Imperfections in atomic arrangements

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1 MME131: Lecture 8 Imperfections in atomic arrangements Part 1: 0D Defects A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s Topics Occurrence and importance of crystal defects Classification of crystal defects Characteristics of 0D defects References: 1. Callister. Materials Science and Engineering: An Introduction 2. Askeland. The Science and Engineering of Materials Lec 08, Page 1/16

2 Crystalline imperfections Crystals are like people, it is the defects in them which tend to make them interesting! Colin Humphreys, Professor and Director Research Department of Materials Science and metallurgy, University of Cambridge, UK Real crystals are never perfect, there are always defects Most atoms are in ideal locations, only small numbers are out of place Most defects cause the periodicity of the crystal to be disturbed over distances of several atomic diameters from the defect Defects have a profound impact on the macroscopic properties of materials Based on the bond strength, most materials should be much stronger than they are. Why? Strength of an ionic bond 10 6 psi, but more typical strength is 40x10 3 psi. Materials do not usually fail by breaking bonds!! Bonding + Structure + Defects Properties Lec 08, Page 2/16

3 The processing of materials determines the nature and amount of defects in it. Composition Bonding Crystal structure Thermo-mechanical processing Defect introduction and manipulation Microstructure Why are defects important? Defects, even in very small concentrations, can have a dramatic impact on properties of material. Without defects: solid-state electronic devices could not exist metals would be much stronger ceramics would be much tougher crystals would have no color Lec 08, Page 3/16

4 Types of crystal defects based on dimension of defects 0D, Point defects vacancies interstitials substitutional 2D, Surface defects external surfaces grain boundaries twin boundaries 1D, Line defects dislocations 3D, Bulk / Volume defects porosity crack foreign inclusion Point Defects Localized disruptions in the lattice involving one or more atoms. All real materials have point defects. More can be added by heating of material processing of material introducing impurities during materials processing intentionally adding during alloying Affect material properties by influencing atomic movement (diffusion process) dislocation movement (strengthening methods) Lec 08, Page 4/16

5 Classes of point defects Vacancy Self-interstitial Intrinsic defects Substitutional Interstitialcy Extrinsic defects (a.k.a. impurity) Frenkel Schottky In compound crystals (e.g., in ceramics) Point defects: (a) vacancy, (b) interstitial atom, (c) small substitutional atom, (d) large substitutional atom, (e) Frenkel defect (vacancy-interstitial pair), (f) Schottky defect (anion-cation vacancy pair). All of these defects disrupt the perfect arrangement of the surrounding atoms. Lec 08, Page 5/16

6 Vacancy A lattice position that is vacant because the atom is missing There are naturally occurring vacancies in all crystals Point defects occur as a result of the periodic oscillation or thermal vibration of atoms in the crystal structure. How many vacancies are there? The equilibrium concentration of vacancies at a given temperature can be expressed as: N V = N exp - Q V k T N V : number of vacancies Q V : activation energy for the formation of a vacancy N: total number of atomic sites k: Boltzmann s constant = 1.38x10-23 J/atom-K T: temperature in Kelvin = 8.62x10-5 ev/atom-k Ideal gas constant, R = Boltzmann s constant, k per mole = k N A = J/mol-K N V = N exp - Q V k T The concentrations of vacancies increase with: increasing temperature, T decreasing activation energy, Q V N V N exponential dependence ln N V N 1 slope - Q V / k defect concentration T T Lec 08, Page 6/16

7 Example 1: Vacancy concentrations in iron An iron crystal has a density of 7.87 g/cm 3. Determine the number of vacancies present in the crystal. The lattice parameter of BCC iron is cm. SOLUTION The expected theoretical density of iron can be calculated from the lattice parameter and the atomic mass. This density is reduced to 7.8 g/cm 3 due to the presence of vacancies. Let us assume that, instead of 2 atoms, there are x iron atoms present in the unit cell. So, for this x iron atoms, the density equation will become Or, there should be = vacancies per unit cell. The number of vacancies per cm 3 is: Lec 08, Page 7/16

