EMA5001 Lecture 8 Review for Diffusion · EMA 5001 Physical Properties of Materials Zhe Cheng...

© 2016 by Zhe Cheng

EMA5001 Lecture 8

Review for Diffusion

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 8 Diffusion – Review

Expectations about Diffusion (1)

Can name major categories and mechanisms of diffusion

Can give practical examples of applications of diffusion

Understand the similarity & difference between down-hill and up-hill

diffusion

Remember Fick’s 1st and 2nd Law and can derive them based on 1-D

geometry

Can derive thermo activation of diffusion and calculate activation

energy and frequency factor, if given diffusion data at different

temperatures

Can derive concentration profile for steady state diffusion in 1-D

Can name major example solutions Fick’s 2nd Law and assumptions

(no need to remember the math)

Remember expression of diffusion length and can explain how its

significance

2


Expectations about Diffusion (2)

Understand the relationship between self diffusion and vacancy

diffusion coefficients

Understand the significance of Kirkendall effect

Understand Darken’s equations

Can derive the relationship between diffusion coefficient and mobility

Can explain the relationship between tracer diffusion, intrinsic

diffusion, and interdiffusion coefficients

Understand why Matano analysis is needed and can explain the

significance of Matano interface

Can describe how diffusion coefficients are obtained in experiments

Can derive relationship between apparent diffusion coefficient for

diffusion through multi-crystalline or single crystalline materials with

dislocations

Can draw the concentration profile for reaction diffusion involving

multi-phases

3


Diffusion coefficient of Carbon

in α-Fe (BCC) is ~100 times higher

than in -Fe (FCC) at 910 oC

“Closer packing” for -Fe is cited

“The reason for this huge difference lies in

the fact that the bcc structure is more open

and the diffusion process requires less lattice

distortion (enthalpy or thermodynamically

favored).” http://www.eng.utah.edu/~lzang/images/lecture-7-experimental-

measurement-interdiffusion-coefficient-kirkendall-effect.pdf

Diffusion Coefficient of Carbon in

α-Fe vs. -Fe (1)

4

Deformation-Mechanism Maps, The Plasticity and Creep of Metals and Ceramics, by Harold J Frost, Dartmouth College, USA, and Michael F Ashby, Cambridge University, UK. Chapter 8 http://engineering.dartmouth.edu/defmech/

Crystal structure D0 (mm2/s) Q (kJ/mol)

C in α-Fe (BCC) 2 84.1

C in -Fe (FCC) 20 142

James P Schaffer, et al. The Science & Design of Engineering Materials, 2nd Ed, McGraw-Hill, (1999), p. 131

http://www.eng.utah.edu/~lzang/images/lecture-7-experimental-measurement-interdiffusion-coefficient-kirkendall-effect.pdf

















http://engineering.dartmouth.edu/defmech/

http://engineering.dartmouth.edu/defmech/



α-Fe vs. -Fe (2)

Size & Number of Interstitials in BCC & FCC structures

Carbon occupy octahedral sites in both α-Fe (BCC) and -Fe (FCC)

despite the size of octahedral site is much smaller in α-Fe (BCC)

“at least 85% of the carbon atoms occupy octahedral interstices” G.K. Williamson, “X-ray evidence for the interstitial position of carbon in α- iron,” Acta Cryst. (1953), p.361

The number (density) of octahedral interstitials per unit cell (or per

atom) is much larger for α-Fe (BCC) than for -Fe (FCC)

5

Crystal structure

Packing density

Ratio of radius of largest

tetrahedral atom to host

Ratio of radius of largest

octahedral atom to host

Number of tetrahedral site

per unit cell (per host atoms)

Number of Octahedral site

per unit cell (per host atoms)

BCC 0.68 0.291 0.155 12 (6) 6 (3)

FCC 0.74 0.225 0.414 8 (2) 4 (1)

James P Schaffer, etc. The Science & Design of Engineering Materials, 2nd Ed, McGraw-Hill, (1999), p. 82 & 87



α-Fe vs. -Fe (3)

The jumping distance in α-Fe (BCC) is much shorter than in -Fe (FCC)

FCC:

BCC:

