Introduction to Engineering Materials

ENGR2000

Chapter 5: Diffusion

Dr. Coates

5.1 Introduction

• Many reactions & processes that are important in

the treatment of materials rely on the transfer of

mass either within a specific solid or from a liquid,

gas or another solid phase.

• Diffusion

– Material transport by atomic motion

Types of Diffusion

• Inter-diffusion or Impurity diffusion

– Atoms of one metal diffuse into another

– Net drift of atoms from high to low concentration

regions.

• Self-diffusion

– All atoms exchanging positions are of the same type:

occurs for pure metals.

You can observe compositional changes in order to

detect inter-diffusion, can you do the same to detect

self diffusion?

5.2 Diffusion Mechanisms

• Diffusion is the stepwise migration of atoms

from lattice site to lattice site.

• You are at a music concert, it’s very crowded

and you are exhausted. You are surrounded by

sweaty people and you can’t see the performance

very well. What physical conditions that would

exist ideally, so you would be motivated to move

from your current spot?

Necessary Conditions

empty adjacent sites

sufficient energy to break existing bonds

• The two conditions needed for diffusion to occur

are

Vacancy Diffusion

• Interchange of an atom from a normal lattice site

to an adjacent vacant lattice site.

Vacancy Diffusion

• How is the vacancy motion related to the diffusing

atom motion?

• Self-diffusion & inter-diffusion occur by this

mechanism.

Interstitial Diffusion

• Atoms migrate from an interstitial position to a

neighboring one that is empty.

• Atoms must be small enough to fit into the

interstitial positions.

• Would this apply to inter-diffusion or self diffusion?

Diffusion mechanisms in

metals & metal alloys

• Which would occur more rapidly, interstitial

diffusion or vacancy diffusion? Why?

Interstitial atoms are smaller & more mobile.

There are more empty interstitial positions than vacancies.

Rate of Mass transfer defined

• What are the units of J?

dt

dM

AJ

form aldifferenti In

At

MJ

(t) time unit per solidof A, area, sectionalcross

unit a to and through diffusing atoms) of number (or M,mass,

the as defined (J), flux Diffusion

1

Kg/m2s or #atoms/m2s

Fick’s First Law

dx

dCDJ

:lawfirst sFick'

t.coefficiendiffusion theis D where

:emperatureconstant tat

direction,x in the state)(steady atoms offlux diffusion Net

KmolJconstGasR

KetemperaturabsoluteT

atomeVormolJdiffusionforenergyactivationQ

RT

QDD

/31.8.

)(

//

exp

:dependence etemperaturt coefficienDiffusion

0

Steady State Diffusion

• Concentrations not changing with time

• What does this imply regarding

– Mass leaving/entering?

Does a constant diffusion flux

value (with time) imply steady

state?

yes

• What then are units of

D?

For diffusion problems it’s sometimes best to express

concentration, C, in terms of mass of diffusing

species per unit volume of solid

m2/s

dx

dCDJ

Fick’s First Law

• Fick’s first law

– restricted to problems for which the concentration

gradient does not change with time (steady state).

– the –ve sign indicates that mass flows from regions of

high concentration to regions of low concentration.

dx

dCDJ

Practical Examples of Diffusion

• The controlled diffusion of P-type or other

dopants into Si wafers.

• The controlled diffusion of oxygen and carbon

dioxide through a membrane in a heart-lung

machine permits surgeons to operate on the heart.

• Carburization of steels.

Example 5.1

• A thin plate of Fe is exposed to a carburizing

atmosphere on one side and a decarburizing

atmosphere on the other side at 700 0C. If a

condition of steady state is achieved, calculate the

diffusion flux of C through the plate if the

concentrations of C at positions of 5 mm & 10 mm

beneath the carburizing surface are 1.2 kg/m3 and

0.8 kg/m3 respectively. Assume a diffusion

coefficient of 3 x 10-11 m2/s at this temperature.

?