8 Example 2: The effect of temperature on vacancy concentrations Calculate the equilibrium number of vacancies per cubic meter for copper at 27 and 1000 C. The energy for vacancy formation is 0.9 ev/atom; the atomic weight and density for copper are 63.5 g/mol and 8.4 g/cm 3, respectively. Number of regular atomic sites per m 3 for copper N = N A r A Cu = (6.023x10 23 # atom/mol) x (8.4 g/cm 3 ) x (10 6 cm 3 /m 3 ) (63.5 g/mol) = 7.96x10 28 # atom/m 3 The number of vacancies at 27 C (= 300 K): N V = N exp (-Q V / kt) = (7.96x10 28 # atoms/m 3 ) = 6.11x10 13 vacancies/m 3 x exp ev/atom (8.62x10-5 ev/k) x (300 K) one vacancy in every 1000 trillion (10 15 ) lattice atoms The number of vacancies at 1000 C (= 1273 K): N V = N exp (-Q V / kt) = (7.96x10 28 # atoms/m 3 ) = 2.18x10 25 vacancies/m 3 x exp ev/atom (8.62x10-5 ev/k) x (1273 K) one vacancy in every 3000 lattice atoms Lec 08, Page 8/16

9 Example 3: Sites for carbon in iron In FCC iron, carbon atoms are located at octahedral sites at the center of each edge of the unit cell (1/2, 0, 0) and at the center of the unit cell (1/2, 1/2, 1/2). In BCC iron, carbon atoms enter the tetrahedral sites, such as 1/4, 1/2, 0. The lattice parameter is nm for FCC iron and nm for BCC iron. Assuming that carbon atoms have a radius of nm, would we expect a greater distortion of the crystal by an interstitial carbon atom in FCC or BCC iron? tetrahedral sites in BCC iron octahedral sites at the edge centre in FCC iron SOLUTION BCC iron atom Atomic radius: Interstitial site at the (1/4, 1/2, 0) location: Lec 08, Page 9/16

10 FCC iron atom The interstitial site in BCC iron ( nm) is smaller than the interstitial site in FCC iron ( nm). Although both are smaller than the size of carbon atom (0.071 nm), carbon distorts the BCC crystal structure more than the FCC crystal. As a result, fewer carbon atoms are expected to enter the interstitial positions in BCC iron than in FCC iron. Self-Interstitial A point defect caused when a normal atom occupies an interstitial site in the crystal distortion of planes Self-interstitials in metals introduce large distortions in the surrounding lattice distortion of planes selfinter selfinterstials The energy of self-interstitial formation is ~ 3 times larger as compared to vacancies (Q i 3 Q v ). Equilibrium concentration of self-interstitials is very low (less than one self-interstitial per cm 3 at room temperature). Lec 08, Page 10/16

11 Impurities and alloys Impurity atoms are different from the host atoms. All real solids are impure. Very pure metals are only about % pure (i.e., ~ one impurity per 10 6 host atoms) May be intentional or unintentional Examples: Carbon added in small amounts to iron makes steel, which is stronger than pure iron (alloying) Sulphur remained in steel due to difficulties in synthesis (impurity) The equilibrium amount of a given impurity in a solid is determined by: How well the impurity "fits" into the structure; any bond distortion raises the energy requirement Similarity of bonding type between the impurity and the host How much of the impurity is available externally In ionic crystals, requirements of charge neutrality Lec 08, Page 11/16

12 Example: NaCl Na + Cl - cation vacancy Ca 2+ Na + Na + initial geometry Ca 2+ Ca 2+ impurity resulting geometry O 2- anion vacancy Cl - Cl - initial geometry O 2- impurity resulting geometry Impurities satisfying charge balance Point defects in alloys Two outcomes, when an alloying element B is added to a host A: Form a solid solution of B in A (i.e., random distribution of point defects) OR Substitutional alloy (e.g., Cu in Ni) Interstitial alloy (e.g., C in Fe) Form a solid solution of B in A plus formation of new phase second phase particle --different composition --often different structure Lec 08, Page 12/16

