2.57 Fall 2004 – Lecture 24
1
2.57 NanotoMacro Transport Processes
Fall 2004
Lecture 24
In the last lecture, we have talked about Einstein’s work on the Brownian motion.
In the left figure above, pressure difference exists in the fluid. For one particle, the
osmotic pressure is determined by
1
BB
PN
k
T
n
k
T
V
==
.
Also, we can use the solution of Stoke’s flow around a sphere to estimate the drag force.
The value is
3
FD
u
π
µ
=
. For area A
c
, the forced balance over the control volume gives
() ( )
()
c
A Px Px d
x
F
dN
−+=
,
30
c
dP
A dx
D udN
dx
πµ
−⋅
−
=
,
dP
Du
n
dx
−−
=
,
B
dn
kT
D un
dx
=
,
in which the product
un
indicates flux. We have
3
B
p
dn
dn
Ja
Dd
x
d
x
=−
,
where a=
3
B
D
is the diffusivity and can be obtained by the diffusion experiment of
some materials. In a time t, the diffusing radius is
6
ra
t
∆=
. We can calculate the
constant
a
and substitute back into a=
3
B
D
, thus D can be obtained.
The relationship
between thermal diffusivity and viscosity is an example of the more general fluctuation
dissipation theory as viscosity is a measure of dissipative process and diffusivity is a
measure of random walk (fluctuation) process.
The relationship between thermal
Stoke’s flow
3
u
=
P(x)
P(x+dx)
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2
diffusivity and viscosity is also called Einstein relation.
In chapter 6, there is a similar
Einstein relation between electron diffusivity and mobility.
But Einstein really worked
on the Brownian motion.
Note: The osmotic pressure can be observed by putting a semipermeable membrane such
that only the base molecules in the solution penetrate the membrane, building up a
concentration gradient on the two sides of the membranes.
In addition to the above approach, Einstein also established another method to determine
the diameter of the solute particles.
He proposed to measure the viscosity of the solvent
and of the solution,
µ
, and
b
,
respectively, and derived, again assuming dilute solute
particles, the following relationship between the viscosities
12
.
5
o
ϕ
=+
,
where
is the volumetric concentration of the solute molecules.
The Einstein relation can also be derived from the stochastic approach developed by
Langevin to treat Brownian motion of particles much larger than that of the surrounding
medium.
The key idea of the Langevin equation is to consider that the motion of a
Brownian particle is subject to a friction force that is linearly proportional to its velocity,
as in Stokes law, and a random driving force (or “noise”), R(t), imparted by the random
motion of the molecules in the bath.
In the absence of an external force, the Langevin
equation that governs the instantaneous velocity of the Brownian particle can be written
as,
()
d
mm
t
dt
η
=−
+
u
uR
,
where
is the friction coefficient, and for Brownian particles in a fluid the Stokes law
gives
3
FD
u
π
=
, so that the random driving force R(t) has the following characteristics:
0
t
=
R
(average of random driving force is zero)
() ()
0
tt
•=
Ru
(random driving force is not correlated to the velocity)
(
)
2
o
ts
s
R t
πδ
+•
=
RR
.
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 Fall '04
 GangChen

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