G. McFiggans et al. Aerosol effects on warm cloud activation 2595 2 Theory of activation of aerosol particles in warm activating particles which may be more or less accurate de pending on the supersaturation, there are numerical approxi- mations such as that associated with the Taylor series expan- The description of the equilibrium size of a droplet with wa- sion of the exponential which limit the range of applicability ter saturation ratio, founded on the early work of Kohler of Eq (2). Figure 1 shows the contribution of the Kelv (1936), is now well-established and can be readily derived and Raoult terms to the activation behaviour of a 200 nm dry from the Clausius-Clapeyron equation modified to give a diameter ammonium sulphate particle general equilibrium relation between an aqueous salt sol This form of the expression shows a single characteris- tion droplet and water vapour tic maximum in supersaturation for a given dry composition and size, known as the critical supersaturation, Sc, associated =awexpKe = awexp with a unique size, denoted the critical radius, rc or diame- RTr (1) ter, De. Using the simplified expression(2), solutions for the critical quantities are the vapour pressure of wa es is the saturation vapour pressure of water, 3B12 (e/es=S, is known as the saturation ratio), A aw is the water activity Ke is the Kelvin factor Ww is the partial molar volume of water, S 27B sol/v is the surface tension of the solution at the composition of the droplet, R is the universal gas constant, For an increasing environmental value of s below Sc there T is the droplet temperature, is a unique equilibrium droplet size. Once the droplet grows is the particle radius beyond its critical size (i.e. as the environmental S increases above Sc) the droplet will exhibit unimpeded growth unless the environmental S reduces below the equilibrium value of This form of the Kohler equation is not generally accessi- S at the instantaneous value of r. In this case, with no fur- Yau, 1989; Pruppacher and Klett, 1997; Chylek and Wong, ther change in S, the droplet will evaporate to its sub-critical 1998: Seinfeld and Pandis, 1998) provide standard deriva- equilibrium size tions to yield the simplified form of the Kohler equation The Kohler expression can be envisaged as the compet tion between the two expressions of component properties a B determining activation of particles; the curvature term and (2) the solute term. The solute terms depends first on the number of solution molecules and then on the dissociation of these where A==dule and B= where v is the num- molecules. The effect can be illustrated for two frequently as- er of dissociated ions per so cule, ms is thethe se sumed cloud condensation nuclei types: ammonium sulphate lute mass and subscripts s and w relate to solute and water and sodium chloride.(NH4)2SO4 has a molecular weight properties, respectively. The term in A is denoted the Kelvin of 132 gMol- while that of NaCl is 58.5 gMol-1. Thus, in or curvature term and that in b. the raoult or solute term the absence of dissociation, a given mass of Nacl in solu- This latter form of the equation assumes that the droplet tion would yield 2. 26 times more dissolved molecules than behaves ideally, i.e. that the practical osmotic coefficient of (NH4)2SO4. Assuming full dissociation(infinite dilution) the salt, =l, where (NH4) SO4 yields 3 ions while NaCl yields 2, so the net effect of the molecular mass and dissociation is that nacl is 2. 26/1.5=1.5 times more active than(NH4)2SO4 for the ( same dry mass of particle(the Se ratio is around 1.22 due to the square root dependence). This is illustrated in Fig. 2 and that the number of ions in solution is independent of so- where the peak supersaturation is plotted versus dry diameter lution concentration. Equation(2)further assumes that the for particles comprising each electrolyte. This figure directly solute is completely soluble and it implicitly follows that the illustrates the significant differences in the critical supersat- solution droplet is assumed homogeneous-that the compo- uration as a function of both the chemical composition and sition is independent of distance from droplet centre. It is dry size of a particle(raoult and Kelvin effect further assumed that the surface tension and density of the The atmospheric aerosol does not solely com growing droplet are equal to those of water. In addition to the pended completely soluble inorganic salt solution assumptions relating to physico-chemical properties of the A modification to the Raoult term was reported www.atmos-chem-phys.net/6/2593/2006/ Atmos. Chem. Phys., 6, 2593-2649, 2006G. McFiggans et al.: Aerosol effects on warm cloud activation 2595 2 Theory of activation of aerosol particles in warm clouds The description of