed. Although our model simplifies the particle morphology with a finite number of distinct spheres and an equivalent radius which is about 5 times smaller than the experimental one, the deviation from spherical shape may be even more pronounced than what is expected for an effloresced particle . Despite these approximations, the simulations reproduce the observations of the general interference structure remarkably well, justifying our procedure to evaluate the radius change with time.
In addition, we provide experimental evidence that our interpretation of the patterns in Fig. 2 and their quantitative evaluation are correct. We consider two exemplifying cases: the first is a comparison between the vapor pressure of solid AN at ambient temperatures with high temperature data, and the second is the case of an evaporating aqueous succinic acid (SA) particle undergoing a physical state transition from liquid to solid. Since an evaporating AN particle dissociates in ammonia and nitric acid through the reaction: NH 4 NO 3(s) ⇋ NH 3(g) + HNO 3(g), the vapor pressure can be determined from radius change via :
where R is the ideal gas constant, T the ambient temperature, ρAN and MAN the density and molar mass of AN, and D the diffusivities of the two species in the ambient air . This holds valid if transport is limited by gas phase diffusion, i.e. in the continuum regime, and evaporative cooling is negligible due to slow evaporation rates. When using Eq. 1 to calculate vapor pressure by taking the radius from measurements as shown in Fig. 2, we assume that evaporation and optical resonance are related to the same characteristic radius. For a highly non-spherical particle this assumption may be no longer valid.
As dr 2/dt = 2rdr/dt, we may conservatively estimate the error in the vapor pressure to be about 35% by assuming our initial radius precise within 25%, dr/dt measured with a 15% accuracy, and the diffusivities known with a 10% uncertainty. Note that the radius rate of change in Fig. 2(d) appears linear because the relative radius change is small. For larger radius changes, only the change in radius square with time is linear, see Fig. 6 for an example.
The resulting vapor pressures of experiments at different temperatures (see Fig. 2 for examples at 283 K and 293 K) together with one datapoint obtained by an independent analysis technique  and literature data  of evaporating solid AN extrapolated to lower temperatures via Eq. (2) are summarized in Fig. 5 and Table 1. A fit of our data to the Clausius Clapeyron relationship:
yields a enthalpy of vaporization of ΔH ⊖ = (196.6 ± 12.2) kJ/mol and a value for ptot ⊖ = (9.11 ± 1.77) × 10−5 Pa at T ⊖ = 298.15 K. This may be compared with the enthalpies of dissociation calculated from the published enthalpies of formation at 298.15 K, for NH 3(g), ΔH ⊖ = (−45.94 ± 0.35) kJ/mol , HNO 3(g), ΔH ⊖ = (−133.9 ± 1.0) kJ/mol , and NH 4 NO 3(s,IV), ΔH ⊖ = (−365.61 ± 0.32) kJ/mol , which yields a value of ΔH ⊖ = (185.1 ± 1.1) kJ/mol.
The equilibrium constant for the dissociation reaction, K, is related to the partial pressure of NH 3 and HNO 3 via K = p NH3 p HNO3 = (ptot ⊖/2)2, and K is related to the standard Gibbs free energy change for the reaction by: ΔG ⊖ = −RT ln K. Our data yield ΔG ⊖ = (95.2 ± 1.1) kJ/mol, which is higher but close within errors to the one calculated from tabulated standard free energies of formation, ΔG ⊖ =94.0 kJ/mol . It is also higher than the one recommended by Mozurkewich  of ΔG ⊖ = (93.4 ± 0.3) kJ/mol, but Fig. 1 in that study shows a clear temperature dependence which yields an extrapolated value close to ΔG ⊖ = (94.0±0.9) kJ/mol at T ⊖ = 298.15 K, which is consistent with our result within errors. In conclusion, we consider the agreement to be excellent, thus not only providing support for our interpretation of the patterns of Fig. 2, but also data for the equilibrium constant for the dissociation reaction and enthalpies of vaporization at temperatures relevant for the atmosphere.
