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.

3

244p

l v pl

RG g g R

Energy of converting vapor molecules to bulk liquid molecules

Energy required to maintain a liquid-gas surface boundary

Xv Xl

Kelvin Effect: Energy of Droplet Formation

Rp*

Radius at which G maximizes and beyond which droplet formation becomes possible

S<1

S>1

G

S = ratio of eq. vapor pressure around a droplet relative to to above a flat surface

For a droplet to exist, S>1.pcurved > pflat

alwaysHow does S vary with Rp?

2 2exp expcurved

lflat

B p p

vp MSp k TR RTR

Kelvin Equation

Relates molecular properties (molecular weight, surface tension, density) to the degree to which v.p. over curved surface is enhanced

Questions1. 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)?

2. Do you have a physical explanation to the above answer?

3. 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?

Rp

Rp

Free Molecular orNon-continuum Regime Continuum Regime

Transition Regime

Continuum versus Free Molecular Dynamics

Mean Free Path

In 1 second: red has swept through a volume 2redr

For N molecules per cm3, red-blue collisions per sec ~ redN

The distance traveled between collisions:1

redN

0.05 0.1 0.15 0.2 0.25 0.30

2

4

6

8

10

12

Mean Free Path (microns)

Alti

tude

(km

)

Mean Free Path Increases with Altitude

For 10 nm particle: Kn <<1

For 1 um particle: Kn >1

For 0.2 um particledepends on altitude…

Free Molecular

Continuum

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 VelocityAfter induction time , drag will balance gravity

Velocity that results from this balance is the settling (or terminal) velocity

v = mpgCc/3pimuDp < 0.01 sec for all atmospheric aerosols

Particle MobilityWe 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

V = Fext/3pimuDp

Fext = 3pimuDpv

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

D = kBTCc/3pimuDp

Mean square displacement in 1-D, by diffusion: <x2> = 2Dt

Gravitational Settling vs. DiffusionIn 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?