Size distribution of droplets nucleated by soluble aerosols

J Aerosol &I

Vol 3 I, Suppl

I, pp. s434.s435.

2000

Pergamon www.elsevier.com/locate/jaerosci

Poster Session I. Nucleation SIZE DISTRIBUTION OF DROPLETS NUCLEATED BY SOLUBLE AEROSOLS A. HEITOR REIS’.’ ‘Centro de Geofisica de fivora, Ap. 94,7001, gvora Codex, Portugal

‘Departamento de Fisica, Universidade de l?vora, R. Romao Ramalho, 59,7000-67 1, Fvora, Portugal

Keywords:

AEROSOL, NUCLEATION, DROPLET, DISTRIBUTION. INTRODUCTION

Droplets in the atmosphere are formed essentially by the condensation of the environmental water vapor on the existing aerosols (CCN). In this way, the size distribution of droplets is closely related with the chemical composition of the CCN and with their size distribution. Equilibrium droplet size distributions after the ascent of the air masses containing the CCN are usually evaluated on the basis of the vertical velocity (Pruppacher and Klett 1997). In this work we present a way of determining equilibrium droplet size distributions of static droplets of soluble CCN based on the minimization of the free energy of the system CCN/water vapor. THEORY In order to evaluate the energy of formation of droplets nucleated by CCN, at a temperature T, we consider, for a volume V, in the atmosphere, the difference in the Helmholtz free energy after and before the formation of a solution droplet of volume V, \ AF =-(Pj -P,),V, -(Pl -P,),sV, +@I -&),N, +@I --cL,),~N, +d (1) where P, V, CT, A, N and p denote pressure, volume, surface tension, interfacial area, number of moles and chemical potential, respectively, the subscripts 1 and v stand for liquid (droplet) and vapour and w and s stand for water and solute respectively. From (1) and taking into account Raoult’s law and the Gibbs-Duhem relation, the energy of formation of a solution droplet of radius r ( N,Vmoles of solute with a mean degree of ionic dissociation, i , dissolved in N,,, moles of water of density p, ) is given by (Reis and Serrano, 1998): AF = (4~ /3)r3p,R,T[-

In S, - (x + l)ln(l+

x) + xln xl+ 4rcr2cr,

where S, = PI POstands for the air saturation at temperature

Withx = iN, lN,

(2),

T , and P, PO are actual and saturation

pressure, respectively. Let Y(N,~) represent the concentration of CCN with N, number of moles and let v.

stand for the overall concentration

of CCN (in the volume V, ). In addiction let w(N,~, S,, r) stand

for the probability (unknown) that a solution droplet with N, moles of solute have radius rat saturation S, . Therefore, the droplet size distribution will be the function: x(S,, r> = I cp(N, hy(N,yl SW,rW,? . (3) At constant temperature, the equilibrium conditions may be determined from the minimum of the Helmholtz free energy of a constant volume system (Callen, 1985). Therefore, in order to determine the size distribution of droplets x(S, , r) we minimize the Helmholtz free energy: Af = I q(N,< )m(N,V,

SW1rMN,y 3

(4)

with respect to the free parameters: droplet radius r and saturation S, subjected to the constraints:

s434

AbsIracts of the 2000 EuropeanAerosol Conference

$435

a) constancy of the total water content in the volume V0 : ~ (arc / 3)r 3p w ( O ( N s ) ~ ( N , , S w, r)dN~dr + PoS~

(5a)

PoSwo,

where Po stand for the density of the saturated water vapor at temperature T and S w, Swo stand for the air saturation in the presence and in the absence of droplets, respectively; b) and constancy of the number of droplets in the volume V0 (equal to the number of CCN) (5b)

Z ( S w , r)dr = ~ q~(N~)dN~ = v o .

The result is the drolglet size distribution: z ( S w , r ) = ~P0Sw [ln S w + (1/v0) ~ q~(U,.)ln(1 + x(N,.,r))dU,, - 2(r /(PwRwTr)]l[(

/ 3 ) p w r 3 ,1

(6) where the constant & is to be determined from the constraint (5b).

CONCLUSIONS Therefore, knowing the distribution of soluble CCN eq. (6) provides the corresponding droplet size distribution at temperature T , and saturation S w . Fig. 1 shows an example of application of (5b) for a simple case: 1,00E+06

g~

U'-,,

1,00E+05 1,00E+04

t,f} 0 0

1,00E+03

o

1,00E+02

o

\

i.

o 1,00E-06

1,00E-05

1,00E-04

D r o p l e t r a d i u s (m)

1,00E+01 ....... 1,00E+O0 1,00E-03

I

Fig. 1 - Droplet size distribution induced by a uniform distribution of soluble CCN, with N s = 10 -16 mole and v 0 = 108 C C N / m 3 The values in Fig. 1 are of the same order of the values obtained by some authors using other methods (Pruppacher and Klett, Ch. 13, 1997). However, more tests are needed in order to validate the present model. REFERENCES Pruppacher, H.R. and Klett, J. D. (1997). Microphysics of Clouds and Precipitation (Kluwer Ac. Pub.), Netherlands. Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics, 2rid Ed. Wiley, N. York. Reis, A. H. and Serrano, C.(1998). Contribution of some natural and anthropogenic CCN to local fogs, Proc. Conf. Air Pollution VI, 929-938, W.I.T. Press, Boston.