Perfect Crystals?A Little ThermoFree Energies & EntropyDefects in a Perfect Crystal Atoms Vibrating in a WellDiffusionHoppingConcentration Gradients
HW #4 Due 2/11/03Exam #1 Date Change to Thurs. 2/27 from Thurs. 3/6
Thursday,Feb. 4, 2003MATS275: INTRODUCTION TO MATERIALS SCIENCE
Crystal StructuresBCCFCCHCPCsClNaClDiamond CubicZinc BlendeFluorite (CaF2)Crystobalite (SiO2)Ice (snowflakes)
DefectsZero DimensionalVacanciesInterstitialsImpuritiesOne DimensionalDislocationsTwo DimensionalPlanarThree DimensionalAmorphous Matls
Zero Dimensional DefectsThese are in order of prevalence in nature from most to least
V: VacancyS: Substitutional AtomI: Interstitial atomJ: Self-Interstitial atom
These can also be divided into intrinsic defects (native to the material) and extrinsic defects (other atoms than the lattice)
Unknown Point Defects on CuD. Eigler, IBM Almaden Research Center
D. Eigler, IBM Almaden Research Center
Clean Si (111) - 7 x 7reconstructionSi surface reacted with Br2(top layer stripped away, Br terminated surface)Courtesy: J.J. Boland, UNC Dept. of Chemistry
VacanciesRemove atom from regular lattice site to surface - requires an activation energy EV
A Little ThermoReactionGibbs Free Energy
- A Little ThermoReactionSpontaneous if G
Minimization of DG
At Equilibriumand note that ni / N is the concentration of defects. (N is the total number of lattice sites) So we can write this expression: where ce is the equilibrium concentration.
VacanciesThe vacancy activation energy for VAl in Aluminum is 0.76 eV. At 400C, the fraction of Al sites that are vacant is 2.29 x 10-5. What fraction are vacant at 660C?
Aluminum Vacancy Concentrationmp Al ~ 940 K
Aluminum Vacancy Concentration
- ImpuritiesAdd a different atom to the latticeForm a solid solutionINTERSTITIALRANDOMORDEREDEx) AuCu3 - at T
Liquid Solutions: Molecular MixingH2O Molecule (Solvent)C2H5OH Molecule (Solute)WaterAlcoholMixing on the Molecular Scale
Solid Solution: BronzeCopper Atoms (Solvent)Tin Atoms (Solute)A 10% Cu / Sn alloy (bronze)
- Hume-Rothery RulesWhat can be substitutional?Atomic size difference
ImpuritiesIonic MaterialsPut Cd in AgClCd wants to be +2, Ag is +1...+2-1
T Dependence of DH in FeC in FeFe in FeAl in Al2O3O in Al2O3
Activation Energyvs. Melting PointPbAlCuFeZnMg0501001502002503000500100015002000Melting Point (C)Self-Diffusion Activation Energy
Why do these form in the first place?When P and T are held constant, equilibrium will be where G is minimized.
So the thermodynamic driving force is to minimize Gibbs Free Energy. This is always true for a constant temperature and pressureguess what. That is often the experimental conditions that one is dealing with in a laboratory, and that is why the Gibbs Free Energy is such an important quantity.Notice that the Gibbs Free Energy is composed of an energy term and an entropy term. How does this play out in the crystal?The energy required to create a defect (breaking bonds, compressing or pulling others) is given the variable ED. In a solid:
Therefore what must be driving the formation of point defects?Increase in entropy. In other words, having atoms not in their correct lattice positions introduces disorder into the structure and the entropy term is larger than the energy term.The energy term linearly increase with each defect formed:For interstitials:For vacancies:
The strategy for finding the defect concentration at equilibrium is to set: Using a statistical mechanics argument to understanding the filling of defects one can arrive at the equilibrium concentration of defects in a solid.
Recall the important result that this is a thermodynamically derived resultit necessarily follows from Gibbs Free Energy that there will be defects in all crystals.
The average energy of formation of point defect in solids is about 1 eV / atom. This leads a defect roughly every 1000-10000 atoms at the melting point, and note that this is an activated process, so the number increases dramatically as the temperature increases.
We will use some real data for Al to illustrate this point.This is about 1 per every thousand sites empty