Clouds without supersaturation

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Traditional Köhler theory1 describes the equilibrium vapour-pressure relationship between liquid solution particles and humid air. Here we present the concept of multiphase-multicomponent Köhler theory, which reveals that stable cloud droplets of size 1-10 μm could exist in air with a relative humidity of less than 100%. This may explain the occurrence of persistent large-droplet fogs or smogs such as previously existed in London and which are now found in various heavily polluted locations, near the exits of chimneys and in the plumes of volcanoes.


Assumptions are usually made, explicitly or implicitly, that the solute mass in a droplet is fixed as it grows in size. The result is that a critical size exists at which the Raoult effect (depression of vapour pressure due to dissolved substance) just balances the Kelvin effect (increase of vapour pressure due to droplet curvature). Above that size, a droplet is said to be activated and will grow spontaneously if the ambient supersaturation remains at or above the respective equilibrium value. However, this ordinary Köhler formulation does not describe droplets in which the solute derives from either slightly soluble aerosol particles2 or soluble gases3,4.

We consider a situation in which aerosol particles containing both hygroscopic matter (salt) and a weakly soluble or insoluble core, are suspended in humid, polluted air. The particles absorb water, highly soluble gases such as nitric acid (which are depleted from the vapour due to the uptake by the particles), and weakly soluble gases (with a roughly constant gas concentration). The weakly soluble core is assumed to keep the surrounding liquid as a saturated solution until it has fully dissolved. At equilibrium, the saturation ratio of water vapour (S) equals the equilibrium vapour pressure (psol,w) of water over a solution droplet of a given radius (r) divided by the water vapour pressure over a plane surface (ps,w), which is given by:

Here r0 is the radius of the core, a/r describes the Kelvin effect, xws and xwg are the mole fractions of the weakly soluble species originating in the aerosol particle and in the gas phase, respectively, bs/(r3−r03) describes the Raoult effect of the soluble salt, and ba(r) is the term accounting for the highly soluble, condensing trace gas5.

Figure 1compares the traditional Köhler curve to the more complete multiphase- multicomponent theory. The particle sizes and compositions are selected as realistic examples as are the concentrations of water-soluble gases. We have not included in the curves the effect of a slightly soluble gas. For simplicity, we assume that the highly soluble substances do not influence the solubility of the weakly soluble substances (a conservative approximation). What becomes evident is that the combination of the amount of solute from the soluble gas and/or the slightly soluble aerosol particle, along with the vapour pressure reduction due to the Kelvin effect for the increased initial size of the slightly soluble particle, may be sufficient to keep the saturation ratio below unity for droplet sizes up to ~10 μm. In retrospect, this result is evident in the traditional simple formulation of the Köhler equation6:

because S < 1 if b/r3 >a/r, which simply depends on the magnitudes of a and b. What we have pointed out is that realistic particle sizes, particle solubilities and gas concentrations exist to cause this to happen. In fogs formed by these processes, the cloud-drop-sized particles are thermodynamically similar to submicrometre ‘haze’ particles. However, without sophisticated physicochemical methods, they are physically indistinguishable from classical, ‘activated’ drops.

Figure 1: Figure 1a, Conventional Köhler curve for a 30 nm dry particle consisting of ammonium sulphate at 298 K.

b, A similar particle containing an additional, insoluble 500 nm core (the amount of ammonium sulphate is the same as with a). The effect of insoluble material has been studied in detail7. c, Particle containing a slightly soluble 500 nm core. The solubility used8 (0.00209 g cm−3) corresponds to that of CaSO4. The sharp minimum of the curve shows the point at which all of the core is dissolved. CaSO4 particles occur commonly in air, and dissolved CaSO4 has been found in fog-water collected in the Po Valley, Italy9. d, Effect of an added, highly soluble gas, nitric acid. Initial gas-phase concentration of HNO3 was 1 p.p.b.v., and the Henry's law constant used10 (mole-fraction scale) was 853.1 atm−1. Because nitric acid is allowed to deplete from the gas as it is absorbed by the droplets, the term ba(r) in equation (1) depends on the aerosol number concentration, which in this case was assumed to be 100 cm−3. Aerosol size distribution was taken to be monodisperse (a qualitatively similar curve would result if the aerosol population was 1,000 cm−3 and initial HNO3 concentration 3 p.p.b.v.). The smooth minimum in the curve is caused by the depletion.


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Kulmala, M., Laaksonen, A., J.Charlson, R. et al. Clouds without supersaturation. Nature 388, 336–337 (1997) doi:10.1038/41000

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