Figure 6 shows an evaporation experiment of an aqueous solution SA droplet efflorescing upon slow drying. Here a physical state transition from a liquid, spherical particle to a solid particle takes place, allowing a comparison between the rate of radius change in both the aqueous phase particle, assumed to be perfectly spherical, and the solid, effloresced, non-spherical particle. The retrieval of radius change with time (automatic tracking of the resonances by the cross correlation technique) yields consistent data for both liquid and solid state (panel (b)): the evaporation rate of the aqueous solution is expected to be much higher than the one of the solid because the aqueous particle at the experimental RH is more concentrated in SA than the saturated solution. Also, note that the radius square changes linearly with time as expected from Eq. (1). The solid state vapor pressure of p = (5.4 ± 0.2) × 10−5 Pa at 298.5 K, determined as discussed above, is in excellent agreement with the one from Bilde et al. . It is interesting to finally note some special features of the resonances spectra. First, the shift of the optical resonances with time, when the particle is solid, exhibits distinct discontinuities which we ascribe to sudden rearrangements processes within the effloresced solid. Second, careful inspection reveals that the spectra exhibits some of the features also seen in the modeled spectra of Fig. 4: Between 175 ks ≤ t ≤ 200 ks the structural resonance starting at ca. 590 nm shows a different gradient compared to the one starting at 595 nm. At t ≈ 128 ks two resonances seem even to cross at ca. 590 nm. At this point we can only speculate that the reason for the features being observed here, whereas they are absent in Fig. 2, is the different size of the particles (the particle of Fig. 6 is about a factor of 4 smaller in radius compared to the ones of Fig. 2).
4. Summary and conclusions
We have presented a technique for precision monitoring the size change of solid, non-spherical aerosol particles through the analysis of their optical resonance spectra. The crystals were levitated and effloresced in an electrodynamic balance, illuminated by a broadband LED source, and their spectra collected in a backscattered geometry. Time series of these spectra, opportunely processed, show evidence of patterns which evolve accordingly to the size of the particles, similar to the well-known behavior of spherical particles treated with Lorenz-Mie theory.
To support this interpretation, simulations consisting of evaporating aggregates of spheres were performed to mimic the behavior of evaporating, effloresced aerosol particles, providing excellent agreement between the size of the aggregate and the one retrieved by the series of optical spectra.
Applying the new technique to measure very low vapor pressures of semi-volatile, solid substances, we conclude that vapor pressures down to 10−5 Pa with an accuracy of better than 35% can be determined without the need of any material specific calibration. For solid ammonium nitrate we were able to obtain vapor pressure data at room temperature and below, which we believe to be the most accurate ones available in this temperature range.
The favorable comparison between high temperature vapor pressure data and basic thermodynamic data to our data suggests that our interpretation of the resonance shift is correct and that the characteristic radius obtained from this resonance shift is the one related to evaporation.
We were also able to observe distinct changes in the morphology of solid, evaporating aerosol particles in their optical resonance spectra. It is conceivable to extend the optical modeling of our study to describe particles undergoing sudden rearrangement steps. A detailed comparison of these kind of modeled optical resonance spectra with experimental ones may yield a better understanding of the microscopic processes governing the morphology of aerosol particles upon evaporation in the future.
We thank G. Videen, T. Peter, C. Marcolli, V. Soonsin, and M. Bilde for helpful discussions. This work was partly funded by EUCAARI (European Integrated project on Aerosol Cloud Climate and Air Quality interactions) No 036833-2.
References and links
1. IPCC 2007. Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [ S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor, and H.L. Miller (eds.)] (Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA, 2007).
