Mixed aerosol particles as effective ice nucleating systems

Atmospheric Research, 31 (1994) 199-233

199

Elsevier Science B.V., Amsterdam

Mixed aerosol particles as effective ice nucleating systems Vladimir Ye. Smorodin Physics Dep., Moscow State University, Moscow, 119899, Russia (Received December 28, 1992; revised and accepted June 2 l, 1993)

ABSTRACT The classical nucleation theory has been developed as an approach describing ice nucleation processes on mixed aerosol particles in supercooled clouds or fogs. In part, based on the Fletcher's nucleation theory, results of Fukuta and Schaller ( 1981 ) and a solution theory, a condensation-freezing mechanism on the weakly soluble mixed aerosols has been considered. Criteria of high crystallizing activity for natural and artificial (modified with silver iodide) mixed aerosols have been established. The theoretical calculations are well verified by cloud chamber experiments. Temperature activation spectra of aerosol ice nuclei for different nucleation processes are analyzed, and the meaning of empirical constants of the Fletcher activation formula, Na=No exp(flAT), has been clarified. Also, the ice-forming mechanism of silver iodide crystals is explained and a hypothesis concerning a crystallizing activity of ice forming bacteria is proposed. RI~SUMI~ On d~veloppe la th6orie classique de la nucl6ation afin de d6crire les processus de nucl6ation de la glace sur des particles mixtes d'a6rosol darts des nuages ou des brouiUards suffondus. On consid~re en particulier un m6canisme de condensation-cong61ationsur des a~rosols mixtes faiblement solubles bas6 sur la th6orie de Fletcher, les r6sultats de Fukuta et Schaller ( 1981 ) et une th6orie des solutions. On 6tablit des crit~res de forte activit~ gla~og~ne pour des a~rosols mixtes naturels et artificiels (modifi6s avec de l'iodure d'argent). Les calculs th6oriques sont bien v6rifi6s par des exp&iences clans une chambre/l nuage. On analyse les temp&atures d'activation des noyaux gla~og~nes pour diff6rents processus de nucl6ation, et on discute de la signification des constantes empiriques de la formule de Fletcher, Na = No exp (flAT). On explique 6galement le m6canisme de formation de la glace sur des cristaux d'iodure d'argent, et on propose une hypoth~se sur l'activit~ de cristallisation des bact&ies glaf,og~nes.

INTRODUCTION

It is known that crystallizing aerosol particles play a fundamental role in atmospheric processes (precipitation, hail and electric phenomena). Introduction of artificial crystallization particles is an effective method, of preventing hailstorms, clearing supercooled fogs and stimulating precipitation processes. 016%8095/94/$07.00 © 1994 Elsevier Science B.V. All fights reserved.

SSDIO 169-8095 (93)E0032-T

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V.YE. SMORODIN

As usual, all atmospheric "nuclei" of condensation and crystallization are the mixed aerosol particles (MAP) having the energetically heterogeneous surface. Atmospheric aerosols have a different origin, state of coagulation, gas adsorption, photochemical surface reactions, etc. That is true of artificial pyrotechnical aerosols of the silver iodide (AgI) used for weather modification. Development of experimental and theoretical works on heterogeneous nucleation of aerosol particles of mixed chemical composition was sufficiently inspired by weather modification application. Finally, studies of MAP began to be important in ecological problems, photochemical smogs (in part, in winter), an ozone heterogeneous deactivation (a middle atmosphere ), etc. The main principle of heterogeneous nucleation theory was formulated by Gibbs, Volmer, Becker and Doering, Frenkel, Turnbull and Vonnegut, and others. A theory of heterogeneous nucleation on aerosol particles was developed by Fletcher on the basis of the classical nucleation theory. At first, Fletcher (1958) analyzed heterogeneous nucleation on water insoluble aerosol particles containing AgI with an energetically uniform surface. Later (1969), he extended these ideas to particles with "active sites" on spherical particle. Also, he (1959) studied the entropy effect on different planes of the AgI crystals and the solubility effect of AgI-KI and (AgI)2"KI aerosols on their freezing nucleation mechanism in 1968. Wholly soluble atmospheric "nuclei" of condensation were studied by Koeler (1936). Meszaros (1969) and Bonis (1969) have analyzed the thermodynamics and kinetics of condensation on the mixed aerosol particles of natural origin that contain both soluble and insoluble parts. Peculiarities of binary nucleation were first investigated by Hamill et al. ( 1982 ). Analyzing condensation-freezing nucleation, Fukuta and Schaller ( 1981 ) determined non-trivial dependence for the probability on respective nucleation rates for condensation and freezing, and time. An interrelation of adsorption, condensation nucleation, wetting and ice nucleation on mixed aerosol particles containing AgI (intended for weather modification applications) was studied in our works (Tovbin et al., 1978; Almazov et al., 1980). A series of new important facts were first established there. These are, in particular, (i) dependence of the equilibrium wetting angel 0 on the average thickness of the condensed film, (ii) thermodynamic instability of condensate on energetically heterogeneous surfaces due to local extremes of the Gibbs (or Helmholtz) free energy and (iii) activation barrier for growth (wetting) of the condensing phase on non-uniform surfaces of the ice-forming crystals like AgI, etc. Later, Gorbunov and Kakutkina ( 1982 ) studied a similar effect of double barrier ice nucleation on a large active site of spherical aerosol particle. According to experimental data (Steele and Krebs, 1967; Federer and Sheider, 1981 ) the pyrotechnically generated AgI particles may be approximated by salt particles (e.g., NH4I) having on their surface small particles ofAgI (a

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

201

spherical cap and not a "patch"). It appeared to be a problem of theoretical description and technological optimization of these systems for the weather modification applications (Pomposiello et al., 1979) Since the Fletcher nucleation theory was unable to explain such systems, it became necessary to develop new, more realistic physical models and to construct a theory for the heterogeneous nucleation of ice on crystalline nuclei with modified surfaces (MAP). This task was resolved in our works (Smorodin, 1983; Bazzaev and Smorodin, 1986; Smorodin, 1990, 199 l, etc. ) that were mainly based on ideas of Fukuta and Schaller ( 1981 ) and their generalization and development. Taking into account that the dissolving processes of mixed aerosol particles provoke a change of mechanism of the ice nucleation, we have analyzed the solubility effects and different mechanisms of ice nucleation. DOUBLE BARRIER NUCLEATION

Let us analyze the main peculiarities of wetting of the energetically nonuniform (or heterophilic) insoluble solid surfaces. A first important factor to be taken into account in this problem is the description (topology) of energetic heterogeneity of solid and film on its surface. For our goals, we distinguish between next simplest topological structures of solid heterogeneity: ( 1 ) a single lyophilic site on lyophobic substrate ("active center"), (2) lyophobic site on lyophilic substrate ("inactive center" ), ( 3 ) lyophilic surface with regularly distributed "inactive centers" and (4) lyophobic substrate with regularly "active centers". Here, the influence of gravitation is neglected in comparison with the capillary forces. Let us introduce a scaling parameter, )., linking the typical dimensions of surface heterogeneities (1), and those of the drop (R): 2 = l / R . Two typical cases can be singled out: 2 << 1 and 2>_ 1. ( 1 ) A micro-scale heterogeneity: 2 << 1. Cassie ( 1948 ) established that the effective value of the wetting parameter for the case of small energetic heterogeneities on wettability, a biphilic support is reef= ~ m l + ~2m2,where ~ and ~2 are the surface fractions occupied by the regions with the wettabilities m ~= cos0~ and m2 = cos02, respectively. (2) A macro-scale (colloidal) heterogeneity: 2 > 1--the case of large heterogeneities. The description of surface phenomena will also depend on the adopted type (model) of energetic heterogeneity. Consider the simple type of biphilic surface, when the profile of the transition zone between high- and low-energy surface regions can be approximated by a stepwise function (Smorodin, 1987 ). We consider that the simple model of condensation (or crystallization) nucleation on a spatially extended, isolated, active "center" of radius Ra on a plane with a "wetting" parameter mc >mo (too = cos0 is the characteristics of the "matrix" surrounding the center). We assume that an embryo with a fi-

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V.YE.SMORODIN

nite contact angle 0 (see Fig. 1 ) forms at the "center". Locating the origin of a cylindrical coordinate system in the middle of the "center" we can, by taking into account the symmetry, write the Gibbs free energy of the liquid (or crystal) cap formation

.'fG[h(r) ]= O'23 823 Jr- 0"1,3 ~ I'2 + O'l,2/t(R 2 -ro2) Jr" a12S12

+VAGv,

ro ~ R d

a23S23 +a~,37tR~ +a~3n(r~ - R ~ ) +am2(S,2 - m ~ ) +VdG~,

ro > R~. (1.1)

Here h (r), $23 and V are respectively height of the embryo, free surface area and volume of the embryo, ro is its external radius, $12 is the area of embryosubstrate interface and a o are the respective specific interfacial free ener#es; where 1' denotes the "center", 1 the matrix, 2 the phase surrounding the embryo, and 3 the embryo. As long as the subcritical embryo is small, the contact angle of the embryo is 0= 0c (this follows from the transversality condition for the functional ziG with a free end point to) and ro=R sin0c
~/IG={2rte23(m~-m)Rd~r°' ~r° < 0 ; (1.2) 21t(Tz3(m--mo)Rdt~ro, ~ro > 0 ; where m = cos0, 8ro = 8h/h'r(ro) is the magnitude of the displacement of the perimeter as the profile is varied, and the prime denotes the operation d/dr. From Eq. (1.2) it follows that AG has a m i n i m u m at 8ro=0 and 0~<0<0o; i.e., a displacement of the embryo perimeter in any direction leads to an in-

L

o

®

;///1

0

Fig. 1. Nucleus of a new phase (drop) at the heterogeneoussurface with a spatially extended, "active center".

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

203

crease in AG. For a fixed perimeter the embryo volume can increase if the contact angle 0 increases and the radius of curvature decreases:

V=l/3rrR~sin-30(1-m)2(2+m),

R=Rdsin-lO,

0c<0<00.

