Imperfections in Solids : Materials are often stronger when they have defects. The mechanical and electrical properties of a material are affected by the presence of defects. The study of defects is divided according to their dimension:

Defects in Solids

Gemstones – Hope Diamond blue color due to boron impurities (ppm)

Metals - ductility, stiffness, brittleness, etc. drastically affected

Examples of the large impact of defects:

Defect Classification:0D – point defects: vacancies and interstitials impurities.1D – linear defects: dislocations (edge, screw, mixed)2D – planar defects: grain boundaries, surfaces.3D – extended defects: pores, cracks.

Bonding+

Structure+

Defects

Properties

The defects have a profound effect on the macroscopic properties of materials

The processing determines the defects

Chemical Composition

Type of Bonding

Crystal Structure

ThermomechanicalProcessing

Microstructure

Point Defects (0D)Vacancies and Self-InterstitialsA vacancy is a lattice position that is vacant because the atom is missing. It is created when the solid is formed. They occur naturally as a result of thermal vibrations. An interstitial is an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity atom. In the case of vacancies and interstitials, there is a change in the coordination of atoms around the defect. This means that the forces are not balanced in the same way as for other atoms in the solid (lattice distortion).

Vacancy - a lattice position that is vacant because the atom is missing.Interstitial - an atom that occupies a place outside the normal lattice position. It may be the same type of atom as the others (self interstitial) or an impurity interstitial atom.

The number of vacancies formed by thermal agitation follows an Arrhenius type of equation:

where NA is the total number of atoms in the solid, QV is the energy required to form a vacancy (per atom or per mole), kB is Boltzmann constant, R is the gas constant and T the temperature in Kelvin.Note that kT(300 K) = 0.025 eV (room temperature) is much smaller than typical vacancy formation energies. For instance, QV(Cu) = 0.9 eV/atom. This means that NV/NA at room temperature is exp(-36) = 2.3 × 10-16, an insignificant number. Thus, a high temperature is needed to have a high thermal concentration of vacancies. Even so, NV/NA is typically only about 0.0001 at the melting point.

⎥⎦

⎤⎢⎣

⎡−=

TkQNN

B

VAV exp ⎥⎦

⎤⎢⎣⎡−=

RTQNN V

AV exp

Example: Calculate equilibrium number of vacancies per cubic meter for copper at 1000oC

Given: Activation Energy per vacancy = 0.9 eV/atom ; atomic weight of copper = 63.5 g/mol ; and density at 1000oC = 8.40 g/cm3

Boltzmann’s constant kB=1.38 × 10-23 J/atom-K=8.62 × 10-5 eV/atom-K

325

5328

0

328

336323

/102.2

)1273)(/1062.8()9.0(exp)/100.8(

exp

)1273(1000,/100.8

/5.63)/10)(/40.8)(/10023.6(

mvacanciesXN

KKeVXeVmatomsXN

kTQNN

toequalisKCatvacanciesofnumbertheThusmatomsXN

molgmcmcmgmolatomsX

ANN

V

V

vv

Cu

a

=

⎥⎦

⎤⎢⎣

⎡=

⎟⎠⎞

⎜⎝⎛ −=

=

==

−

ρ

Solution: Determine N, number of atomic sites per cubic meter for Cu

Impurities in SolidsAll real solids are impure. A very high purity material, say 99.9999% pure (called 6N – six nines) contains ~ 6 × 1016 impurities per cm3. Impurities are often added to materials to improve the properties. For instance, carbon added in small amounts to iron makes steel, which is stronger than iron. Boron impurities added to silicon drastically change its electrical properties. Solid solutions are made of a host, the solvent or matrix) which dissolves the solute (minor component). The ability to dissolve is called solubility. Solid solutions are:

•homogeneous •maintain crystal structure •contain randomly dispersed impurities (substitutional or interstitial)

Solid SolutionSolid Solution Solids dissolve other solids

Solid solutions are made of a host (the solvent or matrix) which dissolves the minor component (solute). The ability to dissolve is called solubility.So, most of engineering materials are solid solutions, i.e., alloys: solvent and solutes

Two ways: depending on the size andhost structure

Substitutional

Ni/Cu

- solvent: usually the element present in greatest amount (sometimes referred to as “host atoms”)- solute: usually the element present in minor concentration.

