Microphysical Processes in the UTLS

1. Microphysical Processes in the UTLS KEY 11

2. Recommended reading • Pruppacher and Klett, Microphysics of clouds and precipitation. Contains almost everything. • Fletcher, Physics of Rainclouds (my favourite, albeit old). • Young, Microphysical Processes in Clouds. • Atkins, Physical Chemistry.

3. Contents • Classical nucleation theory (basics) • Koop’s theory of water activity controlled homogeneous freezing of aqueous solution droplets • Some issues with Koop’s theory • Heterogeneous nucleation • Ice supersaturation within clouds • Volume vs. surface dominated homogeneous nucleation

4. Some outstanding problems(see: Cantrell and Heymsfield, BAMS, June 2005) • Homogeneous nucleation • what role do collective fluctuations in water play? • is freezing only a function of the water activity? • what is the structure of the ice embryo and where does it form? • Heterogeneous nucleation • what are the most important properties of the heterogeneous IN? • what are the mechanisms underlying contact and evaporation nucleation? • what role do organic compounds play in ice nucleation?

5. G(T,RH) r* Classical nucleation theory G(r) =  (4/3) r3 nLkT ln(e/e*) + 4r2

6. Classical nucleation theory Number of critical nuclei: Boltzmann distribution N(r*)=N0 exp(G(r*)/kT) G(r*): energy required to form a critical nucleus: G(r*) = 163/3(nLkT ln(e/e*)2 = (4/3) r*2 e*, e: saturation vapour pressure, actual vapour pressure : surface tension or interfacial energy between droplet and vapour nL: number concentration of water molecules in the liquid r*: radius of a critical germ Note the strong dependence on surface tension and temperature.

7. Classical nucleation theory, cont’d Nucleation rate: rate at which critical germs are impinged by single molecules (or larger clusters) to form supercritical clusters. J = B N0exp(G(r*)/kT), where BN0 is of the order 1025 cm-3sec-1. An accurate value of B is not really required since the process is controlled totally by the exponential function. Note the even stronger dependence of J on T!

8. Classical theory of homogeneous freezing Similarly as before: G(r*) = 16SL3/3[nSkT ln(1/awi)]2 = 16SL3/3(S T)2 with geometrical factor , entropy of fusion per unit volume of ice S supercooling of the liquid T awi=e*liq/e*ice. Here B=(kt/h) exp(-g/kT) with activation energy g for self-diffusion, hence J  (nLkT/h) exp(-g/kT) exp (G(r*)/kT)

9. Supercooling and freezing of pure water • Water can be cooled below its equilibrium melting point Tm. • Supercooled water is in a metastable state. • The maximum possible supercooling (Tf) can be achieved when the water is free of any solid particles that can catalyse ice germ formation. • At about Tf freezing happens as a kinetic (i.e. stochastic) nucleation process, homogeneous nucleation. • Heterogeneous nucleation occurs at T>Tf, actual temperature depends on properties of the solid particles. • Tf is a genuine property of the liquid water alone (not classically). • For pure water, arranged in µ-sized droplets, Tf is about 235 K. • When supercooled water is in equilibrium with its vapour, the vapour must have 100 % RH (wrt liquid water). • Solution droplets have both lower Tm and lower Tf than pure water.

10. Concept of homogeneous nucleation in the UTLS • pure water cannot exist at T<-38°C (supercooling limit) • ice formation via homogeneous freezing of solution droplets • foreign molecules (e.g. H2SO4) in the droplets impede formation of ice lattice • ... until droplets are grown to sufficient size in supersaturated air (rarefies the foreign molecules) • hence solution droplets freeze at RHi>140% (this threshold increasing with decreasing T) • freezing threshold independent of chemical composition of the droplets • homogeneous nucleation is driven by thermodynamics • if homogeneous nucleation is the prevalent pathway to cirrus, then high ice-supersaturation must exist in the clear and cloudy atmosphere

11. Solutions – melting and freezing points Rasmussen suggested a linear relation between melting point depression and supercooling required for homogeneous nucleation of aqueous solution droplets: Tf =  Tm Tf = Tf0–Tf Tf0 is the supercooling limit of pure water (235 K), Tm = Tm0–Tm Tm0 = 273.15 K. The constant  is independent of concentration, but depends on the chemical composition of the solute;  is an empirical constant and cannot be derived from first principles.

12. Classical treatment of solution freezing Jsolution(T) = Jpure water(T+ Tf)

13. Water activity-based nucleation theory (1) Koop, 2004 When melting and freezing temperatures of water and various aqueous solutions are plotted vs. water activity, the data collapse on two single curves, with little scatter in the case of the freezing temperature. This implies that homogeneous freezing is independent of the chemical nature of the solute.