(1.3)

When 0= 0o, the m i n i m u m (and the barrier preventing the displacement of the parameter) disappears and the embryo grows freely throughout the "matrix" with a contact angle of 0o. Thus the radius of curvature of the embryo at the "center" depends non-monotonically on its volume: at first for ro < Rd, 0 = 0c (region I ) and R increases as V increases to a maximum value Rma~= Rd sin- ~0~, and for ro = Rd, 0~< 0 < 0o (region II ) it decreases to R m i n -" R d sin- t 0~ and for ro > Rd, 0 = 0o (region III ) it increases again; see Fig. 2 (where to = 3 V/ nR~, z=R/Rd). Using Eq. ( 1.1 ), we write the expression for the Gibbs free energy of the embryo formation in the dimensionless form

ziF_ - ziG / t R 2 °'23

(ziFI =OtltoE/3--2/32to,

(

to<--tolo

=~ziF2 ot2(O)-2/32to, toloto2o

1.4)

(regionI) ; (regionII) ; (regionIII) ;

where 2=Rd/R*, / ~ = m ¢ - m o > 0 , tolo=sin-30¢(1-mc)2(2+m¢), o920= sin-30o( 1 - m o ) 2 ( 2 - too), or1 = [ 2 - me( 1 + m e ) ] ( 1 - m e ) - 1/3(2 + me)-2/3,

OtE(O)=2(1-m)-m¢(1-m2),

ot3=[2-mo)(l+mo)](1-mo) -~/

3(2+mo)-2/3. Analyzing the function ZIF=ziF(to) at its extremum, we find a number of cases, which are classified by the parameter 2: ( a ) 2 < sin0~, sin0o and me < m S (or m S < mo ) ;

(b) sin0o <2 1, o r 2 < 1 a n d r o s
(1.5)

where m S = ( 1 - 2 2 ) l/2. In single-barrier processes the nucleation processes (a), (b) and (c) have a single maximum: ( ( a ) ziFT =4/32-2foo(mc), tot =4/322foo(m¢) ; * 4 / 3 2 - 2 foo(m_) * ZIF?=~( b ) /IF*2 = 4 /3 ,~-2 f o o ( m* _ ) - ( m ¢ - m _ *) , t o 2 = ~ (e) AF'~=4/32-2fo~(mo)- (me--too), to$ =4/32-2foo(mo) .

;

(1.6) Here the index i = 1, 2, 3 denotes the number of the region and to,.* is the

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V.YE.SMORODIN

aF 0--8o

I

F i

[ °

.-

..'""

.

B

[

0°

Fig. 2. Dependence on volume to of (a) radius of curvature x = R / R a (b) contact angle 0 of an embryo and (c) Gibbs energy AF: 2 = 1.5 (1), 0.95 (2), 0.750 (3), 0.585 (4). In the calculation we have used mc = 0.8 and mo= 0.2; nucleation at a homogeneous surface: 3' - m = 0.8, 2 = 0.750; 4' - m = - 0 . 2 , , ~ = 0 . 5 8 5 .

critical volume. In cases (c) and (d) there are two maxima; these are doublebarrier nucleation processes; in case (c) we have the maxima AFT and AFt, and in case (d) we have the maxima AFT and 3 F $ (the relative height of the maxima depends on ,~). The second barrier, as shown above, is related to the non-monotonic variation of the radius of curvature of the embryo. In each of the extrema, it is easy to see that the critical radius of the embryo satisfies the G i b b s - T h o m s o n formula. Figure 2 shows the different cases of nucleation at a "center" as a function of the magnitude of 2; for comparison, we show the nucleation curves for homogeneous active surface ( 3 ' ) and homogeneous passive surface (4'). In curve 4, one of the maxima (the first one) degenerates at the inflection point. In curve 3, the second m a x i m u m is an order of

MIXED AEROSOLPARTICLESASEFFECTIVEICE NUCLEATINGSYSTEMS

205

magnitude greater than the first, and it is obvious that there are cases which stop the nucleation which begins when the first energy barrier is surmounted: Some of the "centers" may turn out to be ineffective. Since the ratio of the height of the second maximum to that of the first decreases with increasing 2, we can write the condition that the second bander be lower than the first as 2>2* (2=2* corresponds to the case in which the maxima are equal). Strengthening the inequality and, for simplicity, setting 2* = l, we obtain the criterion for single-bander nucleation (e) ( - RdAGv)/2a2a > 1.

(1.7)

An analysis shows also that a minimum of the Gibbs free energy condition is satisfied concerning a variable contact angle 0 of the embryo on the "active site". So, in terms of classical thermodynamics, an additional minimization of the AG on 0 is not required. Using these data, we can thus easily estimate the minimal "driving force" AG,, (supersaturation or supercooling) necessary for the operation of an active "center" with a given radius Rd, or the minim u m radius of a "center" that is activated at a given AG,,. For example, the condensation of water at 0°C on a substrate with a "center" of radius Rd---- 10 - 8 - 10 -6 m begins at the critical supersaturation of J * = 10 -1 - 10-3: only "center" with a radius Rd = ( 1 . 0 - 0.5 )* 10-s m is activated in the heterogeneous deposition of ice in the region - 2 0 ° C to - 4 0 ° C . Clearly, the energy of nucleation at an active "center" is generally determined by the condition JG~et = Max{dF?ltR2a23},

i = l, 2, 3.

(1.8)

IfztG~et depends on ztF~, the contact angle of the critical embryo will be variable (it will depend on AGe): O*=O*(,dG,,) and dG~ will then influence the nucleation rate J ~ exp ( -dG~et/kT), much more strongly than in the case of nucleation on homogeneous surfaces. In light of the results discussed above we can explain the empirical observation that the contact angle of the embryo depends on supercooling, 0"= 0* (AG~) in experiments on the nucleation of ice on ice-forming particles. We conclude that for natural or artificial mixed aerosol nuclei with an active "center", there are restrictions on the development of the nucleation: embryos that do not satisfy condition (1.7) are not activated, despite the formation of regions of macroscopic phase on them. T H E R M O D Y N A M I C INSTABILITY D U R I N G W E T T I N G O F A E R O S O L PARTICLE

Let us investigate the evolution of a droplet of liquid on the surface of a spherical nucleus as its volume increases. We will analyze three cases: ( 1 ) the surface of the nucleus is energetically homogeneous; (2) there are two regions on the nucleus with different "philicity" and (3) the surface of the nucleus has two hydrophobic inclusions sep-

206

V.YE. SMORODIN

arated by a hydrophilic region. The first two cases are very simple, the growth of the droplet in these cases occurs monotonically without instability. The last case is the most interesting, and this is the case which we analyze in more detail.

( 1 ) Wetting of homogeneous spherical particle. Let the wettability of the surface of a spherical solid particle of radius Ro be characterized by the Young angle 0, determined by the familiar formula of Young: cos0= (al 2 - al 3)/a23, where aij are the specific free interface energies; the subscripts 1-3 refer to the solid, the air and the liquid, respectively. It is easy to show that, when neglecting the force of gravity, the free surface of the droplet will have the shape of a spherical cap. Furthermore, since the wetted surface is uniform, the contact angle of the liquid will be equal to the Young angle. With an increase in the volume of the droplet, the surface of a sphere coated with a liquid (and the angle ot with it) will increase. Let us designate the Young angle for the hydrophobic region as 0o, and cOS0o= (al,2- it1,3)/tr23 (here and below, the index 1' refers to the hydrophobic surface, keeping the index 1 for the hydrophilic surface and the symbol 0j for the Young angle). Growth of the droplet can occur without limit. (For larger volume, more precisely the droplet is not located on the surface of the particle, but rather the particle is located in the liquid. ) From geometric consideration, it is easy to see that in the limit V--,00, the angle a becomes equal to n - 0i. Thus, even in the case of a hydrophilic particle, gradual growth of the droplet does not lead to complete coverage of the solid surface by the liquid. We can assume that by "forcing" the particle to become submerged in a sufficiently large droplet, we increase the surface free energy of the system by 41tR2tr23 sin40t/ 2 > 0. (This formula is also valid for a hydrophobic surface). The latter inequality is evidence for the impossibility of such a spontaneous process. (2) Growth of a droplet in presence of a single interface between regions of different philicity. Let us assume that the hydrophobic region has the form of a spherical cap, characterized by the angle a. Let the droplet be formed in the hydrophilic section. During growth of the volume V, its contact line will move along the surface of the sphere with fixed contact angle 0;. As long as the liquid has not yet reached the interface, this phenomenon does not differ from that considered above. For some volume, the liquid fills the entire hydrophilic region. It is not difficult to show that upon further growth of volume of the droplet, its contact line will remain at the interface, and the contact angle begins to increase from its initial value 0i. If the angle ot is less than the Young angle 00, then in the limit ( V--,~ ) , the contact angle becomes equal to a and creepage of the liquid to the hydrophobic section does not occur. For c~> 00, the liquid advances toward the hydrophobic region, and within the limit, the contact angle becomes equal to 00. Thus the presence of a single hydrophobic

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

207

Fig. 3. Droplet on a particlewith two lyophobicsections:I-embryo,II-droplet. inclusion (in contrast to the case of a film on planar surface) does not lead to collapse of the cavity.

(3) Model of a particle with two hydrophobic inclusions. Let a spherical particle have two hydrophobic regions at opposite poles. Their angular dimensions will be characterized by the angle a', and a z (Fig. 3). Let us first consider a situation in which the volume of liquid condensed on the hydrophilic region is such that it completely fills the hydrophilic section, and the contact line passes along the interface. If the contact angles are symbolized as 0,, and 0L, respectively (Fig. 3 ), then 0u, 0L> 0i. Let US determine the shape of the free surface of the liquid h (r). The function of the surface free energy of the considered system has the form (Almazov et al., 1980):

(2.1)

F[h(r) ] = a23S23 +al,3S1,3 "~-0"12S12, where S Uare the surface areas of the interface between phases i, j, rl

rl

j s,,~ = 2 ~R~ (2 - cos~ ° - cos °

{~,

l (2.2)

5;13= 4rcRg - $1,3, r. =Ro s i n a i ,

rl =Ro sinai° ,

(2.3)

and where h+ and h_ are the upper and lower branches of the dependence h (r) where d h / d r - h'. Let us minimize F[ h (r) ] under the condition of constant volume of liquid

208

V.YE. SMORODIN

rl

r2

ru

rl

Ro

Ro

-fry+(r)dr+ ; ry_(r)dr]=const, ru

(2.4)

rl

where y + = y + (r) = +_ (R 02 - r 2 ) ~/ 2. Then we arrive at the Euler equation dr

( l + h ~ ) ~/2 = + # r ,

(2.5)

where/~ is the Lagrange undetermined multiplier. Let us give the boundary condition to (2.5) in the form h+ (ru) = y + (ru),

(2.6)

h_(ru)=y_(ru) .

