6 Cloud droplet formation and Köhler theory

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1 6 Cloud droplet formation and Köhler theory This chapter discusses the details of cloud droplet formation. Cloud droplet formation requires phase changes. Some phase changes require a nucleation process, for instance, ice nucleation or gas-to-particle formation of aerosols. Other phase changes, such as cloud droplet formation, do not require nucleation because cloud droplets form on soluble or hydrophilic aerosol particles, which consists already of condensed water before they have grown to cloud droplets. In this chapter, phase changes and nucleation are discussed prior to cloud droplet formation and measurements of cloud condensation nuclei, which are the subset of aerosols that can form cloud droplets at a given supersaturation. 6.1 Nucleation Cloud droplet formation requires a phase change, i.e. the condensation of water vapor, whereas cloud droplets evaporate when the cloud dissipates. A phase change where a thermodynamically stable phase forms and grows within the metastable parent phase is called nucleation. In the atmosphere, nucleation occurs in the formation of cloud droplets, ice crystals and aerosol particles from precursor gases. As in the example of nucleation of the solid salt within a supersaturated salt solution, the final state of the stable phase has lower energy than the initial state, but the transition is inhibited by an energy barrier. This energy barrier can be visualized by calculating the Gibbs free energy of a volume V containing a quantity of the parent phase and an embryo of the forming new phase as a function of the size of the embryo. In the following, this calculation will be done for the specific case of cloud droplet nucleation from the vapor phase, but the same concept is also valid for ice nucleation, for nucleation of new aerosol particles from precursor gases, and crystallization of liquid solution droplets. In the present chapter, we are concerned with the phase transitions, which in most cases not only occur at constant T, but also at constant p. This means that the system is no longer closed with respect to particle number. For example, condensation of a small cloud droplet within the volume V of water vapor requires 152

2 the entrance of a corresponding number of water molecules into volume V to keep the pressure of the water vapor constant. For such a system, the Gibbs free energy G = U + pv T S = F + pv is better suited to describe the equilibrium state than U or F. In fact, a minimum principle can be proven for G, which states that the equilibrium state of a system kept at constant T and p, is the state of minimum G 3. The change of G as a function of the extensive variables of state of the system (e.g. V ) indicates in which direction a phase change occurs and if it occurs spontaneously or not (in chemistry, the minimum G also determines to what extent the educts will react to the products for a specific reaction). As shown in Fig. 6.1 [Kashchiev, 2000] for each partial pressure, the Gibbs free energy exhibits two relative minima, one corresponding to the substance in the gas phase, and one corresponding to the liquid state. Both states correspond to thermodynamic equilibrium, but the state with higher G corresponds to a metastable state, meaning that the state will change into the state of stable equilibrium (global minimum of G) under given circumstances. For p > p s, where p s is the saturation vapor pressure, the gas is supersaturated, and tends to condense into the liquid state. For p = p 2 = p s (T ) both states have identical G and neither the gaseous nor the liquid state can be identified as the more favorable state. The fact that metastable states are separated from stable states by an energy barrier leads to a kinetic inhibition of the phase transitions, i.e. phase transitions do not occur spontaneously, but only on a time scale which increases with increasing height of the energy barrier. This means that, loosely spoken, a supersaturated vapor phase can exist on average for a time smaller than the timescale of the phase transition to the liquid state. The existence of an energy barrier is common to first order phase transitions such as condensation, evaporation, melting and freezing. Regarding the minimum principle for G, there is thermodynamically no reason why a system in a metastable state should increase its Gibbs free energy and cross the energy barrier to reach the stable state. In fact, the process of phase tran- 3 A similar minimum principle can be proven for the thermodynamic variable F in case of a closed system where p is not constant 153

