Chapter 5: Defects in Solids Perfect crystals don’t exist. Many of the important properties of materials are strongly influenced by defects, or even entirely determined by their presence. In some cases properties are improved, and in others degraded. Depends on material, defect type, and the particular property. Defect types and amounts depend on composition, temperature, pressure, and processing history. Classification of defects by dimensionality Point defects (zero dimensional) Line defects (one dimensional) Surface defects (two dimensional) Volume defects (three dimensional)

Point defects: Interstitials and vacancies in a pure metal A vacancy is an empty lattice site; the atom supposed to be there is missing. An interstitial is an atom located between lattice sites. The concentration of both types of defects increases with temperature, although the number of interstitials in metals is generally negligible. Why? Because the atoms are close together and crowding another of the same size between them severely strains the lattice. Just not enough space!

Vacancy concentration The number of vacancies N V in a given amount of metal is given by N V = N exp(-Q V /kT), where: N is the total number of lattice sites in that amount of metal Q V is the activation energy  energy required to form a single vacancy T is the absolute temperature k = R/N A = 1.38x J/atom.K = 8.62x10 -5 eV/atom.K is the Boltzmann constant R is the gas constant (8.314 J/mol.K) N A is Avogadro’s number (6.02x10 23 atom/mol) An electron volt, eV, is the energy acquired by an electron in moving through a potential difference of 1 V. (1eV = 1.602x J) Near the melting point of metals at 1 atm, N V /N  Example Problem 5.1: How many atoms in 1 m 3 of Cu? A Cu = 63.5 g/mol,  Cu = 8.4 g/cm 3. From the units:

By measuring N V versus T, can get value of Q V.  N v N = exp  Q v kT       Measuring Activation Energy N v N T defect concentration 1/T N N v ln - Q v /k/k slope One method of determining N V is to make very precise measurements of lattice constant and density.

One manifestation of changing the vacancy concentration by changing T Increasing the temperature causes the surface islands to grow. Why? When the equilibrium vacancy concentration is increased, expelled atoms diffuse to the surface and are incorporated at the edges of steps. Reprinted with permission from Nature (K.F. McCarty, J.A. Nobel, and N.C. Bartelt, "Vacancies in Solids and the Stability of Surface Morphology", Nature, Vol. 412, pp (2001). Low energy electron microscope image of a (110) surface of NiAl.

Vacancies -- vacancies exist in ceramics for both cations and anions Interstitials -- interstitials exist for cations -- interstitials are not normally observed for anions because anions are large relative to the interstitial sites Point Defects in Ceramics

Paired defects in compounds with ionic bonds Crystal must maintain charge neutrality. This means that a point defect cannot occur by itself. Frenkel defect: cation vacancy – cation interstitial pair i.e. move a cation from a lattice site to a nearby interstitial site Schottky defect: cation vacancy – anion vacancy pair i.e. remove both an anion and a nearby cation at the same time 7

Concentration of Frenkel and Schottky defects Note that N is the number of lattice sites, not the number of atoms. For example, in NaCl there is one Na + and one Cl - associated with each lattice site. The factor of 2 comes about because there are two ions associated with each defect.

Ions with more than one valence state Charge neutrality can also be maintained via different valence states when these exist. For example, iron can exist as Fe 2+ and Fe 3+. This means that in FeO, the two positive charges lost in forming a Fe 2+ vacancy can be compensated by substituting two Fe 3+ for two Fe 2+ in nearby sites (which leads to departure from stoichiometry).

