Droplet stability: the Köhler theory

The process leading to the formation of an atmospheric cloud entails the presence of aerosols in the air and their growth, from the typical size of aerosols modes (see Sec. 2.1) up to the hundreds of micrometer scale. This dynamics is governed by the atmospheric chemical equilibrium between the condensed phase of a growing droplet and the surrounding gas phase.

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250 1.3 - 1.4 North Atlantic 145-370 0.4 - 0.9 North Atlantic

100 - 1000 - Arctic

140 0.4 Cape Grim(Australia)

250 0.5 North Atlantic

25 - 128 0.4 - 0.6 North Pacific

27 - 111 1.0 North Pacific

400 0.3 Polluted North Pacific

100 0.4 Equatorial Pacific

600 0.5 Continental

2000 0.4 Continental (Australia) 3500 0.9 Continental (Buffalo, NY)

Table 2.2: Empirical parameters for the CCN concentration as a function of the supersaturation. Adapted from [6].

The equilibrium of a pure water droplet is described by the well-known Kelvin equation (see Sec. 2.3.2), from which it can be inferred that a large supersatu-ration is required to establish the equilibrium between a small droplet and the surrounding vapor. The equilibration of such particles at low supersaturation is ensured by the fact that their chemical potential is altered by the Gibbs energy gain due to the dissolution of solids, such as nonvolatile NH₄NO₃, (NH₄)₂SO₄or NaCl salts. Indeed, atmospheric aerosols at high relative humidities are solutions of a variety of chemical compounds, which are responsible for their enhanced sta-bilization with respect to pure water. The inclusion of the solute effect into the assessment of droplet equilibrium was firstly implemented by Köhler [102], thus developing the so-called Köhler theory.

Equilibrium of a pure water droplet

For a pure water droplet, the dependence of the vapor pressure over its curved interface on the droplet diameter is given by the Kelvin equation (Eq. 2.12):

p_w(D_p)

photo-induced processes

Figure 2.12: Kelvin effect: ratio of the equilibrium vapor pressure of water over a droplet to that over a flat surface, as a function of droplet diameter, forT =0^◦C (black) and T =20^◦C (green). Dashed lines indicate the evolution of a particle upon small perturbation of its diameter; these trajectories indicate that the equi-librium on the Kelvin curve is unstable.

The effect of the curvature of the droplet surface on the equilibrium water vapor pressure (p_w(D_p)) is displayed in Fig. 2.12, which clearly shows an increase of such equilibrium pressure with respect to that over a flat water surface. Such effect is relevant for droplets smaller than 0.1 µm in diameter, while for larger sizes p_w(D_p)can be well approximated by the equilibrium pressure calculated in the case of a flat surface.

Each point of this curve is obtained by computing the ratio p_w(D_p)/p^◦ that corresponds to the Gibbs energy barrier for a particle of diameterD_p. In Sec. 2.3 the equilibrium around the Gibbs barrier has been determined by inspecting the relative graph of Fig. 2.3. Adopting the same procedure, one can find that the equilibrium on the Kelvin plot for a pure water droplet is unstable, since even a minor perturbation will result in its complete evaporation or in its indefinite growth, until the water vapor depletion will reduce the saturation ratio close to

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unity.

Equilibrium of an aqueous solution drop

A dissolved solute (e.g. a nonvolatile salt) is virtually always present in atmo-spheric droplets, and its contribution to the stability of the droplet counteracts the effect of the surface curvature (Kelvin effect), by reducing the equilibrium water vapor pressure at the surface.

Consider a solution drop of diameterD_p made by water (molecular weightM_w, densityρ_w) andn_s moles of solute andν ions resulting from dissociation of one solute molecule. For a highly diluted solution, the relation between the water va-por pressure over this droplet and that over a flat surface at the same temperature is expressed by [6]:

Recalling the definition of the saturation ratio asS= p_w(D_p)/p^◦, we can rewrite as follows:

S'1+A_w D_p− B_s

D³_p (2.28)

The parameters A and B are defined as:

