Interfacial transport
• So far, we have considered size and motion of
particles
• In above, did not consider formation of particles
or transport of matter between vapor and
particulate phase
• Interfacial transport
– formation of aerosols by nucleation
– growth by condensation
– loss by evaporation
Definitions
• partial pressure - PA pressure that a vapor in a
mixture of gases would exert if it were to occupy,
(all by itself) the entire volume occupied by the
mixture.
• volume fraction of gas A = PA/Ptotal
• saturation vapor pressure - PS if you had a sealed
container containing liquid or solid A, the partial
pressure of vapor phase A in equilibrium with the
flat surface of liquid or solid at the T of the system
• saturation ratio S = PA / PA, equilibrium also known as
relative humidity for air/water systems
Two types of nucleation
• when the concentration of vapor is greater than the
saturation vapor pressure, formation of the liquid
or solid phase is thermodynamically favorable
• homogeneous nucleation - condensation of a vapor
takes place only on clusters of like molecules
• heterogeneous nucleation - condensation occurs
on a dissimilar cluster
Energy balance on a newly
forming particle
 G   v   L n   d p 
2
fv , f
L
= free - energy potential per molecule in vapor
and liquid phase,
in the drop,
n  total number of molecules contained
g = surface tension
In forming droplet, surface free energy went from zero to d2, a +
contribution to free energy, but phase change of molecules to
favored liquid phase is a (-) contribution to free energy. Imagine
the partial pressure of the vapor near the droplet is changed by a
small amount. droplet of size d in a supersaturated vapor.
–
After some substitutions and manipulations:
3



d
N
2
p
A
 G   d p   kT ln S 
 p 

M  6

S  saturation ratio
N A  Avogadro'
s number
M = molecular weight
Shape of G vs dp
Critical drop size, d*
If another molecule is added by condensation, G will go down
The Kelvin effect
• curvature modifies attractive forces between surface
molecules - the smaller the droplet, the easier it is for
molecules to leave the surface
• to maintain mass equilibrium, the equilibrium vapor
pressure over a curved surface is greater than that for over
a flat surface
• Rearranging to solve for S, for droplets of diameter d*, the
equilibrium vapor pressure over the droplet surface, pd, is
given by:
 4  M
p d  p S exp 

 p RTd


* 

Implications
• A pure liquid drop will always evaporate when S <
1
• Even if supersaturation exists, droplets smaller
than the critical size under those conditions will
evaporate
• Since smaller droplets (< d*) may evaporate under
supersaturated conditions, large droplets may
grow at the expense of small ones
S - 0.9
Capillary condensation Kelvin equation in reverse!!
Simulations for neck region
S = 0.95
between nanoparticles using
lattice gas stat thermo modeling.
S=1
Seonmin Kim, graduate student in
my group
Homogeneous nucleation
• even in unsaturated vapor, attractive forces between
molecules lead to cluster formation, and a distribution of
cluster sizes exists
• with more vapor, this distribution shifts towards larger
sizes
• free energy of droplet is given by:
NA   3 
 G   d   ( k T ln S )
 d 

M 6
2
where  = surface tension, d = droplet, M = molecular weight of liquid in
drop, NA = Avogadro’s number,  = droplet density
More material - probability of larger
clusters increases
Homogeneous nucleation con’t
• thermodynamics says that the system will go towards
direction of decreasing free energy of system
4 M
• recall
*
d 
 RT ln S
• for any given T, S, growth is favorable for clusters with d >
d* (the critical nucleus diameter)
• the greaterthe S, the smaller the critical nucleus diameter
• rate of nucleation given by (“classical theory”):
 2 N A M    p  
  G 
 
 S exp 
 RT 
 kT 
1/2
J 

 d 

2
p  = saturation pressure over a plane of the liquid
  constant,
usually taken as 1
kinetic -vs-activated nucleation
• For some systems, S can be extremely high, and d* <
diameter of a molecule
• example: formation of refractory powders where
chemical reaction is fast, and saturation vapor
pressures are low
• If this is the case, nucleation is said to be kinetic,
limited only by rates of collisions between molecules,
not by formation of clusters of critical size
• nucleation discussed earlier - activated
• kinetic nucleation can lead to some model
simplifications
Example problem: kinetic or activated?
• consider silica at 1720 K, forming by rapid
chemical reaction of a precursor in a flame
• data: flame concentration of silica = 1 x 10-5 moles/liter
flame gas at STP, 0.3 J /m2 surface tension, 60 g/mole, 2.2
g/cm3 density, equilibrium vapor pressure 4 x 10-9 bar
Heterogeneous nucleation
• how raindrops are formed- condensation of water
vapor onto so called ‘condensation nuclei’
• heterogenous nucleation requires much lower
saturation ratios than homogenous nucleation
• free molecular growth - governed by rate of
random molecular collisions between particle and
vapor molecules
• molecules may or may not stick, c is the
fraction that stick, uncertainty as to the value
(sometimes a value of 0.04 used)
Growth laws for condensation
• for growth in free molecular regime
d d 
2 M ( p  p )
p
dt
p

c
o
d
1/2
 p N A (2  mkT )
o is partial pressure of vapor in gas surrounding
droplet, pd is partial pressure of vapor at surface of droplet

• for growth in the continuum regime, growth
depends on rate of diffusion of droplet
molecules to droplet surface
Growth laws for condensation
• rate of particle growth given by:
(obtained for an isolated droplet)
d d p 
dt
where
2 D v M  p o
pd

 
R  p d p T
Td



 = Fuch' s correction factor
=
2 + d p
d p  5 .33  / d p  3 .42 
2
• correction factor is needed because diffusion equation breaks
down within one mean free path of the surface, and growth
becomes controlled by kinetic processes

Sources of condensable species
• Chemical reaction - if species formed has
lower vapor pressure than precursor, and
reaction rate is relatively fast compared to
nucleation process
• Physical - cooling via expansion or mixing
with cold stream
Aerosol formation and growth
• to summarize: processes important for
describing aerosol formation and growth
–
–
–
–
nucleation
condensation/evaporation
coagulation
coalescence
An aerosol generator for production of metal nanoparticles
1.E+19
1.E+18
-1
1.E+16
-1
nuc le a t ion ra t e , log s c a le ( # k gg a s s )
1.E+17
1.E+15
1.E+14
1.E+13
1.E+12
1.E+11
Indium 900C
1.E+10
1.E+09
Indium 1000C
1.E+08
1.E+07
Indium 1100C
1.E+06
1.E+05
1.E+04
1.E+03
1.E+02
1.E+01
1.E+00
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distance (cm)
Predicted nucleation rates as a function of distance accounting for
nucleation, condensation, coagulation (not published)
This assumes 1-D temp and velocity profiles in the tube
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