than s. Conversely, the overlap repulsion becomes large rather quickly
when the distance between the two atomic centers is less than s. Figure
5.9b depicts a graph of V(r) versus r for the soft sphere power law model.
Thus far in our discussion of overlap repulsion, we have operated under
the assumption that the atomic or molecular geometry is essentially
spherical. While this assumption is relatively sound for atoms and some
small molecules (CH 4 , for instance, can be modeled as nearly spherical),
most molecules possess other geometries. The concepts of overlap
repulsion we have developed still apply to these species, but different
calculations of their interaction energy are required to account for their
differing geometries. However, such methods of calculation are beyond
the scope of this text.
5.1.8 Total intermolecular potentials
The previous sections provided a fundamental description of the main
forces involved in van der Waals intermolecular interactions. Ultimately,
the total interaction potential energy between two molecules is the sum of
all the different interactions that we have discussed (as well as a few more
complex interactions).
In a very basic treatment of the interactions between two atoms or molecules, the total intermolecular potential is often modeled by the
Lennard-Jones potential, which is the sum of a soft sphere repulsion term
and an attractive term that goes as 1/r
6 (analogous to the London dispersion attractive interaction). The Lennard-Jones potential is given as
U r
ð Þ = 4e
s
r
12 −
s
r
6
!
(5.22)
where −e is the minimum energy and s is a constant parameter (not the
molecular diameter). A graph of the Lennard-Jones potential is shown in
Figure 5.9.
Although the Lennard-Jones potential is a relatively primitive model of
the total intermolecular potential between two molecules, it provides us
with a qualitatively useful picture of a common interaction between two
molecules. Starting from the far right-hand side of Figure 5.10, we see that
the potential energy decreases as the distance between the molecules
grows smaller, until the energy reaches a minimum value of −e. If r
decreases beyond this minimum energy value, then the potential energy
CHAPTER 5: Intermolecular Interactions and Self-Assembly
152
when the distance between the two atomic centers is less than s. Figure
5.9b depicts a graph of V(r) versus r for the soft sphere power law model.
Thus far in our discussion of overlap repulsion, we have operated under
the assumption that the atomic or molecular geometry is essentially
spherical. While this assumption is relatively sound for atoms and some
small molecules (CH 4 , for instance, can be modeled as nearly spherical),
most molecules possess other geometries. The concepts of overlap
repulsion we have developed still apply to these species, but different
calculations of their interaction energy are required to account for their
differing geometries. However, such methods of calculation are beyond
the scope of this text.
5.1.8 Total intermolecular potentials
The previous sections provided a fundamental description of the main
forces involved in van der Waals intermolecular interactions. Ultimately,
the total interaction potential energy between two molecules is the sum of
all the different interactions that we have discussed (as well as a few more
complex interactions).
In a very basic treatment of the interactions between two atoms or molecules, the total intermolecular potential is often modeled by the
Lennard-Jones potential, which is the sum of a soft sphere repulsion term
and an attractive term that goes as 1/r
6 (analogous to the London dispersion attractive interaction). The Lennard-Jones potential is given as
U r
ð Þ = 4e
s
r
12 −
s
r
6
!
(5.22)
where −e is the minimum energy and s is a constant parameter (not the
molecular diameter). A graph of the Lennard-Jones potential is shown in
Figure 5.9.
Although the Lennard-Jones potential is a relatively primitive model of
the total intermolecular potential between two molecules, it provides us
with a qualitatively useful picture of a common interaction between two
molecules. Starting from the far right-hand side of Figure 5.10, we see that
the potential energy decreases as the distance between the molecules
grows smaller, until the energy reaches a minimum value of −e. If r
decreases beyond this minimum energy value, then the potential energy
CHAPTER 5: Intermolecular Interactions and Self-Assembly
152