(2.7)

The solution to Eq. (2.5), satisfying condition (2.6) and (2.7) is not difficult to find by quadrature X(r)dr

h+(r)=RocoSOtu+

[1 - X 2 ( r ) ] 1/2'

(2.8)

ru

h_(r)=Rocosoq +

i

X(r)dr [1 -X2(r) ] 1/z'

(2.9)

FI

where

X( r) + C~/r-r/ C2 ,

(2.10)

and C2 = - 2/# is the constant of integration, and r2 = (1/2)C2[1 + (1 +4C,/C2)

1/2] .

(2.11)

The constant C~ and the Lagrange multiplier p are determined from the boundary condition h_ (rE) = --Ro COSO~L,

(2.12)

and the expression for the volume (2.4). Substituting into (2.4), (2.8) and (2.9), and integrating by parts, we obtain

V=--

~f

rl

ru

r2

r2X(r) dr ; r2(X(r) dr ] [ l_X2(r) ]l/2 + [ l_X2(r) ]l/2 rl

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

+TrR~[cosa,( 1 - 1/3 cos3a,) ~LCOSO~I( 1 -- 1/3 COSa~) ]

209

(2.13)

In formulas (2.8) and (2.9) it is convenient to perform a change of variables, using the relationship sin z = X ( r ) . Then all the branches of the dependence can be described by the single expression pu

h(r) =Ro cosa, + (C2/2) "t [ ( C + sin2z)l/2-sin z]sin z dz ( C + sin2z) 1/2

(2.14)

fl(r)

where C = 4 C1/ C2, fl (r) = arc sin X(r), fl,,= fl ( r, ), and fl (r), depending on the number of the branch, can vary within the limits: - 27r+ aL < fl (r) < ~-- a,. The change of variables performed is convenient because it eliminates the singularity in the function under the integral sign and thus allows us to do the numerical integration on a computer. Let us transform formula (2.13) for the volume in the same way: Pu

V= ~ ~ C2~2 ~3

f [ ( C + s i n 2 z ) l / 2 - s i n z]3sin z dz ( C+sin2z) 1/2 J Pl

+ 7rR~ [cosa, ( 1 - 1/3 cos3au ) + cosa~ ( 1 - 1/3 cos3al ) ] ,

(2.15 )

and also the boundary condition (2.12 ):

( 1/2 ) C2

d pl

[ ( C + sin2z) l/2 _ sin z ] sin z dz ( C + sin2z) 1/2 + Ro (cosa, + cosa ~) = 0

(2.16) where AL= fl(r~ ). Equations (2.15 ) and (2.16 ) allows us, in principle, to determine the constants C~ and C2 with respect to the specified value of V, i.e., to find C~ (V) and C2 (V). After this, using (2.16 ) it is not difficult to plot the curve h (r). However, in order to avoid solving a system of two transcendental equations, C~ (V) and C2 (V) were determined in parametric form according to the following scheme. Some value of the contact angle 0u (the fitting parameter) was specified beforehand, and 0u could be varied from 0 ° to 180 °; and since flu= 0, + a,, and according to ( 2.8 ) and (2.14 ), sin flu = C~/r, - r,/ C2, we can express C~ in terms of 0, and C2: C~ = r , [ s i n ( 0 , + a , ) +ru/C2] •

(2.17)

Substituting this expression into (2.16), we obtain an equation with respect to the parameter C2. This equation was solved numerically on a computer. Thus, for any specified angle 0, in the range 0-180 °, we found the corresponding root C2 = C2 (0,). Then using formula (2.17 ), for the same value of the parameter 0, we calculated the constant C~ = C~ (0,). The C~ and

210

V.YE.SMORODIN

C2 determined were substituted in (2.15 ) and the volume V(0u) was calculated. For the values of C~ and C2 obtained, using formula (2.14) we could also calculate the shape of the free surface of the liquid h (r) corresponding to the volume V(Ou). The results of the h (r) calculation for several angles au, a~, 0, and 0L with the volume are presented in Fig. 3. In Fig. 4 we present the curves connecting the contact angles 0u and 01 with a volume. Everywhere on the graphs below we use the dimensionless units the volume co= 3 V / 4 n - R 3. For simplicity, we consider the symmetric case of hydrophobic regions which are identical in size ( a ° = a ° ). [Here and below, in the figures we plot co with respect to Ou(co), rather than co (0u). This is convenient, since in fact it is specifically the volume which is the independent variable, determining all the characteristics of the droplet]. Curves 1 and 2 (they are superimposed but for clarity, they are separated in the figure ) illustrate the dependencies Ou(co) and 0,(co) and, as expected, 0~=0L, i.e., for a ° the droplet has a symmetric shape relative to the horizontal plane (see Fig. 3 ). Note the solutions with an asymmetric profile h(r). Thus, curves 1' and 2' correspond to a droplet skewed toward the upper pole (0,, > 0L). and the curves 1 and 2 superimposed on them are skewed as much toward the lower pole (0u < 0L). AS shown by an investigation of the stability, these solutions do not correspond to the minimum in the free energy, but rather to extremes of the "saddle point" and consequently should not be considered. (4) Criteria for collapse of the cavity. In the above treatment we obtained formal solutions to the Euler equation (2.5) with fixed boundaries. Just as was done in Almazov et al. ( 1980 ) by analyzing the first variation o f F [ h (r) ],

eu. , ~ ¢t.

g °

~o

o

¢J

I

0

&

l

Fig. 4. Dependences of the angles 0u, 01 on the droplet volume (co= 3 V/4;rR3o ).

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

211

we can show that the contact line of the liquid coincides with the interface between the sections of different philicity only when the contact angle has a value intermediate between the Young angles 0i and 0o, i.e., when

Os
(2.18)

When inequality (2.18 ) is satisfied, small shifts of the contact line in a specific direction relative to the interface upon variation of the shape of the droplet will "run into" a potential barrier. The left-hand inequality in (2.18 ) in this problem will not be of interest to us. It is not satisfied except for small droplet volumes, when its contact line no longer reaches the interface. During gradual increase in volume, if one of the contact angles (for example 0,) attains the value of the Young angle 0o for the hydrophobic section, the potential barrier disappears. At this instant we use COofor the volume of the liquid. With further increment in the volume Jco, the contact line can be displaced some distance and occupy a new equilibrium position corresponding to the volume CO+JCO. (As we see below, such a stable equilibrium position does not always exist. ) The contact condition h ( r, ) = y ( ru ) for CO> CO*is replaced by the transversality condition h'~(r, ) =tan0o or O. =0o

(2.19)

The program for calculating the constants C1 and C2 for boundary condition (2.19 ) is not much different from what has been discussed above. Thus, starting from COo,the contact angle 0u or 0L (for example, 0,.) which first attained the value 0o was fixed and taken to be equal to 0o. The angle ~,, describing the position of the contact line (~ < ~o ), was considered as the parameter of the problem. The ~u was gradually decreased from the value ~o down to zero. In formula (2.17 ), instead of Ouwe substituted cx°, and then CI (au) and C2 (ol,) were determined by the procedure used previously and the corresponding volume CO(a~) was determined. We note that in the symmetric case of identical hydrophobic section, both 0~ and 0L simultaneously reach the values 0o. Nevertheless, in the procedure described above, only one contact line was shifted. This is justified by the fact that in reality, absolutely identical sections do not exist and therefore realistically one of the angles 0u and 0o first reaches the value 0,. As a result of analysis, we may conclude that for some critical value of volume, CO*,equilibrium states of the liquid with a cavity do not exist. If for CO* we use dCO,this leads to the perimeter of the contact line and the liquid fills in the upper hydrophobic section and closes up over it. Collapse of the cavity occurs. For smaller sizes of the hydrophobic regions, the droplet will be even less stable. Let us summarize the results obtained above. For gradual increase in the volume of the liquid, the contact line of the liquid at first coincides with the

212

V.YE. SMORODIN

interface line between sections of different philicity. When one of the contact angles reaches the Young angle, for the hydrophobic region the process follows one of two routes, depending on the sizes of the hydrophobic sections (and 0o). When they are relatively small in size, at this moment collapse of the cavity occurs immediately (i.e., au*-au,o t o = t o . ) . If the hydrophobic region is sufficiently large, with an increase in 09 the liquid begins to gradually move into the hydrophobic section, and only then, for some volume to> to* and a < c~° does an abrupt collapse of the cavity occur. ON HYDRODYNAMICSAND HEATOF THE HYDROPHOBICWETTING

Estimation of time of hole collapse A process of hole collapse in the unstable drop situated on a heterophilic substrate and the subsequent wetting of hydrophobic site under the influence of surface forces may be called the "hydrophobic wetting" (Smorodin, 1990a ). From a law of energy conservation, it follows that a change of system free energy,/IF, after the cavern collapse equals the sum of the kinetic energy,/IEc, and dissipated portion, Ed: l t~

-/IF=/IEc + t (dEJdt)dt.

(3.1)

0

After making a number of reasonable physical assumptions, a non-linear equation of film front movement may be obtained from this general expression. It contains two characteristic times: z~ = [2¢723RdJ~ 2 sin (0o/2)]/~/and z2 = (tr23~4/pR 2 ) 1/2, where r/is the film viscosity and p is its density; 2/~ is the mean distance between centers of lyophobic sites on lyophilic solid matrix. Evidently, a presence of two characteristic times mean two hydrodynamic regimes of wetting. The equation was solved numerically. Under an isothermal condition, Rd = 10-s m , / ~ = 10-6 m and the water film at T= 273 K, we obtained the full cavern collapse time, z= 10 -8 s.

