1. Review of stability regions along “Köhler curve” Scanned from Lamb & Verlinde, 2011

2. “Kappa-Köhler” (Petters and Kreidenweis, ACP, 2007) Let’s go back to the first form of the Köhler equation: We substituted x w for a w (assuming Raoult’s law applied), and immediately wrote an expression for x w for a “dissociating salt” Let’s go back to this point and find an alternative parameterization: Here the  term can account for the nature of the solute, its dissociation, plus even nonidealities of the solution that is formed. If we want to relate it strictly to the model we already had, we find where

3. These curves were computed from the basic equation with a w parameterized using , without making the approximations leading to the “classic” Köhler equation (so slopes can deviate from -3/2) We also propose this as a “grid” for plotting experimental data to determine a best-fit 

4. Slopes deviate because the assumption of very dilute solutions (volume water >> volume dry particle) breaks down 100 nm Also why classic equation blows up if you try to apply it to insoluble but wettable particle

5. Utility of  When  = 0, a w = 1  pure water Can see that  “scales” the water content at a given a w (for a bulk solution, a w = RH)  values for “real” atmospheric particles range from 0 to 1.2 Shaded area = reported range for ammonium sulfate

6. Equivalent to classic Köhler approximation Recall we found For “large enough” , then we can use similar approximations with the relationship to note that So what exerts a stronger control on critical supersaturation: size or composition?

7. This results suggests that a plot of ln s c vs. ln r d, for a chosen composition of dry particle, should yield straight lines, with slopes of -3/2 This line is for pure water (insoluble particle) This line is for a “salt” similar to NaCl dry diameter, cm

9. A caution about fitting data (lab OR field) to the model (any model…) Range of instrument s usually achieved

10. Back to ideal solutions (Raoult’s Law) and handling nonidealities Raoult’s Law: Applying this to water, as we had in previous slide: 10 (ideal behavior!) REAL solutions (solid curves) typically deviate (positively or negatively) from Raoult’s Law (dashed lines) Generally, we need lab measurements to trace out these real activity curves The modified water activity equation is  is called an “activity coefficient” Note that as x i  1, Raoult’s Law applies and as x i  0, Henry’s Law applies (linear, but different slope)

11. Representations of water activity We saw that if solution behaves ideally, a w = x w Write the mole fraction of water in terms of the number of moles of water n w and of solute, n s : If the solute dissociates (many atmospheric aerosol components are salts, so this applies), then the number of moles of the solute in solution INCREASES. If there are n s total moles of undissociated solute, and each salt is composed of ions, then However, many salts do not fully dissociate –or, put more correctly, the ions in solution do not act as if there were completely independent of one another. The molal osmotic coefficient  is used to make corrections for nonideality (more typical for ionic species, rather than  ) 11 m is the “molality” (moles of A per kg water)

12. Representations of water activity (cont’d) Write the defining equation for  as follows, and then do a series expansion: Since we are now in a position to compare this expression to the definition of the van’t Hoff factor, i: From which we notice that Expand on definition of n s (similarly for n w ): so 12 Typically used to describe solutions in the Köhler equation V s is the volume of the solute (dry particle) This is the definition of , the hygroscopicity parameter (applies at any RH that supports a solution)

13. The  parameter If we ignore the Kelvin effect for a moment and assume equilibrium, such that a w =RH, Convert into wet and dry diameters: Suppose the particle is composed of several solutes, each with its own  i. What is the total water content? The “ZSR assumption” says that each solute brings its own water with it. 13 Think about: what does a small  mean? What about  ~O(1)? “volume additivity” has been applied volume fraction in the dry particle

14. Solubility ≠ Hygroscopicity Solubility: Think about putting salt into pure water in a beaker. Is there a limit as to how much salt we can dissolve? 14 Salt in A saturated solution The composition of the saturated solution is associated with a particular a w of that solution. Under the criterion of equilibrium, RH = a w So a w,sat = RH D, the relative humidity of deliquescence RH  DRH The amount of water here is determined by the composition of the saturated solution See Table, previous slide Aside: if a w,sat = x w (ideal), then a w,sat = 1-x s,sat (mole fraction salt) or rather a w,sat = 1- x s,sat

15. How much water does this correspond to? 15 Kreidenweis et al., ERL, 2008

16. Note that assuming  =constant is not accurate below ~90% RH 16

17. Metastable solutions It is often observed that a dry particle deliquesces at the expected DRH; but if the same solution particle is dried out, the wet particle can persist to RH well below the DRH. What’s going on? Similar to the situation for gas-to-particle conversion, there is a barrier to nucleation. Interestingly, aerosol lab studies often find a generally reproducible “efflorescence” RH at which the particle finally completely dries out (e.g., for ammonium sulfate ERH ~ 40%) Argue that most atmospheric particles have some water on them (water persists to low RH) 17 This corresponds to the formation of a metastable solution The thermodynamics for the metastable solutions for many common salts have been measured and documented so that e.g. AIM can predict water contents even in this regime

18. “diameter hygroscopic growth factors” 18 GF (RH) = diameter at RH / dry diameter

19. 19 Example experiments on inorganic aerosol: potassium chloride (KCl) Deliquescence (formation of saturated solution) Parameterization: Relative Humidity, RH (%) humidified diameter dry diameter (from biomass burning)

20. Comparative water contents (relationship to “acidity”) 20 Scanned from Lamb & Verlinde, 2011

21. Back to the Kelvin equation and water contents Definitions: The criterion for equilibrium between the particle (that consists of an aqueous solution) and the environment is 21 We started with this equation to get to the “Köhler curves”. How are they different?

