1

CHAPTER 7CHAPTER 7Structure and Properties of MaterialsStructure and Properties of Materials

Defects and PropertiesDefects and Properties ：：Point Defects and DiffusionPoint Defects and Diffusion

2

7-I. Introduction : Defect and Properties7-I. Introduction : Defect and Properties

Many material properties are determined or related

to defects of structures:

• Diffusion point defects

• Electrical conductivity of ceramics point defects (vacancies)

• Mechanical strength microstructure (grain size)

• Plastic deformation dislocation

• Flexural strength of ceramics volume defects (pores and cracks)

3

A. IntroductionA. Introduction

◎ Diffusion : self-movement of an atom or a molecule

F5.1 F5.2

◎ movement of an atom or a molecule :

(A) bulk flow : e.g., a flowing fluid, local flow of fluid

by convection (due to density difference)

occur only in fluids,

i.e., gases and liquids.

(B) diffusion : movement of single atoms or molecules

occur in all materials,

i.e., gases, liquids and solids.

7-II. Diffusion: General Principles and 7-II. Diffusion: General Principles and Diffusion in Metallic CrystalsDiffusion in Metallic Crystals

4

◎ example of movement of atoms or molecules,

(1) spreading of a gas in air

• air with convection : bulk flow + diffusion

• air is still : diffusion (only)

(2) spreading of a drop of ink in water

• water with convection or stirring : bulk flow + diffusion

• still water : diffusion (only)

(3) solution of salt or sugar in water

(4) diffusion of oxygen into jet engine components :

diffusion severe degradation in mechanical properties.

5

◎ Practical uses of diffusion

(1) microelectronics industries : doping of P, B or other

dopants into Si wafers.

(2) heart-lung machine : diffusion of O2 + CO2 through

silicon-based rubber membrane.

(3) ceramic gases senors : diffusion of O2 through ZrO2

ceramics.

(4) carburization of steels : to improve wear resistance.

f4.4-10

(5) coating of turbine blades : to improve oxidation

resistance.

6

◎ diffusion and point effects

• diffusing species is a type of point defect F5.3

• diffusion processes are related to movement of point defects.

7

B. Mechanisms of Diffusion in (Covalent and Metallic B. Mechanisms of Diffusion in (Covalent and Metallic Crystals)Crystals)

(1) interstitial diffusion

• activation energy : qi(or Qi)

(2) vacancy (exchange) diffusion

• occur only when there is a vacant lattice site adjacent to it.

• activation energy : qv(or Qv)

(both to form a vacancy and to move the atom)

• usually, qv > qi.

f4.4-4

◎ Two main mechanisms : F5.3

8

◎ Other Diffusion PathsOther Diffusion Paths

Atomic migration may also occur along dislocations,

grain boundaries, and external surfaces:

“short-circuit” diffusion paths: much faster than for

bulk diffusion, but insignificant because the cross-

sectional areas of these paths are extremely small.

9

C. A Physical Description of Diffusion (Fick’s First Law)

◎ rate of diffusion J : net number of atoms(or mass) moving across a plane in a particular direction per unit area per unit time ( J ≡ flux, 流通量 , atoms/cm2 –s or Kg/m2-s) f4.4-9

At

MJ

dt

dM

AJ

1

(5.1a)

(5.1b)

10

• concentration gradient (濃度梯度 ):

the driving force (驅動力 ) for diffusion

concentration gradient ＝BA

BA

X XX

CC

X

C

dx

dc

0

XBXA

◎ diffusion : random movement of an atom or a molecule or more precisely for solids, a random jump of an atom.

/J dc dt

11

J = D[(C1 – C2)/ΔX] F5.4

J = -D(dC/dx) Fick’s First law (5.3)

D : diffusion coefficient

D = D0 exp (-Q/RT) (5.8)

InD = 1n D0 + (-Q/R)(1/T) F5.7 T5.2 (5.9b)

◎ Steady State Diffusion

(steady state: unchanged with time; constant with time)

12

◎ Nonsteady State Diffusion: Fick’s Second LawNonsteady State Diffusion: Fick’s Second Lawused when concentration varies with time and position

(i.e., C = C(x, t) C = C(x) only : Fick’s First Law)

f4.4-9

by material balance

rate of input – rate of output = rate of accumulation

(JindA – JoutdA)dt = (Ct+dt – Ct)dxdA

dC/dt = -dJ/dx

J = -D(dC/dx) (Fick’s first law, E.q. 5-3)

dC/dt = D (d2C/dx2) Fick’s second law

13

cases study

for diffusion into a thick plate, concentration at the

surface = Cs, initial bulk concentration = C0

• case A

Cs = const, i.e., with a continuously replenished source

(e.g., carburization of Fe) ; and C0 = const.

solution of Fick’s second law yields :

(4.4-7a)

f4.4-10

)]2/([1][]),([ 00 DtxerfCCCtxC s

14

• Case B

C0 = 0, initial Cs = , i.e., not continuously replenished

(e.g., doping in semiconductor industry)

(4.4-7b)

Effective Penetration Distance

Xeff , defined as the point where

(4.4-8)

)4/exp()2/(),( 2 DtxDttxC

2/)(),( 0 seff CCtXC

15

substituting into Eq. (4.4-7) and using Fig. 4.4-11

(4.4-10)

or (4.4-11)

: geometry factor (=1 for a plate and 2 for cylinders)

D ~ T (D = D0 exp (-Q/RT)

(D = const, i.e., T = const)

(t = const)

DtX eff

DtX eff

tX eff

TDX eff ~

16

• diffusing species: size, bonding strength

D. Factors Influencing Rate of Diffusion

J = -D(dC/dx) Fick’s First law (5.3)

dC/dt = D (d2C/dx2) Fick’s second law

D = D0 exp (-Q/RT) (5.8)

Factor influencing D0 and Q:

JD andor dxdC

JT ,Q ,Do

17

fraction of atoms or molecules with sufficient energy to overcome the energy barrier (i.e., the activation energy, Q)

N = N0 exp (-Q/RT) Boltzmann’s distribution

Effect of temperature

• openness of structure :

(1) gases > liquids > solids

(2) type of crystal structure

• diffusion path (mechanism) : activation energy of

diffusion (Q). f4.4-4

T5.2 F5.7

18

Diffusion for Different Levels of ConcentrationDiffusion for Different Levels of Concentration

• self-diffusion

self-diffusion coefficient or tracer diffusion coefficient

• impurity diffusion (interdiffusion)

impurity diffusion coefficient

• chemical diffusion

chemical diffusion coefficient

f4.4-5 t4.4-1

19

7-IIIDiffusion in Ionic Materials 7-IIIDiffusion in Ionic Materials

Usually occurs by a vacancy mechanism (Figure 5.3a).

In order to maintain localized charge neutrality in the vicinity of this moving ion, it is necessary that another species having an equal and opposite charge accompany the ion’s diffusive motion.

Rate of diffusion is limited by the diffusion rate of the slowest moving species.

When an external electric field is applied across an ionic solid, the electrically charged ions migrate (i.e., diffuse), ionic motion gives rise to an electric current. Electrical conductivity is a function of the diffusion coefficient.

F 7.1

F 7.2

20

f01_05_pg110

21

f02_05_pg111

22

23

24

25

f04_05_pg113

26

f07_05_pg120

27

28

29

30

31

32