Environmental Engineering Reference
In-Depth Information
As our particle is moving in a fl uid that is composed of identical par-
ticles, this diffusion coeffi cient is called the self-diffusion coeffi cient.
Interestingly, one can label a particle experimentally using NMR spec-
troscopy methods, and thus NMR has become the principal method
used to measure the self-diffusion coeffi cient. Also, in a molecular simu-
lation one can follow a single particle and from the mean squared dis-
placement get the self-diffusion coeffi cient directly from simulations.
To derive
D
MS
, the Maxwell-Stefan diffusion coeffi cient, rather than
label a single particle, we monitor the movement of the center of mass of
the entire system. The center of mass is defi ned as:
N
N
1
1
()
()
()
∑
∑
zt
=
mzt
=
zt
,
cm
i
i
mN
N
i
=
1
i
=
1
where
m
is the mass of a single molecule and N is the total number of
molecules. The Maxwell-Stefan diffusion coeffi cient is related to the
mean squared displacement of the center of mass by:
2
11
()
( )
MS
∑∑
D
lim
z
t
z
0
=
−
i
i
2
tN
t
→∞
i
i
Let us look at the term in brackets:
2
(
)
(
)
(
)
()
( )
()
( )
()
( )
()
( )
∑∑
...
zt
−
z
0
=
zt
−
z
0
zt
−
z
0
+
zt
−
z
0
+
i
i
1
1
1
1
2
2
i
i
(
)
(
)
(
)
()
( )
()
( )
()
( )
...
...
+
zt z
−
0
zt z
−
0
+
zt z
−
0
+
+
2
2
1
1
2
2
(
)
(
)
2
(
)
()
( )
()
( )
()
( )
∑
∑ ∑
=
zt z
−
0
+
zt z
−
0
zt z
−
0
i
i
i
i
j
j
i
i
j
≠
i
We see that the fi rst term gives the expression for the self-diffusion coef-
fi cient. The second term accounts for the correlations between the mol-
ecules. If the molecules
i
and
j
do not infl uence each other this term will
be zero. In other words, correlations between molecules cause the
Maxwell-Stefan coeffi cient to differ from the
self-diffusion coeffi cient.
Such correlations are far more likely at high concentrations of diffusing
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