14. aw Water activity-based nucleation theory (2) water activity aw: saturation vapour pressure over solution saturation vapour pressure over pure water the vapour pressure over the solution equals that of pure ice at the melting temperature Tm: e*ice (Tm) = e*liq (Tm) aw or awi (Tm) = aw Tf (aw) = Tm (aw -  aw)  aw independent of chemistry.  aw = 1 - awi (Tsc, max) = 0.305 The locus of the Tf curve is probably a determined by the perturbation to the hydrogen bonding network induced by the foreign molecules. (Koop 2004)

15. Water activity-based nucleation theory (3) saturation vapour pressure over water saturation vapour pressure over ice e* liq(Tm) / e* ice (Tm) (upper inverse aw scale) melting curve (lower aw scale): e*ice (Tm) / e*liq (Tm) = awi critical supersaturation for homogeneous freezing (upper inverse aw scale), red curve  blue curve: e* liq(Tm) / e* ice (Tm)  aw (Tf) freezing curve

16. Water activity-based nucleation theory (4) water saturation critical supersaturation fit: Si,crit = 2.352 - 3.88310-3 T J = 5.5 109 cm-3 s-1 e-folding freezing time 43 s for 1 µ droplets, or 1 s for 3.5 µ droplets Threshold supersaturation for homogeneous nucleation increases with decreasing temperature

17. Why does water activity control hom. freezing ? • Solutes affect the equilibrium and non-equilibrium properties of water substance. • Ice nucleation is affected by the solute molecules, • increasing solute concentration  increasing supercooling necessary for freezing. • Peculiar properties of supercooled water • interactions between water molecules via hygrogen bonds. • nearly tetrahedal arrangement of the two H atoms and the two free electron pairs around the central O atom • preference of tetrahedral co-ordination in the local water structure. • Mechanical pressure and foreign (solute) molecules change the preferred interatomic distances, hence the water structure.

18. How can the state of water’s hydrogen bonding network define the locus of the Tf curve? There are several theories: • Stability limit theory (Rasmussen and coworkers) Proximity of the freezing curve to a postulated stability limit bounding a region where isothermal compressibility is positive. • The singularity-free scenario (Archer and Carter) • Existence of a second critical point (Baker and Baker)

19. Theory of the 2nd critical point • initiation of freezing in pure water • liquid compressibility and density fluctuations reach maxima. • Temperature of the onset of freezing is an equilibrium property of the liquid phase alone. (Remember strong influence of surface tensions in classical theory. Not so here!) • Analytic model of liquid water: thermodynamic response functions have extrema at atmospheric pressure and 235K. • predominance of weak H-bonds at higher temperatures • predominance of strong H-bonds at lower temperatures • locus of the extrema is a region of a significant change in the character of the H-bonding network • loci of the compressibility maxima and the freezing curve are nearly the same at atmospheric pressures.

20. Heuristic argument and However….. As T approaches Tf density fluctuations rise. So the probability rises to find in the liquid regions where the density approaches that of ice. However, as a function of T at atmospheric pressure the extrema of compressibility etc. are much weaker then the sharpness of the sudden increase of the freezing rate. This makes this explanation somewhat unconvincing. See Baker and Baker, GRL, 2004

21. test of Koop‘s theory in the AIDA cloud chamber Good agreement between measurements and model results show that Koop’s parameterisation is able to predict correctly homogeneous nucleation of H2SO4/H2O solutions in the AIDA chamber. Non-equilibrium effects lead to slightly higher critical supersaturations as in Koop’s equilibrium theory. Haag et al., ACP, 2003

22. Some issues with Koop’s theory • derivation of Tf(aw) needs assumption that aw does not depend on temperature. This is indeed often the case above Tm where the water activity can easily be measured. • Below Tm, aw must often be determined using models or extrapolations. • For sulphuric acid, aw is nearly T-independent. • However, there are exceptions, e.g. ammonium nitrate NH4NO3. • aw(NH4NO3) increases with decreasing T • decreasing interaction of NH4NO3 with H2O at lower T • more and more ion-ion recombination (NH4+ with NO3), which makes it “invisible” for the water molecules. • decreasing solubility of ammonium nitrate in water upon cooling.

23. issues with Koop’s theory: ammonium sulfate behaves differently from Paul DeMott

24. measurements of very large supersaturation, influence of organics? no organics with organics Jensen et al., ACP, 2005, report on measurements of very large supersaturations in very cold air, Si being much larger than Si,crit De Mott et al., PNAS, 2003

25. 200 K 215 K 230 K Kärcher & Koop, 2005, show that organic material within the solution droplets is able to impede nucleationsuch that the peak supersaturation can be much higher than Si,crit

26. possible impedence of freezing by organic materials or surfactants Kärcher & Koop, 2005 • At a certain temperature it needs a certain activity for freezing (big dots). • Different solutions reach that activity at different solute mass fractions (W). • Solutions containing organics generally have higher solute mass fractions than inorganic solutions when the critical activity is reached. • This implies • less water, • smaller particle volume (1/W), • smaller freezing nucleation rates.