Endothermic wetting effect During this wetting process, over a transition zone between the lyophilic matrix and lyophobic site, the molecules of polar liquid (water) change their dipole orientation. This must cause a deformation, weakening or partial rupturing of hydrogen bonds liquid like water and, as a consequence, a local increasing of the entropy of boundary layer (AS> 0) and its enthalpy (AH> 0). So, it is possible that the wetting heat, Q = - / I H , is negative. The maximal estimate (upper limit) of the endothermic effect may be found assuming that the process of wetting is thermodynamically reversible. In this

MIXED AEROSOLPARTICLESAS EFFECTIVEICE NUCLEATINGSYSTEMS

213

case, assuming that a change of the Gibbs free energy of system, ,JG = 0, in the Gibbs-Helmholtz equation: AG = z l H - TAS, we find (Smorodin, 1990a): -Q=AI-7=Nd TA,~°(TtR2/to ) ( h*/zlh ) ,

(3.2)

calculated for unit heterogeneous surface area and - Qo =a//o = Tz~,~'°(~rR2/to) ( h * / d h ) ,

(3.3)

for one hydrophobic section. There d g ° is the mean change in entropy in the transition zone; to is the "landing area" of one molecule (water or polar liquid); h* is the critical film thickness corresponding to cavern closing up; zth is the thickness of the liquid monolayer. Considering the temperature relaxation of the endothermic effect, we may write the equation for heat transfer: - Qo = - z t T C p n R 2 h * ,

(3.4)

where C is the heat capacity of the layer averaged with respect to thickness; p is it density, and - A T i s the decrease in temperature of the layer owing to the endothermic effect. Comparing the fight-hand sides of last two equations, we obtain a limiting value for the temperature effect of"heterophobic" wetting: - A T , , , TAS°(Cptozth)

-

1

.

(3.5)

Let us consider an example of wetting of the heterogeneous surface of a single crystal of AgI by a supercooled aqueous film. Let the hydrophobic sections of the surface consist of~Agl phase in the {10i0} prismatic plane, where Ag + and I - ions alternate. The hydrophilic sections on this surface are clusters of silver formed as a result of photolysis. Assuming to= 1.2.10-19 m 2, Ah=3" 10 -1° m, Cv=4.18.10 3 J/kg.K, p = 103 kg/m 3, Ra= 10-s m, N d = 10 m -2, T=263 K, and estimating d~°__0.3 k (where k is the Boltzmann constant), we find Q_~ - 3 . 1 0 - 2 J and -zIT_~ 30 K. Therefore, an additional cooling of drop (film) during the wetting of heterogeneous surface may provoke its freezing. These preliminary considerations need thorough experimental studies. TM

G E N E R A L A P P R O A C H IN N U C L E A T I O N O N M I X E D A E R O S O L PARTICLES

Model o f nucleation on mixed aerosol particles Depending on the external conditions and the properties of the aerosol, heterogeneousice nucleation in the atmosphere can proceed by various mechanisms; (I) contact nucleation; (II) deposition (or desublimation) of ice from vapor and (III) the vapor-liquid-crystal (VLC) mechanism. If condensation nuclei immersed in water are transported from a warm region of the cloud into a cold region, we may distinguish an immersion-freezingmechanism from a condensation-freezing mechanism; actually, the former is a retarded (or

2 !4

V.YE. SMORODIN

"extended" into two stages) form of the vapor-liquid-crystal nucleation mechanism. In the latter, the freezing mechanism takes place immediately after the condensation nucleation. In addition to ordinary condensation nucleation on insoluble particles in supersaturated vapor (when PIPs> 1 ), a major role can also be played by adsorptive growth of condensate films (when pips < 1 ), and we may therefore distinguish especially the "adsorption-freezing" and "adsorption-condensation-freezing" mechanisms as variation of type III. An important type of ice-forming nuclei consists of insoluble or nearly insoluble particles that maintain their crystallographic individuality. We investigate the thermodynamics of the nucleation of a new phase in its general form and also consider as a special case the mechanisms of deposition nucleation of ice from vapor onto an inactive nucleus containing "active sites" of AgI. We use the following model, which expresses the principal features of real conglomerates of mixed composition, e.g., formed by combustion, condensation and coagulation of the components in a supercooled atmosphere. We assume (Fig. 5 ) that on the surface of an insoluble particle ( 3 ) of radius Ro there occur non-overlapping zones of different character (3'), i.e., active nucleation sites, with radius of curvature RI,, "edge" angle ~o, a surface concentration (of active centers), n, a coefficient of wetting of active site with new phase (an embryo ) ml = cos0~, and a coefficient of wetting of the main (inactive) particle with the embryo, mo = cOS0o. We assume that the spherical embryo of stable phase, with radius R2, nucleates from the metastable phase (supersaturated vapor, supercooled liquid, or melt) on the surface of the particle, simultaneously covering N > 0 active "centers". In general, the centers may be active (in which case embryo will form on them) or inactive; an exact criterion of activity will be formulated below. Depending on the properties of the centers, we can distinguish instances of nucleation of liquid or of crystalline phase on the heterogeneous surface (Smorodin, 1983 ).

Active centers. Two cases are possible: (a) the embryo is located entirely on

.[

0

Fig. 5. Model of embryo on a surface of mixed aerosol particle.

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

215

the active site and (b) the embryo overlaps a center and its periphery zone is on the substrate particle. The criterion for occurrence of condition (a) can be written in the form Og----(COS01 --COS~2)gl g2 =gl (Zl --ml ) --g2 (12 --m0) < 0 ,

(4.1)

fl=m~ + # - l [ (1 --g) (g2 +g~ -~ ) - m o ] > 0 .

(4.2)

Here, cos~l = (I1-rnl )/gl, cos~2= (x2-mo)/g2; gi = ( 1 +I12 -2m111 )I/E, g2 = ( 1 +12 --2moX2)1/2; 11 =Ro/RI, X=Ro/R2, g=RI/R2, ml = cOS~o. Condition (4.2) states that the height of the embryo OH~ (from the center of the substrate particle) exceeds the height of the active center OH2 (see Fig. 5 ); this excludes annular embryo from consideration. Case (b) occurs when a > 0 (¢2 > ~i ), i.e., the size of the embryo is greater than that of the center. When ce< 0, the embryo is located entirely on the surface of the center functioning as a crystallization nucleus.

The case of inactive centers. The embryo is not located on any of the centers ifRo sin¢2 < 2R, ot < 0 or Ro sin~2 < 2R, c~> 0. The embryo either overlaps the center when Ro sin¢2 > 2R, c~> 0 or is located between centers and partially overlaps them as well as the substrate particle when ot < 0, Ro sin~2 > 2R. In this latter case, the surface of the embryo is nonspherical which will not be considered here. Mixed cases. When a condensate crystallizes by a two-stage condensationfreezing mechanism, the active centers of crystallization are inactive with respect to condensation or adsorption, i.e., are lyophobic. The liquid phase will appear initially on the lyophilic part of the crystallization substrate particle (on the condensation or adsorption centers) and then on the lyophobic centers, where, if the degree of supercooling is sufficient, crystal will nucleate. Energy of heterogeneous nucleation Consider the Gibbs free energy zlG~et required for the formation of a critical embryo of new phase in a single-stage nucleation mechanism (condensation, freezing or deposition) on a spherical mixed aerosol particle (MAP). According to the classical theory of nucleation, which can be applied in the thermodynamic conditions of clouds with relatively low degrees of supersaturation and supercooling, the energy of the embryo (Hobbs, 1974) is equal to Z~Ghe t = S 1 2 0 - 1 2 "3t- (0"23 - 0-13)$23 -Jr- (0"23, - 0-13,)$23 "~

From geometric consideration (see Fig. 5),

VAGv.

(4.3)

216

V.YE.SMORODIN

$12 =22tR2x~- l ( 1 - - q 4 ) , ) $23 =21tR~ [ ( 1 - q 2 ) - N ( 1 - q , )] , S23, = 2ztREyi- I N( 1 -q3 ) , , V2= 1/3~tRo3{2~-3(2 - 3q4 +q43) - (2 - 3q2 +q3 ) _ N [ x i - 3 ( 2 _ 3q3 + q 2 ) _ ( 2 - 3 q l + q 3 ) ] } ,

(4.4)

where ql =cos¢1, q2-~COS~2, 13=COS~1 = ()Clml - - 1 )/g2. The radius of the critical embryo is found from the condition (dAGhet/dR2) IR2=R~=0. From this follows the Gibbs-Thompson equation: R~ = - 2tr~E/AGv .

(4.5)

Using Eqs. (4.3) through (4.5), the expression for the free energy of the critical embryo can, after certain manipulations, be written as AGhet = AGhom t ~ .

(4.6)

Here, AG~om = (161t/3 )tr32/AGZv is the energy of homogeneous nucleation. The dimensionless function ~ = qb(m~, mo, m~, Zl, X2, N) or "heterogeneity factor" is a quantitative expression of the effect of surface curvature and wettability of the crystallization substrate particle q~= 1/2{1 _q3 +jr3 ( 2 _ 3qz +q23) _ 3m22~2z( 1 --q2) - N [ z 3 ( 2 - 3ql +q3) + 3rn~/tz ( 1 - q 3 ) - 3moX23( 1 - q l ) _/~3( 1 -q3 - q ~ ) ]}.