3 G gas p 3 p 2 = p s liquid p 1 V Figure 6.1: Schematic of the Gibbs free energy of a substance as a function of the volume that the substance occupies for three different pressure levels at a given T. Adopted from Kashchiev [2000].. sition cannot be described only by thermodynamics, but concepts from statistical mechanics describing fluctuations of the system on a molecular scale are equally required because kinetic aspects of a process cannot be derived from thermodynamics. In thermodynamics, the state of a quantity Q of a substance is described by its variables of state, S, V and N 4. Theoretically, the transition from a metastable to a stable state can occur via 4 In reality, Q consists of a large number of molecules N whose instantaneous state can be described by 6N coordinates (3 for each position and 3 for the impulse of each molecule). Such a state is also called a microstate. For most practical purposes, it is way too complicated to describe a system via its microstates, and it makes more sense to merge all the microstates leading to the same macrostate described by (S, V, N) and regard them as one state. This is actually the purpose of thermodynamics. However, taking the continuous fluctuations of the microstate of a system into account is required to understand that also quantities attributed to the actual macrostate of the system, such as G, can fluctuate. For example, fluctuations can lead to three molecules building a small cluster of the condensed phase, and separating again in the next moment. Of course, the higher the energy barrier of a phase transition, the less likely it is that fluctuations lead to a corresponding microstate, and the longer it takes on average for the transition from the metastable to the stable state to occur. 154

4 various pathways. Following the curve for p = p s in Fig. 6.2 [Kashchiev, 2000] corresponds to a uniform change of density transferring the substance as a whole from the gaseous to the liquid state. However, as this involves all molecules of the system, the energy barrier impeding this pathway is very high. Nucleation of a new phase within the old phase is an alternative pathway of a phase transition. It is by far the more likely pathway, since the new phase is initiated locally via a small cluster of the new phase, resulting in a much lower energy barrier. In fact, phase transitions of the first order generally occur via a nucleation process. In the following, we derive an expression for the energy barrier impeding the formation of cloud droplets within water vapor for the case of homogeneous nucleation, where no foreign surface is available to catalyze the nucleation process and to lower the energy barrier. This provides information on the conditions under which a small cloud droplet can exist, and also indirectly leads to the understanding how in the atmosphere the presence of soluble aerosol particles opens up a pathway for cloud droplet formation without even involving a nucleation process. Figure 6.2: Old phase of M molecules before (state 1) and after (state 2) homogeneous formation of a cluster of n molecules. From Kashchiev [2000]. Consider the transition from an initial state consisting of the uniform old phase (state 1) to a state where a cluster of n molecules of the new phase has formed within the old phase. State 1 has a Gibbs free energy G 1 = Mµ old, where µ old is the Gibbs free energy per molecule (or chemical potential) of the old phase, and state 2 has G 2 (n) = (M n)µ old + G(n), where G(n) is the total Gibbs free energy of the cluster. G(n) can be written as G(n) = nµ new + G ex (n), where 155

5 µ new is the chemical potential of the bulk new phase, and G ex (n) arises due to the condensed phase appearing in a cluster as opposed to the bulk phase. The work for cluster formation is therefore: W (n) = G 2 (n) G 1 = n µ + G ex (n) (6.1) where µ = µ new µ old. For the condensation of a vapor phase into a liquid phase, it can be shown that µ can be approximated by: µ(p, T ) = kt ln ( p ) p s (T ) (6.2) where p s (T ) is the saturation vapor pressure. Introducing the saturation ratio as S = p p s(t ) µ(p, T ) = kt ln S (6.3) It can be shown that the excess energy for cluster formation, G ex (n), can be written as G ex (n) = n(µ new,n µ new ) (p n p)v n + ϕ(v n ) (6.4) where p n is the pressure within the cluster, ϕ(v n ) is the total Gibbs free energy of the interface between the cluster and the old phase, and µ new,n is the chemical potential of the molecules in the cluster. The above expression shows that G ex (n) accounts for three effects: 1) the difference in chemical potential of the molecules within the cluster as opposed to molecules in the bulk new phase. 2) the difference in pressure within the cluster as compared to outside the cluster and 3) the existence of an interface between the cluster and the old phase. Since p n p also µ new,n µ new. From eq. (2.54) it follows that dg = dµ = αdp sdt = αdp = 1 n V n(p)dp and therefore µ new,n (p n ) = µ new (p) + 1 n pn p V n (p )dp (6.5) In the case of condensation of vapor into the liquid phase, the equation of state V n = V n (p n, T ) of the cluster is simple. As the liquid phase is approximately incompressible, V n (p n ) nv 0, where v 0 is the volume of an individual molecule 156