Interstitial

C/Fe

For fcc, bcc, hcp structures the voids (or interstices) between the host atoms are relatively small � atomic radius of solute should be significantly less than solvent.Normally, max. solute concentration ≤ 10%, (2% for C-Fe)

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…

•Similar atomic size (to within 15%) •Similar crystal structure •Similar electronegativity (otherwise a compound is formed) •Similar valence

Composition can be expressed in weight percent, useful when making the solution, and in atomic percent, useful when trying to understand the material at the atomic level.Example Ni is completely miscible in Cu (all rules apply)Zn is partially miscible in Cu (different valence, different crystal structure)

Factors for high solubility in Substitutional alloys (Hume-Rothery Solubility Rules)

Ni - Cu binary isomorphousalloy

Ni Cucrystal structure FCC FCCatomic radius 0.125 0.128

2.4%electronegativities 1.8 1.8valence 2 + 2 +

Solubility Cu in Ni 100%

Pb CuFCC FCC0.175 0.128

36.7%1.6 1.82+, 4+ 2 +

Limited solubility (eutectic) alloys

Solubility Cu in Pb 0.1%

Solid Solution:homogeneousmaintain crystal structurecontain randomly dispersed impurities(substitutional or interstitial)Second Phase: as solute atoms are added, new compounds / structures are formed, or solute forms local precipitates

ExamplesCalculate the critical radius (in nanometers) of a homogeneous nucleus that forms when pure liquid copper solidifies. Assume ΔT(undercooling) = 0.2Tmelt. For Cu Tm=1083oC; Heat of fusion (ΔHf) = 1826J/cm3; Surface Energy (γ) =177x10-7

J/cm2; Lattice parameter of FCC copper = a=0.361nmCalculate the number of atoms in the critical-sized nucleus at this undercooling.

We make use of the equation for a spherical nucleus to calculate the size of the critical nucleus ( )

( )( ) nmcmxTcmJ

TcmJxTH

Tr

m

m

f

m 969.01069.92.0.1826

.1017722 83

27

0 ===ΔΔ

= −−

−−γ

Then, the volume of the critical nucleus is

The volume of an FCC cell is

The number of cells in the critical nucleus is

As the number of atoms in an FCC cell is 4 then the total number of atoms is atoms.xser of AtomTotal Numb 32534814 ==

( ) 333 82.397.034

34 nmrnucleuscritVol ===−− ππ

( ) 333 047.0361.0 nmnmacellVol ===−

cellsnm

nmcellVol

nucleusVol 34.81047.082.3

3

3

==−

−

Calculate (a) the equilibrium number of vacancies per cubic meter in pure magnesium at 450oC. Assume that the energy of formation of a vacancy in pure magnesium is 0.89eV. (b) What is the vacancy fraction at 600oC? (Boltzmann’s constant kB=8.62x10-5eV/K) ; Atomic weight of Mg = 24.31g.mol-1 ; Density of Magnesium = 1.74g.cm-3

Solution:First, we calculate the number of magnesium atoms in a cubic meter

322

15328

0

328

336323

10712

7231062889010314

72345010314

312410741100236

−

−−−

−

=

⎥⎦

⎤⎢⎣

⎡−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

=

==

mvacanciesxN

KKeVxeVmatomsxN

TkQNN

toequalisKCatvacanciesofnumbertheThusmatomsxN

molgmcmcmgmolatomsX

ANN

V

V

B

vv

Mg

a

..))(..(

).(exp)..(

exp

)(,/.

/.).)(/.)(/.(ρ

Vacancy fraction at 600oC

615 1036.7

)873)(.1062.8()89.0(exp −−− =⎥

⎦

⎤⎢⎣

⎡−= x

KKeVxeV

NNV

Dislocations—Linear DefectsDislocations are abrupt changes in the regular ordering of atoms, along a line (dislocation line) in the solid. They occur in high density and are very important in mechanical properties of material. They are characterized by the Burgers vector, found by doing a loop around the dislocation line and noticing the extra interatomic spacing needed to close the loop. The Burgers vector in metals points in a close packed direction.Edge dislocations occur when an extra plane is inserted. The dislocation line is at the end of the plane. In an edge dislocation, the Burgers vector is perpendicular to the dislocation line. Screw dislocations result when displacing planes relative to each other through shear. In this case, the Burgers vector is parallel to the dislocation line.