27. freezing temperature and nucleation rates • Koop’s theory is able to predict freezing temperatures or critical supersaturation for homogeneous freezing. • it is not able to predict freezing rates. • Freezing rates are parameterised in Koop’s paper as a function of awawi. • In the classical theory it is relatively straightforward to envisage • critical germ • attack frequency by single molecules •  nucleation rate. • The notion of an ice germ does not exist in Koop’s theory. • Hence difficult to see how a nucleation rate could be derived within the framework of this theory.

28. heterogeneous nucleation Classical framework: Energy for germ formation (contact angle ) G(het) = G(hom) × f(cos ) with 0  f(cos )  1  G(het)  G(hom), i.e. lower critical supersaturation or higher critical temperatures (less supercooling) for heterogeneous nucleation.

29. heterogeneous nucleation Water activity based framework: Zuberi at al. 2002: it may be possible to compute freezing temperatures for solution droplets with insoluble inclusions in a way analogous to the description of homogeneous nucleation by Koop et al. 2000. However, the scatter in the measured freezing temperature in the aw-T diagram is large and the fit is not perfect. It could be that such an analogy is indeed there, but if there is not a single value of aw (a single value of maximum supercooling of pure water drops with insoluble inclusions) such an analogy does not help much.

30. Measurements of het. freezing in the AIDA chamber The AIDA chamber at IMK in Karlsruhe is a large (84 m3) cloud chamber. Freezing is initiated by quasi- adiabatic expansion. Ice crystals appear and start to grow as soon as the critical supersaturation characteristic for the IN is reached. Different species have different thresholds. See Stefanie Schlicht’s poster!

31. observations of nucleation thresholds in data of RHi ambient RHi in cloud RHi Haag and Kärcher, 2003 onset of homogeneous freezing onset of heterogeneous freezing

32. Ice supersaturation within clouds The usual thinking is that after a (short) while the relative humidity in a cloud should approach saturation. However, this while, the so-called relaxation time, can last very long, depending on temperature and crystal number concentration.  g= [(4/3) N D(T,p)]-1 When the updraught goes on after cloud formation, saturation is not reached because of the ongoing decrease of the saturation pressure. Instead a residual supersaturation of a few percent will be the stable situation. sasympt.= g/(u g) with updraft time scale u= (Rv cp T2) / (Lgw), provided u > g

33. Sometimes the relaxation time is longer than other relevant time scales within a cloud, e.g. the sedimentation time scale. Then saturation will never be reached within a cloud. Altitude (m) Ni = 5L-1, w = 4.5 cm/s, RHihet = 130 % Time (min)

34. Ice supersaturation within clouds – examples Ovarlez et al. (INCA data) Comstock et al. (ARM data)

35. All CRYSTAL-FACE RHi within clouds: Physical-chemical effects may also cause persistent supersaturation within clouds, e.g. Delta-ice or cubic ice. Mean RHi binned by T: open circles. 13 July 2002 contrail: triangles. 19 July 2002 contrail: diamonds. (taken from Figure 1 of: R. S. Gao et al., Science, 2004) From R. Herman, 2004

36. Where does homogeneous nucleation occur?Drop Volume or Surface? Traditionally the homogeneous nucleation process is described as a process that occurs somewhere in the bulk of the droplet that then freezes completely. • Freezing rate is then proportional to the droplet volume • Freezing of an ensemble obeys: P{unfrozen at time t} = exp(-JVt) with [J] = cm-3s-1 Djikaev et al. (JPCA, 2002) and Tabazadeh et al. (PNAS, 2002) have found indications and given arguments that homogeneous nucleation (germ formation) should instead proceed close to the droplet surface.

37. wetting criterion Condition: at least one facet of the crystal is only partly wetted by liquid water (could be valid for the ice-water system). vs vl < ls surface energies extrapolated to T= 40°C vs = 102 to 111 mJ/m2 vl  87 mJ/m2 ls = 15 to 25 mJ/m2 Hence vs vl is approximately in the range 15 to 26 mJ/m2. The surface energies are only measured for systems with macroscopic dimensions, not for the small clusters containing only some tens of molecules.

38. If homogeneous nucleation is a surface process, freezing experiments with droplets in oil-emulsions can be affected by the oil. Is this also relevant for droplets in an atmospheric environment? Tabazadeh et al., PNAS, 2002

39. Duft and Leisner (ACP, 2004): • Test of the surface nucleation hypothesis using an electrodynamic • droplet levitation apparatus. • droplets of 19 and 49 µm radius have the same (volume-) nucleation rates. • At least for such large droplets freezing seems to occur preferentially in the bulk. • Relevant for freezing of droplets in Cb or fog. • Experiments do not exclude that in sub-micron droplets ice germs form preferentially at the surface.