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CHAPTER 6:

DIFFUSION IN SOLIDS

Chapter 5-


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ISSUES TO ADDRESS...ISSUES TO ADDRESS...ISSUES TO ADDRESS...ISSUES TO ADDRESS...

How does diffusion occur?

CHAPTER 5:

DIFFUSION IN SOLIDS

Chapter 5-

Why is it an important part of processing?

How can the rate of diffusion be predicted forsome simple cases?

1

How does diffusion depend on structureand temperature?


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DIFFUSION DEFINED :

The phenomenon of material transport by

movement of atoms.

Glass tube filled with water. At time t = 0, add some drops of ink to one end

of the tube.

Chapter 5-

Measure the diffusion distance, x, over some time. Compare the results with theory.

to

t1

t2

t3

xo x1 x2 x3 mm

s

ec


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Interdiffusion: In an alloy, atoms tend to migratefrom regions of large concentration.

Initially After some time

Adaptedfrom Figs.5.1 and 5.2,

DIFFUSION: THE PHENOMENA (1)

Chapter 5-

100%

Concentration Profiles0

Cu Ni

3

100%

Concentration Profiles0

a s er e .


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DIFFUSION MECHANISMS

Takes place in presence of vacancies and

interstitial voids of suitable size.

The diffusing atoms have enough energy tobreak the bonds with neighboring atoms

Chapter 5-

ere are vacan s es w ere e us ng a omscan move.

Diffusion rate increases at elevated temperature.

mechanisms: self, vacancy, interstitial etc.


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Self-diffusion: In an elemental solid, atomsalso migrate.

Label some atoms After some time

CC

DIFFUSION: THE PHENOMENA (2)

Chapter 5- 4

A

B

DA

B

D

Activation energy for self diffusion = act. Energy of

vacancy formation + activation energy of movementof vacancy


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Substitutional Diffusion:

applies to substitutional impurities

atoms exchange with vacancies rate depends on:

--number of vacancies

--activation energy to exchange.

DIFFUSION MECHANISMS

Chapter 5- 5

increasing elapsed time


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Simulation ofinterdiffusion

across an interface:

Rate of substitutional

DIFFUSION SIMULATION

Chapter 5-

--vacancy concentration--frequency of jumping.

(Courtesy P.M. Anderson)


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Applies to interstitialimpurities.

More rapid thanvacancy diffusion.

-- shows the jumping of a

INTERSTITIAL DIFFUSION

Chapter 5- 7

one interstitial site to

another in a BCCstructure. The

interstitial sites

considered here areat midpoints along the

unit cell edges.


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FACTORS INFLUENCING DIFFUSION

Temperature

size of solute atoms

Chapter 5-

melting point of solvent

packing efficiency of solvent

number of vacancies


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Flux:

J =

1

A

dM

dt

kg

m2s

or

atoms

m2s

Directional Quantity x-direction

MODELING DIFFUSION: FLUX

Chapter 5- 10

Flux can be measured for:--vacancies--host (A) atoms

--impurity (B) atoms

Jx

y

Jz xz

Unit area Athroughwhichatomsmove.


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Concentration Profile, C(x): [kg/m3]

Concentration

of Cu [kg/m3]

Concentration

of Ni [kg/m3]

Cu flux Ni flux

CONCENTRATION PROFILES & FLUX

Chapter 5- 11

Fick's First Law: Position, x

The steeper the concentration profile,the greater the flux!

Jx = D

dC

dx

Diffusion coefficient [m2/s]

concentration

gradient [kg/m4]

flux in x-dir.

[kg/m2-s]


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Steady State: the concentration profile doesn'tchange with time.

Jx(left)=Jx(right)

Steady State:

Jx(right)Jx(left)

x

STEADY STATE DIFFUSION

Chapter 5- 12

Apply Fick's First Law:

Result: the slope, dC/dx, must be constant(i.e., slope doesn't vary with position)!

oncen ra on, , n e ox oesn c ange w me.

