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3. Solidification & Crystalline Imperfectionssolidification (casting process) of metals

divided into two steps(1) nucleation – formation of stable nuclei in the

melt(2) growth of nuclei into crystals(3) formation of a grain structure

formation of stable nuclei two main mechanisms

a. homogeneous nucleationliquid metal is cooled below freezing point, slow-moving atoms bond together to create homogeneous nucleinucleus – larger than critical size, can grow

into a crystalembryo – smaller than critical size,

continuously being formed and redissolved in the molten metal 1

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two kinds of energies involved in homogeneous nucleation:(1) the volume free energy ΔGV – free energy

difference between the liquid and solidΔGV = 4/3πr3ΔGv

(2) the surface energy ΔGs – the energy neededto create a surface for the spherical particles

ΔGs = 4πr2γγ: specific surface free energy of the particle

total free energy ΔGT = ΔGV + ΔGsΔGT = 4/3πr3ΔGv + 4πr2γ

r* : critical radius ΔGT reaches the maximum

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d(ΔGT)/dr = 0 when r = r*2γ

r* = - ──ΔGv

critical radius versus undercooling

critical nucleus size mainly determined by ΔGVamount of undercooling increases, the critical nucleus size decreases

the relationship is2γTmr* = ────

ΔHf ΔTγ: surface free energy Tm: freezing temperatureΔHf : latent heat of fusionΔT: amount of undercooling

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ex. calculate the critical radius of homogeneous nucleus forms from pure liquid Cu. Assume ΔT = 0.2ΔTm (γ = 1.77 × 10-7 J/cm2, Tm = 1083oC, ΔHf = 1826 J/cm3)calculate the number of atoms in critical-sized nucleus at this undercoolingΔT = 0.2ΔTm = 1356 K × 0.2 = 271 K

2γTm 2(1.77 × 10-7 J/cm2)(1356 K )r* = ─── = ─────────────

ΔHf ΔT (1826 J/cm3)(271 K)= 9.70 × 10-8 cm

volume of nucleus = 4/3 π (9.70 × 10-8 cm)3

= 3.82 × 10-21 cm3

Cu: FCC structure, unit length a = 3.61 × 10-8 cm4 atoms per unit cellvolume of unit cell = (3.61 × 10-8 cm)3

= 4.70 × 10-23 cm3

3.82 × 10-21 cm3

number of atoms = ─────── × 4 = 325 atoms4.70 × 10-23 cm3

b. heterogeneous nucleationoccurs in a liquid on the surface of its container, insoluble impurities and other structural materials that lower the critical free energy required to form a stable nucleus

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the surface energy to form a stable nucleus on the nucleating agent is lowermuch smaller amount of undercooling is required to form a stable nucleus

crystal growth and grain formationnuclei crystals grainspolycrystalline – solidified metal containing

many crystalsgrains – crystals in solidified metalgrain boundaries – the surfaces between the

grainstwo major types of grain structures:(1) equiaxed grains – crystals grow about

equally in all directions, commonly found adjacent to a cold mold wall

(2) columnar grains – long, thin, coarse grains, created when metal solidifies 5

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rather slow in the presence of a steep temperature gradientcolumnar grains grow perpendicular to the mold surface

solidification of single crystalssolidification takes place around a single nucleus, no other crystals are nucleated and growthe interface temperature between solid and liquid must be slightly lower than m.p. of solid and the liquid temperature must increase beyond the interface

Al ingot

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single-crystal gas turbine airfoil

Si single crystal Czochralski method

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metallic solid solutionmetal alloy – mixture of two or more metals or a

metal (metals) and a nonmetal (nonmetals)ex. cartridge brass: 70% Cu + 30% Zn

Ni-base superalloy Inconel 718: about 10 elements in its nominal composition

solid solution – solid that consists o two or more elements atomically dispersed in a single-phase structure

(1) substitutional solid solutionsolute atoms can substitute for parent solvent atoms in a crystal lattice

ex. (111) plane in an FCC crystal lattice

Hume-Rothery rules: the following conditions are favorable for extensive solid solubility of one element in another: 8

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(a) the diameters of the atoms of the elements must not differ by more than about 15%

(b) the crystal structures of the two elements must be the same

(c) no appreciable difference in the electronegativites of the two elements

(d) two elements should have the same valenceex. Using the following data, predict the

relative degree of atomic solid solubility of the elements in copper

atomic crystal electro-radius structure negativity valence

Cu 0.128 nm FCC 1.8 +2Zn 0.133 nm HCP 1.7 +2Pb 0.175 nm FCC 1.6 +2, +4Si 0.117 nm diamond 1.8 +4

cubicNi 0.125 nm FCC 1.8 +2Al 0.143 nm FCC 1.5 +3Be 0.114 nm HCP 1.5 +2

Cu-Zn Cu-Pb Cu-Si Cu-Ni Cu-Al Cu-BeΔr +3.9% +36.7% -8.6% -2.3% +11.7% -10.9%Δχ 0.1 0.2 0 0 0.3 0.3predict B E C A C Csolubilityexp. 38.3% 0.1% 11.2% 100% 19.6% 16.4%

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(2) interstitial solid solutionthe solute atoms fit into the spaces between the solvent atoms, the spaces are called intersticesinterstitial solid solution can form when one atom is much larger than another