22. Particle growth continuum 22 Notice the scale Notice how deliquescence differs by size Scanned from Lamb & Verlinde, 2011 Are these lines straight?

23. Particle growth continuum 23 Scanned from Lamb & Verlinde, 2011

24. Measuring “CCN” A longstanding measurement technique is the thermal diffusion chamber (can also generate supersaturations by expansion, but harder to control) A supersaturation is produced in a chamber allowing small droplets to form that are counted optically. Assuming a one-to-one correspondence between droplets and CCN, a measure of the CCN concentration as a function of supersaturation can be obtained. To understand measurement principle, consider a static system with plates held at different temperatures. Heat transfer occurs by diffusion through the medium (air) H z T2T2 T1T1 T(z) e v (z) e s (z) Wetted plate The temperature at any point in the chamber is given by  2 T/  2 z = 0 This is the diffusion equation for steady state conditions, and no sources, and assuming horizontal homogeneity. Boundary conditions are the following T = T 1 at z=0 and T = T 2 at z=H. Imagine ∞ plates in all directions, including out of and into page

25. Measuring “CCN” A longstanding measurement technique is the thermal diffusion chamber (can also generate supersaturations by expansion, but harder to control) A supersaturation is produced in a chamber allowing small droplets to form that are counted optically. Assuming a one-to-one correspondence between droplets and CCN, a measure of the CCN concentration as a function of supersaturation can be obtained. To understand measurement principle, consider a static system with wetted plates held at different temperatures (moist air between): The top and bottom surfaces are kept moist at temperatures T 1 and T 2 (warmer T on top, T 2 > T 1 ). Heat transfer occurs by conduction, and vapor transfer occurs by diffusion. Remember Fick’s Law: these processes establish a linear T or concentration profile (approximately, and in the absence of flows or other disturbances). Imagine ∞ plates in all directions, including out of and into page

26. Linear profiles across chamber Reminder: Fick’s Law is  2 T/  2 z = 0 [written here for T, also applies to concentration] This is the diffusion equation for steady state conditions, and no sources in the volume, and assuming horizontal homogeneity. When solved, get T(z) = c 1 z + c 2 (the classic linear T profile) Need two boundary conditions, e.g., T = T 1 at z=0 and T = T 2 at z=H. After application of the boundary conditions, we get the expected linear profile, T(z) = z (T 2 -T 1 )/H + T 1 The vapor density gradient is (r v (T 2 ) - r v (T 1 ))/H, where r v is the saturation vapor density, and both gradients are constant with vertical position. In steady state, heat and vapor flow from the warm upper surface to the cooler lower surface, and the profiles of temperature and vapor density are linear. So there is a (steady-state) flow of water mass (molecules) across the space. Note this means that we have to keep wetting the warm wall, since vapor keeps flowing from it to the cool wall…. 26

27. Computing supersaturation It is often stated that the water vapor pressure (or partial pressure) profile across such a chamber is linear. At first glance this doesn’t seem correct, but it is approximately true: So now we can compare the actual vapor pressure through the chamber, set up by the diffusion process, with the calculated saturation vapor pressure, which is only a function of temperature. We know the partial pressure of water vapor is linear with height. What do we expect for the saturation vapor pressure? Recall from Clausius Clapeyron, with p in mbar and T in K: 27

28. Saturation vapor pressure as f(T) Plotting this and putting into linear-linear space, we get the relationship below. How would you find the saturation ratio? 28

29. Pruppacher and Klett, 1978 Continental air masses are richer in CCN than maritime air masses. Concentrations of CCN increase with supersaturation. Some ocean measurements are influenced by continents. Remote ocean regions have the lowest CCN concentrations. At 1 % SS, N is about 1000 cm -3 for continental air masses. At 1% SS, N is about 100 cm -3 for maritime air masses. Relationship between CCN and supersaturation S can be expressed as a power law of the form N ccn = cS k typical units are cm -3, where S is supersaturation in percent. Continental air masses, c = 600 cm -3, k = 0.5 Maritime air masses, c = 100 cm -3, k = 0.7 Reflects “classic” picture – before recent development of commercial instrument

30. Andrea and Rosenfeld, Earth Science Reviews, 89 (2008), 13-41.