Jx = D

dC

dx

dC

dx

left

=

dC

dx

right IfJ

x)left= Jx)right, then


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Steel plate at700C withgeometry

shown:Adaptedfrom Fig.5.4,Callister 6e.

C1=1.2

kg/m3

C2=0

.8kg/m

3

Carbonrichgas Carbon

deficient

Steady State =straight line!

EX: STEADY STATE DIFFUSION

Chapter 5- 13

Q: How much

carbon transfersfrom the rich tothe deficient side?

J = DC2 C1x2 x1

= 2.4 109 kg

m2s

10mm

x1 x205m

m

D=3x10-11m2/s


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Diffusion Flux computationDiffusion Flux computationDiffusion Flux computationDiffusion Flux computation A plate of iron is exposed to carburizing (carbon-rich)

atmosphere on one side and a decarburizing

atmosphere on the other side at 700

0

C. If a conditionof steady state is achieved, calculate the diffusion fluxof carbon through the plate if the concentrations ofcarbon at ositions 5 and 10 mm beneath carburizin

Chapter 5-

surface are 1.2 and 0.8 kg/m3, respectively. J = -D(cA-cB/xA-xB)

= -(3 x 10-11m2/s){(1.2-0.8)kg/m3/ (5 x 10-3-10-2)m}

= 2.4 x 10-9 kg/m2s


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NON-STEADY STATE DIFFUSION

The flux is not same at different cross-sections

perpendicular to the diffusion

The flux through a given cross-section varies withtime

There is a net accumulation or de letion of

Chapter 5-

diffusing species.

Cs area=unity

x

x

Jx Jx+x=(C/t)x

C/t = -Jx/x


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C/t = -/x(-D(C/x)

If D does not depend on concentration

C/t = D(2C/x2)

Ficks second Law

NON STEADY STATE DIFFUSION

Chapter 5-

The solution to 1D equation is

C(x,t) = A B erf(x/2Dt)

A,B are constants determined from initial andfinal boundary conditions of the problem.


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Steady State: the concentration profile doesn'tchange with time.

Jx(left)=Jx(right)

Steady State:

Jx(right)Jx(left)

x

STEADY STATE DIFFUSION

Chapter 5- 12

Apply Fick's First Law:

Result: the slope, dC/dx, must be constant(i.e., slope doesn't vary with position)!

oncen ra on, , n e ox oesn c ange w me.

Jx = D

dC

dx

dC

dx

left

=

dC

dx

right IfJx)left= Jx)right, then


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Steel plate at700C withgeometry

shown: Adaptedfrom Fig.5.4,Callister 6e.

C1=1.2

kg/m3

C2=0

.8kg/m

3

Carbonrichgas Carbon

deficient

Steady State =straight line!

EX: STEADY STATE DIFFUSION

Chapter 5- 13

Q: How much

carbon transfersfrom the rich tothe deficient sideper unit area per unit time

(Diffusion Flux, J)?

J = DC2 C1x2 x1

= 2.4 109 kg

m2s

10mm

x1 x205m

m

D=3x10-11m2/s


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NON-STEADY STATE DIFFUSION

The flux is not same at different cross-sections

perpendicular to the diffusion

The flux through a given cross-section varies withtime

There is a net accumulation or de letion of

Chapter 5-

diffusing species.

Cs area=unity

x

x

Jx +x Jx= (C/t)x

C/t = -Jx/x


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C/t = -/x(-D(C/x)

If D does not depend on concentration

C/t = D(2C/x2)

Ficks second Law


Chapter 5-

The solution to 1D equation is

C(x,t) = A B erf(x/2Dt)

A,B are constants determined from initial andfinal boundary conditions of the problem.


