diffusion

Chapter 5-

ISSUES TO ADDRESS...

• How does diffusion occur?

• Why is it an important part of processing?

• How can the rate of diffusion be predicted for

some simple cases?

1

• How does diffusion depend on structure

and temperature?

CHAPTER 5:

DIFFUSION IN SOLIDS

Chapter 5- 2

• Glass tube filled with water.

• At time t = 0, add some drops of ink to one end

of the tube.

• Measure the diffusion distance, x, over some time.

• Compare the results with theory.

DIFFUSION DEMO

Chapter 5- 3

• Interdiffusion: In an alloy, atoms tend to migrate

from regions of large concentration.

Initially After some time

100%

Concentration Profiles0

Adapted

from Figs.

5.1 and 5.2,

Callister 6e.

Cu-Ni Diffusion Couple

Chapter 5- 3

The couple is heated at extended period at elevated

temperature

The concentrations of nickel and copper is a function of

position across the couple.

After some time

100%


Adapted from Figs. 5.1 and 5.2, Callister 6e.

Interdiffusion or Impurity Diffusion

The concentration profile can

be determined by elemental

analysis.

This indicates that Cu atoms

have diffused into the Ni and

that Ni atoms have diffused into

the Cu

Chapter 5- 4

Occurs for pure metals, but all atoms exchanging positions

are of the same type

Not normally subject to observation by noting composition

change

Initially

Note the labeled atomsAfter some time

Note the final positions

A

B

C

D

Self-diffusion

100%


Cu100%


Cu

Chapter 5- 5

Stepwise migration of atoms from lattice/interstitial

site to lattice/interstitial site

Required conditions

• there must be an empty adjacent site

*vacancy or interstitial

• sufficient energy to break bonds with its neighbor

atoms and then cause some lattice distortion during the

displacement

DIFFUSION MECHANISMS

Chapter 5- 5

VACANCY Diffusion:

• applies to both

interdiffusion (substitutional impurities) andself-diffusion

• atoms exchange with vacancies

• rate depends on:

--number of vacancies

--activation energy to exchange.

Metallic Diffusion Mechanisms

Chapter 5- 5

VACANCY Diffusion:

• applies to both

interdiffusion (substitutional impurities) andself-diffusion

• atoms exchange with vacancies

• rate depends on:

--number of vacancies

--activation energy to exchange.


Initially After some time

A

B

C

D

Chapter 5- 6

• Simulation of

interdiffusion

across an interface:

• Rate of substitutional

diffusion depends on:--vacancy concentration

--frequency of jumping.

(Courtesy P.M. Anderson)

DIFFUSION SIMULATION

Chapter 5- 7


Interstitial Diffusion

• Applies to interstitial impurities.

• In most metal alloys, interstitial

diffusion occurs more rapidly than

vacancy mode.

• There are more interstitials the vacancies

• Applications

Steel manufacturing (C and Fe)

Purification of gas using thin sheet of

metal


Chapter 5- 7


Interstitial Diffusion

Simulation:--shows the jumping of a smaller atom (gray)

from one interstitial site to another in a BCC

structure. The interstitial sites considered here

are at midpoints along the unit cell edges.


Chapter 5- 7

SUBSTITUTIONAL vs INTERSTITIAL

Chapter 5-

• Case Hardening:--Diffuse carbon atoms

into the host iron atoms

at the surface.

--Example of interstitial

diffusion is a case

hardened gear.

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

"lock" planes from shearing.

--hard to crack: C atoms put

the surface in compression.

8

Fig. 5.0,

Callister 6e.

(Fig. 5.0 is

courtesy of

Surface

Division,

Midland-

Ross.)

PROCESSING USING DIFFUSION

Chapter 5-

• Doping Silicon with P for n-type semiconductors:

• Process:

9

1. Deposit P rich

layers on surface.

2. Heat it.

3. Result: Doped

semiconductor

regions.

silicon

silicon

Fig. 18.0,

Callister 6e.

PROCESSING USING DIFFUSION

Chapter 5-

Diffusion is a time dependent process.

Rate of mass transfer is frequently expressed as FLUX (J).

In differential form

Definition:

amount of M (mass or number of atoms)diffusing through and perpendicular to aa unit cross sectional area A of solid per unitof time t

10

DIFFUSION: FLUX

Chapter 5- 10

• Directional Quantity

• Flux can be measured for:--vacancies

--host (A) atoms

--impurity (B) atoms

DIFFUSION: FLUX

Unidirectional

Chapter 5-

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

11

• Fick's First Law:

Concentration

of Cu [kg/m3]

Concentration

of Ni [kg/m3]

Position, x

Cu flux Ni flux

• The steeper the concentration profile,

the greater the flux!