A_w= 4M_wσ_w RTρw

B_s=6νn_sM_w π ρw

(2.29) where σ_w is the air-water surface tension. Eq. (2.27) and (2.28) are equivalent forms of the Köhler equation [102]. These equations feature two terms which represent two opposite effects that compete to determine the saturation ratio of the solution droplet at a temperature T. The curvature termA_w/D_pexpresses the Kelvin effect, which tends to increase the water vapor pressure of the droplet, hence the saturation. Obviously, in a pure water drop only the Kelvin term is present, and this results in higher vapor pressures as compared to a flat surface, especially for small diameters (see Fig. 2.12). Conversely, the presence of a so-lute dissolved in the drop yields an opposite effect on the vapor pressure, that can be significantly lower than for the pure water, depending on the magnitude of the solute termB_s/D³_p with respect to the curvature term. Both terms increase as the droplet size decreases, but the solute effect scales as 1/D³_p thus features a stronger increase. Tracing the supersaturation, defined asS−1, as a function of the diameter results in the Köhler curves displayed in Fig. 2.13. All the Köhler curves feature a maximum which occurs at a specific diameterD_c calledcritical droplet diameter, for which one can also identify the correspondingcritical satu-ration S_c. Left of this maximum, the droplet thermodynamics is dominated by the

photo-induced processes

Figure 2.13: Köhler curves for nonvolatile NaCl (ν =2) and (NH₄)₂SO₄ (ν =3) particles, for different dry diameters. Dashed lines indicate the evolution of a particle upon small perturbation of its diameter. Beyond of the critical diameter, the trajectories indicate that the equilibrium of the particle is unstable. Adapted from [6]

solute effect, which results in the steep rise of the curve; this feature can easily be explained by recalling that such plots are obtained for a fixed amount of salt core, thus the smaller the drop, the higher the solute concentration and hence its effectiveness. As long as the diameter increases, the relative importance of the Kelvin effect also increases, and becomes dominant beyond the critical diameter.

All the Köhler curves approach the Kelvin equation for droplets in the tens of mi-crometer range, as the solute is so diluted that the solution behaves as they were purely watery.

These curves are specific of a given chemical composition of the droplet: the po-sition and the height of the maxima depend on the dissolved solute.

Köhler plots can also be read as the equilibrium size of a droplet for different

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ambient supersaturation, hence the relative humidity. Therefore, a way to inter-pret Köhler plots is to look at them as size-RH equilibrium curves. A droplet of diameterD_p, containing n_s moles of solution and surrounded by an atmosphere whose vapor partial pressure isp_w, is in equilibrium with the environment if it sat-isfies the Köhler equation, thus lying on the Köhler curve. While for a pure water droplet it has been shown that such equilibrium state is unstable at any sizes, for an aqueous solution drop the stability is a function of its position on the Köhler plot with respect to the critical diameterD_c, as it can be inferred from the trajec-tories drawn as dashed lines on Fig. 2.13.

Let us consider a droplet whose diameter isD_p<D_c, thus lying on the rising por-tion of the curve, at a fixed supersaturapor-tion S. If a droplet experiences an infinites-imal increase in its diameter,its equilibrium water vapor pressure will be higher than the fixed ambient water pressure, and the droplet will evaporate some water until it returns to its original size. If the initial perturbation causes a temporary loss of water molecules, the original equilibrium will be restored through the op-posite process, that is the condensation of water vapor on the droplet. Therefore, droplets lying on the rising front of the Köhler curves are in a stable equilibrium with the environment.

Conversely, beyond the critical diameter D_c, the equilibrium of the droplet is driven by the Kelvin effect, already examined in the previous paragraph.

From these considerations, it emerges that, for ambient supersaturation below zero, i.e. a relative humidity lower than unity, there exists one equilibrium state, which constrains the droplet on the rising edge of the Köhler curve. If the en-vironment supersaturation is such that 0<S<S_c, two possible equilibrium can be attained, at different diameters: the entailing the smaller diameter is stable, while the other one is unstable, since is located in the Kelvin-like portion of the Köhler curve. When the ambient supersaturation happens to exceedS_cthere is no equilibrium size for the droplets, leading to an indefinite growth for any starting diameter. The situation depicted in Fig. 2.11a, where only a fraction of an aerosol population of uniform chemical composition was activated at a given supersat-uration, corresponds to the two-equilibrium case defined above. The activation diameter results from the intersection between the ambient supersaturationSand the descending branch of the Köhler curve. If the aerosols have different chemical composition, each type of aerosol will behave according to its specific Köhler plot.