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General solution:

"error function"

C(x,t) Co

Cs Co

= 1 erfx

2

Dt


Boundary conditions

Chapter 5- 15

the two media are considered to be semi-infinite concentrations on either side of the interfacechange abruptly and are uniform

x=0 at the interface

t=0 when diffusion begins

Tabulation of Error function Table 5.1


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Co

Cs

C(x,t)

tot1

t2t3

Chapter 5-

At t=0(to) C(x,0) = Co for 0


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Case Hardening:--Diffuse carbon atoms

into the host iron atoms

at the surface.--Example of interstitial

diffusion is a case

PROCESSING USING DIFFUSION (1)

Chapter 5-

.

Result: The "Case" is--hard to deform: C atoms

"lock" planes from shearing.

--hard to crack: C atoms putthe surface in compression.

8


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Carburization of steel

Carburizing atmosphere Steel

Cs

C1

0

x

C(x,0)=c1 x> 0

C(0,t)=cs

C(x,0)=c x< 0

Chapter 5-

Decarburizing atmosphere

C2

Steel

Cs

0

x

C(0,t)=cs


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PROCESSING USING DIFFUSION (3)

Corrosion resistance of duralumin

Alloy of Al with 4% CuMuch higher strengththan Al

Chapter 5-

Corrosion resistancepoor compared to Al

Duralumin sheets covered with thin Al sheets on bothsides ALCLAD


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Experimental determination of D

Can be determined using a diffusion couple

Bar 2 Bar 1

t = 0t2>t1 t1>0

c2

c

C

oncen

Chapter 5-

0

Distance x

c1

tratio

n

, 1

c2 x


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ATOMIC MODEL OF DIFFUSION

Hm

P.

E

Atom jumps to a vacant site

Needs enough energy

Breaking of bond in initial site

Formation of bond at final site

Chapter 5-

1 2 Hm enthalpy of motion

Probability of atom to have sufficient vibrational

energy to overcome the P.E. barrier

exp(- Hm /RT)


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DIFFUSION AND TEMPERATURE

Diffusivity D=Do exp(-Hm/RT) (for interstitialdiffusion)

D also depends on the probability of finding avacant site (for vacancy/substitutional diffusion)

n/N=exp(-Hf/RT)

Chapter 5-

So D=Do exp(-(Hm +Hf)/RT)

Activation energy Qd = (Hm +Hf)

Is the energy required to diffuse from one site to thenext


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Diffusivity increases with T.

Experimental Data:

00

00

0 0

pre-exponential [m2/s] (see Table 5.2, Callister 6e)activation energy

gas constant [8.31J/mol-K]D= Doexp

Qd

RT

diffusivity

[J/mol],[eV/mol](see Table 5.2, Callister 6e)

Chapter 5-

1000K/T

D (m2/s) Cin-Fe

Cin-Fe

AlinAl

Cuin

Cu

ZninC

uFein

-F

e

Fein

-F

e

0.5 1.0 1.5 2.010-20

10-14

10-81 1 6 3

D has exp. dependence on TRecall: Vacancy does also!

19

Dinterstitial >> Dsubstitutional

C in -Fe

C in -Fe Al in Al

Cu in Cu

Zn in Cu

Fe in -FeFe in -Fe

Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from

E.A. Brandes and G.B. Brook (Ed.) Smithells Metals ReferenceBook, 7th ed., Butterworth-Heinemann, Oxford, 1992.)


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DIFFUSION IN COMPOUNDS

Diffusion process is helped by presence of pointdefects.

Diffusion coefficients are different for cations and

anions

Chapter 5-

Qlattice > Qgrain boundary > Qsurface

D lattice < D grain b < D surface

C.S. area of mass transport usually smaller inspecial diffusion paths.

At higher temp diffusion through bulk prevails.


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Diffusion FASTERfor...

open crystal structures

lower melting T materials

Diffusion SLOWERfor...

close-packed structures

higher melting T materials

SUMMARY:

STRUCTURE & DIFFUSION

Chapter 5- 20

materials w/secondarybonding

smaller diffusing atoms

cations

lower density materials

materials w/covalentbonding

larger diffusing atoms

anions

higher density materials

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