Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Homogeneous and heterogeneous nucleation IN vs. soluble aerosols IN Cziczo et al., 2004 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 1 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Energy of i-mer formation at 0 ◦ C Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 2 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Energy barrier of hom. freezing vs. hom. nucleation Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 3 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei Lowest observed supercooling IN requirements Theory stochastic vs. singular (P&K, Fig.7-7) Figure : Temp. at which 99.99% of a uniform-sized drop population freezes at R T0 cooling rates of (1) 1 K min−1 and (2) 1 K s−1 : T99.99 J(T )dT = 9.21γc /Vd Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 4 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Variation of the homogeneous nucleation rate and activation energy for ice in supercooled water with temperature Temperature (◦ C) -29 -32 -35 -38 -41 -44 J (cm−3 s−1 ) 46 × 10−11 65 × 10−3 20 × 104 50 × 108 20 × 1012 20 × 1017 Activation energy (kcal mole−1 ) 10 9.6 8.2 7.7 6.9 4.5 Pruppacher and Klett, 1997, Tables 7.2&7.3 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 5 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Surface or volume nucleation rate? I Assume that the total nucleation rate JT is the sum of a volume JV and a surface JS nucleation rate: JT = JV Vt + JS St I (1) Here Vt and St = total volume and total surface area of all droplets in a unit volume of air: v kT −∆Gact JV = NL exp (2) h RT s r kT −∆Gact JS = JV = NS exp (3) 3 h RT where r = droplet radius, NL , NS = number of water molecules in solution and at the surface (following Tabazadeh et al., 2002). Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 6 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Surface vs. volume nucleation rate Tabazadeh et al., PNAS, 2002 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 7 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Surface vs. volume nucleation rate Figure : Total nucleation rate for one droplet as a function of its radius Kuhn et al., ACP, 2011 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 8 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Nucleation of drops on water-insoluble CCN I Consider a wettable, but completely water-insoluble AP. Assume that the germ nucleated in supersaturated vapor on a water-insoluble, partially wettable surface forms a spherical cap. I The surface nucleation rate for a germ growth by vapor deposition is given as: −∆FS JS = J0 exp (4) kT Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 9 / 30 Hom. nucl. I Volume vs. sfc. nucl. where Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular 4 3 J0 = n · A · Z πrIN 3 (5) where A (s−1 ) = molecule flux to the ice germ, Z = Zeldovitch factor, n (m−2 ) = number density of water molecules in contact with the ice germ, rIN = radius of the ice nucleus I Now modify ∆F for heterogeneous nucleation on a surface (∆FS ) with m = cos Θ = contact angle: ∆FS = ∆Ff (mw ,v ) = ∆F (2 + mw ,v )(1 − mw ,v )2 /4 (6) I Complete wettable substrate: m = 1, f = 0 and there is no energy barrier to nucleation I Completely non-wettable surface: m = −1, f = 1 and ∆FS = ∆F , i.e. energy of germ formation for homogeneous nucleation Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 10 / 30 Hom. nucl. I Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular In general: 0 ≤ f (m) ≤ 1, confirming that the presence of a foreign substance serves to lower the free energy barrier to nucleation Figure : Critical supersaturation for water nucleation (JS = 1 cm−2 s−1 ) on a planar insoluble substrate as a function of contact angle based on eqs. (4) & (6) Fig. 9.11 Pruppacher&Klett, 1997 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 11 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular IN concentration as a function of T I IN concentration increases exponentially with decreasing temperature: NIN = A exp(−bT ) (7) where A ∼ 10−5 l−1 and b ∼ 0.6 C−1 I What is missing here? Fig. 9.17 Pruppacher&Klett, 1997 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 12 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Different freezing modes Dashed line: hom. freezing of solution droplets (Koop et al., 2000). Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Hoose and M¨ ohler, 2012 Nov 11, 2014 13 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular organics Discussion Paper Onset T and RH for different freezing modes 1 | Discussion Paper black carbon Discussion Paper dust | sulfate bioaerosols | Discus updated from Hoose and M¨ ohler, 2012 Fig. 2. (IACETH) Overview Ulrike Lohmann of iceHomogeneous nucleation onset temperatures and saturation ratios. Data sources and heterogeneous nucleation Nov 11, 2014 are 14 / 30 C Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Preactivation of IN in deposition mode Fig. 9.23 Pruppacher&Klett, 1997 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 15 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Deactivation of IN? I There are some suggestions that gases, such as SO2 , NH3 , NO2 , hinder an aerosol’s IN capability because they may occupy the active sites or modify the IN surface I IN exposed to small particles with low ice nucleation potential (e.g., organics, sulfates), which are common especially near urban areas could deactivate the IN I If it is true that contact freezing is the most efficient way of ice crystal formation, then deactivation of a contact IN may convert the IN into an immersion nuclei, where it could still be activated but at lower temperatures. Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 16 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Requirements of IN - insolubility&size I Phase: IN need to be solid/crystalline particles as soluble material cannot provide a rigid substrate I Size: Larger particles are more likely to act as IN, because IN should have a size ≥ that of the critical ice embryo. IN > 0.1µm are necessary for having active sites (Marcolli et al., 2007) Fig. 9.28 Pruppacher&Klett, 1997 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 17 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Requirements of IN - chemistry&crystallography I I Chemical bonds: free hydrogen bonds on surface (like ice) are desired. Hence many organics are excellent IN Crystallography (importance not clear): I I I Atomic arrangement (in at least one plane) that mimics ice Ability to deform in order to match ice structure AgI and kaolinite (P&K Fig. 9-30 below) satisfy this requirement Fig. 9.30 P&K Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 18 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Requirements of IN - active sites I Need local sites on the aerosol that can capture or adsorb water molecules - helps to form critical nucleating ice embryo I The number of active sites increases with decreasing temperature I Cracks, impurities and cavities appear to act as active sites I Active sites explain memory of previous nucleation, because a thin water/ice layer may remain Figure : Fletcher, JAS, 1974 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 19 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Classical nucleation theory (CNT) for heterogeneous ice nucleation I CNT assumes that nucleation is a stochastic process. However time is less important than temperature I The nucleation energy barrier for deposition nucleation and immersion freezing can be obtained analogous to homogeneous droplet nucleation, resp. homogeneous freezing I There is currently no theory for contact freezing I CNT has to specify the contact angle I It is used to specify the lowering of the nucleation energy barrier within CNT I It is based on parameterizations that fit laboratory data Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 20 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Nucleation rates according to CNT I I I Heterogeneous nucleation rate Jhet (m−2 s−1 ): kT ∆Fdiff (T ) ∆G (T )fhet (α) Jhet (T , α) = n exp − exp − (8) h kT kT where k = Boltzmann constant, h = Planck’s constant, n (m−2 ) = number density of water molecules at the surface of the IN Compatibility factor fhet (α) (cf. eq.6): 1 fhet (α) = (2 + cos α)(1 − cos α)2 (9) 4 where α = contact angle Change in Gibbs free energy for immersion freezing: 2 3 16π vi (T )σi,w (T ) ∆G (T ) = 3 [kT ln Si (T )]2 (10) where vi (T )=volume of a water molecule in the ice embryo Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 21 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Immersion/deposition nucleation rates using CNT I Diffusion energy barrier of a water molecule to cross the water/ice embryo interface: ∆Fdiff (T ) = ∂Dl (T ) 2 kT ∂T (11) where Dl (T ) = diffusivity of water. I Deposition nucleation rate Jhet (m−2 s−1 ): ∆Fdiff (T , Si ) ∆G (T )fhet (α) Jhet (T , Si , α) = A·Z ·n·exp − exp − kT kT (12) where A (s−1 ) = molecule flux to the ice germ, Z = Zeldovitch factor, ∆Fdiff = activation free-energy barrier for molecules to diffuse into the ice germ Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 22 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Deposition nucleation rates using CNT I Change in Gibbs free energy for deposition nucleation: 2 3 16π vi (T )σi,v ∆G (T , Si ) = 3 [kT ln Si (T )]2 I (13) Activation energy barrier ∆F has two parts: diffusion energy barrier for adsorbed water molecules to diffuse on the surface of the substrate to the ice embryo (∆Fdiff ,sub ) and the gain in energy by desorption of molecules from the ice embryo (∆Fdes,ice ): ∆F = ∆Fdiff ,sub − 2∆Fdes,sub I (14) Contact angles for deposition nucleation (5-40◦ ; Hoose et al., 2010) are smaller than for immersion freezing (70-105◦ ; Pinti et al., 2012; Welti et al., 2012) Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 23 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Heterogeneous nucleation homogeneous freezing 90° contact angle 45° contact angle T = 253 K -18 1.0x10 -19 8.0x10 critical embryo ice germ ∆G [J] -19 nucleation ice embryo 6.0x10 ∆G* -19 4.0x10 -19 2.0x10 0.0 0.0 -9 1.0x10 -9 2.0x10 rc -9 3.0x10 -9 4.0x10 -9 5.0x10 Ice embryo radius [m] Energy barrier to nucleation of ice in water at -20◦ C for hom. and two cases of het. freezing. Dashed vertical: critical radius rcrit (L¨ u¨ ond, 2009). Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 24 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei Empirical nucleation rates IN requirements Theory stochastic vs. singular (Vali, JAS, 1994) Nucleation rates can be determined from experiments such as this: J = J0 exp(−aT ) Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation (15) Nov 11, 2014 25 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Immersion freezing rates from CNT Hoose et al., JAS, 2010 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 26 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei Stochastic freezing hypothesis I I I IN requirements Theory stochastic vs. singular (Bigg, 1953) “Freezing results from a random formation of a critical size embryo, and the presence of foreign particles increases the probability of the nucleation without disturbing its stochastic nature” The theory for immersion freezing is then analogous to homogeneous freezing (see eq. (19) from last lecture): 1 − dNu = Vd J(T )dt (16) Nu where Nu is the number of unfrozen drops. Using J(T ) ∼ aBi exp(aTs ) (17) where Ts = T0 − T , T0 = 0◦ C , a = 1◦ C −1 and Bi = ice nucleation efficiencies in the immersion freezing mode of an insoluble particle per unit volume of liquid (Diehl and Wurzler, JAS, 2004) to obtain an expression for immersion freezing. Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 27 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Singular (deterministic) hypothesis Theory stochastic vs. singular (Langham & Mason, 1958) “Every particle contained inside a drop has one characteristic temperature at which freezing will be initiated in the drop” Figure: Random freezing temperatures observed for one ice-forming nucleus in 3 different physical conditions (Shaw et al., JPC, 2005) Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 28 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Stochastic vs. singular freezing Theory stochastic vs. singular (P&K, Fig. 9.43) Figure : Fraction of drops from distilled water, frozen per unit time interval as a function of time. Vd = 0.01 cm−3 , t = 0 at 0◦ C. At -20◦ C, the temperature was held constant for 15 minutes. Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 29 / 30 Hom. nucl. Volume vs. sfc. nucl. Het. nucl. theory Ice Nuclei IN requirements Theory stochastic vs. singular Comparison of freezing modes Durant & Shaw, GRL, 2005 Ulrike Lohmann (IACETH) Homogeneous and heterogeneous nucleation Nov 11, 2014 30 / 30
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