Physical Metallurgy

The broad topic of physical metallurgy provides a basis that links the structure of materials with their properties, focusing primarily on metals.

Crystal BindingIn our discussions of how and why structure determines physical and chemical properties of solidswe will sometimes discuss atomic cohesion and the internal energy or free energy of a solid.

We do this more or less from a heuristic point of view so we often neglect entropic effects. Heuristically,we consider a solid that has been assembled by a combination of interatomic attractive and repulsive forcesof the general form shown below.

Repulsive forces

Attractive forces

Ener

gy

Interatomic distance, r

Eo

Generally the attractive and repulsiveinteractions occur over distancesgreater than nn. Often approximationsare made that only consider nn inter-actions.

roro is the nn spacing

Eo is the bondenergy of nn.

Ionic crystalsTo a high level of approximation, ionic crystals are held together by attractive Coulombic forces and“hard sphere” (electron) core-core repulsive forces.

The attractive potential energy takes the form

The repulsive potential energy takes the form

Then the total potential energy is the sum

In an ionic crystal like NaCl, these Coulombic potentials are summed over 1st, 2nd, 3rd, etc., neighborsso that the attractive potential assumes the form

Here, the zi are the charges (+1, -1, +2, -2, etc.) on the ions.

Ionic crystals

Generally both B and the exponent n are not known, however one of these can be solved for with respectto the other using the known crystal structure of the ionic compound.

The lattice potential energy Uois the amount of energy released when one mole of the crystal is formed from gaseous ions which are at infinite separation. So by definition

Example: NaCl

Example: NaCl

Summing these terms and noting that the zi = 1 in NaCl,

B in the repulsive part of the potential is connected to the compressibility defined as

The compressibility is connected to the curvature at ro (or Vo)in the energy-distance curve

Example: NaClUsing these relations one can show that in the case of NaCl or other ionic solids with the same crystal structuren=9. Then in evaluating the Uo one obtains 756 kJ/mole. The experimental value is 788 kJ/mole.

Dipole-Dipole, Dipole-Induced Dipole, London-Dispersion Bonds

Van der Waals Bonds

Dipole-Dipole Bonds

A molecule that contains a polar covalent bond has a permanent electric dipole moment due to charge separation. When2 such molecules are in close proximity they will attract one another forming a weak bond ~ 0.1 eV.

+C O +C O

Dipole-Induced Dipole Bonds

When a molecule with a permanent dipole approached a 2nd molecule that is purely covalent, it can induce atime-dependent (temporary) dipole moment in this 2nd molecule.

Van der Waals Bonds

+C O +C C

Induced Dipole-Induced Dipole or London-dispersion Bonds

Electrons and nuclei are in a constant state of motion. Fluctuations can result in instantaneous dipolemoments causing the molecules to attract one another if they are close enough.

R C C R

R C C R

+

+

The bonds that exist within molecules, such as covalent bonds, ionic bonds and polar covalent bonds, are part of a group of intramolecular bonds known as strong bonds. They are called strong bonds to distinguish them from another type of bond known as the weak bond. Strong bonds have a bond energy that ranges from about 2 eV to 5 eV of energy. However, weak bonds have energies that vary from 0.04 eV to 0.3 eV. Weak bonds are the bonds that exist between molecules and are therefore known as intermolecular bonds. There are three major types of weak bonds and together these weak bonds are known as van der Waals bonds (or van der Waals forces). Dipole-dipole bonds are the weak bonds that exist between two molecules as a result of their permanent dipole moments. A special type of dipole-dipole bond is the hydrogen bond, which also happens to be the strongest type of weak bond. Hydrogen bonds are dipole-dipole interactions in which at least one of the atoms involved is a hydrogen atom. Dipole-induced dipole bonds are those electric bonds that exist between a molecule with a permanent dipole moment and a molecule in which a temporary dipole moment has been induced (by the other molecule). London-dispersion bonds are those bonds that exist between molecules as a result of their instantaneous dipole moments. Instantaneous dipole moments arise in all molecules as a result of the fact that electrons are in constant state of motion.

Van der Waals Bonds

Van der Waals BondsElectric Dipole Interactions

Van der Waals BondsInduced Electric Dipole Interactions

External fieldSpherically symmetrice- distribution for a barenon polar atom. The centerof the negative and positive charge coincide.

Distorted e- distribution; the centersof charge no longer coincide.

The magnitude of the induced dipole moment is given bywhere is the polarizability.

+ +-

The force on the induced dipole is

Induced Dipole-Induced Dipole or London-dispersion Bonds

Van der Waals Bonds

Inert-gas solids

We want to calculate how the attractive interaction energy decays with distance forA pair of induced dipoles.

The interaction energy is

Metallic BondingWe were able to get a good picture of bonding and cohesive or lattice energy in ionic and rare-gas solidsfrom classical physics considerations. Not so for metals. Quantum mechanics is necessary to get an accurateview of cohesion and metallic properties and this is beyond the nature of the topics to be discussed in thisclass. The simplest metals to understand are the “one electron” (a single valance electron, i.e., the alkali metalssuch as sodium and potassium), whereas, in other metals modern approaches such as density function theory(DFT) need to be used.

Cohesive energy in metallic bonding. Na metal is used as an example. The curve 𝜀𝜀o(r ) represents the lowest energy of electrons with the wave vector k= 0, while the curve WF(r) represents an average kinetic energy per electron. 𝜀𝜀I represents the ionization energy needed to remove the outermost 3s electron in a free Na atom to infinity and 𝜀𝜀c is the cohesive energy. The position of the minimum in the cohesive energy gives an equilibrium interatomic distance ro .