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

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 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 )

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* =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

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 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