6 in the bulk liquid phase. Inserting this into eqs. (6.4) and (6.5), we end up with a simple expression for G ex : G ex = ϕ(v n ) (6.6) i.e. for the formation of a cluster of a condensed phase within the vapor phase, the work of cluster formation is: G = n µ + ϕ(v n ) (6.7) It can further be shown that for spherical clusters of a condensed phase ϕ(v n ) = ϕ(r) = 4πσR 2, where R is the cluster radius and σ is the free energy associated with the formation of a unit surface area of interface between the condensed and the vapor phase. In the end, we obtain: G = nkt ln S + 4πσr 2 = 4πr3 3α l R v T ln S + 4πσr 2 (6.8) Note that no latent heat release has appeared in the formulation of nucleation so far. We have assumed that nucleation occurs at constant temperature, i.e. our system has to be thought of as coupled to a heat reservoir which absorbs any latent heat released by molecules entering into the condensed phase. Therefore latent heat does not appear in the Gibbs free energy. As soon as the new phase begins to grow by vapor diffusion assuming a constant temperature is no longer justified and latent heat release has to be taken into account (see chapter 7). From a molecular point of view, the release of latent heat is maybe not straight forward to understand. Macroscopic latent heat release arises from the fact that molecules in the condensed phase on average are in a lower energetic state than molecules in the vapor phase. If an individual molecule in the vapor phase is to change to the condensed phase, its kinetic energy has to be low enough that it can enter a bound state with other molecules. If this is not the case, it has to reduce its kinetic energy first by collisions with other molecules in the vapor phase. If we remove a certain number of molecules from the lower energy tail of the Maxwell- Boltzmann distribution of the vapor (and transfer them to the condensed phase), the Maxwell-Boltzmann distribution of the remaining vapor phase will shift to higher energies, i.e. to a higher temperature. 157

7 It is important to note that the above derivation of the work for cluster formation has been performed under the assumption of pure substances, i.e. a cluster of the condensed phase forming within its pure vapor phase. However, it is obvious that this is not the case of cloud droplets forming in the atmosphere, as water vapor is only a minor constituent of the atmosphere. Nevertheless, adding air at atmospheric p air does not change the result, as p air cancels out in the first and second term of equation 6.4. Accordingly, the total absolute pressure does not appear in the final expression for W (n), eq. (6.5). This, however, only holds for the case of condensation. For the opposite case of evaporation, the work for the formation of a small steam bubble within the old liquid phase is much more complicated and depends on the absolute pressure. 6.2 Kelvin equation This equation is graphically evaluated in Figure 6.3. It shows that if S < 1, both terms are positive and G increases with r. If S=1, the first term on the right hand side is zero and G still increases with r. An increase of G with increasing radius means that it is always more likely for a cluster to evaporate than to grow. Only if S > 1, G contains both positive and negative contributions. At small values of r, the surface term dominates and the behaviour of G is similar to the S < 1 case. As r increases, the first term on right hand side, the volume term, becomes more important, so that G reaches a maximum at r = r c and then decreases. At r = r c, G = 0, i.e. the cluster is in equilibrium with the vapor r so that a cloud droplet can exist at this size or any larger size. Note that 2 G < r 2 0, the Gibbs free energy has a local maximum, i.e. the cluster is in an unstable equilibrium with the vapor phase at r = r c. Whereas in a situation of a stable equilibrium, the system deviating slightly from its equilibrium configuration is pushed back towards the equilibrium, a cluster deviating slightly from r = r c will either shrink to zero or grow (theoretically) to infinite size. This unstable equilibrium at r = r c together with the monotonous decrease of G for r > r c is the reason why in the atmosphere supersaturated vapor leads to macroscopic cloud and rain drops. 158

