Page 65 - Academic Press Encyclopedia of Physical Science and Technology 3rd Chemical Engineering

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Encyclopedia of Physical Science and Technology En001c-14 May 7, 2001 18:25

288 Aerosols

by (1) an adiabatic expansion of a gas, (2) the mixing of production rate of embryos in a homogeneous supersatu-
a warm, moist gas with a cool gas stream, or (3) chemical rated vapor is
reactions producing condensable species. In the absence
∗
I = C exp(
G /kT )
of particles in a condensable vapor, particles are formed
by homogeneous nucleation on molecular clusters in a where
G is the free energy of formation of critical-sized
∗
supersaturated vapor. When vapor supersaturation takes embryos that do not evaporate and C is a rate factor. Ac-
place in a multicomponent system, mixed particles may cording to the theory as presented in 1935 by Becker and
2
2
form from heteromolecular nucleation. When particles ex- Doring,
G =−16πσ 3 m /3(kT ) ln S. The rate factor
∗
ist in a supersaturated vapor mixture, they act as nuclei is
for condensation. Perhaps the best known of these pro- 2/3 1/2
2p 0 (n A m ) σ m
cesses is the formation of clouds in the earth’s atmosphere C = 1/2
by water condensation on dust or water-soluble aerosol (2πm A kT ) kT
particles. wheren A ismolesofcondensateAandm A isthemolecular
The theory of nucleation begins with the consideration mass of condensate A.
of vapor–liquid equilibrium. The vapor pressure p 0 over a The heteromolecular production of particles in a vapor
flat liquid surface at equilibrium is given by the Clapeyon mixture is estimated from a model similar to the homoge-
equation, neous case above. However, the production rate depends
H on the energy of formation of mixed embryos, the compo-
d ln p 0
= sition of which depends on the thermodynamic properties
dT T 2
of the mixture.
where H is the molar heat of vaporization and the gas If particles (or ions) are already present in a supersat-
constant. At saturation, the vapor pressure p 0 is its equi- urated vapor, nucleation will take place preferentially on
librium value at the temperature of the liquid beneath the these particles at supersaturations far smaller than for the
vapor. Condensation may take place when the vapor is homogeneous vapor. In this case, nucleation takes place
supersaturated or when the supersaturation ratio heterogeneously on the existing nuclei at a rate depen-
dent on the free energy of a condensate cap forming on or
S = p/p 0 > 1
around the nucleus. Heterogeneous nuclei always occur in
The vapor pressure over a pure liquid droplet at equi- the earth’s atmosphere. They are crucial to the formation
librium p s depends on its radius of curvature. The Kelvin of water clouds and to the formation of ice particles in
equation gives this relationship as: supercooled clouds.

p s 2σ m
ln =
p 0 RkT 2. Growth Laws
Thus, a supersaturation ratio greater than unity is ex- The droplet current I calculated by nucleation models
pected for small droplets at equilibrium with a condens- represents a limit of initial new phase production. The
able vapor. The logarithm of this ratio is proportional to initiation of condensed phase takes place rapidly once a
the product of the particle surface free energy σ times the critical supersaturation is achieved in a vapor. The phase
molecular volume of the liquid ( m ) and inversely propor- change occurs in seconds or less, normally limited only
tional to the particle radius R. by vapor diffusion to the surface. In many circumstances,
The rate of formation of new particles by nucleation is we are concerned with the evolution of the particle size
given in terms of a theory for the production of clusters of distribution well after the formation of new particles or the
molecules (embryos) in a supersaturated vapor. The pro- addition of new condensate to nuclei. When the growth or
duction of embryos follows from considering a population evaporation of particles is limited by vapor diffusion or
of molecules and agglomerates of molecules in a homoge- molecular transport, the growth law is expressed in terms
neous vapor. There is an energy barrier to producing large of vapor flux equation, given by Maxwell’s theory, or
agglomerates that depends on the free energy of formation 2
n∂ n4π D AB R m (p ∞ − p s )
of a cluster containing a given number of molecules. As
I( , t) = =
the supersaturation of a vapor is increased, the free energy ∂t kT
of formation passes through a maximum value such that Other growth processes are also derived from theory.
a critical level is attained. Beyond the critical value, em- They include those associated with chemical reactions to
bryos of critical size are generated that are stable and grow. form condensed species taking place either at the parti-
The “steady” rate of production of embryos of critical size cle surface or within the particle volume. The growth by
represents the expected production of new particles. The surface reactions or a vapor diffusion-limited process is