(4.7)

If a particle is entirely absent, (Ro = R ~= 0 ), q~= 1, the situation corresponds to homogeneous nucleation. For geometrically and energetically uniform crystallization substrate particle (N = 0 ), • is converted into the Fletcher factor: q~(mo= m, jt2- x, N = 0 ) ---f(m, x). Rate of nucleation and criterion of activity of centers When considering the rate or frequency of nucleation J, it must be borne in mind that the wettability of the MAP by the embryo governs not only the exponent (the nucleation energy) in the expression for J, but also the coefficient preceding it. Summarizing data on the dependence of J on the wetting angle of uniform surfaces (Hirs and Moazed, 1970) we can write the following equation for the rate of nucleation on an MAP: J~ =4rtR2gk'l exp( - A G ~ e t / k T ) ,

l= 1, 2, 3,

(4.11)

where l= 1 represents desublimation and l= 2 represents condensation: ~fl, = f12 = sin0 (deposition or condensation ) fl( O) = fll ( O) ----~fl3 =sin20/2 (freezing) ( l = 3 ) ; k~flt- k:

217

MIXED AEROSOLPARTICLESAS EFFECTIVE ICE NUCLEATINGSYSTEMS

~fl(0o)Y2 =,0(0o){1 + l / 2 N [ (1 - q 3 ) x i -u Ji-qi"1 ]}, Z=].,8(O,)y, = 112fl(O,)N( 1 --q3)zi -2 , o r < 0 .

a>O

;

For deposition, k'~,01- kl -- 10 29 m-2 s-l, and for freezing f13k'3-k3 --- 10 31 m - z s -1. When a < 0 , the factor 71 =Si/4rd~ (the fractional surface area of all centers on the MAP) expresses the fact that the seeds are formed only in active zones; when a > 0 , the factor 7z=S/4nRo (where S is the total surface area of the MAP) expresses the fact that the entire surface of the particle can be involved in nucleation. Making use of Eq. (4.11 ), the criterion of activity of the centers, indicating that the rate of nucleation on them is higher than that on the rest of the surface of the MAP, can be written in a general form as

AG~et[f(m2,z2)-f(ml,lt) ]>kTln( AGent[f(m2, X2) -

l--y~

fl(Oo) )

?~ #(01)

ot<~0;

' a>0;

(4.12)

~ ] > k T l n [ ( 1 - ? [ )/?'1 ] ,

where Y'~= ( 1/2)N( 1 - q 3 ) . If instead of Eq. (4.11 ) we use the opposite inequalities, the centers are inactive. For relatively smooth surface, on which the relief of the centers may be neglected and the fraction of the surface occupied by the centers is compared with the rest of the MAP so that the role of the coefficient is minor compared with the exponential expression itself, Eq. (4.11 ) is converted to the simple condition m 1> mo. ICE N U C L E A T I O N BY D E P O S I T I O N M E C H A N I S M

Optimum characteristicsof MAP's and maximum potential of surface modification. The probability of a one-stage nucleation process P ( t ) (deposition, condensation, or freezing) on a single particle is given (in the isothermal, quasi-steady-state regime ) by the equation (Hobbs, 1974 ):

P(t) =

1- e x p (

-Jt),

(5.1)

where t is the time. When Jt << 1, we can expand the exponential expression and obtain, as a first-order approximation in Jt,

R(t)~-Jt

(5.2)

The probability P ( t ) can be interpreted not only as a random measure of an individual event (relating it to the mean or ensemble mean, i.e., P( t ) = Na ( t ) / N~, where Na (t) is the concentration of activated particles (of monodisperse aerosol), and Ny is their total concentration. The effect of the modified surface of the crystallization particle on heterogeneous nucleation is conveniently analyzed by comparing the nucleation probabilities on the MAP pro(t) and on a homogeneous nucleus po (t):

218

V.YE. SMORODIN

/x../I /

._ Z-2.----_-z_-Z//"

"/"

""'_

--

i---

-t0

-2o

.....

~

g

........

'3 -a0qoc |

b',

a

8

Fig. 6. Gain in specific yield of activated particles with modified surfaces compared with ideal Fletcher case as a function of temperature t ° (a) and of particle size Ro (b); m~=0.9 (I); 0.8 (II); 0.45 (II); -0.5 (IV); Ro= 5"10-s (1); 10-7 (2); 10-6 m; ( 3 ) ; "levilites" (4); t ° = - 10° (5); -20 ° (6); -30°C (7).

pm( t ) /P°( t ) =N~a ( t ) ° ( t ) = J m / j ° = c o n s t a n t

(5.3)

For comparison, we choose the o p t i m u m nuclei defined by Fletcher, i.e., uniform particles o f the m i n i m u m radius R~ on which a single embryo arises in 1 second; then, the effect o f modification is described by the function

tl= [pm( t)P°( t) ] ( Vo/VI ) ~- ( j m / j o ) ( Vo/V, ) ~-Nma / N ° ,

(5.4)

where j o = 1 s -a, jm has been given above in Eq. (4.1 1 ), and the factor Vo/ V~) expresses the fact that the activities are compared for the same a m o u n t o f activating component. Numerical analysis o f the multiparameter function r/= q (ml, m l, Z1, Z2, N) for inactive deposition nuclei activated by addition of AgI indicated that r/( VI ) has a m a x i m u m at some o p t i m u m volume of modifying additive VI = VT (mo, ml, ml, ...). Thus, there exist optimal parameters o f modification. The envelope o f the m a x i m a for the o p t i m u m MAP's, which indicates the m a x i m u m capabilities o f the modification technique, is shown for several cases o f practical interest in Figs. 6 and 9. In the calculation, we assume that the ice embryos are formed by deposition, that they have a constant value trl2 =0.1 J / m 2, and that the parameter for the wet-

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

219

ting of the active center by ice ml =0.99. The contribution of elastic deformation of the AgI lattice to the nucleation energy (the "incoherence" effect associated with differences between the crystallographic characteristics of the ice embryo and the substrate) is accounted for by the Fletcher method (Fletcher, 1958) and shifting the plots of RT is the radius of the optimum center. The calculations indicate that for optimum MAPs and chosen characteristics of the inactive nuclei, i.e., ct < 0, that the size of the active zone is greater than the area on which the embryo forms. To improve wetting of the inactive base material of the particle (mi--, 1 ), the size of the active zone can be decreased. Figure 6 shows that r/increases monotonically with decreasing temperature and with increasing radius of the principal particle material. For particles with the realistic parameter value ml <0.99, at temperatures between - 10 o and - 30 ° C, r/reaches 5* 102. Interestingly, the optimum MAP's are more effective than particles of the critical Fletcher radius R~ with continuous coverage with a monolayer of AgI (so-called "levilites"), as indicated by curve 4 in Fig. 6a. The specific yield of active optimum MAP's N ~ (per kg of AgI) at temperatures between 10 ° and - 3 0 ° C may be as great as 1018--10 i9 kg -1 (see Fig. 9). -

Temperature activation spectrum of deposition ice nuclei in supercooled clouds. Numerous measurements of the concentration of activated ice-forming nuclei (Na) versus the degree of supercooling in an atmosphere cloud according to Fletcher ( 1962 ) may be approximated by the equation N, ~-No exp (flAT) ,

(5.5)

where fl and No are constants; at temperatures between - 15 ° and - 30 oC, on average r = 0.6. The simplicity of Eq. (5.5) suggests that a single mechanism of activation dominates in this temperature interval; we assume that it is heterogeneous deposition nucleation of vapor. Under conditions typical of the quasi-equilibrium droplet and phases of the cloud, the supersaturation of water vapor with respect to ice is well approximated by the expression J_~ ln(pw/ Pi), where Pw is the saturation vapor pressure of water and Pi is that of ice. Suppose that the nucleation probability is outside of saturation region, P~-Jt << 1. Since experimental data indicate that the principal contribution to P is made by rather large nuclei (Ro> 10 - 7 m), we consider only large particles; in this case, the surface curvature plays practically no role. The "heterogeneity factor" in this case is ~ - f ~ ( m ) = 1 / 4 ( 2 + m ) ( 1 - m ) 2, where m = cos0 is the wetting parameter for ice. Averaging P over the particle-size spectrum W(Ro), we obtain for a system of polydisperse particles in a unit volume P=N./Nz =

PW(Ro)dRo ~o

Jto W( dRo) ~-~kt to e x p ( - A G ~ t / k T ) o

220

V.YE. S M O R O D I N

where S is the mean surface area of a single particle and to is some standard observation time. Taking the logarithms of the last equation, we obtain In ( Na/ Nz ) "~In (,~k, to ) - AG~et/ k T .

(5.6)

We now expand the second term on the fight as a Taylor series in the neighborhood of some temperature To < 273.16 K, where To may then represent the maximum temperature for a given 8 at which deposition can still occur. In the linear approximation, after simple manipulations and substitution of the Clausius-Clapeyron relation, we obtain .2

.g2 i~3.-/-, 3

t~ s ~ 0 n ,

.l o

++ 2(Li +Lw)n (273.16-To)tr3~o 2 .g3/.4-r 5 s O O n . .t o

(5.8)

~ 2 .g3 I~4"/-' 5 r t s u'Or~ I 0

where A T = 273.16- T; Li and Lw are the heat of sublimation of ice and the heat of evaporation of water, respectively, ns is the number of molecules of water per cm 3, and the subscript 0 indicate that the values refer to the temperature To. The criterion of rapid convergence is reduced to 2 ( L i - L ) A T / (n~k3TTo) << 1, which at To= 258 Kis satisfied for AT< 30 K. If we introduce the notation [3=327r/3(Li-L,~)tr3~o/(nsSoke

3 4To),

(5.9)

167t[ 1 2 ( L i - L w ) (273.16- To)-] 0"3 ~ 0 , lnNo=ln(Nzgklto)--~n2~2(-kTo) 3 t nsk283oT 5

I

(5.10) then Eq. (5.6) becomes equivalent to (5.5). The meaning of the constant fl is revealed by Eq. (5.9). The treatment of the constant No, on the basis of (5.5), as the concentration of crystallization nuclei activated as AT--,0 is ruled out by classical thermodynamics: as AT--, O, .4G'~et ---,oo, P ~ O. The true meaning of No is indicated by Eq. (5.10). Assuming To=258.16 K, ao-0.1 J / m 2, ~o_-_6.3"10-2, we obtain from Eq. (5.9) f l - 2 9 . 5 7 f ° ; comparing this figure with the empirical value fl= 0.6, we f i n d f o _ 1.00.10-2, so that m-~ 0.9: the wetting parameter for ice is close to the ideal value of m_-_1. This figure is consistent with Fletcher's statement ( 1969 ) that the site of preferential generation of ice nuclei on natural crystallization nuclei is active zones of the surface.