Adapted

from Fig.

5.2(c),

Callister 6e.

CONCENTRATION PROFILES & FLUX

Chapter 5-

• Steady State: the concentration profile doesn't

change with time.

12

• Apply Fick's First Law:

• Result: the slope, dC/dx, must be constant

(i.e., slope doesn't vary with position)!

Jx D

dC

dx

dC

dx

left

dC

dx

right

• If Jx)left = Jx)right , then

STEADY STATE DIFFUSION

Chapter 5- 13Adapted from Fig. 5.4, Callister 6e.


Linear Concentration Profile

BA

BA

xx

CC

x

C

dx

dC

gradientionConcentrat

Chapter 5-

PROBLEM: A plate of iron is exposed to a carburizing

(carbon-rich) atmosphere on one side and decarburizing

(carbon deficient) on the other side at 700°C. If a

condition of steady state is achieved, calculate the

diffusion flux of carbon through the plate.

Refer to the figure for

Additional information

about the system.

13



Chapter 5-

• Steel plate at

700C with

geometry

shown:

13

• Q: How much

carbon transfers

from the rich to

the deficient side?

Adapted

from Fig.

5.4,

Callister 6e.



Chapter 5- 13


Purification of Hydrogen Gas

Mixture of

gases

LowConcentration of

H2

x1 x20

Steady State = straight line!

HighConcentration of

O2

H2O

H2

N2

H2

Palladium

Hydrogen gas selectively diffuses through the Palladium metal sheet

from the region of high to low concentration of hydrogen gas.

Chapter 5-

PROBLEM: Compute the kg of H2

that pass per hour through a 6

mm thick sheet of palladium

having an area of 0.25 m2 at

600°C. Assume the diffusion

coefficient of 1.7x10-8 m2/s, that

the concentrations at the high

and at the low pressure sides of

the plates are 2.0 and 0.4 kg

H2/m3 of palladium, and that

steady state conditions have

been attained.

13



Chapter 5-

GIVEN:

A = 0.25 m2

D = 1.7x10-8 m2/s @ T = 600°C

C1 = 2.0 kg H2/m3 of Pd

C2 = 0.4 kg H2/m3 of Pd

Δx = 6 mm

Compute the kg of H2 that pass per hour or rate of transport

13



PROBLEM: When -Fe is subjected to an atmosphere of

N2, the CN (in wt%) is a function of N2 pressure PN2 (in

MPa), and absolute temperature (T) according to

Furthermore, the values of D0 and

Qd for this diffusion system are

3.0x10-7 m2/s and 76.25 kJ/mol,

respectively. Consider a thin iron

Membrane 1.5 mm thick that is at

300°C. Compute the flux through this membrane if PN2 on

one side is 0.10 MPa and on the other side 5.0 MPa.13



RTPC mol

kJ

NN

6.37exp1090.4

2

3

Chapter 5-

• Diffusivity increases with T.

D has exp. dependence on T

Recall: Vacancy does also!

19

DIFFUSION AND TEMPERATURE

pre-exponential [m2/s]

activation energy

gas constant [8.31J/mol-K]

D Doexp Q

d

RT

diffusivity

[J/mol],[eV/mol]

(Absolute Temperature)

Chapter 5-

• Experimental Data: D has exp. dependence on T

Recall: Vacancy does also!

19

Dinterstitial >> Dsubstitutional

C in -FeC 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 Reference Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)

DIFFUSION AND TEMPERATURE

Chapter 5-

PROBLEM: Using the data below, compute the value of D

for the following:

a. magnesium in aluminum at 500°C

b. zinc in copper at 1000K

c. copper in nickel at 0°C

13

DIFFUSION and TEMPERATURE

Solute Host metal Do (m2/s) EA or QD, (kJ/mol)

Zn Cu 2.4x10-5 189

Cu Ni 2.7x10-5 256

Mg Al 1.2x10-4 131

D Doexp Q

d

RT

Chapter 5-

PROBLEM: Using the data below, compute the value of D

for the following:

a. magnesium in aluminum at 500°C (Ans:1.7x10-13 m2/s)

b. zinc in copper at 1000K (Ans:3.2x10-15 m2/s)

c. copper in nickel at 0°C (Ans:2.8x10-54 m2/s)

13


Solute Host metal Do (m2/s) EA or QD, (kJ/mol)

Zn Cu 2.4x10-5 189

Cu Ni 2.7x10-5 256

Mg Al 1.2x10-4 131

D Doexp Q

d

RT

Chapter 5-

PROBLEM: The diffusion coefficient for carbon and

nickel are given at two temperatures

a. Determine the values of Do and QD.

b. What is the magnitude of D at 900°C?