6

http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_1/illustr/t1_3_3.html

CFeFeFeC rra3

1

3

2

3

4

2

1

2

1

FeFeFeC rra 22

4

2

1

2

1

FCC BCC


Successful Jump Frequency,

Vibration Frequency

Thermal vibration frequency

- Thermal vibration frequency

(Successful) Jump frequency

In text book, p. 72-73, (successful) jump frequency was written slightly

differently:

’ is for “in the x direction it makes ’ attempts per second to jump…” (i.e.,

vibration frequency in x direction)

z is the number of such sites (e.g., acceptable interstitials) in 3D

Naturally,

It does not help further understanding, and we did not adopt it here

7

RT

GmB exp

RT

Gz m

B exp'

x

G

Gm

' z


Concentration Profile & Darken’s Equation

http://www.matsceng.ohio-state.edu/~gupta/mse732/quiz2.pdf

8






Movement for Kirkendall Interface

Displacement of Kirkendall Interface l satisfy l2 = At, if at that time t, the

interface has moved l, what is the velocity of the Kirkendal interface ?

9

Atl 2

t

l

t

l

t

At

t

A

dt

Atd

dt

dlu

222

Adtldl 2

l

A

dt

dlu

2


Darken’s Equation & Flux due to

Brownian Motion & Interface Movement

The problem and the original solution given is awkward and, most

likely, wrong.

I would use

JA’ = 100 mol/(m2sec); JB’ = -60 mol/(m2sec)

C0 = 5 mol/m3

CB = 5 x 60% = 3 mol/m3 ; CA = 5 x 40% = 2 mol/m3

JA = JA’ + uCA ; JB = JB’ + uCB

JA + JB = 0

Therefore,

JA = 84 mol/(m2sec)

u = -8 m/sec http://www.matsceng.ohio-state.edu/~gupta/mse732/quiz1.pdf

10







(a) Steady state constant flux

Assuming constant diffusion

coefficient Linear profile

Changing diffusion coefficient

Curve

(b) D changes with concentration

Constant flux

Porter 3rd Exercise 2.1

11

Surface 1 for

carburization

Surface 2 for

decarburization

x

C

x

CDJ B

BB

21

)()( 21

xx

BB

xx

BBB

x

CxxD

x

CxxDJ

32.07.7

5.2

)(

)(

1

2

2

1

xxD

xxD

x

C

x

C

B

B

xx

B

xx

B

Larger D, Smaller dC/dx

Smaller D, Larger dC/dx



Using average diffusion coefficient D = 5.1x10-11 m2/sec

Flux

The answer given in the book is too complicated and end up with the

same answer

12

m

mkgsm

x

CDJ B

BB002.0

/60%8.0/%)15.0%4.1(/101.5

3211

126104.2 skgmJ B



Concentration profile for “spin on dopant” type

At constant time, we have

Plotting lnCB versus x2,

slope will be -1/(4Dt)

Plotting the data, we have

13

Dt

x

Dt

NtxCB

4exp,

2

Dt

x

Dt

NxCB

4lnln

2

81035.94

1

Dt

smD /101.31035.93600244

1 215

8

y = -9.35E+08x + 4.57E+00 R² = 9.96E-01

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 1E-09 2E-09 3E-09



Free energy Chemical potential

14

A B

G

(1) (2)

G1

G2

Gsys_eq

1

A

2

A

2

B

1

B

Gα Gβ

(1’) (2’)

eq

A

eq

BGsys_i

A rich B rich

A

B

(1) (2)

eq

AAA '2'1

eq

BBB '2'1



Interdiffusion coefficient

As Zn concentration increases, all three diffusion coefficients increase due to lower

melting point of Zn

15

sec/105.4~ 213 mDXDXD CuZnZnCu

sec/106.2 8 mmx

XDDu Zn

CuZn

22.0ZnX 78.0CuX mmx

X Zn /089.0

sec/105.422.078.0 213 mDD CuZn

sec/1092.2/089.0

sec/106.2 2138

mmm

mmDD CuZn

sec/101.5 213 mDZn

sec/102.2 213 mDCu

EMA5001 Lecture 8 Review for Diffusion · EMA 5001 Physical Properties of Materials Zhe Cheng...

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