Biomedical Engineering Reference
In-Depth Information
language of quantum mechanics. Nevertheless, the following qualitative discussion is to
be found in all elementary texts:
The electrons in an atom or molecule are in continual motion, even in the ground state. So,
although on average the dipole moment of a spherically symmetrical system is zero, at any
instant a temporary dipole moment can occur. This temporary dipole can induce a further
temporary dipole in a neighbouring atom or molecule and, as in the case of the inductive
interaction, the net effect will be attractive.
Paul K. L. Drude gave a simple quantum mechanical description of the effect, and his
theory suggests that the dispersion contribution can be written
D
6
R
6
D
8
R
8
D
10
R
10
U
disp
=−
+
+
+···
(2.16)
The first term (that I have written
D
6
) is to be identified with the instantaneous dipole-
induced dipole mechanism. The higher terms are caused by instantaneous quadrupole-
induced quadrupoles, etc. According to Drude's theory
3α
2
ε
1
4 (4πε
0
)
2
D
6
=−
(2.17)
In this expression, ε
1
is the first excitation energy of the atomic or molecular species con-
cerned. Once again I apologise for the clash of symbols, for ε
0
is of course the permittivity
of free space.
The dispersion energy is again seen to be attractive and to fall off as 1/
R
6
.
2.8 Repulsive Contributions
When two molecular species approach so closely that their electron clouds overlap, the
positively charged nuclei become less well shielded by the negative electrons and so the
two species repel each other. The repulsive term is sometimes written
U
rep
=
A
exp (
−
BR
)
(2.18)
where
A
and
B
are specific to the particular molecular pair and have to determined from
experiment. The precise form of the repulsive term is not well understood; all that is certain
is that it must fall off quickly with distance, and the exponential function is therefore a
possible suitable candidate.
The total interaction is
U
=
U
rep
+
U
dip...dip
+
U
ind
+
U
disp
, which we can write
C
R
6
=
−
−
U
A
exp(
BR
)
(2.19)