Effect of a soluble trace gas

With respect to a pure water droplet, the inclusion of a solute leads to a strong modification of the size-RH equilibrium curve, now described by the classical Köhler plots. In the classical Köhler theory, the aqueous solution is assumed to

photo-induced processes be entirely formed prior to the droplet growth, i.e. the available solute is already within the droplet. This is a reasonable hypothesis when the solute is a highly soluble solid substance such as a nonvolatile salt, or a highly soluble liquid such as H₂SO₄.

Laaksonen et al. [103] developed a more general reformulation of the Köhler the-ory that considers other cloud active chemicals which don’t fit this assumption.

HNO₃, for instance, is a highly soluble trace gas, which can be absorbed by grow-ing droplets, and is potentially ubiquitous in the atmosphere as it is produced by oxidation of nitrates (see Sec. 2.3.5). Therefore, as suggested by Kulmala et al. [104], it is more reasonable to assume that a multi-component condensation process is more likely to occur, rather than the two-steps process. This implies that water and nitric acid condense at the same time, when the droplets are still grow-ing. Therefore, as nitric acid is present within the droplet already from the early stage of its evolution, its contribution will reenforce the solute effect, assisting the stabilization of the aerosol at even lower saturation ratio. This generalization was justified by "observations of large but unactivated cloud or fog droplets [105]"

(see below in the text) and "..observations of the existence in the atmosphere of soluble gases like HNO₃in appreciable concentrations [106]".

The extended version of the Köhler theory is extremely useful to our purpose, since HNO₃is produced in large amounts by the laser filaments and may poten-tially strongly influence the stability of laser-generated aerosols [19].

Laaksonen and co-authors considered a dilute ionic solution with n_a moles of soluble gas per droplet and v_a ions resulting after dissociation of one molecule;

analogous to (Eq 2.29) the parameterB_acan be defined as follows:

B_a= 6ν_an_aM_w

π ρwD³_p (2.30)

The modified Köhler equation now features an additional term and reads:

S'1+A_w D_p− B_s

D³_p−B_a

D³_p (2.31)

The first three terms of Eq. (2.31) are unchanged with respect to the classical Köhler equation Eq. (2.28), while the fourth term accounts for the dissolved trace gas effect and adds up to the∼B_s/D³_psolute term. As the coefficientB_adepends on the number of dissolved moles n_a, it is function of the temperature and the droplet diameter.

If the soluble gas concentration in the vapor phase is not affected by the absorption into the droplet, the respective term in (Eq. 2.31) simplifies. Therefore, under the assumption of undepleted trace gas, the Köhler equation reads as follows:

S'1+A_w D_p− B_s

D³_p−ν_ap

p_aH_a (2.32)

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Figure 2.14: Effect of the presence of 10 ppb of HNO₃ inside a dry ammonium sulphate particle of 0.05µm diameter, at 20^◦C and 1 atm. Reprint from [6].

The Köhler size-RH equilibrium curves for dry ammonium sulphates particles containing 10 ppb of HNO₃are displayed in Fig. 2.14. The maximum featured by the classical Köhler curve is lowered by the presence of nitric acid, thus becoming a local maximum when HNO₃depletion is considered. The barrier for the activa-tion is now fixed by the second maximum, which corresponds to the original curve at large sizes, since the available nitric acid concentration will have significantly decreased owing to the uptake by the growing particle. This entails that stable droplets as large as few micron in diameter can exist without being activated.

The general trend in the equilibrium of atmospheric aerosols consists in a easier activation as the soluble trace gas concentrations gets higher. As it will be pre-sented in Chapter 3, the laser filaments were proved to generate extremely high amounts of HNO₃, up to the ppm level [19]; therefore, we developed a slightly modified version of the extended Köhler theory, more pertinent to the chemical composition of the laser-affected atmosphere, that will be briefly discussed in

photo-induced processes Sec. 3.4.