C in γ-Fe max. 2.08% C can dissolve

ex. calculate the radius of the largest interstitial void in FCC γ-Fe lattice

2R + 2r = a2R = √ 2 (R + r)√ 2 R = R + rr = (√ 2 - 1) R = 0.414 R

= 0.414 × 0.129 nm = 0.053 nm

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crystalline imperfectionzero-dimensional or point defectsone-dimensional or line defects (dislocation)two-dimensional or planar defectsthree-dimensional or volume defects

(1) point defectsvacancy – atom is missing, may be created by

‧ local disturbances during the crystal growth‧ atomic arrangements in an existing crystal‧ plastic defromation, rapid cooling‧ bombardment with energetic particles

interstitialcy or self-interstitial – an atom in a crystal can occupy an interstitial site between surrounding atomscan be introduced by irradiation

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Schottky imperfection – two oppositely chargedions are missing form an ionic crystala cation-anion divacancyFrenkel imperfection – a cation moves into an interstitial site, and a cation vacancy is createdvacancy-interstitialcy pairthe presence of these defects in ionic crystals increases their electrical conductivity

impurity is also a type of point defect

(2) line defects (dislocations)crystalline solids are defects that cause lattice distortion centered around a lineformed by plastic deformation, vacancy condensation, and atomic mismatch 12

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two types of dislocationsa. edge dislocation – created by the insertion of

an extra half plane of atoms above ┴

slip or Burgers vectordisplacement distance of atoms around the dislocation and is perpendicular to the edge-dislocation lineregion of compressive strain where the extra half plane is, region of tensile strain below the extra half plane

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b. screw dislocationformed by applying upward and downward shear stress to a perfect crystal that have been separated by a cutting plane

the slip vector is parallel to the dislocation linemost dislocations are of the mixed type

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(3) planar defects․external surface – the most common type of

planar defect, the atoms on the surface are bonded to other atoms only on one side, the higher energy associated with these atoms makes the surface susceptible to reaction

․grain boundaries – a narrow region between two grains of about 2~5 atomic diameters in width and a region of atomic mismatch between adjacent grains

the higher energy of grain boundaries and more open structure make them more favorable for nucleation and growth of precipitates

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․twin or twin boundary – a region in which a mirror image of the structure exists across a plane or a boundaryex. twin boundaries in the grain structure of

brass

․small-angle tilt boundarysmall-angle twist boundary

misorientation angle θ is generally < 10o

small-angle boundaries are regions of high energy and tend to strengthen a metal

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․stacking faults or piling-up faultsone or more of the stacking planes may be missing, give rise to another two-dimensional defectex. ABCABAACBABC in FCC crystal

ABAABBAB in HCP crystal(4) volume defects

a cluster of point defects join to form a three-dimensional void or a porea cluster of impurity atoms join to form a three-dimensional precipitatethe size from a few nm to cm

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identification of microstructure and defects(1) optical metallography technology․ at the μm level (magnification 2000×)․ information about grain size, grain boundary,

existence of various phases, internal damage, defects may be extracted

․ sample preparation: i grinding stageii polishing stage

smooth, mirror-like surface must be produced after polishing stage

iii etching processthe atoms at the grain boundary will be attacked at a much more rapid rate than the atoms inside the graintiny groves along the boundaries are produced

․ grain size and average grain diameterAmerican Society for Testing and Materials(ASTM) method N = 2n-1

n : grain-size numbern < 3 : coarse-grain, 4 < n < 6 : medium-grain7 < n < 9 : fine-grainn > 9 : ultra-large grain 18

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average grain diameterd = C/(nL M)

C = 1.5 constant M : magnificationnL : the ratio of the number of grains

intersected by a line of known lengthschematic diagram of optical microscope

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microstructure observed in optical microscope

low-carbon steel magnesium oxide(magnification 100×) (magnification 225×)

ex. 64 grains/in2 in a photomicrogarph of a metal are observed at a magnification of 100×. What is the ASTM grain size number?

N = 2n-1 64 = 2n-1

n = 7

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(2) scanning electron microscopy (SEM)

․ used for microscopic feature measurement, fracture characterization, microstructruestudies, thin coating evaluation, surface contamination examination, failure analysis

․ wide range of magnification 15 ~ 100,000×․ resolution to about 5 nm․ easy to prepare sample, often coated with Au

SEM of intergranular corrosion fracture

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schematic diagram of SEM

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(3) transmission electron microscopy (TEM)․ important technique for

studying defects and precipitates (secondaryphase) in materials

․ resolved features in thenm range

․ samples have a thicknessof several hundred nmand sample preparation is complex

․ bright-field image – the direct electrons dark-field image – the scattered electrons

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schematic diagram of TEM

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(4) high-resolution transmission electron microscopy (HRTEM)

․ has a resolution of about 0.1 nm․ allowing viewing of crystal structure and

defects at the atomic level․ sample must be significantly thin 10 ~ 15 nm

HRTEM image of several dislocations forming a small-angle boundary

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(5) scanning probe microscopes (SPM)tools allow scientist to analyze and image materials at the atomic level

(a) scanning tunneling microscope (STM)․ invented by Binnig and

Rohrer in early 1980s ․ extremely sharp tip (made

of W, Ni, Pt-Ir, Au, or carbon nanotube) to probe the surface

․ constant current modeconstant height mode

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(b) atomic force microscope (AFM)

contact mode – short-range repulsive forcenoncontact mode – long-range attractive forceAFM can be applied to all materials even nonconductor

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