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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
€
ΔG = φv −φL( )n + πdp2γ
fv, fL = free - energy potential per molecule in vapor
and liquid phase, n = total number of molecules contained
in the drop, 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:
€
ΔG = πdp2γ − kT lnS( )
NAM
πdp3
6ρ p
⎛
⎝ ⎜
⎞
⎠ ⎟
S = saturation ratio
NA = Avogadro' s number
M = molecular weight
Shape of ΔG vs dp
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:
€
pd = pS exp4γM
ρ pRTd*
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
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
Capillary condensation -
Kelvin equation in reverse!!
Simulations for neck region
between nanoparticles using
lattice gas stat thermo modeling.
Seonmin Kim, graduate student in my group
S - 0.9
S = 0.95
S = 1
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:
where = surface tension, d = droplet, M = molecular weight of liquid in drop, NA = Avogadro’s number, = droplet density
ΔG d kT SN
MdA= −
⎛⎝⎜
⎞⎠⎟π γ
πρ2 3
6( ln )
Homogeneous nucleation con’t
• thermodynamics says that the system will go towards direction of decreasing free energy of system
• recall
• 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”): €
d* =4γM
ρRT lnS
€
J =α
ρ d
2NAMγ
π
⎛
⎝ ⎜
⎞
⎠ ⎟
1/ 2p∞
RT
⎛
⎝ ⎜
⎞
⎠ ⎟2
S exp -ΔG
kT
⎛
⎝ ⎜
⎞
⎠ ⎟
p∞ = saturation pressure over a plane of the liquid
α = 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
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
€
d dp( )
dt=
2Mα c (po − pd )
ρ pNA (2πmkT)1/ 2
po
Growth laws for condensation
• rate of particle growth given by:
(obtained for an isolated droplet)
€
d dp( )
dt=
2DvM
Rρ pdp poT∞
−pdTd
⎛
⎝ ⎜
⎞
⎠ ⎟φ
where φ = Fuch's correction factor
= 2λ + dp
dp + 5.33λ2 /dp + 3.42λ• correction factor is needed because diffusion equation breaksdown 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
1.E+001.E+011.E+021.E+031.E+041.E+051.E+061.E+071.E+081.E+091.E+101.E+111.E+121.E+131.E+141.E+151.E+161.E+171.E+181.E+19
34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
distance (cm)
nucleation rate, log scale (# kg
gas
-1 s-1)
Indium 900C
Indium 1000C
Indium 1100C
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