Membranes - Introduction to Carbon Capture and Sequestration - page 319

Because a random walk by defi nition has uncorrelated sequential steps,
we have:

2

II = a and II = 0

ii i j ij

≠

This gives the mean squared displacement:

2

() ()

2

z N − z 0 = Na ,

where N is the number of steps in our random walk and a is the distance
between two neighboring lattice points. This number of steps is equal to
the hopping rate k (in number of steps per unit time) times the time t , or:

2

() ()

2

z N − z 0 = kta

A comparison with the relation of the diffusion coeffi cient and the
mean squared displacement shows that we can relate our self-diffusion
coeffi cient to the hopping rate, or:

1
Dka

s 2

=

This is a very useful relation as we can now relate the hopping of a mol-
ecule from one site to another to the diffusion coeffi cient.
Now, looking at the system where we have many molecules, we can
tag a particle and let that particle hop with a hopping rate k . If the parti-
cles do not see each other, we can use this relation to obtain both the
self- and Maxwell-Stefan diffusion coeffi cients from the hopping rate.
However, when the particles interact we see differences. One example
comes from the assumption that the particles only successfully hop to an
open lattice site. As a consequence, the effective hopping rate decreases
with increased loading. We observe that the self-diffusion coeffi cient
decreases as a function of the fraction of occupied sites, θ :

( )

s 0

DD = 1 − θ

It is interesting to look at the situation where we have a high loading.
If molecule i successfully jumps, then the molecules surrounding the site