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 ratioNA = Avogadro' s numberM = molecular weight

Shape of ΔG vs dp

Critical drop size, d*

If another molecule is added by condensation, ΔG 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:

€

pd = pS exp 4γMρ 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 SNM

dA= − ⎛⎝⎜

⎞⎠⎟π γ

πρ2 3

6( ln )

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• 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* = 4γMρRT lnS

€

J =αρ d

2NAMγπ

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/ 2 p∞

RT ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

S exp - ΔGkT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

p∞ = saturation pressure over a plane of the liquidα = 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

= 2DvMRρ 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

An aerosol generator for production of metal nanoparticles

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 50distance (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