Interfacial transport So far we have considered

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 In forming droplet, surface free energy went from zero to pd 2 g, 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: Shape of DG vs dp

Critical drop size, d* If another molecule is added by condensation, DG 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:

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: where g = surface tension, d = droplet, M = molecular weight of liquid in drop, NA = Avogadro’s number, r = 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 • 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”):

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 /m 2 surface tension, 60 g/mole, 2. 2 g/cm 3 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, ac 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

Growth laws for condensation • rate of particle growth given by: (obtained for an isolated droplet) • 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

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