Physical Metallurgy

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

2. Crystal Binding In our discussions of how and why structure determines physical and chemical properties of solids we 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 forces of the general form shown below. Repulsive forces Energy Interatomic distance, r Eo is the bond energy of nn. Eo Generally the attractive and repulsive interactions occur over distances greater than nn. Often approximations are made that only consider nn inter- actions. ro is the nn spacing Attractive forces ro

3. Ionic crystals To 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 Here, the ziare the charges (+1, -1, +2, -2, etc.) on the ions. In an ionic crystal like NaCl, these Coulombic potentials are summed over 1st, 2nd, 3rd, etc., neighbors so that the attractive potential assumes the form The repulsive potential energy takes the form Then the total potential energy is the sum

4. Ionic crystals Generally both B and the exponent n are not known, however one of these can be solved for with respect to the other using the known crystal structure of the ionic compound. The lattice potential energy Uo is the amount of energy released when one mole of the crystal is formed from gaseous ions which are at infinite separation. So by definition

5. Example: NaCl

6. 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

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

8. Van der Waals Bonds Dipole-Dipole, Dipole-Induced Dipole, London-Dispersion Bonds Dipole-Dipole Bonds A molecule that contains a polar covalent bond has a permanent electric dipole moment due to charge separation. When 2 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 a time-dependent (temporary) dipole moment in this 2nd molecule.

9. 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 dipole moments causing the molecules to attract one another if they are close enough. + R C C R + R C C R

10. Van der Waals Bonds 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.

11. Van der Waals Bonds Electric Dipole Interactions

12. Van der Waals Bonds Induced Electric Dipole Interactions External field Distorted e- distribution; the centers of charge no longer coincide. Spherically symmetric e- distribution for a bare non polar atom. The center of the negative and positive charge coincide. + + - The force on the induced dipole is The magnitude of the induced dipole moment is given by where is the polarizability.

13. Van der Waals Bonds Induced Dipole-Induced Dipole or London-dispersion Bonds Inert-gas solids We want to calculate how the attractive interaction energy decays with distance for A pair of induced dipoles. The interaction energy is

14. Metallic Bonding We were able to get a good picture of bonding and cohesive or lattice energy in ionic and rare-gas solids from classical physics considerations. Not so for metals. Quantum mechanics is necessary to get an accurate view of cohesion and metallic properties and this is beyond the nature of the topics to be discussed in this class. The simplest metals to understand are the “one electron” (a single valance electron, i.e., the alkali metals such 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 o(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. I   represents the ionization energy needed to remove the outermost 3s electron in a free Na atom to infinity and cis the cohesive energy. The position of the minimum in the cohesive energy gives an equilibrium interatomic distance ro .