2. Brownian motion, colloidal suspensions

1
BROWNIAN MOTION
1. Diffusion equation
At the simplest level the movement of a particle in a liquid is
described by a Markovian random walk. The particle’s
displacement distribution (PDD)—probability density of a jump,
∆r during time interval τ:
P( ∆r
, τ)
⎡2π
⎤
=
⎢
w( τ)
⎣ 3 ⎥
⎦
−3 / 2
⎡
exp⎢−
⎢⎣
2
3∆r
⎤
⎥
w( τ)
⎥⎦
Mean squared displacement,
w( τ)
=
∆r
2
( τ)
=
6Dτ
P(∆r,τ) derives from the diffusion equation
∂
∂t
P 2
= D∇
P
with the initial condition P(∆r,0)=δ(∆r)
D is the diffusion coefficient

2
Displacement in time interval τ;
∆
τ
r ( τ)
= ∫ v(t)dt
, leads to [BP];
0
∆r
2
( τ)
=
=
2
τ
∫
0
∞
2τ
dt( τ − t) v(0)v(t)
∫
0
dt
v(0)v(t)
(B.1)
(since τ can be arbitrarily large)
and
1
D = Z(t)dt with Z(
τ ) = v(0)v( τ)
3
∫ ∞
0
(B.2)
the velocity auto-correlation function.

3
2. Langevin equation
Force on particle is a delta correlated random force, R(t), and
friction, -ζv;
dv
M
dt
=
−ςv
+
R(t)
R(t)
= 0
R(0)
R(t)
v(0)
.R(t)
=
=
6kTζδ(t)
0
B.3
Formal solution;
t
⎡ ⎛ ζ ⎞ ⎤
⎡ ⎛ ζ ⎞ ⎤
v(t)
= v(0)exp⎢−
⎜ ⎟t⎥
+ ∫ dt' R(t')exp⎢−
⎜ ⎟(t
− t' ) ⎥
⎣ ⎝ M ⎠ ⎦
⎣ ⎝ M ⎠ ⎦
0
∴
Z( τ)
=
v
2
(0)
⎡ ⎛ ζ ⎞ ⎤
exp⎢−
⎜ ⎟τ⎥
⎣ ⎝ M ⎠ ⎦
=
3kT ⎡ ⎛ ζ ⎞ ⎤
exp⎢−
⎜ ⎟τ
M
⎥ ⎣ ⎝ M ⎠ ⎦
(B.4)
And
D
∞
kT ⎡ ⎛ ζ ⎞ ⎤
= ∫ dt exp⎢−
⎜ ⎟t
M
⎥
⎣ ⎝ M ⎠ ⎦
0
=
kT
ζ
.

4
The last result
D = kT is the Einstein relation
ζ
From B.1 and B4;
∆r 2 ( τ)
⎧ ⎡ ⎛ ζ ⎞ ⎤⎫
= 6Dτ−
6D(M / ζ)
⎨1
− exp
⎢
− ⎜ ⎟τ
⎥⎬
⎩ ⎣ ⎝ M ⎠ ⎦⎭
One recovers the following limits
(i) ballistic
∆ r
2
( τ
>
(M / ζ))
= 6Dτ−
6MkT
ζ
2
= 6Dτ−
cons tan t
What does all this mean
1. Brackets, ........ , imply ensemble average, OR if all particles
are statistically equivalent, average over initial times “0”.
This expresses ergodicity.
2. Impulsive thermal force in L.E. is inconsistent with steady
motion. So only the asymptotic results are correct.
3. The above applies only freely diffusing particles.

5
Some numbers
Particle of radius a=50nm, M=10 -19 kg. in water’
Persistence time τ l = M/ζ = M/(6πηa) ≈ 10 -9 s.
3kT
2M
2
1 1 6
RMS thermal velocity; v RMS = v = ≈ 10
− −
ms ≈ 10 radii,
But, distance moved is v RMS x τ l ≈ 10 -10 m ≈ 10 -3 radii.
Thus, the particle gets nowhere due to viscous friction.
Consequences
1. Motion strongly overdamped; use low Reynolds number
hydrodynamics
2. For τ>>τ=M/ζ particles have lost inertia (individually), but
not memory of instantaneous momenta.
3. More useful time scale; time to diffuse RMS distance of one
radius;
τ b = a 2 /(6D)
4. Above all, Einstein’s reasoning; thermal motion of liquid
molecules, and the attendant hydrodynamic flows in the
liquid, must be transmitted to (visible) particles in the
liquid.
Q. What if the particle concentration is increased so that
packing constraints become significant

6
HYDRODYNAMICS
A local spontaneous, thermally activated, momentum fluctuation
dissipates by sound and viscous flow, respectively propagating
longitudinal and diffusing transverse momentum currents.
In the limit, τ>>τ=M/ζ, sound propagates instantaneously. This
subjects particles to delta correlated thermal force as per L.E.
Momentum diffusion (simple dimensional argument); after time
t a given particle’s initial velocity, v(0), is shared (equally)
among ρV(t) particles in volume V(t). So, at time t all particles,
including the given particle, have increased their velocity by 1/(
ρV(t)). V(t) increases by diffusion; V(t)~t 3/2 . Thus, memory of
a particle’s instantaneous velocity decays as t -3/2 , and
Z( τ )
=
hτ
−3/
2
is the form that characterizes fully developed viscous flow—
diffusing vorticity.

Statistically averaged velocity field around a central disc from
MD (large arrows) compared with that for the hydrodynamic
model (small arrows). From B.J. Alder and T.E. Wainwright,
Phys. Rev. A 1, 18 (1970).
7