Download DIFFUSION IN AN EXTERNAL POTENTIAL Here we consider

DIFFUSION IN AN EXTERNAL
POTENTIAL
Here we consider diffusion in an external potential. This is an important issue in physical
chemistry and biology. The external potential can be gravity in connection with sedimentation or the electric potential in connection with electrolytes etc. We also derive
Einsteins relation in another way.
The diffusing particles are subject to an external force fext(x) and a viscous drag force
fd.
The external force is given by
fext(x) = −∇U (x)
where U (x) is the external potential.
The drag force is given by Stokes law
fd = −αvd
Here vd is the drift velocity and the friction
constant is given by
α = 6πηR
where η is the viscosity of the medium and R
the radius of the sphere.
In a steady state the external force and the
drag force must balance, i.e.,
fd + fext = 0
or
−αvd + fext = 0
or
f
1
vd = ext = − ∇U
α
α
The drift current is given by
jd = nvd
where n is the density.
The diffusion current is given by Fick’s law
jdiff = −D∇n
The total current is then given by the diffusion current plus the drift current, i.e.,
jtot = jdiff + jd
or inserting
1
jtot = −D∇n − n ∇U
α
In thermodynamic equilibrium jtot = 0 and
the equilibrium density neq(x) is given by the
Boltzmann distribution
neq(x) ∝ e−U (x)/kT
i.e.,
1
−D∇neq − neq∇U = 0
α
or
1 −U/kT
D
−U/kT
∇U e
− e
∇U = 0
kT
α
This equation is satisfied provided
kT
kT
D=
=
α
6πηa
which is Einstein’s relation connecting D to
microscopics (Einstein 1905)