13


Temperature, °C D (m2/s)

600 5.5x10-14

700 3.9x10-13

Chapter 5-




13



600 5.5x10-14

700 3.9x10-13

TR

EDD A 1

3.2loglog 0

Chapter 5-

SOLUTION:

13



600 5.5x10-14

700 3.9x10-13

KR

EDx

KR

EDx

A

A

973

1

3.2log)109.3log()2(

873

1

3.2log)105.5log()1(

0

13

0

14

Chapter 5-

SOLUTION:

Linear Regression:

13



600 5.5x10-14

700 3.9x10-13

b

AA

DDbercepty

RslopeER

Eslope

xinT

vsyinDplot

10logint

)3.2(3.2

)(1

)(log

00

Chapter 5-




Ans: Do = 1.04x10-5 m2/s and

Ans: QD = 138 kJ/mol

13



600 5.5x10-14

700 3.9x10-13

TR

EDD A 1

3.2loglog 0

Chapter 5-

• Concentration profile,

C(x), changes

w/ time.

14

• To conserve matter: • Fick's First Law:

• Governing Eqn.:

NON STEADY STATE DIFFUSION

∂C

∂t= D

∂2C

∂x2

Chapter 5- 15

"error function"

Values calibrated in Table 5.1, Callister 6e.

Co

Cs

position, x

C(x,t)

tot1

t2t3

Adapted from

Fig. 5.5,

Callister 6e.


Diffusion of Cu in Al

x1

Chapter 5- 15


Diffusion of Cu in Al

For semi-infinite solid (none of the diffusing atoms reaches the bar end during the time over which diffusion takes place),

at t<0 (before diffusion takes place)

C = C0 = uniform concentration of the solute

at t = or > 0 (during diffusion)

C x = 0 = Cs = constant surface concentration (Cs>C0)

C x > 0 = C(x,t) = unsteady concentration profile

pre-existing conc., Co of copper atomsbar

Chapter 5- 15

"error function”

ERROR FUNCTION

General solution:

Chapter 5-

PROBLEM: Consider a carbon iron alloy that initially has a

uniform carbon concentration of 0.25 wt% and is to be

treated at 950C. If the carbon concentration at the

surface is suddenly brought to and maintained at 1.20

wt%, how long will it take to achieve a carbon content of

0.80 wt% at the position 0.5 mm below the surface?

Assume D at this temperature is 1.6x10-11 m2/s and the

alloy is semi-infinite.

13


Carburizing

During diffusion, C is a function of x and t

Surface conc., Cs of C atoms bar

Before diffusion, C0 = conc of C atoms

bar

Chapter 5-

• Copper diffuses into a bar of aluminum.

• 10 hours at 600C gives desired C(x).

• How many hours would it take to get the same C(x)

if we processed at 500C?

16

• Result: Dt should be held constant.

• Answer:Note: values

of D are

provided here.

Key point 1: C(x,t500C) = C(x,t600C).

Key point 2: Both cases have the same Co and Cs.

PROCESSING QUESTION

Chapter 5- 17

• The experiment: we recorded combinations of

t and x that kept C constant.

• Diffusion depth given by:

C(xi, t i )Co

Cs Co 1 erf

xi

2 Dt i

= (constant here)

DIFFUSION DEMO: ANALYSIS

Chapter 5-

• Experimental result: x ~ t0.58

• Theory predicts x ~ t0.50

• Reasonable agreement!

18

DATA FROM DIFFUSION DEMO

Chapter 5- 20

Diffusion FASTER for...

• open crystal structures

• lower melting T materials

• materials w/secondary

bonding

• smaller diffusing atoms

• cations

• lower density materials

Diffusion SLOWER for...

• close-packed structures

• higher melting T materials

• materials w/covalent

bonding

• larger diffusing atoms

• anions

• higher density materials

SUMMARY:

STRUCTURE & DIFFUSION

diffusion

Documents

vacancy diffusion

vacancy concentration

diffusion distance

diffusion demochapter

concentration profiles0adapted

concentration profiles0cu100

diffusion flux ofcarbon

cuni diffusion couplechapter