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11MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystals“Crystals are like people, it is the defects in them which tend to make them interesting!”- Colin Humphreys.• Defects in Solids¾ 0D, Point defects9 vacancies9 interstitials9 impurities, weight and atomic composition¾ 1D, Dislocations9 edge9 screw¾ 2D, Grain boundaries9 tilt9 twist¾ 3D, Bulk or Volume defects¾ Atomic vibrationsOptional reading (Parts that are not covered / not tested):4.9 – 4.10 Microscopy 4.11 Grain size determinationChapter Outline2MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsWhy are defects important?Defects have a profound impact on the various properties of materials:Bonding+Crystal Structure+DefectsPropertiesProduction of advanced semiconductor devices require not only a rather perfect Si crystal as starting material, but also involve introduction of specific defects in small areas of the sample. Defects are responsible for color (& price) of a diamond crystal.Forging a metal tool introduces defects … and increases strength of the tool.3MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsCompositionBonding Crystal StructureThermomechanicalProcessingMicrostructureDefects can be introduced/removed during processingdefects introduction and manipulationProcessing allows one to achieve the required propertieswithout changes in composition of the material, but just by manipulating the crystal defects.Control (and intentional introduction) of defects is in the coreof many types of material processing.4MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsA 2D representation of a perfect single crystal with regular arrangement of atoms. But … structures of real materials can be better represented by the schematic drawing to the left.Schematic drawing of a poly-crystal with defects by Helmut Föll, University of Kiel, Germany. Defects in CrystalsReal crystals are never perfect, there are always defects25MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsTypes of DefectsDefects may be classified into four categories depending on their dimension:¾ 0D, Point defects: atoms missing or in irregular places in the lattice (lattice vacancies, substitutional and interstitial impurities, self-interstitials)¾ 1D, Linear defects: groups of atoms in irregular positions (e.g. screw and edge dislocations)¾ 2D, Planar defects: the interfaces between homogeneous regions of the material (e.g. grain boundaries, stacking faults, external surfaces)¾ 3D, Volume defects:extended defects (pores, cracks)6MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsHow many vacancies are there?Vacancy = absence of an atom from its normal location in a perfect crystal structurePoint Defects: VacanciesVacancies are always present in crystals and they are particularly numerous at high temperatures, when atoms are frequently and randomly change their positions leaving behind empty lattice sites (vacancies). Statistical thermodynamics predicts that the number of vacancies, Nv, have very strong dependence on temperature, T, and can be estimated using the Boltzmann distribution: where Nsis the number of regular lattice sites, kBis the Boltzmann constant, Qvis the energy needed to form a vacant lattice site in a perfect crystal, and T the temperature in Kelvin (note, not in oCor oF).⎟⎠⎞⎜⎝⎛−=TkQexpNNBvsv7MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsNote, that the above equation gives the lower end estimation of the number of vacancies in real materials, a large numbers of additional (non-equilibrium) vacancies can be introduced in a growth process or as a result of further treatment (plastic deformation, quenching from high temperature to the ambient one, etc.)⎟⎠⎞⎜⎝⎛−=TkQexpNNBvsvEquilibrium number of vacancies (continued)¾ Vacancies are required to be present in a crystal at any temperature!¾ The concentration of vacancies increases sharply with temperature. We can estimate for copper: • at room temperature - one vacancy pet 1015lattice sites• at high temperature, just below the melting point - one vacancy for every 10,000 atoms.8MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsExample: number of vacancies in Cu at room TThe Boltzmann’s constant kB= 1.38 × 10-23 J/atom-K = 8.62 × 10-5eV/atom-KThe temperature in Kelvin T = 27oC + 273 = 300 K. kBT = 300 K × 8.62 × 10-5eV/K = 0.026 eVThe energy for vacancy formation Qv= 0.9 eV/atomThe number of regular lattice sites Ns= NAρ/AcuNA= 6.023 × 1023atoms/molρ = 8.4 g/cm3Acu= 63.5 g/mol⎟⎠⎞⎜⎝⎛−=TkQexpNNBvsv()322323scmatoms108molg5.63cmg4.8molatoms10023.6N ×=⎟⎠⎞⎜⎝⎛××==⎟⎠⎞⎜⎝⎛−×=atomeV026.0atomeV9.0expcmatoms108N322v37cmvacancies104.7 ×=39MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsSelf-interstitials in metals introduce large distortions in the surrounding lattice ⇒ the energy of self-interstitial formation is ~ 3 times larger as compared to vacancies (Qi~ 3×Qv) ⇒ equilibrium concentration of self-interstitials is very low (less than one self-interstitial per cm3at room T).Self-interstitials:12345Other point defects: self-interstitials, impuritiesSchematic representation of different point defects:(1) vacancy;(2) self-interstitial;(3) interstitial impurity;(4,5) substitutional impuritiesThe arrows show the local stresses introduced by the point defects.Due to the local stresses introduced by point defects, they can feel each other (interact) and feel external stresses.The interactions can give a directionality to otherwise random jumps of atoms.10MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsImpuritiesImpurities - atoms which are different from the host¾All real solids are impure. Very pure metals 99.9999% - one impurity per 106atoms ¾ May be intentional or unintentionalExamples: carbon added in small amounts to iron makes steel, which is stronger than pure iron. Boron added to silicon change its electrical properties.¾ Alloys - deliberate mixtures of metalsExample: sterling silver is 92.5% silver – 7.5% copper alloy. Stronger than pure silver.interstitial impuritiessubstitutional impurity11MSE 2090: Introduction to Materials Science Chapter 4, Defects in crystalsSolids with impurities - Solid SolutionsSolid solutions are made of a host


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