Page 59 - Materials Science and Engineering An Introduction

2.5 Bonding Forces and Energies • 31

which is also a function of the interatomic separation, as also plotted in Figure 2.10a.
When F A and F R are equal in magnitude but opposite in sign, there is no net force—
that is,

F A + F R = 0 (2.4)
and a state of equilibrium exists. The centers of the two atoms remain separated by the
equilibrium spacing r 0 , as indicated in Figure 2.10a. For many atoms, r 0 is approximately
0.3 nm. Once in this position, any attempt to move the two atoms farther apart is coun-
teracted by the attractive force, while pushing them closer together is resisted by the
increasing repulsive force.
Sometimes it is more convenient to work with the potential energies between two
atoms instead of forces. Mathematically, energy (E) and force (F) are related as

Force–potential
energy relationship E = 3 F dr (2.5a)
for two atoms
And, for atomic systems,

E N = 3 F N dr (2.6)
r

= 3 F A dr + 3 F R dr (2.7)
r r
= E A + E R (2.8a)

in which E N , E A , and E R are, respectively, the net, attractive, and repulsive energies for
two isolated and adjacent atoms. 4
Figure 2.10b plots attractive, repulsive, and net potential energies as a function
of interatomic separation for two atoms. From Equation 2.8a, the net curve is the
sum of the attractive and repulsive curves. The minimum in the net energy curve cor-
bonding energy responds to the equilibrium spacing, r 0 . Furthermore, the bonding energy for these
two atoms, E 0 , corresponds to the energy at this minimum point (also shown in Figure
2.10b); it represents the energy required to separate these two atoms to an infinite
separation.
Although the preceding treatment deals with an ideal situation involving only two
atoms, a similar yet more complex condition exists for solid materials because force and
energy interactions among atoms must be considered. Nevertheless, a bonding energy,
analogous to E 0 above, may be associated with each atom. The magnitude of this bond-
ing energy and the shape of the energy–versus–interatomic separation curve vary from
material to material, and they both depend on the type of atomic bonding. Furthermore,

4 Force in Equation 2.5a may also be expressed as

dE
F = (2.5b)
dr
Likewise, the force equivalent of Equation 2.8a is as follows:
F N = F A + F R (2.3)

= dE A + dE R (2.8b)
dr dr