EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion...EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law Nonsteady-State

EMA5001 Lecture 3

Steady State &

Nonsteady State Diffusion -

Fick’s 2nd Law & Solutions

EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law

Steady State

Steady State = Equilibrium?

− Similarity: State function (e.g., , C) does NOT change with time

− Difference: Net flux (or net reaction rate)

• Zero (0) net flux for equilibrium state vs. non-zero net flux for steady state

Fick’s 1st Law

If steady state diffusion

CB does NOT change with time

JB does NOT change with time

For 1-D, if DB constant, what is the

concentration profile under steady state?

Steady-State Diffusion

2

No!

x

CDJ B

BB

dxD

JxC

B

BB )(

xD

JxCxC

B

BBB )0()( Linear concentration profile x

CB

CB (x=0)

Slope = B

B

D

J

0

dx

dCDJ B

BB


Nonsteady-State Diffusion

Nonsteady State

Concentration changes with both

− Location (x, y, z)

− Time (t)

Take a small slice at location x

− δx : Thickness of the slice

− J1 : Flux into the slice

− J2 : Flux out from the slice

In small time period δt, the change

of concentration in that slice

We have

3

xA

tAJtAJC B

21

x

C

0

x

J

0

x x+δx

x+δx x

J2

J1

J1 J2 Area A

δx x

JJ

t

C B

21


Fick’s 2nd Law

Continued from p. 3

As δx and δt 0, we have

Therefore

Invoking Fick’s 1st Law

We have

i.e.,

If DB constant, simply to

4

x

J

x

JJ

t

CB

B

21

x

CDJ B

B

B

x

CD

xx

CD

xt

C BB

BB

B

x

J

t

CB

B

x

CD

xt

C BB

B Fick’s 2nd Law

2

2

x

CD

t

C BB

B


Implications of Fick’s 2nd Law

Two concentration profiles

Does the concentration in the specified region increase or decrease with time?

5

x

CB

0 x

CB

0

02

2

x

CB 02

2

x

CB

CB increases with time CB decreases with time

02

2

x

CD

t

C BB

B0

2

2

x

CD

t

C BB

B


Special Case – Homogenization

Diffusion to eliminate to local

concentration variation

Simplest case: t = 0, CB varies sinusoidally

Assumption:

− DB constant

Solution to Fick’s 2nd Law takes the form

in which is relaxation time

The amplitude of the variation

6

x

C

0

t = 0

t = τ

High CB

Low CB

Low CB

High CB

l

02

2

x

CB02

2

x

CB

l

xCCB

sin0

C

β0

t

l

xCtxCB expsin, 0

BD

l2

2

t

exp0

2

2

x

CD

t

C BB

B


Special Case – Spin-on Dopant for

Silicon Wafer

Diffusion in Semi-Infinite Bar w/ Fixed Amount of Total Dopant

Doping silicon surface with boron or

phosphorous spin-on dopants and

diffuse at 800-1000 oC

Fick’s 2nd Law

Assumption:

− DB constant

Initial condition

− CB (x =0, t = 0) = ∞; CB (x >0, t = 0) = 0

Boundary condition

− Zero concentration far away from surface

CB (x ∞) = 0

If the total amount of dopant is fixed of N, then

7

x

CD

xt

C BB

B

Dt

x

Dt

NtxCB

4exp,

2

x

C

0

t1 t2 t3

t1 < t2 < t3


Special Case – Infinite Diffusion Couple

Diffusion in Infinite Diffusion Couple

Two dilute alloys of B in A welded together

Fick’s 2nd Law

Assumption:

− DB constant

Initial condition

− CB (x > 0, t = 0) = C2; CB (x < 0, t = 0) = C1

Boundary condition:

