IMPERFECTIONS IN SOLIDS

Week 3 1

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 nuclei crystals growing grain structure liquid • Crystals grow until they meet each other 2

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 3

Grains can be

Solidification

- equiaxed (roughly same size in all directions) columnar (elongated grains) ~ 8 cm heat flow 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.

4

Imperfections in Solids

There is no such thing as a perfect crystal. • What are these imperfections? • Why are they important? Many of the important properties of materials are due to the presence of imperfections.

Crystalline defect

-> a lattice irregularity having one or more of its dimensions on the order of an atomic diameter 5

Types of Imperfections

• Vacancy atoms • Interstitial atoms • Substitutional atoms • Dislocations • Grain Boundaries

Point defects Line defects Area defects

6

Point Defects

• Vacancies : -vacant atomic sites in a structure.

Vacancy distortion of planes • Self-Interstitials : -"extra" atoms positioned between atomic sites.

self interstitial distortion of planes 7

Equilibrium Concentration: Point Defects

• Equilibrium concentration varies with temperature!

No. of defects No. of potential defect sites.

N N v

Activation energy = exp     -

Q v k T

Temperature Boltzmann's constant (1.38 x 10 -23 J/atom-K) (8.62 x 10-5 eV/atom-K) Each lattice site is a potential vacancy site 8

Measuring Activation Energy

• We can get

Q v

an experiment.

from • Measure this...

N N v

= exp     -

Q v k T

• Replot it...

Nv N

exponential dependence!

defect concentration

T

ln

Nv N

slope -

Qv

/

k

1/

T

9

EXAMPLE PROBLEM 4.1

Calculate the equilibrium number of vacancies per cubic meter for copper at 1000C . The energy for vacancy formation is 0.9 eV/atom ; the atomic weight and density (at 1000C) for copper are 63.5g/mol and 8.4g/cm 3 , respectively 10

Estimating Vacancy Concentration

Find the equil. # of vacancies in 1m 3 • Given: r = 8.4 g /cm 3

A

of Cu at 1000 Cu = 63.5 g/mol  C.

Qv

= 0.9 eV/atom

N

A = 6.02 x 10 23 atoms/mol

N

r =N.A

cu /V.N

A

N v

= For 1 m 3 ,

N

= exp r x     -

k Q T N

A

v A

Cu  = 2.7 x 10 1273K -4 8.62 x 10 -5 eV/atom-K x 1 m 3 = 8.0 x 10 28 sites • Answer:

Nv

= (2.7 x 10 -4 )(8.0 x 10 28 ) sites = 2.2 x 10 25 vacancies 11

Observing Equilibrium Vacancy Concentration .

• Low energy electron microscope view of a (110) surface of NiAl.

• Increasing

T

causes surface island of atoms to grow.

• Why?

The equil. vacancy conc. increases via atom motion from the crystal to the surface, where they join the island.

Island grows/shrinks to maintain equil. vancancy conc. in the bulk.

12

IMPURITIES IN SOLIDS

• • Impurity or foreign atoms will always be present, and some will exist as crystalline point defects

Alloys ->

impurity atoms have been added intentionally to impart specific characteristics to the material • Alloying with copper significantly enhances the mechanical strength without depreciating the corrosion resistance appreciably • The addition of impurity atoms to a metal will result in the formation of a

solid solution

13

Point Defects in Alloys

Two outcomes if impurity (B) added to host (A): • Solid solution of B in A (i.e., random distribution of point defects) OR Substitutional (e.g., Cu solid soln.

in Ni ) Interstitial solid soln.

(e.g., C in Fe ) • Solid solution of B in A plus particles of a new phase (usually for a larger amount of B) Second phase particle --different composition --often different structure.

14

Imperfections in Solids

Conditions for substitutional solid solution (S.S.) • W. Hume – Rothery rule – 1. 

r

(atomic radius) < 15% – 2. Proximity in periodic table • i.e., similar electronegativities – 3. Same crystal structure for pure metals – 4. Valency • All else being equal, a metal will have a greater tendency to dissolve a metal of higher valency than one of lower valency 15

Substitutional Solid Solution – Cu-Ni

• The atomic radii for copper and nickel are 0.128

and 0.125nm

, respectively • Both have the FCC crystal structure • Their electronegativities are 1.9

and 1.8

• Valencies for Cu and Ni are +2 16

Imperfections in Solids

Application of Hume –Rothery rules – Solid Solutions

Element Atomic Crystal Radius Structure (nm) Electro nega tivity

1. Would you predict more Al or Ag to dissolve in 2. More in Cu ?

Zn or Zn Al ? Cu C H O Ag Al Co Cr Fe Ni Pd Zn 0.1278

0.071

0.046

0.060

0.1445

0.1431

0.1253

0.1249

0.1241

0.1246

0.1376

0.1332

FCC FCC FCC HCP BCC BCC FCC FCC HCP 1.9

1.9

1.5

1.8

1.6

1.8

1.8

2.2

1.6

Valence

+2 +1 +3 +2 +3 +2 +2 +2 +2 17

Conditions for Interstitial Impurity

• The atomic diameter of an interstitial impurity must be substantially smaller than that of the host atoms • The maximum allowable concentration of interstitial impurity atoms is low • Even very small impurity atoms are ordinarily larger than the interstitial sites, and as a consequence they introduce some lattice strains on the adjacent host atoms • Different crystal structures can fill interstitials 18

Specification of Composition

– weight percent

C

1 =

m

1

m

1 

m

2 x 100

m 1

= mass of component 1 – atom percent

C

1 ' =

n m

1

n m

1 

n m

2 x 100

n m1

= number of moles of component 1 19

Dislocations - Line Defects

Dislocations : • are line defects, • slip between crystal planes result when dislocations move, • produce permanent (plastic) deformation.

Schematic of Zinc (HCP): • before deformation • after tensile elongation slip steps 20

Imperfections in Solids

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

Burgers vector ( b )

represents the magnitude and direction of the distortion of dislocation in a crystal lattice

Dislocation Line

-> A curve running along the center of a dislocation. • Edge dislocation: – extra half-plane of atoms inserted in a crystal structure –

b

 to dislocation line • Screw dislocation: – spiral planar ramp resulting from shear deformation –

b

 to dislocation line 21

Imperfections in Solids

Edge Dislocation

22

Motion of Edge Dislocation

• 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.

23

Imperfections in Solids

Screw Dislocation

Screw Dislocation Dislocation line Burgers vector b (a) b (b) 24

Edge, Screw, and Mixed Dislocations

Mixed Edge Screw 25

Imperfections in Solids

Dislocations are visible in electron micrographs 51,450 magnified 26

Planar Defects in Solids

• One case is a

twin boundary (plane)

– Essentially a reflection of atom positions across the twin plane .

• A

twin boundary

is a special type of grain boundary across which there is a specific mirror lattice symmetry •

Annealing twins

are typically found in metals that have the

FCC

crystal structure, while

mechanical twins

are observed in

BCC

and

HCP

metals 27

Planar Defects in Solids

• •

Stack Fault

are found in FCC metals when there is an interruption in the

ABCABCABC . . .

stacking sequence of close-packed planes

Phase boundaries

exist in multiphase materials across which there is a sudden change in physical and/or chemical characteristics 28

Bulk or Volume Defects

• Other defects exist in all solid materials that are much larger than those discussed • These include pores, cracks, foreign inclusions, and other phases • They are normally introduced during processing and fabrication steps.

29

Numerical Problems

• Problems

4.1

to

4.5

and

4.7

to

4.25

30