13 Solid solutions Solid solutions are made by dissolving the minor component (solute) to a host (the solvent or matrix) material. The ability to dissolve is called solubility. homogeneous maintain host crystal structure, no new structure formed contain randomly dispersed impurities (substitutional or interstitial) Second Phase: as solute atoms are added, new compounds / structures are formed, or solute forms local precipitates. Whether the addition of impurities results in formation of solid solution or second phase depends the nature of the impurities, their concentration and temperature, pressure Substitutional solid solution For impurities, certain conditions favour substitutional solution formation (Hume-Rothery rules) 1. R (atomic radii) <15% 2. Same crystal structures 3. Similar electronegativities (DE 0.6, preferably 0.4) 4. Same valence (if multivalent, must have at least one valence state in common) all four rules must be satisfied to form a complete substitutional solid solution Cu-Ni Yes (2.34%) Yes (FCC) Yes (0.0) Yes (2+) Complete solid solution Pb-Ni No (28.57%) No (Pb FCC; Ni BCC) Yes (0.2) No (Pb - 2 +, 4 + ; Ni -1 + ) Limited solid solution Lec 08, Page 13/16

14 Interstitial solid solution Metallic materials have relatively high APF, making interstitial positions relatively small. An atom must be fairly tiny to fit into the interstitial holes. The maximum allowable concentrations of interstitial impurities is ~10% (only 2% max. C in Fe-C system) [ atomic radius of C = nm; radius of BCC interstitial void = nm ] Even very small impurity atoms are larger than interstitial sites, so all interstitial impurities introduce lattice strains on the adjacent host atoms. Point defects in ceramic materials Ceramic materials are usually compounds and contain ions of at least two kinds, and point defects such as interstitial, substitutional, and vacancy for each ion type may occur. Substantial strains will be introduced to the crystal lattice by these point defects. Schematic representations of cation and anion vacancies and a cation interstitial Because the atoms exist as charged ions in ceramics, when defect structures are considered, conditions of electroneutrality must be maintained. As a consequence, defects in ceramics do not occur alone. Lec 08, Page 14/16

15 Frenkel defect (a cation-vacancy and a cation-interstitial pair ) a pair of point defects produced when an ion moves to create an interstitial site, leaving behind a vacancy Schottky defect (a cation-vacancy and a anion vacancy pair) a point defect in ionically bonded materials. In order to maintain a neutral charge, a stoichiometric number of cation and anion vacancies must form Assuming that we are dealing with a AX-type ceramic crystal (e.g., NaCl), the state of electroneutrality is still maintained since the cation maintains the same positive charge as an interstitial, and both cations and anions have the same charge. When a divalent cation impurity replaces a monovalent parent cation (e.g., Ca in place of Na), to maintain charge neutrality, a second monovalent parent cation must also be removed, creating a vacancy Lec 08, Page 15/16

16 If the ratio of cations to anions is not altered by the formation of defects, the material is said to be stoichiometric. Stoichiometry: A state for ionic compounds wherein there is the exact ratio of cations to anions as predicted by the chemical formula. A ceramic compound is nonstoichiometric if there is any deviation from the exact cation-to-anion ratio. Nonstoichiometric may occur for some ceramic materials in which two valence states exist for one of the ion types. Iron oxide (FeO) is one such material, for the iron can be present in both Fe 2+ and Fe 3+ states; the number of each of these ion types depends on temperature and the ambient O 2 pressure. Fe 3+ ions disrupt the electroneutrality of the crystal, i.e., extra positive charge. Excess + charge offset by formation of Fe 2+ vacancy. Formation of one Fe 2+ vacancy for every two Fe 3+ that are formed. The crystal is no longer stoichiometric because there are more O than Fe (i.e., deficiency of Fe); however still remains electrically neutral. Chemical formula becomes: Fe 1-x O, where x is some small fraction much less than 1. Next Class MME131: Lecture 9 Imperfections in atomic arrangement Part 2: 1D 3D defects Lec 08, Page 16/16

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