Particle Dynamics: Brownian Diffusion
Prof. Sotiris E. Pratsinis
Particle Technology Laboratory
Department of Mechanical and Process Engineering,
ETH Zürich, Switzerland
www.ptl.ethz.ch
1
or
or
or
or
nucleation
inception
condensation
surface growth
v
evaporation
flocculation
coalescence
aggregation
sintering
chemical bonding
Aerosol-based Technologies in Nanoscale Manufacturing:
from Functional Materials to Devices through Core
Chemical Engineering, AIChE J., 56, 3028-3035 (2010)
agglomeration
attachement
physical adhesion
v
or
Particle Dynamics
Coagulation
Fragmentation
Convection
in
Shrinking
Growth
by evaporation
or dissolution
by condensation
or chemical reaction
Convection
out
Diffusion
Settling
3
Theory: Population Balance Equation
n
t
  n u
   Dn
convection diffusion


  dv 
n 
v  dt 
growth
  cn
external force

1v ~
~
~
~
~








v
,
v

v
n
v
n
v

v
d
v
 v, ~v nv n~v d~v

20
0
coagulation

 Sv nv     v, ~
v Sn~
v d~
v
v
fragmentation
u x , u y , uz
u
= gas velocity vector
D
c

= particle diffusivity
S
= fragmentation rate

= fragment size distribution
  n u  u n  n  
u

= velocity of particles of size v (e.g. settling)
0
continuity
= coagulation rate
4
Mean Free Path: Continuum vs. Free-molecule regime
The mean free path of a gas, , is the average distance
traveled by gas molecules between their collisions.
When particles are much larger than  (e.g. dp > 10), they
do not sense individual collisions with molecules feeling a
“continuum” so particle motion takes place in the so-called
continuum regime and described by the standard or classic
Navier-Stokes equations, the gospel of engineering.
When particles are much smaller than (e.g. dp < ),
they are in the so-called free-molecule regime and their
motion is described by the kinetic theory of gases.
Inbetween, interpolations are devised specific to the
process (e.g. for diffusion, coagulation or condensation)
5
1. DIFFUSION
Particles suspended in a fluid medium exhibit a haphazard dancing motion
Botanist Robert Brown discovered that this motion was a general property
of matter regardless of its origin (dust vs. pollen)
Hands-on experiment resembling the motion of oil droplets in water with ball
bearings and metal rings on a vibrating table at the Museum of Fine Arts at
the San Francisco Exploratorium organized by Dr. Frank Oppenheimer
(brother of Robert the father of the Atomic Bomb)
6
Diffusion is the net migration of particles from regions of HIGH to LOW concentration
Net rate of transport into that element:
Friedlander, S.K., Smoke, Dust and Haze, Chapter 2, Oxford Press, 2nd Edition, New York, 2000
7
The rate of change of the number of particles per unit
volume (& size), n, in the elemental volume δxδyδz is:
From experimental observations:
Fick’s first law
Substituting in the above gives second Fick’s law:
Coefficient of Diffusion or Diffusivity, D
8
D = f (particle size and gas properties)
Consider particle transport in one dimension, x
Release equally sized particles, N0, at t=0
and observe the n distribution in space and time
For the boundary conditions at x = 0, x = ∞ & t =0,
the particle concentration distribution in x,t is
9
The mean square displacement of the particles from x=0 at time t is:
We can measure
chequered glass.
by putting spheres in a liquid and follow their motion through a
10
The goal is to relate the mean square displacement of a particle with the energy
required for this “job”.
Force balance on a particle in Brownian motion:
Now multiply both sides of eq. (5) by the displacement x and divide by m.
For a single particle:
x
du
f
F (t )
  ux 
x
dt
m
m
(6)
11
define as β = f/m and A = F(t)/m and remember that:
Using these expressions eq. (6) becomes
Integrate from t=0 to t and obtain:
where t′ is a variable of integration representing time.
12
Average over all particles:
Since there is no correlation between displacement x and “kick”, A, the second term
of eq. (7) vanishes:
You can also write:
13
Because the derivative of the mean over particles with respect to time is equal to the
mean of the derivative:
From eq. (8) & (9):
Integrate over time from t = 0 to t
for t >> 1/β (or β t >> 1):
14
Invoke the equipartition of energy, meaning that the kinetic energy of particles is
equal to the kinetic energy of the surrounding gas molecules:
This is the Stokes-Einstein expression for D.
It relates D to the properties of the fluid and the particle through the friction coefficient.
15
Perrin’s (1910) study allowed calculation of the Avogadro number NAV . By observing
the motion of an emulsion he calculated the number of (attacking) molecules:
where R is the gas constant
He gave an experimental proof of the kinetic theory by measuring the net displacement
Modern methods show that NAV=6.023×1023 molecules/mol
16
Friction coefficient
mean free path of gas medium
with ρ : density of the medium (e.g. air)
m1: molecular mass of the medium
In the continuum regime (dP >> λ ): f = 3dP
In the free molecular regime (dP << λ ):
with a: accomodation coefficient ≈ 0.9
In the entire range:
with C: Cunningham correction factor
17
Coefficients A1, A2, and A3 are empirical constants that have been obtained by
measuring the settling velocities of particles in various gases. Table 1 gives these
constants for various gases (Rader, D. J., Momentum slip correction factor for small
particles in nine common gases, J. Aerosol Sci., 21 (1990), 161-168)
The ratio of the mean free path of the gas and the particle radius is the Knudsen number
Kn = 2λ/dp. The Cunningham correction factor does not change very much with different
gases for the same Kn (Table 2).
18
Diffusion and sedimentation dominate the particles‘
motion at opposite size regimes (Table 3).
19
DIFFUSION during LAMINAR PIPE FLOW
Entrance length: L = 0.04dRe (laminar flow)
d : pipe diameter
Re : Reynolds-number
The momentum boundary layer develops rapidly while the concentration boundary
layer follows:
20
Separation of variables:
Result:
So the average particle concentration is defined as:
And in general it is given as:
Where Gk and λk2 are given in Table 3.1 in the book by Friedlander (1977).
The above ratio is called also the particle penetration, P, and it is defined as:
21
Penetration curves for particle diffusion to pipe walls
Penetration versus deposition parameter  for circular tubes and rectangular cross section
channels: D: particle Diffusivity, L: tube or channel length, Q: gas flowrate, h: interplate
distance, W: channel width (Hinds, 1982)
22
23
Rowell, J. M., Scientific
American, October 1986, 147.
MacChesney, J. B., O’Connor, P. B. and Presby, H. M., 1974, A
new technique for preparation of low-loss and graded index optical
fibers. Proc. IEEE 62, 1280-1281.
J. Aerosol Sci., 20,
101-111 (1989)