Biomedical Engineering Reference
In-Depth Information
Q
A
Q
B
R
1
R
2
(4πε
0
)
U
AB
=
+
(
Q
B
p
A
cos θ
A
−
Q
A
p
B
cos θ
B
)
p
A
p
B
R
3
−
(2 cos θ
A
cos θ
B
−
sin θ
A
sin θ
B
cos ϕ)
2
R
3
Q
A
q
B
3 cos
2
θ
B
−
1
+
Q
B
q
A
3 cos
2
θ
A
−
1
+···
1
+
(2.7)
The physical interpretation is as follows. The first term on the right-hand side gives the
mutual potential energy of the two charged molecules A and B. The second term gives
a contribution due to each charged molecule with the other dipole. The third term is a
dipole-dipole contribution, and so on.
If A and B correspond to uncharged molecules, then the leading term is seen to be the
dipole-dipole interaction
p
A
p
B
R
3
(4πε
0
)
U
AB,dip...dip
=−
(2 cos θ
A
cos θ
B
−
sin θ
A
sin θ
B
cos ϕ)
(2.8)
The sign and magnitude of this term depend critically on the relative orientation of the two
molecules. Table 2.1 shows three possible examples, all of which have φ
=
0.
Table 2.1
Representative dipole-dipole terms for two diatomics
θ
A
θ
B
Relative orientations
Expression for dipole-dipole
U
2
p
A
p
B
4
πε
0
R
3
0
0
Parallel
−
2
p
A
p
B
4
πε
0
R
3
0
π
Antiparallel
+
0
π
/2
Perpendicular
0
2.5 Taking Account of the Temperature
We now imagine that the two molecules undergo thermal motion; we keep their separation
R
constant but allow the angles to vary. The aim is to calculate the average dipole-dipole
interaction. Some orientations of the two dipoles will be more energetically favoured than
others and we allow for this by including a Boltzmann factor exp(
U
/
k
B
T
), where
k
B
is
the Boltzmann constant and
T
the thermodynamic temperature. It is conventional to denote
mean values by angle brackets < ... >and the mean value of the dipole-dipole interaction
is given formally by
−
U
AB
exp
dτ
U
AB
k
B
T
−
U
AB
dip...dip
=
exp
dτ
(2.9)
U
AB
k
B
T
−