10 Two common outcomes when impurity (B) added to host (A): 1. Solid solution of B in A (usually randomly distributed). Point defects. 2. Solid solution of B in A plus particles of a new phase (volume defects) OR Substitutional solid solution (e.g., Cu in Ni) Interstitial solid solution (e.g., C in Fe) Second phase particle -- different composition -- usually a different structure Impurities in Metals Definition: alloy = mixture

Electroneutrality must be maintained Example: Ca 2+ substituting for 2Na + in NaCl Impurities in ionic compounds Example: O 2- substituting for 2Cl - without impurity O 2- impurity O 2- Cl - anion vacancy Cl - with impurity Na + Cl - without impurity Ca 2+ impurity with impurity Ca 2+ Na + + Ca 2+ cation vacancy

Impurities in compounds with covalent bonds When a substitutional impurity has an extra valence electron the extra electron can move in an electric field. Since the current is carried by negative charges, the semiconductor is called "n-type." For example, P in Si: When the "dopant" has one fewer electrons, the missing electron in a covalent bond is called a "hole." An electron from an adjacent covalent bond can move into this hole in an electric field. Since the electric current is carried by what appears to be positive charges, the semiconductor is called "p-type." For example, B in Si: More in section in the text. conduction electron

Variation in solubility Solid solubility depends strongly on temperature At high concentrations can get very complex behavior: Chapter 10 When solubility is higher at elevated temperature, a solid solution can survive to room temperature if the cooling rate is sufficient The solubility can depend strongly on the presence of other components -- which is normal in engineering alloys

Specification of alloy compositions Three common methods: 1.Molar concentration (moles per volume) or mass concentration (mass per volume) 2.Mass fraction or weight percent 3.Atom fraction of atom percent (mole fraction and mole percent for compounds) Need to be able to convert between these. Can’t remember all of the equations, and they are a pain to derive. So it’s better to pick a basis and then rely on the units. Best shown by examples.

Conversion of weight percent to atom percent Example Problem 5.5 without equations Given: Alloy of 97 wt% Al, 3 wt% Cu Atomic weight of Cu: g/mol Atomic weight of Al: g/mol Basis: 100 g Contains 97 g of Al and 3 g of Cu Moles of Al = (97g Al)(mol Al / g Al) = mol Al Moles of Cu = (3 g Cu)(mol Cu / g Cu) = mol Cu Total moles = moles Atom % of Cu = ( mol Cu / total mol)X100% = 1.30 at% Cu Atom % of Al = (3.595 mol Al / total mol)X100% = 98.7 at% Al Notice that the number of significant figures in the final result of a calculation must not be excessive – you invite ridicule if you copy everything down from the computer or calculator. If you have more than two components, this type of calculation is best done using Excel.

Conversion of mole percent to weight percent Problem 5.16: An alloy consists of 6 atom% Pb and 94 atom% Sn. What is its composition in weight %? Need atomic weights: for Pb and for Sn. Basis: 100 mol In 100 mol of alloy we have 6 mol of Pb and 94 mol of Sn. Mass of Pb = (6 mol Pb)(207.2 g Pb / mol Pb) = 1,243.2 g Pb Mass of Sn = (94 mol Sn)( g Sn / mol Sn) = 11, g Sn Total mass of 100 mol of alloy: 12, g Weight % of Pb = (1,243.2 g Pb)/(12, g alloy)(100%) = 10.0 % Pb Weight % of Sn = (11, g Sn)/(12, g alloy)(100%) = 90.0 % Sn

Conversion to molar concentration Need the density of the alloy. If this isn’t known, estimate from the densities of the pure components by assuming that the alloy is an ideal solution, i.e. when the components are mixed to form the alloy the total volume doesn’t change. Consider the previous example, 10 weight % Pb and 90 weight % Sn. Basis 100 g, so have 10 g of Pb and 90 g of Sn. Density of Pb at room T is g·cm −3 Density of white Sn at room T is g·cm −3 Volume of Pb: (10 g Pb)(cm 3 Pb /11.34 g Pb) = cm 3 Pb Volume of Sn: (90 g Sn)(cm 3 Sn/7.365 g Sn) = cm 3 Sn Total alloy volume if alloy forms ideal solution: cm 3 Mass concentration of Pb: (10 g)/( cm 3 ) = g Pb/cm 3 alloy Mass concentration of Sn: (90 g)/( cm 3 ) = g Sn/cm 3 alloy Density of alloy if ideal solution: 7.63 g/cm g/cm 3 given for this alloy at That is, when you mix lead and tin the volume expands slightly! Few liquid or solid solutions are exactly ideal, so use this method only when you don’t know the actual density and you don’t want to measure it.