Imperfections in Solids

Imperfections in Solids

• Solidification- result of casting of molten material• 2 steps

• Nuclei form

• Nuclei grow to form crystals – grain structure

• Start with a molten material – all liquid

• Crystals grow until they meet each other

nuclei crystals growing grain structureliquid

Polycrystalline Materials

Grain Boundaries

• regions between crystals

• transition from lattice of one region to that of the other

• ‘slightly’ disordered

• low density in grain boundaries

• high mobility

• high diffusivity

• high chemical reactivity

SolidificationGrains can be - equiaxed (roughly same size in all directions)

- columnar (elongated grains)

Columnar in

area with less

undercooling

Shell of

equiaxed grains

due to rapid

cooling (greater

T) near wall

Grain Refiner - added to make smaller, more uniform, equiaxed grains.

heat

flow

~ 8 cm

Point defects

Point defects, as the name implies, are imperfect point-

like regions in the crystal. Typical size of a point defect is

about 1-2 atomic diameters.

A vacancy is a vacant lattice position from where the atom

is missing. It is usually created when the solid is formed

by cooling the liquid. There are other ways of making a

vacancy, but they also occur naturally as a result of

thermal excitation, and these are thermodynamically

stable at temperatures greater than zero.

• Vacancies:-vacant atomic sites in a structure.

• Self-Interstitials:-"extra" atoms positioned between atomic sites.

Point Defects

Vacancy

distortion

of planes

self-interstitial

distortion of planes

7

Point Defects: Frenkel and Schottky

• Frenkel Defect

To maintain the charge neutrality, a cation vacancy-cation interstitial

pair occur together. The cation leaves its normal position and moves to the interstitial site.

• Schottky Defect

To maintain the charge neutrality, remove 1 cation and 1 anion;

this creates 2 vacancies.

Schottky

Defect

Frenkel

Defect

SELF-INTERSTITIAL: very rare occurrence

An interstitial atom or interstitialcy is an atom that

occupies a place outside the normal lattice position. It may

be the same type of atom as the rest surrounding it (self

interstitial) or a foreign impurity atom. Interstitialcy is most

probable if the atomic packing factor is low.

Another way an impurity atom can be fitted into a crystal

lattice is by substitution. A substitutional atom is a foreign

atom occupying original lattice position by displacing the

parent atom.

In the case of vacancies and foreign atoms (both

interstitial and substitutional), 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, which results in lattice distortion around

the defect.

SELF-INTERSTITIAL: very rare occurrence

• This defect occurs when an atom from the crystal occupies

the small void space (interstitial site) that under

ordinary circumstances is not occupied.

• In metals, a self-interstitial introduces relatively (very!)

large distortions in the surrounding lattice.

POINT DEFECTS

• The simplest of the point defect is a vacancy, or vacant lattice site.

•All crystalline solids contain vacancies.

• Principles of thermodynamics is used explain the necessity of the

existence of vacancies in crystalline solids.

• The presence of vacancies increases the entropy (randomness) of

the crystal.

• The equilibrium number of vacancies for a given quantity of

material depends on and increases with temperature as

follows:

Nv= N exp(-Qv/kT)

Equilibrium no. of vacanciesTotal no. of atomic sites Energy required to form vacancy

T = absolute temperature in Kelvin

k = gas or Boltzmann’s constant

Example Problem 4.1

Calculate the equilibrium number of vacancies per cubic meter for copper at

1000°C. The energy for vacancy formation is 0.9 eV/atom; the atomic weight and

density (at 1000 ° C) for copper are 63.5 g/mol and 8.4 g/cm3, respectively.

Solution.

Use equation 4.1. Find the value of N, number of atomic sites per cubic meter for

copper, from its atomic weight Acu, its density, and Avogadro’s number NA.

toequal ie )1273( 1000at vacanciesofnumber theThus,

/8.0x10

/5.63

)/10)(/4.8)(/10023.6(

3 28

336323

KC

matoms

molg

mcmcmgmolatomsx

A

NN

Cu

A

325

5

328

/m vacancies2.2x10

)1273)(/1062.8(

9.0(exp)/(8.0x10

exp

KKeVx

eVmatoms

kT

QNN v

v

Continuing:

And Note: for MOST MATERIALS just below

Tm Nv/N = 10-4

Here: 0.0022/8 = .000275 = 2.75*10-4

Two outcomes if impurity (B) added to host (A):• Solid solution of B in A (i.e., random dist. of point defects)

• Solid solution of B in A plus particles of a new

phase (usually for a larger amount of B)

OR

Substitutional solid soln.

(e.g., Cu in Ni)

Interstitial solid soln.

(e.g., C in Fe)

Second phase particle

--different composition

--often different structure.

Point Defects in Alloys

And:

• slip between crystal planes result when dislocations move,

• this motion produces permanent (plastic) deformation.