8 Figure 6.3: Gibbs free energy for the formation of a droplet of radius r from a vapor with a saturation ratio S = 0.99 (solid line), S = 1 (dotted line), S = 1.01 (dashed line) and S = 1.02 (dot-dashed line). The critical radii of 0.12 µm for S = 1.01 and of 0.06 µm for S = 1.01 are shown as dashed resp. dot-dashed lines together with the critical Gibbs free energy of J and J, respectively. 159

9 r c can be calculated from differentiating eq. (6.8) with respect to r and setting the derivative to zero: G r = 0 = 4πr2 cr v T ln S + 8πr c σ (6.9) rc α l This yields: r c = 2σα l (6.10) R v T ln S This equation defines the minimum supersaturation that is required that a droplet with radius r c can exist and subsequently grow. The supersaturation S can be understood as the ratio of the water vapor pressure e s (r) at which a droplet with radius r is in equilibrium to the vapor pressure at which a flat water surface is in equilibrium. Replacing S with es(r) e s( ) and re-arranging terms leads to: Replacing α l by 1 ρ w ln e s(r) e s ( ) = 2σα l R v T r and solving for e s (r) this equation yields: e s (r) = e s ( ) exp ( 2σ ) ρ w R v T r (6.11) (6.12) Equation (6.12) is called the Kelvin equation. It describes that the equilibrium vapor pressure is larger over a droplet with radius r than over a plane or bulk surface. The surface tension is that of pure water, which is given at 273 K as N m 1. The increase of equilibrium vapor pressure due to the curvature of a small droplet has been derived from thermodynamics so far. The surface tension has been introduced as the free energy per unit surface area of the liquid, i.e. the work per unit area that is required to extend the surface of a liquid at a constant temperature. It can also be understood in a phenomenological way. Because a molecule at the surface of the droplet has fewer neighboring molecules than a molecule in the interior of the droplet, it experiences a net force towards the interior of the droplet (see Figure 6.4). Therefore work is required to bring a molecule from the interior to the surface as at least one binding to a neighboring molecule has to be 160

10 broken. Hence, less energy is required to break the remaining bindings and to release the molecule into the gas phase as compared to a molecule in the bulk. This difference is more important for a molecule at a curved water surface as compared to a place surface, and the probability that a molecule evaporates from the surface is larger for a curved surface. A curved surface can therefore only be in equilibrium with the water phase at a higher vapor pressure as compared to a plane surface. Figure 6.4: Schematic picture of surface tension. Some critical radii for droplet formation in the absence of CCN (eq. 6.10) are given in Table 6.1 as a function of the necessary supersaturation. Only for saturation ratios between 5 and 10, the number of molecules necessary for droplet formation is small enough that random cluster formation could lead to nucleation. However, under typical atmospheric conditions, the saturation ratio seldom exceeds 1.01 due to the presence of CCN. This renders homogeneous nucleation of water droplets directly from the vapor phase highly unlikely. Instead cloud droplet nucleation in the atmosphere occurs via heterogeneous nucleation where cloud droplets form in the presence of a foreign substance, i.e. cloud condensation nuclei (CCN). 161

11 Table 6.1: Critical radius and number of molecules for a given saturation ratio at a temperature of K and a surface tension of N m 1. Saturation ratio Critical radius Number of molecules S r c (µ m) n x x x x x x Hygroscopic growth Aerosols can also change phase. The phase change of the solid soluble aerosol to a liquid aerosol is called deliquescence and does not require a nucleation process. The opposite process, the formation of a solid aerosol, is called efflorescence or crystallization and requires a nucleation process. Nucleation requires overcoming an energy barrier, which can only be achieved if prior to nucleation the nucleating substance is in a supersaturated state. I.e. the atmosphere needs to be supersaturated with respect to water/ice before water droplets/ice crystals form or needs to contain supercooled water before it can freeze. As cloud droplets form on aqueous aerosol particles, we will first discuss the different phases of soluble aerosol particles (liquid and solid) before we study cloud droplet formation. When aerosols take up water as described in eq. (5.12) they can change their phase. The growth of the aerosol is described in terms of the growth factor, which is the ratio of the actual diameter to the diameter at 0% relative humidity (RH). It is only defined for RHs below 100%. The growth factor depends on particle size because of Kelvin s law. Some aerosols such as sulfuric acid (H 2 SO 4 ) remain liquid at all relative humidities (Figure 6.5) at atmospheric temperatures and pressures and just change their size according to relative humid- 162