2. S. Chattopadhyay, H. J. Tobias, and P. J. Ziemann, “A Method for Measuring Vapor Pressures of Low-Volatility Organic Aerosol Compounds Using a Thermal Desorption Particle Beam Mass Spectrometer,” Anal. Chem. 73, 3797–3803 (2001). [CrossRef] [PubMed]
3. D. J. Rader and P. H. McMurry, “Application of the Tandem differential Mobility Analyzer to studies of droplet growth or evaporation,” J. Aerosol Sci. 17, 771–787 (1986). [CrossRef]
4. P. F. Paradis, T. Ishikawa, and S. Yoda, “Non-contact measurement technique of the vapor pressure of liquid and high temperature solid materials,” Eur. Phys. J. AP 22, 97–101 (2003). [CrossRef]
5. G. O. Rubel, “Evaporation of Single Aerosol Binary Oil Droplets,” J. Colloid Interface Sci. 85, 549–555 (1982). [CrossRef]
6. A. Ashkin and J. M. Dziedzic, “Observation of resonances in the radiation pressure on dielectric spheres,” Phis. Rev. Lett. 38, 1351–1354 (1977). [CrossRef]
7. J. L. Huckaby, A. K. Ray, and B. Das, “Determination of size, refractive index, and dispersion of single droplets from wavelength-dependent scattering spectra,” Appl. Opt. 33, 7112–7125 (1994). [CrossRef] [PubMed]
8. A. A. Zardini, U. K. Krieger, and C. Marcolli, “White light Mie resonance spectroscopy used to measure very low vapor pressures of substances in aqueous solution aerosol particles,” Opt. Express 14, 6951–6962 (2006). [CrossRef] [PubMed]
9. C. Braun and U. K. Krieger, “Two-dimensional angular lightscattering in aqueous NaCl single aerosol particles during deliquescence and efflorescence,” Opt. Express 8, 314–321 (2001). [CrossRef] [PubMed]
10. K. B. Aptowicz, R. G. Pinnick, S. C. Hill, Y. L. Pan, and R. K. Chang, “Optical scattering patterns from single urban aerosol particles at Adelphi, Maryland, USA: A classification relating to particle morphologies,” J. Geophys. Res. 111, D12212 (2006). [CrossRef]
11. J. R. Janesick, in Scientific Charge-Coupled Devices (SPIE press, Washington, USA, 2001), pp. 204–207.
12. B. Berge, K. Sudholz, B. Steiner, J. Rohmann, and E. Rühl, “In situ size determination of single levitated solid aerosols,” Phys. Chem. Chem. Phys. 1, 5485–5489 (1999). [CrossRef]
13. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990). [CrossRef] [PubMed]
14. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996). [CrossRef]
15. T. Ichikawa, “The Assembly of Hard Spheres as a Structure Model of Amorphous Iron,” Phys. Status Solidi A 29, 293–302 (1975) [CrossRef]
16. J. C. Maxwell, Encyclopedia Britannica, Vol. 2, 82 (1877).
17. U. K. Krieger and A. A. Zardini, “Using dynamic light scattering to characterize mixed phase single particles levitated in a quasi-electrostatic balance,” Faraday Discuss. 137, 377–388 (2008). [CrossRef] [PubMed]
18. J. D. Brandner, N. M. Junk, J. W. Lawrence, and J. Robins, “Vapor Pressure of Ammonium Nitrate,” J. Chem. Eng. Data 7, 297–298 (1962). [CrossRef]
19. H. Xue, A. M. Moyle, N. Magee, J. Y. Harrington, and D. Lamb, “Experimental Studies of Droplet Evaporation Kinetics: Validation of Models for Binary and Ternary Aqueous Solutions,” J. Atmos. Sci. 62, 4310–4326 (2005). [CrossRef]
20. W. J. Massman, “A Review of the molecular diffusivities of H2O,CO2,CH4,CO,O3,SO2,NH3,N2O,NO, And NO2 in air, O2 and N2 near STP,” Atmos. Environ. 32, 1111–1127 (1998). [CrossRef]
21. J. D. Cox, D. D. Wagman, and V. A. Medvedev, CODATA Key Values for Thermodynamics, (Hemisphere Publishing Corp., New York, 1989).
22. L. V. Gurvich, I. V. Veyts, and C. B. Alcock, “Thermodynamic Properties of Individual Substances,” 4th ed. (Hemisphere Publishing Corp., New York, 1989).
23. J. D. Cox, D. Harrop, and A. J. Head, “The standard enthalpy of formation of ammonium nitrate and of the nitrate ion,” J. Chem. Thermodyn. 11, 811–814 (1979). [CrossRef]
24. CRC Handbook of Chemistry and Physics, 88th ed. (CRC Press, Boca Raton, 2007).
25. M. Mozurkewich, “The dissociation constant of ammonium nitrate and its dependence on temperature, relative humidity and particle size,” Atmos. Environ. 27A, 261–270 (1993).
26. M. Bilde, B. Svenningsson, J. Mønster, and T. Rosenørn, “Even-Odd Alternation of Evaporation Rates and Vapor Pressures of C3-C9 Dicarboxylic Acid Aerosols,” Environ. Sci. Technol. 37, 1371–1378 (2003). [CrossRef]