MIXED AEROSOLPARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS THE CONDENSATION-FREEZING

221

MECHANISM

Basic model and the description Investigating the "condensation-freezing"nucleation mechanism, it is reasonable to separate two characteristic cases: (a) nucleation on the "weakly soluble" MAP and (b) that on the "strongly soluble" ones. Let us determine that the "weakly soluble" MAP's are those having parameters of the wetting by condensing phase, m - c o s 0 < 1, less than 1: i.e., the stable contact angle 0> 0. We determine that the "strongly soluble" MAP are those not having the stable contact angle 0. If an inactive hydrophilic part of MAP is dissolving faster than an ice embryo nucleates, ice-forming hydrophobic parts of MAP (active "ice centers") in contact with the solution act as independent particles, immersed or floating. Since at the weak solubility of the "ice centers", its crystallographic individuality and ice-forming activity are being better preserved, we will deal with these particular cases. As a basic model we take the case of the "weakly soluble" spherical hydrophilic particles having the hydrophobic "ice centers" in the form of spheres ("cap"-shape). Let the main hydrophilic particle characterized by a radius Ro, have Arc= 4~Ro2nc hydrophobic "ice centers" with radius Re, forming with the main particle the geometric angle ~ (that correlates with the parameter of "pseudo-wetting" r~ ~cos0); here nc is the surface concentration of the ice centers. An average distance between the centers equals 2R ~_n ~-m/2.Let the surface of the particle be characterized by phenomenologlcalparameters of wetting: for a liquid phase, m~l=COS0~ (the ice center), mo~=COS0o~(the main particle), for ice embryo, mci----cos0ci (the ice center), moi--cOS0oi (the main particle); besides that, mci
222

V.YE. SMORODIN

When active ice centers are covered with the solution, as a result of its reaction with a solution, it might form an additional "field of solubility", ~M. We may suppose that ~M<< M.

A condensation stage. Usually, vapor pressure of liquid lowers as the concentration of solute increases: P~/P~o=F(M) < 1, where for Raoult's solution F(M) = 1 - M . A critical radius of liquid embryo, taking into account the solubility, equals: R* = -2oc/AGvc = (2trdn~kt){~l + l n [ 1 - F ( M ) ]} - 1 ,

(6.1)

where ~ =In(pips) and p~ is the saturation vapor pressure "over water", AGv~= -nlkTt~ is the free energy difference for a unit volume of liquid (without taking in account capillary effects: let us limit ourselves to relatively "large" sizes of drops, and Ro and Re); ~= t~ - In [ 1 - F(M) ]; n l is the molecular concentration in a solution. The critical embryo radius in the solution is less than that in pure water ("driving force" of condensation, AGv~ increases). If ~-~In [ 1 - F(M) ], in part t~ = 0, the solubility effect dominates and condensation may begin even under subsaturation. Let us find a minimal solubility of MAP, for condensation nucleation to take place at the rate of Jc = 1 s- 1. After the simple transformation of last condition for a case of ideal solutions, one can arrive at M~*= e x p { t ~ - ( 4 / n l kT) I

rttr3@c 3kTln( Kc4rtR2 Sc ] 1/2}

(6.2)

At M
A freezing nucleation stage. The reduced vapor pressure over the drop solution leads to decrease in "driving force" for freezing nucleation: AG~r= - ASAT{ 1+ ( nsk T) / ( ASAT) In [ 1 - F ( M) ]}

(6.3)

where AS is the average entropy within AT and ns is the water molecule concentration in ice. A critical ice embryo radius increases:

R'~=-2trf/AG~f=(2tr/ASAT){I+(nskT)/(ASAT)In[1-F(M)]}.

(6.4)

The temperature of solution freezing decreases:

dTf = ( nskT) / (AS)In[ 1 - F ( M ) ],

(6.5)

for ideal solutions: A Tf ~ (n~kT) / (AS) In M. From a condition dGv*r< 0 follows a next criterion

1+ (nskT) / (ASzlT)ln [ 1 - F ( M ) ] > 0,

(6.6)

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

223

determining a minimal supercooling ( J T ? ) at a known solubility, M, (or a minimal M~ at a known ATf), that is required for the start of freezing. E.g., for ideal solutions, one can find a crystallization criteria:

M>M'~ = e x p [ - (3SAT)/(nskT) ] - - - e x p ( - 2 . 5 J T / T )

(6.7)

In view of the cryogenic effect (dTf), a value M* may be confronted with a some minimal "crystallization barrier", ATe. Let us consider an example: a large aerosol particle (Ro = 1 0 - 7 m ), having a wetting parameter (me = 0.99 ), consists o f the small active sites of crystallization (Sf ~*:4 ~R 2 ). At T= 271 K, ~ = 0 and K~= 102~ s -~ m -2 one can calculate M*---0.98. A value of AT~ corresponding with this M*, in a case of ideal solution, equals 2.8 K. It correlates with an empirical value of the "temperature barrier" of freezing nucleation for AgI. One may conclude that in a case of ice nucleation on the AgI particle by the "condensation-freezing" mechanism, the concentration of its soluble impurities is no less than 2%.

A dissolving processes. If characteristic dissolving times Zdo=poRo/2CoDo,2 o Zdc=pcR~/2C°Dc >~ z~, Zf, ~'co and some realistic physical conditions are satisfied, the general problem could be reduced to a nucleation one. (Here % and zf are the condensation and freezing nucleation times; Z~ois the time for covering of the lyophobic center by drop; Pc and P0 are the densities of "center" and particle, D~, Do are the diffusion coefficient for its molecules within the solution, C o and C o are its saturation concentrations). So, the problem reduces to that of condensation-freezing nucleation and we may use our general thermodynamic method which has earlier been developed for nucleation on the MAP.

The probability of condensation-freezing. The heterogeneous ice nucleation probability for the "condensation-freezing" mechanism in quasi-stationary classical approximation for the "point" aerosol particles was derived by Fukuta and Schaller ( 1981 ): Pcf(t) = 1 + (Jc - J r ) -~[Jc e x p ( - J f t ) - J r e x p ( - J e t ) ] ,

(6.8)

where t is the time. This important formula was obtained for the dimensionless ("point") particles without taking into account the surface heterogeneity. We may develop this result for the case of MAP, using our general theoretical approach for the one-stage heterogeneous nucleation on MAP and the solution theory, interpreting accordingly the values Jc and Jr. Depending on the external (thermodynamic) conditions and MAP parameters, it is possible to distinguish between four cases: (a) The ice embryo is totally placed on the ice center, and the liquid embryo is placed on the hydrophilic part of MAP (between the ice centers), i.e., on the condensation center:

224

V.YE. SMORODIN

R? sinOidRd < 1 , R*~ sinOwo/(Zq--Rd) < 1 ,

(6.9)

where Rd=R1 sin¢/l (see Fig. 5); R? and R* are the radii of the critical embryo for freezing and condensation nucleations, respectively. (b) Both critical embryos cannot be placed on their centers: R? sin0iJRd < 1,

R~ sin0wo/(l~--Rd) > 1 .

(6.10)

(C) The ice embryo is placed on its center and the liquid one is overlapping its condensation center: R? s i n 0 i J R d < l ,

R ~ s i n 0 w o / ( l ~ - R d ) > 1.

(6.11)

(d) The ice embryo is overlapping its center, while the liquid one doesn't (a case of small crystallization centers): R? sin0ic/R~> 1,

R* sin0wo/(l~--Rd) < 1.

(6.12)

Depending on these conditions the processes of condensation and ice nucleation each one separately, may happen either in one-barrier or double-barrier nucleation. In the latter case if the second barrier is larger than the first one, the nucleation already can be stopped: the center happens to be insufficiently active. The criterion (a) is necessary and sufficient conditions of onebarrier activation: nucleation inevitably goes to condensation and crystallization. An analytical presentation of Jc and Jf is different for cases (a)-(d ). Thus, in the case (a) for J~ and Jr, we may use results of the Fletcher theory [with the "geometric factor" f ( m , x ) ] and the solution theory. For other cases we must take the more complicated analytical formulas based on the "heterogeneity factor" O. Characteristic times. For process modeling, it is important to compare the different characteristic times: %, zf, %f, %c, Zso,etc. For example, if % :,~ zcf the ice nucleation (by the condensation-freezing) may happen on the same hydrophilic sites. In a case of % << %f, the freezing nucleation may take place on hydrophobic ice centers. In both of these cases we can use the Fukuta-SchaUer formula applying proper rates of freezing (on hydrophilic or hydrophobic sites). Other cases are more complicated because we must consider hydrodynamic transition processes. Firstly, we limit ourselves to a simpler case ( % ~ ~cf). Let us investigate a dependence Y ~ f = f ( M ) limited to large aerosol particles satisfying the condition (a) and the Raoult law. From the condition (dPcf/ d M ) IM. = 0 at t-~ O, we obtain

[ (AG~f/AGvc) -- (of/at) (Of/Oc) 1/3 ] IMo = 0 ,

(6.13 )

where the "heterogeneity factors" O~ = ~t~ and Of= fl~° . Solving this equation with respect to M*, one can find

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

225

(ac/of)(~f/~c),/3d_(AS/iT)l(nskT) } M*=exp -

1 -I- (¢7c/o'f) (¢~f/~c) 1/3

(6.14)

Evidently, M* < M* < M~. In view of condition 0 < M < 1, for existence of the extremum, as can be seen from the upper equation, it is necessary that

(/iT/T) /~>O.1

(6.15)

Here it was taken into account that approximately tTf/a~_~0.25 and nsk/ /IS---0.40, and it was supported approximately that for the large activity centers, • f/~c--- 1. In the supercooled clouds where, as a rule, ~ is small, the last criterion may be satisfied for the active natural ice nuclei: the extremum exists. An infringement of this condition means an absence of M*: dissolving always decreases the P~y. At a high value of (/IT/T)/t~ the "condensationfreezing" nucleation is changed by deposition nucleation mechanism.