the equilibrium size of a droplet with wa￾ter saturation ratio, founded on the early work of Kohler ¨ (1936), is now well-established and can be readily derived from the Clausius-Clapeyron equation modified to give a general equilibrium relation between an aqueous salt solu￾tion droplet and water vapour: e es = awexpKe = awexp 2vwσsol/v RT r  (1) where e is the vapour pressure of water, es is the saturation vapour pressure of water, (e/es=S, is known as the saturation ratio), aw is the water activity, Ke is the Kelvin factor, vw is the partial molar volume of water, σsol/v is the surface tension of the solution at the composition of the droplet, R is the universal gas constant, T is the droplet temperature, r is the particle radius. This form of the Kohler equation is not generally accessi- ¨ ble to analytical solution and a number of texts (Rogers and Yau, 1989; Pruppacher and Klett, 1997; Chylek and Wong ´ , 1998; Seinfeld and Pandis, 1998) provide standard deriva￾tions to yield the simplified form of the Kohler equation: ¨ S = e es ≈ 1 + A r − B r 3 (2) where A= 2Mwσw/v RT ρw and B= νmsMw Ms(4/3πρw) , where ν is the num￾ber of dissociated ions per solute molecule, ms is the the so￾lute mass and subscripts s and w relate to solute and water properties, respectively. The term in A is denoted the Kelvin or curvature term, and that in B, the Raoult or solute term. This latter form of the equation assumes that the droplet behaves ideally, i.e. that the practical osmotic coefficient of the salt, φ=1, where aw = exp − νns nw φs  (3) and that the number of ions in solution is independent of so￾lution concentration. Equation (2) further assumes that the solute is completely soluble and it implicitly follows that the solution droplet is assumed homogeneous – that the compo￾sition is independent of distance from droplet centre. It is further assumed that the surface tension and density of the growing droplet are equal to those of water. In addition to the assumptions relating to physico-chemical properties of the activating particles which may be more or less accurate de￾pending on the supersaturation, there are numerical approxi￾mations such as that associated with the Taylor series expan￾sion of the exponential which limit the range of applicability of Eq. (2). Figure 1 shows the contribution of the Kelvin and Raoult terms to the activation behaviour of a 200 nm dry diameter ammonium sulphate particle. This form of the expression shows a single characteris￾tic maximum in supersaturation for a given dry composition and size, known as the critical supersaturation, Sc, associated with a unique size, denoted the critical radius, rc or diame￾ter, Dc. Using the simplified expression (2), the analytical solutions for the critical quantities are: rc =  3B A 1/2 (4) Sc = 4A3 27B !1/2 (5) For an increasing environmental value of S below Sc there is a unique equilibrium droplet size. Once the droplet grows beyond its critical size (i.e. as the environmental S increases above Sc) the droplet will exhibit unimpeded growth unless the environmental S reduces below the equilibrium value of S at the instantaneous value of r. In this case, with no fur￾ther change in S, the droplet will evaporate to its sub-critical equilibrium size. The Kohler expression can be envisaged as the competi- ¨ tion between the two expressions of component properties determining activation of particles; the curvature term and the solute term. The solute terms depends first on the number of solution molecules and then on the dissociation of these molecules. The effect can be illustrated for two frequently as￾sumed cloud condensation nuclei types: ammonium sulphate and sodium chloride. (NH4)2SO4 has a molecular weight of 132 gMol−1 while that of NaCl is 58.5 gMol−1 . Thus, in the absence of dissociation, a given mass of NaCl in solu￾tion would yield 2.26 times more dissolved molecules than (NH4)2SO4. Assuming full dissociation (infinite dilution), (NH4)2SO4 yields 3 ions while NaCl yields 2, so the net effect of the molecular mass and dissociation is that NaCl is 2.26/1.5=1.5 times more active than (NH4)2SO4 for the same dry mass of particle (the Sc ratio is around 1.22 due to the square root dependence). This is illustrated in Fig. 2 where the peak supersaturation is plotted versus dry diameter for particles comprising each electrolyte. This figure directly illustrates the significant differences in the critical supersat￾uration as a function of both the chemical composition and dry size of a particle (Raoult and Kelvin effects). The atmospheric aerosol does not solely comprise sus￾pended completely soluble inorganic salt solution particles. A modification to the Raoult term was reported by Hanel ¨ www.atmos-chem-phys.net/6/2593/2006/ Atmos. Chem. Phys., 6, 2593–2649, 2006