Are called Dislocations:

Schematic of Zinc (HCP):

• before deformation • after tensile elongation

slip steps which are

the physical

evidence of large

numbers of

dislocations

slipping along the

close packed plane

{0001}

Line Defects

Linear Defects (Dislocations)• Are one-dimensional defects around which atoms are misaligned

• Edge dislocation:• extra half-plane of atoms inserted in a crystal structure

• b (the berger’s vector) is (perpendicular) to dislocation line

• Screw dislocation:• spiral planar ramp resulting from shear deformation

• b is (parallel) to dislocation line

Burger’s vector, b: is a measure of lattice distortion and is

measured as a distance along the close packed directions in

the lattice

Edge DislocationEdge Dislocation

Fig. 4.3, Callister 7e.

• Dislocation motion requires the successive bumping

of a half plane of atoms (from left to right here).

• Bonds across the slipping planes are broken and

remade in succession.

Atomic view of edge

dislocation motion from

left to right as a crystal

is sheared.

(Courtesy P.M. Anderson)

Motion of Edge Dislocation

Screw Dislocation

MICROSCOPIC EXAMINATION

Applications

• To Examine the structural elements and defects that influence the

properties of materials.

• Ensure that the associations between the properties and structure (and

defects) are properly understood.

• Predict the properties of materials once these relationships have been

established.

Structural elements exist in ‘macroscopic’

and ‘microscopic’ dimensions

MACROSCOPIC EXAMINATION: The shape and average size or

diameter of the grains for a polycrystalline specimen are large

enough to observe with the unaided eye.

• Useful up to 2000X magnification (?).

• Polishing removes surface features (e.g., scratches)

• Etching changes reflectance, depending on crystal

orientation since different Xtal planes have different

reactivity.

Micrograph of

brass (a Cu-Zn alloy)

0.75mm

Optical Microscopy

Adapted from Fig. 4.13(b) and (c), Callister 7e. (Fig. 4.13(c) is courtesy

of J.E. Burke, General Electric Co.

crystallographic planes

Grain boundaries...

• are planer imperfections,

• are more susceptible

to etching,

• may be revealed as

dark lines,

• relate change in crystal

orientation across

boundary.Adapted from Fig. 4.14(a)

and (b), Callister 7e.(Fig. 4.14(b) is courtesy

of L.C. Smith and C. Brady,

the National Bureau of

Standards, Washington, DC

[now the National Institute of

Standards and Technology,

Gaithersburg, MD].)

Optical Microscopy

ASTM grain size number

N = 2n-1

number of grains/in2

at 100x magnification

Fe-Cr alloy(b)

grain boundary

surface groove

polished surface

(a)

GRAIN SIZE DETERMINATION

The grain size is often determined when the properties of

a polycrystalline material are under consideration. The

grain size has a significant impact of strength and

response to further processing

Linear Intercept method

• Straight lines are drawn through several

photomicrographs that show the grain

structure.

• The grains intersected by each line segment are

counted

• The line length is then divided by an average

number of grains intersected.

•The average grain diameter is found by dividing this

result by the linear magnification of the

photomicrographs.

ASTM (American Society for testing and Materials)

VISUAL CHARTS (@100x) each with a number

Quick and easy – used for steel

ASTM has prepared several standard comparison charts, all having different

average grain sizes. To each is assigned a number from 1 to 10, which is termed

the grain size number; the larger this number, the smaller the grains.

N = 2 n-1No. of grains/square inch

Grain size no.

NOTE: The ASTM grain size is related (or relates)

a grain area AT 100x MAGNIFICATION

Determining Grain Size, using a micrograph taken

at 300x

• We count 14 grains in a 1 in2 area on the image

• The report ASTM grain size we need N at 100x not 300x

• We need a conversion method!

2

12100

M is mag. of image

N is measured grain count at M

now solve for n:

log( ) 2 log log 100 1 log 2

log 2log 41

log 2

log 14 2log 300 41 7.98 8

0.301

n

M

M

M

m

MN

N M n

N Mn

n

For this same material, how many Grains would I expect /in2 at 100x?

1 8 1 2

2 2

8 1

2 2

2 2 128 grains/in

Now, how many grain would I expect at 50x?

100 100N 2 128*

50

N 128*2 512 grains/in

n

M

M

N

M

Electron Microscopes

beam of electrons of

shorter wave-length

(0.003nm) (when

accelerated across large

voltage drop)

Image formed with

Magnetic lenses

High resolutions and

magnification (up to

50,000x SEM); (TEM up

to 1,000,000x)

Summary

• Point, Line, and Area defects exist in solids.

• The number and type of defects can be varied and controlled

• T controls vacancy conc.• amount of plastic deformation controls # of dislocations• Weight of charge materials determine concentration of substitutional

or interstitial point ‘defects’

• Defects affect material properties (e.g., grain boundaries control crystal slip).

• Defects may be desirable or undesirable • e.g., dislocations may be good or bad, depending on whether plastic

deformation is desirable or not.• Inclusions can be intention for alloy development