12 Figure 6.5: Water uptake of sulfuric acid (H 2 SO 4 ), ammonium bisulfate (NH 4 HSO 4 ) and ammonium sulfate ((NH 4 ) 2 SO 4 ) aerosols expressed in terms of their growth factor, which is the ratio of the actual diameter D p to the dry diameter at 0% RH (D p0 ), as a function of relative humidity [Seinfeld and Pandis, 1998]. ity. Other aerosols, predominantly salts such as ammonium sulfate or ammonium bisulfate, change their phase. At which relative humidity this phase change occurs depends on the direction of the change in RH. Let us take the example of ammonium sulfate starting at 10% RH, where it is solid. If we increase RH its diameter remains constant until 80% RH, where it suddenly increases by 50%. The RH at which the aerosol suddenly takes up water and changes from being solid to being liquid, i.e. where the salt dissolves in water, is called the deliquescence relative humidity (DRH). Only this RH provides a sufficient density of water molecules to dissolve the salt into a saturated solution, which is determined by the solubility of the salt in water. At higher RHs it rapidly takes up water and approaches a similar growth factor as sulfuric acid at 95% RH (Figure 6.5). If we now decrease RH, ammonium sulfate remains in the liquid phase until 163

13 we reach 40% RH, where it suddenly solidifies. This RH is called the crystallization relative humidity (CRH). Obviously the growth factor of ammonium sulfate exhibits a hysteresis, meaning the value of the growth factor is not unambiguously determined by RH, but depends also on the history of the particle. Between 40% and 80% RH the liquid phase of ammonium sulfate is metastable. This means that energetically the aerosol would prefer to be in the solid state. This is best illustrated in terms of the Gibbs free energy (cf. chapter 2) and discussed in the next subsection. Figure 6.6 shows the Gibbs free energy as a function of RH for an aerosol with a DRH of 80% such as ammonium sulfate. The Gibbs free energy decreases with increasing RH for an aqueous solution, which is in equilibrium with the corresponding RH, but is constant for a solid. A system always tends to minimize its Gibbs free energy. Thus at RH higher than the DRH the liquid phase is the preferred state and below the DRH the solid is the preferred state. Nevertheless, the ammonium sulfate aerosol remains liquid down to the CRH when RH decreases. Whether a process like a phase transition or a chemical reaction occurs spontaneously not only depends on the Gibbs free energies of the initial and final state, but on all the intermediate states that the system goes through as well. In the case of a salt solution droplet at RH below DRH, transition to the solid state involves the formation and growth of a solid embryo within the aqueous solution. As the Gibbs free energy of this intermediate state is higher than the one of the initial and the final state, the process of crystallization is inhibited by an energy barrier (Fig. 6.6). In the case of an ice crystal melting, the ongoing process continuously decreases the Gibbs free energy, i.e. there is no energy barrier between the initial and the final state. This is because a quasi-liquid layer forms on ice crystals at temperatures warmer than -25 C [Li and Somorjai, 2007]. Its role is to minimize the surface energy and it causes the melting process to be a continuous process starting with a small quasi-liquid layer at -25 C that increases in thickness until all the ice is melted at 0 C. The same argument holds for solid hygroscopic aerosols that experience deliquescence because also on those a quasi-liquid layer forms already at relative humidities below the deliquescence relative humidity. 164