− CB (x ∞, t > 0) = C2 ; CB (x -∞, t > 0) = C1

Solution is

in which error function, erf is given by

8

(1) (2) C1 C2

x

CD

xt

C BB

B

Dt

xerf

CCCCtxC

222, 2121

dyyzerfz

0

2 )exp(2

)(

0

1

erf(x)

x

-1

C

0

t1 t2

x

C1

C2

t0

t0 < t1 < t2


Special Case – Carburization &

Decarburization of Steel

Diffusion in Semi-Infinite Bar w/ Constant Surface Concentration

Increase/decrease carbon concentration in surface

− Carburization: CH4/CO atmosphere

at elevated temperature (for FCC γ-Fe)

− Decarburization: vacuum at elevated

temperature

Boundary/Initial conditions:

− Carburization:

CB (x = 0) = CS; CB (x ∞) = C0

− Decarburization:

CB (x = 0) = 0; CB (x ∞) = C0

Solutions are

− Carburization

− Decarburization

9

x

C

0

C0

CS

t1 t2 t3

Dt

xerfCCCtxC SS

2)(, 0

Dt

xerfCtxC

2, 0 x

C

0

C0

t1 t2 t3

Carburization of steel

Decarburization of steel

t1 < t2 < t3

t1 < t2 < t3


Diffusion Length

Example: Carburization of steel

For error function, if erf (z) = 0.5, z ≈ 0.5

Therefore, for concentration profile

When

Indicating

Therefore,

10

x

C

0

C0

CS

t1 t2 t3

Carburization of steel

5.02

Dt

xerf

2

, 0CCtxC S

5.02

Dt

xz

Dtx

Dt

xerfCCCtxC SS

2)(, 0

2

0CCS

x1 x2 x3

If 32

321

xxx

What is the relationship between

t1, t2, and t3 assuming D constant?

32

32

1

DtDtDt

2

3

2

21

32

ttt

Diffusion Length - characteristic length in a material

within which it experiences significant change due to diffusion


Microscopic View of Diffusion Length (1)

For an interstitial atom

Each random jump with displacement of

After n jumps, total displacement vector is

To obtain absolute displacement length after n jumps, we have

11

Original

position

Position after

n jumps

Rn

n

i

inn rrrrrR1

321 ...

n

i

i

n

i

inn rrRR11

1

1

)1(1

2

1

)2(2

2

1

22

1

1

11

1

22...22i

inn

i

inn

n

i

i

n

i

ii

n

i

i rrrrrrrrrr

ir

1

1 11

2 2n

j

ij

jn

i

j

n

i

i rrr

r1

r4

r5 r3 r2

r1 rn

r6



Continue from p. 11

Successful jumping occurs only to the nearest neighbor, then for i = 1, 2, … n,

in which is the jumping distance for an (interstitial) atom to its nearest neighbor.

Consider

in which is the angle between and

The displacement after n jumps will be

Now consider random jumping of a large amount of atoms:

Each atoms jumps for n times, and the “average” displacement among all atoms

12

ir

1

1 1

,

2222

n

j

jn

i

ijjn CosnR

1

1 11

2 2n

j

ij

jn

i

j

n

i

inn rrrRR

ijjijj Cosrr ,

2

ijj ,jr ijr

1

1 1

,

2222

n

j

jn

i

ijjn CosnR



Continued from p. 12

For average over a large amount of atoms, we have

Therefore,

If is the successful jump frequency, and if the n jumps take time t,

From earlier derivation about diffusion of interstitial atoms,

Therefore,

The “average” (root mean square) displacement after time t for random walk is

13

01

1 1

,

n

j

jn

i

ijjCos

tn

ttnRn

2222)(

2

6

1BBD

tDtR Bn 6)( 22

tDR Bn 4.22

21

1 1

,

2222 nCosnR

n

j

jn

i

ijjn

BB D62


Homework

Porter 3rd Ed, Exercise 2.1, 2.3, 2.6

Due Feb 3 class

14

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EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion...EMA 5001 Physical Properties of Materials Zhe Cheng (2016) 3 Nonsteady State Diff – Fick’s 2nd Law Nonsteady-State