21/1/03: Types of bonds: Ionic, Metallic and Van der Walls bonds

Ionic bonding

Conceptually the simplest type of bond to understand - occurring between atoms from Group I and those from Group VII of the periodic table.
In order to get to the energetically favorable configuration of an inert atom, the Group I atom needs to lose one electron while the Group VII atom needs to gain one electron.
Both criteria are satisfied if an electron is transferred from the Group I atom to the Group VII atom.
But is this exchange energetically favorable ? Let us look at the case of NaCl

Ionization energy of Na atom ~ 5.1 eV (energy needed to separate the outermost electron)
Electron affinity of Cl atom ~ 3.6 eV (energy reduction on gaining extra electron)

So - a net input of 1.5 eV (~ 5.1 - 3.6 ) is required to create a NaCl bond.
We now need to consider the energy reduction due to Coulomb attraction between the Na+ and Cl- ions, seperated by a distance of r:
E = - e2 / (4*pi*epsilon0*r)
Assuming the distance between the two ionic nuclei to be 0.236 nm, we get E ~ - 6.1 eV
So the reduction in energy due to Coulomb attraction is more than enough to allow for electron transfer between Na and Cl.

But - a typical grain of salt contains about 1021 atoms of Na and Cl. How are we sure that this entire complex will be stable.

First, look at a 1D arrangement of Na and Cl atoms - the energetically favorable configuration will be one in which the Na and Cl atoms alternate along a chain (to maximise attraction between like ions and minimise repulsion between unlike ions). If we calculate the potential energy of any one ion due to its interaction with all the other ions,
E = - {2 e2/ 4*pi*epsilon0*a0} [ 1 - 1/2 + 1/3 - 1/4 + ...]
or, E = - {e2/ 4*pi*epsilon0*a0} * 2 ln(2)  where the factor 2 ln(2) is called the Madelung constant for the 1-D chain (it has the same numerical value for any kinds of atoms forming ionic bonds placed in a 1-D chain).

Extending this to 3 dimensions we get a Face Centered Cubic (FCC)-like arrangement
(as verified by X-ray crystallography). The Potential Energy of an ion in such a structure is
E = - (1.748)*{e2/ 4*pi*epsilon0*a0}
The Madelung constant for all crystals having the NaCl structure is 1.748
Note that ionic crystals having other kinds of structures ( e.g., CsCl which has a BCC like structure) will have a different Madelung constant (e.g., CsCl has a Madelung constant value of 1.763)

Putting the value of a0 = 0.281 nm for NaCl, we get E = -8.95 eV
This is almost twice the experimentally observed value of the cohesive energy since we have double counted the contribution of each bond.
So the potential energy should be ~ - 4.48 eV
Contribution of each of Na and Cl ion to the energy required to transfer an electron from Na to Cl is (~ 1.5 eV / 2) ~ 0.75 eV
The cohesive energy is -4.48 + 0.75 ~ -3.73 eV
compared to the experimentally observed value of -3.28 eV

This is because we have neglected the repulsive interaction between the ions (due to Pauli exclusion principle operating when the two electron clouds overlap). This is considered by introducing a short range (compared to the Coulumb interaction) force whose potential energy contribution is
Erep = constant/ rn
Therefore, the total potential energy at equilibrium seperation (~ a0) is
E = - (1.748)*{e2/ 4*pi*epsilon0*a0}*(1 - 1/n)
From observed compressibilities of ionic crystals, we estimate n ~ 9.
[Putting n=10 instead of 9, changes E by only 1%]

So, the cohesive energy of NaCl is E ~ -4.48*(1 - 1/9) + 0.75 ~ -3.21 eV which is close to the experimentally observed value.

Most ionic solids are brittle, have high melting point, insulators and can be dissolved in polar liquids (e.g., water). The characteristic bond energy is 3 - 4 eV.

Metallic bonding

It is possible to regard metallic bonding as an extreme case of covalent bonding in that the electrons are accumulated between the ion cores - but, in contrast to covalent bonding, the electrons in metals have wavefunctions that are very extended compared to the seperation between atoms. The "free" or delocalized electrons can be said to form an electronic "sea" in between the atoms.

An alternative picture is to look at the energy levels of the electrons as the atoms are brought close to each other. Interaction between atoms leads to splitting of energy levels - the large number of atoms in solids leads to the formation of a quasicontinuous scale of energy levels or "energy band". In metals, these bands overlap - e.g., in Li and Be, the 2s and 2p bands overlap - making several energy states available to the valence electrons which thereby become delocalised. The bonding between metallic atoms can be seen as reduction in electronic energy due to level broadening. The "free" electrons also imply that metallic solids can conduct electric current (unlike the other types of bonding).

Classes of metals: simple or sp metals (only s or p shell valence electrons), transition metals (partially filled d shell) and post-transition metals (filled d shell with eiether 1 or 2 s valence electrons).

For simple metals, binding energy is likely to be considerably smaller (~ 1 eV) than in an ionic solid. In transition metals, the d-shell electrons (which are more localised than the s electrons) provide a kind of covalent framework, which forms the main contribution to the binding energy. As a result transition and post-transition metals are as hard as covalent bonded materials.

Van der Walls bonding

In principle, Van der Walls bond is always present - but it is only significant in cases where other types of bonding are not possible (e.g., inert gas atoms like Ne, Ar, Kr, Xe which have closed electron shells). Typical energy of such bonding is ~ 0.1 eV (i.e., comparable to thermal energy at room temperature).

The physical source of this bond is the attractive induced dipole-dipole interactions from fluctuations in the electronic charge distribution of atoms due to zero-point motion. The bonding energy is dependent on the polarizability of the atoms involved.

Explicit calculation of potential energy due to induced dipole-dipole interaction between two linear harmonic oscillators shows that the interaction reduces the energy of the combined system by a factor of (Constant)/ r6.
There is also a repulsive interaction when the atoms are brought very close to each other (from Pauli exclusion principle) which is approximated by an interaction of the form (Constant)/ r12.
So total potential energy of two atoms at a seperation of r is:
E = 4 * epsilon * [ (sigma/r)12 - (sigma/r)6 ]
where sigma and epsilon are parameters characterising different substances. The above expression for interaction potential is known as Lennard-Jones potential. This potential expression is used very commonly in molecular dynamics simulations.

Relationship between the type of bond and the physical properties of a solid