Burgers vector b. It help us to describe the size and the direction of the main lattice distortion caused by a dislocation.Dislocations shown above have Burgers vector directed perpendicular to the dislocation line. These dislocations are called edge dislocations.There is a second basic type of dislocation, called screw dislocation. The screw dislocation is parallel to the direction in which the crystal is being displaced (Burgers vector is parallel to the dislocation line).

DislocationsDislocations

Edge dislocation

Screw dislocation

Burgers vector b

Dislocationline ζ

bζb//ζ⊥b

Mixed/partial dislocations: The exact structure of dislocations in real crystals is usually more complicated than the ones shown. Edge and screw dislocations are just extreme forms of the possible dislocation structures. Most dislocations have mixed edge/screw character. To add to the complexity of real defect structures, dislocation are often split in "partial“ dislocations that have their cores spread out over a larger area.

Interfacial Defects External Surfaces : The environment of an atom at a surface differs from that of an atom in the bulk, in that the number of neighbors (coordination) decreases. This introduces unbalanced forces which result in relaxation (the lattice spacing is decreased) or reconstruction (the crystal structure changes). Surface atoms have unsatisfied atomic bonds, and higher energies than the bulk atoms � Surface energy, γ (J/m2)• Surface areas tend to minimize (e.g. liquid drop)• Solid surfaces can “reconstruct” to satisfy atomic bonds at surfaces.

External SurfaceExternal SurfaceFree surface can be modeled as a simple termination of the bulk crystal on low-index planes (those having the lowest energy). The resultant picture is the so-called terrace-ledge-kink (TLK) model.

Surfaces and interfaces are very reactive and it is usual that impurities segregate there. Since energy is required to form a surface, grains tend to grow in size at the expense of smaller grains to minimize energy. This occurs by diffusion, which is accelerated at high temperatures.

Grain Boundaries : Polycrystalline material comprised of many small crystals or grains. The grains have different crystallographic orientation. There exist atomic mismatch within the regions where grains meet. These regions are called grain boundaries.Surfaces and interfaces are reactive and impurities tend to segregate there. Since energy is associated with interfaces, grains tend to grow in size at the expense of smaller grains to minimize energy. This is accelerated at high temperatures.The density of atoms in the region including the grain boundary is smaller than the bulk value, since void space occurs in the interface.

Interfacial defects

Interfacial defects form either between different phases, or between different crystals.

Interfaces between crystals are grainboundaries, as discussed earlier.

Microscopy

Visible light: 0.4~0.7m

Resolution limit ~ 0.5λ~ 0.3μ

Brass (annealing twins)FCC - Cu/Zn alloy

Bronze (Cu/Sn alloy)

Examined by the optical properties of the surface: need etching

Depth of field is important: needs a flat surface(polishing)

Up to 2000x

Grain Size

100x

3.5”

ASTM (American Society for Testing And Materials) grain size number n

12 −= nNN = no. of grains/in2

46 grains + 22 on circumference = 46 + 22/2 =57 grains

N=57 / π (3.5/2)2 = 6 grains/in2

6=2n-1, n=3.6n N

With

Scanning Electron Microscope

Electron gun

Condenserlens

Scan coils

Objectivelens

Specimen Detector

Display &storage

Display &storage

Scan generator

High depth offield !!!

2000x

Modes: Emissive, reflective, absorptive, X-ray, etc..

10~50,000x

Transmission Electron Microscope - TEM

Electron gun

Condenserlens

Objectivelens

Specimen

Camera

In TEM, sample thickness < 1000A for electron transmission

Magnifyinglenses

Diffraction, Microscopy, Spectroscopy are all possible in the same column. Applications: determine dislocation, defects, micro-precipitates, etc..

Up to1,000,000x