/103700

/8.0

/2.1

10

5

700

:Given

2110

3

3

J

smCD

mkgC

mkgC

mmx

mmx

CT

B

A

B

A

smkg

xx

CCD

dx

dCDJ

BA

BA

29 /104.2

:lawfirst sFick' Using

5.4 Non Steady-State Diffusion

• Fick’s first law

– restricted to problems for which the concentration

gradient does not change with time.

• Diffusion through a solid bar…

Fick’s Second Law

x

txJ

xx

txJtxJx,tC

xxdAdxdAdV

dAtxJdAtxJdVx,tC

dAtxJdAtxJdVtt

txCtxC

inout

outoutinin

inout

outoutinin

outoutinin

ii

ii

,,,

t

: Using

,,t

,,,,

out rate flow -in rate flow on accumulati of rate

:dV volumeelementalan in solute theof balance' mass' Using

1

1

Fick’s Second Law

.(geometry) conditionsboundary and

conditions initial theon depends solution - ,

t

:law second sFick'

,,:)dependence time(withlaw first s Fick'Using

2

2

x

txCD

x,tC

x

txCDtxJ

Fick’s Second Law solution

(semi-infinite solid)

law. second sFick' osolution t -

21

,

:source dreplenishely continuous awith

and plate, thick a intodiffusion 0,

x0 ion,concentrat initialconstant 0,

surface, at theion concentratconstant 0,0

:constant held is surface at the conc.

hein which t solid infinite semi a intodiffusion For

0

0

0

0

Dt

xerf

CC

CtxC

CtxC

CtxC

CtxC

s

s

z

dyyzerf0

2exp2

What are dimensions of erf (z)?

Questions?

• What if you want to attain a fixed composition ?

–𝑥2

𝐷𝑡= constant, Prove this!

• What about a fixed composition at a particular

distance?

– 𝐷𝑡 = constant, Prove this!

Example 5.2

• For some applications, it is necessary to harden

the surface of a steel (Fe & C alloy) above that of

its interior. One way this may be accomplished is

by increasing the surface concentration of C in a

process termed carburizing; the steel is exposed at

an elevated temperature, to an atmosphere rich in a

hydrocarbon gas such as methane.

Cs

C0

• Consider one such alloy that initially has a

uniform C concentration of 0.25 wt. % and is to be

treated at 950 0C. If the concentration of the C at

the surface is suddenly brought to and maintained

at 1.20 wt. %, how long will it take to achieve a C

content of 0.80 wt. % at a position 0.5 mm below

the surface? The diffusion coefficient for C in Fe

at this temperature is 1.6 x 10-11 m2/s. Assume that

the steel plate is semi-infinite.

Example 5.2

Diffusion of C into a large slab of Fe…

Example 5.2

?

/106.1

950

5.0

%80.0,

%2.1

%25.0

:Given

211

0

0

t

smTD

CT

mmx

txC

C

C

s

hst

Dt

x

Dt

xerf

Dt

xerf

Dt

xerf

CC

CtxC

s

1.7400,25

392.02

:5.1 Table Using

4215.02

21

25.020.1

25.080.0

21

,

:law second sFick' oSolution t

0

0

Example 5.3

• The diffusion coefficients for Cu in Al at 500 0C

and 600 0C are 4.8 x 10-14 m2/s and 5.3 x 10-13

m2/s, respectively. Determine the approximate

time at 500 0C that will produce the same

diffusion result (in terms of concentration of Cu at

some specific point in Al) as a 10 h heat treatment

at 600 0C.

How do you know whether this is a

steady state scenario or not?

Example 5.3

?

