“Perfect Crystals?”

• A Little Thermo– Free Energies & Entropy

• Defects in a “Perfect” Crystal Atoms Vibrating in a Well

• Diffusion– Hopping– Concentration Gradients

• HW #4 Due 2/11/03• Exam #1 Date Change to Thurs. 2/27 from

Thurs. 3/6

Thursday,Feb. 4, 2003

MATS275: MATS275: INTRODUCTION TO INTRODUCTION TO

MATERIALS SCIENCEMATERIALS SCIENCE

Crystal Structures

• BCC• FCC• HCP• CsCl• NaCl• Diamond Cubic• Zinc Blende• Fluorite (CaF2)• Crystobalite (SiO2)

• Ice (snowflakes)

Defects

• Zero Dimensional– Vacancies

– Interstitials

– Impurities

• One Dimensional– Dislocations

• Two Dimensional– Planar

• Three Dimensional– Amorphous Mat’ls

Zero Dimensional Defects

These are in order of prevalence in nature from most to least

• V: Vacancy• S: Substitutional Atom• I: Interstitial atom• J: Self-Interstitial atom

These can also be divided into intrinsic defects (native to the material) and extrinsic defects (other atoms than the lattice)

V

s

J

I

Unknown Point Defects on Cu

D. Eigler, IBM Almaden Research Center

D. Eigler, IBM Almaden Research Center

Clean Si (111) - 7 x 7reconstruction

Si surface reacted with Br2

(top layer stripped away, Br terminated surface)

Courtesy: J.J. Boland, UNC Dept. of Chemistry

Vacancies

Remove atom from regular lattice site to surface - requires an activation energy EV

A Little Thermo

• Reaction

KM

V

XVXM

M

M

kTG

eK

Gibb’s Free Energy

kTGK ln

A Little Thermo

• Reaction– Spontaneous if G<0– Exothermic if H<0

G = H - T S

Gibb’s Free Energy

Enthalpy

Entropy

kTH

TS

kTH

kTG

CeeeK

Minimization of G

Num ber of Defects

Ene

rgynED

G

-T S

n i

At Equilibrium

kT

ENn D

i exp

and note that ni / N is the concentration of defects. (N is the total number of lattice

sites) So we can write this expression:

kT

E

N

nc Die exp

where ce is the equilibrium concentration.

Vacancies

• The vacancy activation energy for VAl in Aluminum is 0.76 eV. At 400°C, the fraction of Al sites that are vacant is 2.29 x 10-5. What fraction are vacant at 660°C?

241110292C

eN

nCCe

N

n

Al

V

K673

11604eV7605

Al

eVK

kTH

kTH

.exp..

4K933

11604eV760108282411Ce

N

n eVK

kTH

.exp.

.

Aluminum Vacancy Concentration

0.E+00

1.E-04

2.E-04

3.E-04

4.E-04

5.E-04

6.E-04

7.E-04

8.E-04

9.E-04

0 200 400 600 800 1000

Temperature (K)

[V]/

[Al]

mp Al ~ 940 K

Aluminum Vacancy Concentration

1E-1911E-1811E-1711E-1611E-1511E-1411E-1311E-1211E-1111E-101

1E-911E-811E-711E-611E-511E-411E-311E-211E-11

0.1

0 0.01 0.02 0.03 0.04 0.05 0.06

1/T

ln([

V]/

[Al]

)

Defect Complexes

Schottky Frenkel

ClNaClNa VVClNa iAgClAg AgVClAg

Impurities

• Add a different atom to the lattice– Form a solid solution

• INTERSTITIAL• RANDOM• ORDERED

Ex) AuCu3 - at T<390°C Cu occupies face centers of FCC, above that Cu and Au are randomly distributed

SUBSTITUTIONAL

Liquid Solutions: Molecular Mixing

H2O Molecule (Solvent)

C2H5OH Molecule (Solute)

Water Alcohol

Mixing on the Molecular Scale

Solid Solution: Bronze

Copper Atoms (Solvent)

Tin Atoms (Solute)

A 10% Cu / Sn alloy (bronze)

Hume-Rothery Rules

• What can be substitutional?– Atomic size difference <15%– Same crystal structure– Similar electronegativities– Same valence

• EXAMPLE - Cu and Ni– rCu=0.128nm, rNi=0.125 (2.3% diff)

– both fcc

– ENCu=1.90, ENNi=1.91

– Valences: Cu +1 +2, Ni +2

ImpuritiesIonic Materials

• Put Cd in AgCl– Cd wants to be +2, Ag is +1...

Cl2

AgAgClAg ClCdVClAgCd

+2

-1

T Dependence of D

1.E-27

1.E-25

1.E-23

1.E-21

1.E-19

1.E-17

1.E-15

1.E-13

1.E-11

1.E-09

1.E-07

1.E-05

1.E-03

1.E-01

4.0E-04 6.0E-04 8.0E-04 1.0E-03 1.2E-03 1.4E-03

1/T

ln(D

)

H in Fe

C in Fe

Fe in Fe

Al in Al2O3

O in Al2O3

Activation Energyvs. Melting Point

PbAl

Cu

Fe

Zn

Mg

0

50

100

150

200

250

300

0 500 1000 1500 2000

Melting Point (C)

Sel

f-D

iffu

sio

n A

ctiv

atio

n

En

erg

y