14 Figure 6.6: Schematic of the Gibbs free energies over a saturated aqueous solution and over a solid particle as a function of relative humidity [Seinfeld and Pandis, 1998]. 165

15 6.4 Raoult s law Given that homogeneous nucleation from the vapor phase is highly unlikely for cloud droplet formation, we now discuss the role of CCN. Even though the aerosol particles that activate to form cloud droplets are already liquid particles, this process is nevertheless referred to as heterogeneous nucleation to illustrate that aerosol particles are necessary for it. Whereas the curvature of a water surface increases its equilibrium vapor pressure, soluble material dissolved in water can reduce its equilibrium vapor pressure. In cloud formation, this effect is important because a sufficient amount of soluble material is present in the atmosphere to substantially reduce the supersaturation required for cloud droplet formation. This reduction in vapor pressure is called Raoult s law. It is given for a plane surface of water as: e ( ) e s ( ) = n 0 n + n 0 (6.13) where e is the equilibrium vapor pressure over a solution consisting of n 0 water molecules and n solute molecules. This equation shows that if the vapor pressure of the solute is less than that of the solvent, in our case water, and if the total number of molecules remains constant, the vapor pressure over the solution is reduced in proportion to the amount of solute present. One can visualize this reduction in vapor pressure by recalling that the saturation water vapor pressure is only related to the exchange of water molecules between the droplet and the vapor phase. Assume that we have an ideal solution of water droplets and solute molecules in which the water droplets and the solute molecules do not interact with each other. Thus, if the solution possesses solute molecules at the surface, the exchange of water molecules between the water surface and the overlying vapor is limited to those places where water molecules occupy the surface. This picture also holds for non-ideal solutions except that instead of having solute molecules at the surface we would have hydrates of the dissociated solute molecules with water instead of non-interacting salt molecules that occupy places at the surface. Although the probability per unit time for a single water molecule at the surface 166

16 to evaporate is the same as in pure water, the flux of molecules from the surface is lower in the solution. If the solution is in equilibrium with the water vapor, this is equivalent to a lower density of water molecules in the vapor, i.e. to a lower saturation vapor pressure. For solutions in which the dissolved molecules are dissociated n must be multiplied by the degree of ionic dissociation. This degree of ionic dissociation is frequently expressed with i, the van t Hoff factor. For example, the van t Hoff factor for NaCl is i=2, for (NH 4 ) 2 SO 4 i=3 and for many organics i=1 [Broekhuizen et al., 2004]. Raoult s law describes the reduction in equilibrium vapor pressure over a solution consisting of n 0 water molecules and n solute molecules as compared to the saturation vapor pressure of pure water. It is shown in Figure 6.7 for two arbitrary substances a and b. In case that only substance a is present, the total vapor pressure over the solution is the same as the vapor pressure of a (cf. eq. 6.13). As the number fraction of the substance b increases, the total vapor pressure over the mixture decreases. It is smallest if only substance b is present. The dotted lines refer to a solute where the ions dissociate (i=3). They show an even larger reduction in vapor pressure over the mixture because in this case the molecular bindings between substances a and b are larger than the intramolecular bindings within substance a and within substance b. In this case, the minimum total pressure for a mole fraction of substance b of around If the molecular bindings were weaker between a and b then within substances a and b, positive deviations from solid curves would originate and the molecules could more readily escape from the mixture. In the atmosphere we typically have droplets with only dilute solutions (i.e. n n 0 ). If we associate substance a in Figure 6.7 with water, then only the curves in the left part of the plot are relevant. For dilute solutions, eq. (6.13) can be simplified using Taylor series expansion: e ( ) e s ( ) = 1 n/n n (6.14) n 0 167

17 Figure 6.7: Raoult s law for a system containing two liquid substances a and b. The left y-axis shows the vapor pressure of substance a and the right y-axis the vapor pressure of the substance b. The x-axis shows the mole fraction of substances a and b, where substance a decreases from left to right and substance b increases from the left to right. The solid lines refer to eq. (6.13) and the dotted lines to a solute with a van t Hoff factor of 3. The total pressure p tot is the sum of p a and p b. 168