10

,

/103.5600

/108.4500

:Given

500

600

0

2130

2140

t

ht

samex

sametxC

sameC

sameC

smCD

smCD

s

ht

tDtD

constDt

constDt

x

constDt

xerf

Dt

xerf

CC

CtxC

:law secondsFick' to Solution

s

4.110

2

2

21

,

500

600600500500

0

0

5.5 Factors that Influence Diffusion

- Temperature

constant. gas universal theis

andKelvin in re temperatu theis

constant, a is

diffusion,for energy activation theis where

exp

:t coefficienDiffusion

0

0

R

T

D

Q

RT

QDD d

5.5 Factors that Influence Diffusion

- Diffusing Species

– Diffusing species/host

• Crystal structure (APF)

• Atomic radius

– Temperature

Compare the self-diffusion of Fe and the

inter-diffusion of C in Fe at 500 0C

Example 5.4

- Using the data in Table 5.2, compute the

diffusion coefficient for Mg in Al at 550 0C.

Example 5.4

sm

KKmolJ

molJsmCD

RT

QDD

/108.5

273550./31.8

/000,131exp/102.1550

exp

:t coefficienDiffusion

213

240

0

Design Example 5.1

• The wear resistance of a steel gear is to be improved byhardening its surface. This is to be accomplished byincreasing the carbon content within an outer surface layeras a result of C diffusion into steel; the C is to be suppliedfrom an external C-rich gaseous atmosphere at an elevatedand constant temperature. The initial C content of the steelis 0.20 wt%, whereas the surface concentration is to bemaintained at 1.00 wt %. For this treatment to be effective,a C content of 0.60 wt% must be established at a position0.75 mm below the surface. Specify an appropriate heattreatment in terms of temperature and time fortemperatures between 900 C and 1050 C. Use data inTable 5.2 for the diffusion of C in -iron.

?

?

1050900

75.0

%60.0,

%00.1

%20.0

Fe-in C -

:Given

00

0

t

T

CTC

mmx

txC

C

C

s

27

0

0

1024.6

4747.02

:5.1 Table Using

5.02

21

,

:law second sFick' oSolution t

mDt

Dt

x

Dt

xerf

Dt

xerf

CC

CtxC

s

range. given the in etemperatur

specified somefor calculated bemay time required the -

s

T

t

mRT

QtDDt

RT

QDD

:tcoefficien Diffusion

molJQ

smD

:5.2 Table Using

d

d

d

810,17exp

0271.0

104.6exp

exp

/000,148

/103.2

270

0

250

What R should we use

(units)?

Diffusion in Semi-Conductors

• Semiconductor Integrated Chips (IC)

– Base material is single crystal silicon

– Two heat treatments

• Predisposition Step: impurity atoms (gas phase) are diffused

under pressure where surface composition remains constant

• Drive in diffusion: used to transport impurity atoms farther

without increasing impurity content

– Higher temperature

– Oxide layer on surface (few impurity atoms diffuse out)

Diffusion in Semi-Conductors

law second sFick' osolution t -

4exp,

:source) dreplenishely continuous a(not consumed issupply theif

and plate, thick a intodiffusion ,

ion,concentrat initialconstant 0,

surface, at theion concentrat ,0

:plate thick a intodiffusion For

20

0

0

Dt

x

Dt

QtxC

CtxC

CtxC

CtxC s

pp

s

tDC

Q

2Q where

tors.semiconduc of doping -

.)(mass/area species diffusing theofsupply theis where

0

0

Surface Concentration

Diffusion coefficient for the predisposition step

predisposition treatment time

5.7 Other Diffusion Paths

• Atomic migration can also occur along

– dislocations

– Grain boundaries

– External surfaces

• Short-circuit diffusion paths

– Faster than bulk (normal) diffusion

– their contribution is insignificant because the cross-

sectional areas of these paths are small.

Example

• The diffusion of H in FCC Fe

– r(H) << r(Fe)

– interstitial mechanism

• The self-diffusion in FCC Fe

– vacancy mechanism

– higher activation energy

Example

• Explain each of these observations

(b) The activation energy for the diffusion of H in BCC

Fe is less than that for diffusion in FCC Fe.

Example

• The diffusion of H in BCC Fe

– interstitial mechanism

– BCC structure is more open (lower APF)

– low activation energy

• The diffusion of H in FCC Fe

– interstitial mechanism

– FCC structure is a close-packed structure (high APF)

– high activation energy