Numerical calculations. A system of equations describing the "condensation-freezing" mechanism was numerically investigated for different sets of parameters that are characteristic of weakly soluble mixed crystallization nuclei and the thermodynamic constants prevailing in a cloud (/IT, ~). In the calculation process, as the solubility diminishes ( M ~ 1 ), the probability Pcf and Pd (deposition) were compared; in the case of Pcf< Pd the program was automatically switched over to a subprogram the "deposition mechanism". The cases of large hydrophilic nuclei was considered, on whose surfaces were deposited AgI sectors. It was considered that such sectors might still be active while their thickness exceeds a monolayer (as a result, the crystallographic structure is retained). (The temperature dependencies of the surface are retained. ) The temperature dependencies of the surface tensions of liquid and crystalline nuclei, a ( T ) , were taken into consideration. Figure 7 illustrates the concurrence of the effects of dissolving and nucleation of ice in time. It was used for the calculations: d=10-2; /IT=15 °, M = 0 . 8 8 , R o = 1 0 -7 m, Rco=3.89-10 -8 m, m~o=0.999, mc~=-0.9, mf~= 0.999, mf~=- 0 . 9 , m=0.8. In Fig. 8 are represented the calculation results ofecf function (t= ls) for one set of parameters. The nuclei are activated in the case where at a temperature - 15 °C the solubility gets into the range of values: 0.88 < M < 0.96. (In this scale, a slight maximum within those limits does not show. ) The value of P~f sharply decreases outside that range of values. At a high solubility, the decrease in P~f is associated with the retardation of the ice nucleation rate; whereas at a low solubility with a decrease in the condensation rate. The optimum values of M (with regard to the nucleation of ice) strongly depend on the values of~ and/IT.

226

V,YE. SMORODIN

-t

-3

t

l

t0 °

t,

$ec

Fig. 7. Concurrenceof resolvingand ice nucleationin the time: t= 1 s; M= 0.88; the lifetimeof a "center": 17 s (l) 25 s (2) and 50 s (3). This phenomenon was reported in 1985 (Reports of the Central Aerological Observatory, 1985). Later, Kim and Shkodldn (1986) received experimental verification on copper acetylacetonate aerosols modified with hygroscopic impurities (the Russian Institute of Experimental Meteorology, Obninsk). In reality, they proposed a mistaken explanation of the P~f(C) decrease after a maximum. In Fig. 9 is presented the dependence of the maximum yield, Na, of the activated mixed AgI-NH4I particles (curve 2 ), and the dependence contouring the maximum values, Na(T), at a fixed temperature (as calculated per kg AgI). Curve 1 illustrates the experimental data for the same aerosols that are obtained at the Central Aerological Observatory by Sosnikova and Plaude (both the experiment and the theoretical calculations were carried out there in 1984). The adequacy of the theory and model has been proven by a verification.

Temperature activation spectrum for the condensation-freezing. In a range of slight supercooling ( A T e 0 ) for particles having the soluble components, a mechanism of ice nucleation must change. A number of experimental data

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

t/;

f

/

-6

227

J

-~'0 ~x I

0.~

I

0.~

I

age

,-

-

M

Fig. 8. D e p e n d e n c e o f t h e p r o b a b i l i t y o f ice f o r m a t i o n , P~f (t= 1 s), a n d t h e n u c l e a t i o n rates, J~, Jr, o n solubility, M . F o r calculations, w e h a v e a s s u m e d : t ° = - 15 ° C, ~ = 1 0 - 2; Ro = 1.5 10 - 7 m , R e = 3.9 10 - s m.

show increase of the • constant in the Fletcher ice nucleation formula. Using the general theory of ice nucleation on MAP for the mechanism "condensation-freezing", we can use the same theoretical approach, as in a case of the deposition mechanism, expanding the probability function (the FukutaSchaller formula) into the Taylor series on AT. At a suggestion that Jet, Jtt << I, Pcf(t) - 1 / 2 J j f t 2. Let us suppose that the conditions of case (a) are satisfied. For the sake of simplifying, we suggest also that:

(ol) (P-Ps)/Ps << 1 ; (~) AT/T<< I .

(1~) 1 >>~C;

(y) zlSztT>>~CnskT; (6.16)

Then, we choose a temperature, To < 273.16 K (here the condensation-freezing is possible), as a point of the Taylor expansion, analogously the deposition case. We replace the solubility parameter M by C: l n M = - yC, where y is the activity coefficient for an ideal solution (in the Raoult law). Remembering that C = Co e x p ( - L d / k T ) (where Ld is the heat of dissolving and Co is the constant) and an equilibrium pressure of saturated vapor as the temperature function equals: p~ =P~o exp ( - L v / k T ) , after a number of simple trans-

228

V.YE.SMORODIN

~3

~0

,/"/t....I

o°

-~*

J

t

-~o °

-i¢*

I

_,co"

p

/o, c

Fig. 9. Dependence of the specific yield of activated MAP, modified by AgI sites, on a temperature: 1-3 indicate the condensation-freezing mechanism (2. is the theoretical dependence, 1. indicates the experiment by Sosnikova and Plaude on NH3-AgI aerosols), 3. is the calculation according to Fletcher; 4. is the desublimation (deposition) mechanism: the limit (maximum) yield of inactive particles modified by additions of AgI.

formations, taking into account the Clausius-Clapeyron theorem, finally, one can find: N~f- No exp (fiefAT) ,

(6.17)

where No= 81t2S~fK~rto exp{- [ G*(To) + G.f(To) ]/kTo - fief(273.16To)}, 16no'~F¢o fief= n2 k2T4 ( 6+ ~Co) {(2/3)Lv/kTo-TC[l+(2/3)LJkTo]} 16n°f3°Ff°

{

+ kTo(AS)2(273.16_To ) (1-91/To)+yCnsk/AS

MIXED AEROSOLPARTICLESASEFFECTIVEICE NUCLEATINGSYSTEMS

To-91 1} • X I l+(2/3)LJkTo+32--~-3Z-~o

229

(6.18)

Here ~c and ~f are the average part hydrophilic and hydrophobic sites on the aerosol particle surface, accordingly; K¢ and Kf are pre-exponential multipliers that appear in the nucleation rate equations of condensation and freezing (J¢ and Jr) on the MAP, respectively. According to the simplifying conditions shown above: [AGc*(To)+ ZIG~(To) ]/kTo >> flAT. When ZIG~ >> ziG* and it is possible to neglect the solubility effects in a particular case, from the formula expressed above, we retrieve the formula that was found earlier (Smorodin, 1991 d). As one can see from the formula for the condensation-freezing mechanism, the fiefactivation coefficient is larger than fld (the deposition one); that is in accordance with the experimental data. Depending on the manner in which the activation mechanism dominates the condensation or the freezing, the influence of solubility (Co) may both increase the resulting fiefand decrease it. A saturation effect leads to the fiefdecrease: dflcf/d~< 0. I C E - N U C L E A T I O N M E C H A N I S M O F AgI A N D E F F E C T I V E ICE N U C L E A N T S

The surface of AgI has been modified to achieve better pyrotechnical mixtures for weather modification. Study of the ice-forming mechanism of the AgI has a long and intriguing history. [Here it is impossible to review this subject wholly, and we must eliminate a discussion on such interesting subjects like the scale effects correlating with the Debye screen radius (Layton, 1973), the different relative activity of or-, fl-, ~,-modifications of AgI, and simulation of water adsorption on surface of AgI (Fukuta, 1975; Hale et al., 1983; etc.) ]. As it is well known, ice-forming ability of AgI was found by Vonnegut (1947) to be a perspective crystallizing reagent for weather modification by reason of its crystallographic isomorphism with ice. It was formerly believed that the main factor responsible for the effectiveness of heterogeneous ice nucleation on AgI particles was a crystallographic match and affinity with ice (in the plane of the growth). Later, the fact that crystallographic isomorphism is not a unique factor of ice nucleation on AgI was reported by Coulter and Candella (1952) who found an increase of ice-forming activity of AgI by introducing hygroscopic impurities. In experiments (Karasz et al., 1956), it was found that AgI is essentially hydrophobic and that its surface and ice are probably energetically incompatible. This view was then supported by measurements of Zettlemoyer et al. ( 1961 ). Based on adsorption measurements, authors suggested that the fundamental role in the crystallization of water on the surfaces of ice-forming

230

V.YE, SMORODIN

particles is played by non-uniformity of the energy or chemical composition of the surface. [The idea was confirmed in practice by a technique of heterophilization (modifying) of inactive nuclei: Zettlemoyer et al., 1963 ]. Interpreting their results, they postulated that the ice first develops on AgI at a few isolated impurity sites which are hydrophilic in character. On the contrary, other researchers, e.g., Edwards and Evans ( 1962 ) believe that ice initiates on hydrophobic sites ofAgI. Pruppacher and Pflaum ( 1981 ) have found that ice is better nucleated at the boundary between positive and negative charged sites of substrate surface. Corrin and Nelson (1968) concluded that the surface of "pure" AgI must more strongly resemble the bulk structure than the surface of the contaminated materials resembles the bulk structure. Yet the "pure" material is a much poorer nucleant when contaminated with hygroscopic impurities. This is a strong evidence against the epitaxial growth (of ice on AgI) argument. Then, the variations in the nucleating ability of AgI exposed to ultraviolet light are explicable in terms of these ideas about heterophilicity. The sum total of all these previous attempts to unravel the AgI ice-forming mechanism, as we have, follows really from the results aforesaid. From a phenomenological point of view, all the effective ice nucleant crystals, including the AgI, must have two main characteristics: (a) the heterophilicity with a determinedgeometry (the hydrophilic condensation sites, in part, soluble ones and hydrophobic ice active sites) and (b) the crystallographic iso-(or homo)morphism of ice centers with the ice. A heterophilic surface of the AgI presents the hydrophobic matrix (a "pure" AgI crystal) with hydrophilic silver (or its oxides ) clusters to be easily formed under photolysis. Usually, researchers, simulating processes of adsorption or nucleation on the AgI surface, neglect its real heterophilicity. The presence of separate condensation and crystallization sites on the substrate surface provides optimization of two different stage of ice nucleation: ( 1 ) water condensation or adsorption on the hydrophilic sites and (2) ice nucleation on the hydrophobic crystallization sites. The same surface may not unite both contrary properties with a high ice-forming activity (see, e.g., Isaka, 1966). The second criterion, the isomorphism (a slight misfit), provides minimization of the ice nucleation energy. The AgI crystals unite both heterophilicity and isomorphism factors. Besides that, at the surface boundary of hydrophilic and hydrophobic sites of AgI crystal, the endothermic wetting effect that is possible at some conditions (Smorodin, 1990a) may stimulate the ice nucleation (this corresponds to the Pruppacber and Pflaum experiments). But this idea needs further study. Two criteria (a) and (b) concern both inorganic substrates (AgI, PbI2, CuS ) and organic ones (glycine, metaldehyde etc. ). It is reasonable to suggest a hypothesis that the nucleation mechanism of the ice-forming bacteria (like the Pseudomonas Syringae, Erwinia Herbicola etc. ) are also due to its heter-