18 The number of effective ions in a solute of mass M s is given by: n = in 0M s m s (6.15) where N 0 is the Avogadro s number and m s is the molecular weight of the solute. The number of water molecules in mass M w may likewise be written as: n 0 = N 0M w m w (6.16) We now apply Raoult s law to a solution droplet of radius r. Its size is related to the masses of water and the solute as follows: M tot = M s + M w = 4 3 πr3 ρ s (6.17) where ρ s is the density of the solution. For dilute solutions, we can neglect the mass of the solute M s in eq. (6.17) and approximate the density of the solution droplet with that of water so that M tot M w 4 3 πr3 ρ w. Inserting the above three expressions in eq. (6.14) and evaluating it for droplets with radius r instead of for bulk solutions, we obtain: where e (r) e s (r) = 1 3iM sm w 4πm s ρ w r 3 = 1 b r 3 (6.18) b = 3iM sm w i M s [m 3 ] (6.19) 4πρ w m s m s Writing Raoult s law in this way shows its dependence on the droplet radius for a given solute mass M s and molecular weight m s. I.e. the larger the droplet radius, the more dilute is the solution droplet. The transition from a small, more concentrated droplet to a large, diluted droplet corresponds to atmospheric cloud droplet formation where a cloud droplet forms heterogeneously on a salt particle with mass M s. The smaller the droplet, the larger is the effect of the solute molecules in reducing the ambient water vapor pressure required for the droplet to be in equilibrium with the ambient air. On the other hand, the curvature of the droplet, via the Kelvin effect, progressively increases the equilibrium vapor pressure for decreasing droplet radius. The balance between the two effects will be discussed in the following. 169

19 6.5 Köhler curve In order to see the competition between Raoult s law and the Kelvin equation, we look for an expression for the equilibrium saturation ratio for a solution droplet with radius r, taking into account both effects, i.e. where e (r) e s (r) a = ( e s (r) e s ( ) = e (r) e s ( ) = 2σ w ρ w R v T 1 b r 3 e (r) e s( ) : ) ( ) a exp r (6.20) [m] (for water) (6.21) T where σ w is the surface tension of water. If the radius is not too small, a good approximation is: exp ( ) a r 1 + a. For a droplet of 1 µm in radius, the error is r less than 5%. This simplifies eq. (6.20) to: e (r) e s ( ) = S = 1 + a r b (6.22) r 3 Here the term ab has been neglected because it is much smaller than the other r 4 terms. Equation (6.22) is known as the Köhler equation. The Köhler equation combines Kelvin s equation and Raoult s law and describes the ratio between the equilibrium vapor pressure over a solution droplet with radius r and that of a plane surface of water. The vapor pressure over the solution is increased because of the Kelvin effect and reduced due to the Raoult effect. The Köhler equation governs the growth of solution droplets that usually originate from deliquescence of salt aerosol particles. Fig. 6.8 displays the Köhler curves, i.e. the equilibrium vapor pressure as a function of the droplet radius, for droplets containing different amounts of salt. Note that an individual Köhler curve can be associated to every salt particle given its dry radius r d. However, the transition from the dry particle to the solution droplet, i.e. the deliquescence of the particle is not described by Köhler theory. For the aerosols starting at smaller dry sizes the peak in the saturation ratio can clearly be seen. These so-called critical saturation ratios (S c ) and the corresponding critical radii (r c ) can be obtained by differentiating the Köhler equation 170

20 Figure 6.8: Köhler curves, i.e. the equilibrium vapor pressure as a function of the droplet radius, for NaCl (solid lines) and (NH 4 ) 2 SO 4 (dashed lines) for droplets originating from salt particles with different dry radii r d. In addition, the Kelvin curve (common for all particles) is shown in brown and Raoult s law for (NH 4 ) 2 SO 4 as dot-dashed lines. 171

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