Kelvin Effect: Physical Cartoon Equilibrium vapor pressure is higher over a curved surface than a flat one. Important for nucleation of new particles, lifetime of small droplets.
Kelvin Effect: Energy of Droplet Formation Energy of converting vapor molecules to bulk liquid molecules Energy required to maintain a liquid-gas surface boundary Xv Xl
S = ratio of eq. vapor pressure around a droplet relative to to above a flat surface S<1 S>1 G For a droplet to exist, S>1. pcurved > pflatalways Rp* Radius at which G maximizes and beyond which droplet formation becomes possible How does S vary with Rp?
Kelvin Equation Relates molecular properties (molecular weight, surface tension, density) to the degree to which v.p. over curved surface is enhanced
Questions • Some organic compounds are highly surface active. That is, they prefer to reside at the gas-liquid interface, and lead to a lower surface tension. By how much would S change if the surface tension of a droplet changed from 75 dynes (pure water) to 35 dynes (surfactant coated water)? • Do you have a physical explanation to the above answer? • What is the surface tension of a cluster of 10 H2SO4 molecules and 10 H2O? Is it the same as the surface tension for a 50 wt% H2SO4 bulk solution?
Continuum versus Free Molecular Dynamics Transition Regime Rp Rp Free Molecular or Non-continuum Regime Continuum Regime
Mean Free Path In 1 second: red has swept through a volume For N molecules per cm3, red-blue collisions per sec ~ The distance traveled between collisions:
Mean Free Path Increases with Altitude For 10 nm particle: Kn <<1 Free Molecular For 1 um particle: Kn >1 Continuum For 0.2 um particle depends on altitude… Continuum to transition
Question What is a physical explanation for the mean free path being related to diffusivity and gas viscosity? Does the atmosphere’s viscosity depend on pressure? Temperature? Does the diffusivity of air depend on pressure? Temperature?
Terminal Settling Velocity After induction time , drag will balance gravity Velocity that results from this balance is the settling (or terminal) velocity • < 0.01 sec for all atmospheric aerosols v = mpgCc/3pimuDp
Particle Mobility We can generalize the gravitational settling problem. A particle experiencing an external force will accelerate until its velocity leads to a drag force opposing the external force. A steady-state terminal velocity can be found by balancing the external forces with the drag force Fext = 3pimuDpv V = Fext/3pimuDp Fext: gravity, electric field, random collisions by fluid…
Brownian Motion Collisions between particle and randomly moving molecules leads to irregular-”jerky” particle motion As Kninf: random walks are more the norm Mean square displacement proportional to time
Particle Diffusivity Mean square displacement in 1-D, by diffusion: <x2> = 2Dt D = kBTCc/3pimuDp
Gravitational Settling vs. Diffusion In 1 second, how far does a 1 micron particle move due to gravity? How far does it move by diffusion? What about for a 0.1 micron particle? What do you conclude about the relative importance of gravitation settling for big and small particles?