MIXED AEROSOL PARTICLES AS EFFECTIVE ICE NUCLEATING SYSTEMS

231

ophilicity. [As it is known, the cell membranes consist of lipid (hydrophobic) sites and protein (heterophilic) ones ]. On the contrary, the antipodes of ice-forming bacteria must be characterized by homophilic surfaces. The aforesaid ideas and presented data may be used as an heuristic way and basis for more detailed studies. SUMMARY In summary, we have used the classical theory to investigate the principal properties of ice nucleation on insoluble and weakly soluble mixed aerosol particles. We have considered, in particular, the deposition of ice on artificial crystallization nuclei with ultra-low additions of AgI. For the condensationfreezing mechanism on weakly soluble mixed aerosol particles, our investigations were based on ( 1 ) the classical nucleation theory, expanded on mixed aerosols, (2) the Fukuta-Schaller probability formula for the condensationfreezing mechanism and (3) a solution theory. This approach yields a logical model explaining the experimental temperature activation spectra of natural and artificial crystallizing nuclei and reveals their relationship to ice generation mechanisms in clouds. The theoretical method could be easy generalized for more complicated and realistic cases. But the main result is very important for cloud microphysics: there is an optimal solubility range of effective atmospheric ice forming aerosols. Optimization of MAP enables us to evaluate the maximum capability of the modification technique and to make quantitative estimates which are of practical value in developing effective ice-generating agents. The maximum gain in the yield of optimum modified AgI-based crystallization nuclei (compared with the case of uniform particles of AgI with the Fletcher critical radius) in the temperature interval from - 10 o to - 30 oC reached 5-102. The optimal modification of weakly soluble aerosol particles with AgI (when the condensation-freezing mechanism is effected) permits increases to a maximum gain in the yield of activated ice nuclei to the upper limit of order 1021 per kg of AgI. At least, an explanation of the AgI ice-forming activity and a hypothesis concerning crystallization mechanism of ice-forming bacteria were advanced. Other results of our studies on the MAP [in part, an adsorption-freezing, peculiarities of ice nucleation on bacteria and scale (cluster) effects ] will be discussed in future reports. ACKNOWLEDGMENT I am very grateful to Prof. Norihiko Fukuta (the University of Utah) for kind support and careful wording of this article.

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REFERENCES Almazov, L.A., Smorodin, V.Ye. and Tovbin, M.V., 1980. Water wetting of ultramicroheterogeneous surfaces of solids. Colloid J. USSR, 42:17 l - 178. Bazzaev, T.V. and Smorodin, V.Ye., 1986. Heterogeneous nucleation of ice on aerosol particles with modified surfaces. Tr. TsAO, 162:119-128. Bonis, K., 1969. On the kinetics of condensation of atmospheric water vapor on soluble nuclei. Idojaras, 73: 65-71. Cassie, A.B., 1948. Contact angles. Discuss. Faraday Soc., 3 ( l ): 14-16. Corrin, M.L. and Nelson, J.A., 1968. Energetic of the adsorption of water vapor on "pure" silver iodide. J. Phys. Chem., 72: 643-645. Coulter, L.V. and Candella, G.A., 1952. A preliminary report on the adsorption of water vapor by silver iodide. Z. Electrochem., 56: 449-452. Edwards, G.R. and Evans, L.F., 1962. Effect of surface charge on ice nucleation by silver iodide. Trans. Farrady Soc., 58: 1649-1695. Federer, B. and Scheider, A., 1981. Properties of pyrotechnic nucleants used in grossversuch. J. Appl. Meteorol., 20: 997-1005. Fletcher, N.H., 1958. Size effect in ice crystal nucleation. J. Chem. Phys., 29: 572-576. Fletcher, N.H., 1959. Entropy effect in ice crystal nucleation. J. Chem. Phys., 30:1476-1482. Fletcher, N.H., 1962. Physics of Rainclouds. Cambridge University Press, 386 pp. Fletcher, N.H., 1968. Ice nucleation behaviour of silver iodide smokes containing a soluble components. J. Atmos. Sci., 25:1398-1403. Fletcher, N.H., 1969. Active sites and ice crystal nucleation. J. Atmos. Sci., 26:1266-1271. Fukuta, N., 1975. Molecular mechanisms of the nucleation. In" I. Gaivorousky (Editor), Proc. 8th. Int. Conf. Nucleation. Gidrometeoizdat, Moscow, pp. 26-36. Fukuta, N. and Schaller, R.C., 1981. Ice nucleation by aerosol particles: Theory of condensation-freezing nucleation. J. Atmos. Sci., 39: 648-655. Gorbunov, B.Z. and Kakutkina, N.A., 1982. Ice crystal formation on aerosol particles with a non-uniform surface. J. Aerosol Sci., 13: 21-28. Hamill, P., Turko, R.P., Took, O.B. and Whitten, R.C., 1982. An analysis of various nucleation mechanisms for sulfate particles in the atmosphere. J. Aerosol Sci., 13:561-586. Hirs, J.P. and Moazed, K.L., 1970. Nucleation in the crystallization of thin films. In: Physics of Thin Films (Russian translation), Vol. 4, Mir Press, Moscow, pp. 123-166. Hobbs, P.V., 1974. Ice Physics. Clarendon Press, Oxford, 837 pp. Isaka, H., 1966. Interrelation between ice nucleation and condensation. J. Rech. Atmos., 2: 383-385. Karasz, F.E., Champion, W.M. and Harsley, G.D., 1956. The growth of ice layers on the surfaces of anatase and silver iodide. J. Phys. Chem., 60: 376-378. Kim, N.S. and Shkodkin, A.B., 1986. Study of an ice-forming activity of the copper acetylacetonate in a supercooled double phase flow. Meteorol. Hydrol. (Russian), 2: 28-31. Koeler, H., 1936. The nucleus in and growth of hygroscopic droplets. Trans. Faraday Soc., 2: 1152-2261. Layton, R.G., 1973. Ice nucleation by silver iodide: A new size effect. J. Colloid Interface Sci., 45: 215-216. Meszaros, E., 1969. On the thermodynamics of the condensation on water soluble and mixed particles. Idojaras, 73: l - l I. Pomposiello, M.C., Trigubo, A.B., Giordano, M.C. and Vallana, C.A., 1979. Density effect on heterogeneous nucleation. J. Rech. Atmos., 12: 286.

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Pruppacher, H.R. and Pfiaum, J.C., 1975. Some characteristics of ice nuclei. J. Colloid Interface Sci., 52: 543-552. Smorodin, V.Ye., 1983. Heterogeneous nucleation on aerosol particles with modified surfaces. In: S. Ferrousky (Editor), Tsirkulatsiya atmosfery i vlagoperenos had tsentral'noy i vostochnoy Yevropy (Atmospheric Circulation and Moisture Transport above of the International KAPG Symposium), Moscow, 12-13. Smorodin, V.Ye., 1987. Double barrier nucleation on "active center" of an energetically inhomogeneous surface. Soy. Phys. DOE., 32: 405-407. Smorudin, V.Ye., 1990a. Endothermic wetting effect and the mechanism of the ice-forming action of AgI. Colloid J. Russian Acad. Sci., 53: 249-256. Smorodin, V.Ye., 19901). Mechanisms of heterogeneous ice nucleation on mixed ice nuclei in the atmosphere. J. Aerosol Sci., 21, Suppl. 1,249-253. Smorodin, V.Ye., 199 l a. The "adsorption-freezing" mechanism on mixed aerosol particles in supercooled clouds. Report Series in Aerosol Sci., 17:111- l 13. Smorodin, V.Ye., 199 lb. The "condensation-freezing" mechanism on mixed aerosol particles in supercooled clouds. Ibid., 114-114. Smorodin, V.Ye., 199 l c. Investigation of the thermohydrodynamic instability of a drop on an aerosol particle. In: G. Matsui, A. Serizawa and Y. Tsuji (Editors), Proc. of the Int. Conf. Multiphase Flows '91-Tsukuba'. Tsukuba, 1: 169-172. Smorodin, V.Ye., 199 ld. The temperature activation spectrum of atmospheric ice nuclei and mechanisms of heterogeneous ice nucleation in supercooled clouds. J. Aerosol Sci., 22, Suppl. 1: 553-555. Steele, R.L. and Krebs, F.W., 1967. Characteristics of AgI ice nuclei originating from anhydrous ammonia-silver iodide complexes. J. Appl. Meteorol., 6:105-113. Tovbin, M.V., Smorodin, V.Ye. and Golovchenko, O.L., 1978. Form of water films on a mosaic surface. Ukr. Fiz. Zh. (Russian), 24:415-418. Turnbull, D. and Vonnegut, B., 1952. Nucleating catalysis. Ind. Eng. Chem., 44:1292-1295. Vonnegut, B., 1947. The nucleation of ice formation by silver iodide. J. Appl. Phys., 18: 593595. Ward, R.C., Hale, B. N. and Terrazas, S., 1983. A study of the critical cluster size for water monolayer clusters on a model AgI basal substrates. J. Chem. Phys., 78:420-431. Zettlemoyer, A.C., Theurekdjian, N. and Chessick, J.J., 1961. Surface properties of silver iodide. Nature, 192 (4803) 653. Zettlemoyer, A.C., Theurekdjian, N. and Hosler, C.L., 1963. Ice nucleation by hydrophobic substrates. Z. Ann. Math., 14: 496-502.