Lecture: Diffusion in Metals and Alloysnikolai.priezjev/papers/Lecture_5_Ch_2_diffusion.pdfDiffusion Flux The flux of diffusing atoms, J, is used to quantify how fast diffusion occurs

Textbook: Phase transformations in metals and alloys

(Third Edition), By: Porter, Easterling, and Sherif (CRC

Press, 2009).

Diffusion and Kinetics

Lecture: Diffusion in Metals and Alloys

Nikolai V. Priezjev

Diffusion in Metals and Alloys

► Diffusion mechanisms

(1) Vacancy diffusion (2) Interstitial diffusion (3) Impurities

► The mathematics of diffusion

(1) Steady-state diffusion (Fick’s first law)

(2) Nonsteady-State Diffusion (Fick’s second law)

► Factors that influence diffusion

Diffusing species, Host solid, Temperature, Microstructure

Reading: Chapter 2 of Porter, Easterling, Sherif

https://www.slideshare.net/NikolaiPriezjev

What is diffusion?

Diffusion is material transport by atomic motion.

Inhomogeneous materials can become homogeneous by diffusion. For an active

diffusion to occur, the temperature should be high enough to overcome energy

barriers to atomic motion.

Free energy and

chemical potential

changes during

diffusion

Oil/water

Miscible Why diffusion?

Interdiffusion and Self-diffusion

Interdiffusion (or impurity diffusion) occurs in response to a concentration

gradient.

Self-diffusion is diffusion in one-component material, when all atoms that

exchange positions are of the same type.

Diffusion Mechanisms (I)

To jump from lattice site to lattice site, atoms need energy to break bonds with

neighbors, and to cause the necessary lattice distortions during jump. This energy

comes from the thermal energy of atomic vibrations (Eav ~ kBT)

The direction of flow of atoms is opposite the vacancy flow direction.

Diffusion Mechanisms (II)

Interstitial diffusion is generally faster than vacancy diffusion because bonding of

interstitials to the surrounding atoms is normally weaker and there are many more

interstitial sites than vacancy sites to jump to.

Requires small impurity atoms (e.g. C, H, O) to fit into interstices in host.


Diffusion Flux

The flux of diffusing atoms, J, is used to quantify how fast diffusion occurs. The

flux is defined as either the number of atoms diffusing through unit area per unit

time (atoms/m2-second) or the mass of atoms diffusing through unit area per

unit time, (kg/m2-second).

For example, for the mass flux we can write

where M is the mass of atoms diffusing through the area A during time t.


Steady-State Diffusion: Fick’s first law

Steady-state diffusion: the diffusion flux does not change with time.

Concentration profile: concentration

of atoms/molecules of interest as

function of position in the sample.

secondper jumps ofnumber B

2

6

1BBD

1n

2n

x

CDJ B

BB

42

2

m

atoms,

sm

atomsJ ,

s

m

x

CDB

Fe/C example: Co1000at Fe

nm26.020.37/ 0.37nm;~ lattice

/sm105.2 211CD jumps/s102 9B

mpattemps/ju10 :10~ freqvibr 413

Dilute solution!

Random walk of a single atom

n isnt displacemenet the

,length of stepsn After

tr

:atoman ofnt displaceme

average the t,After time

2

6

1BBD

Dtr 4.2


m10~only isnt displacemenet but

0.5m~ moves C second 1Every

Einstein relation

Mean square displacement

Random walk of a single atom

n isnt displacemenet the

,length of stepsn After

tr

:atoman ofnt displaceme

average the t,After time

2

6

1BBD

Dtr 4.2Anomalous diffusion


m10~only isnt displacemenet but

0.5m~ moves C second 1Every

Effect of temperature – thermal activation (I)

Interstitial atom, (a) in equilibrium position, (b) at the position of maximum lattice distortion, (c) Variation of the free energy of the lattice as a function of the position of interstitial.

energy activation m G

2

6

1BBD

RT

GvzB

mexp

frequency lvibrationav

numberon coordinatiz

secondper jumps ofnumber B

Effect of temperature – thermal activation (II)

2

6

1BBD

RT

GvzB

mexp


RT

QDD BB

exp0

TR

QDD BB

1

3.2loglog 0

Such plots are called Arrhenius plots.

enthalpy activationQ

RT

Q

R

SzvD m

B

expexp

6

1 2


Effect of temperature – thermal activation (III)

2

6

1BBD

RT

GvzB

mexp


RT

QDD BB

exp0

TR

QDD BB

1

3.2loglog 0

enthalpy activationQ

Low THigh T

Effect of temperature – thermal activation (IV)

TR

QDD BB

1

3.2loglog 0

Diffusion of

interstitials is typically

faster as compared to

the vacancy

diffusion mechanism

(self-diffusion or

diffusion of

substitutional atoms).Low THigh T

Effect of temperature – thermal activation (V)

Diffusion of a cluster of 10 atoms on a substrate.


Example: Temperature dependence of D

Example: Temperature dependence of D

50C = 2x diffusion!

Nonsteady-State Diffusion: Fick’s second law

In many real situations the concentration profile and the concentration

gradient are changing with time. The changes of the concentration profile

can be described in this case by a differential equation, Fick’s second law.

Solution of this equation is concentration

profile as function of time, C(x,t):

Nonsteady-State Diffusion: Fick’s second law


Nonsteady-State Diffusion: Homogenization

The effect of diffusion on a sinusoidal variation of composition.

l

xCC o

sin

t

l

xCC o

expsin

BD

l2

2

Relaxation time

to

exp

Initial concentration profile:

increasestion homegeniza of rate decreases, As llength

B B

Nonsteady-State Diffusion: Carburization of Steel


Nonsteady-State Diffusion: Carburization of Steel

Concentration profiles at successive times for diffusion into a semi-infinite bar when the surface concentration is maintained constant.

Dt

xCCCC oss

2erf)(

sC oC

1C 2C

dyyz

z

0

2)exp(2

erf

C1000at min 17in mm 0.2 :E.g. o

Dt

xCCCCC

2erf

22

2121

s/m104D 2-11

Nonsteady-State Diffusion: Analytical solution for plane source

Analytical solution for plane source






Dtx ~

Diffusion: Role of the microstructure (I)

Self-diffusion coefficients

for Ag depend on the

diffusion path. In general

the diffusivity if greater

through less restrictive

structural regions – grain

boundaries, dislocation

cores, external surfaces.

Diffusion: Role of the microstructure (II)

The plots are from the computer simulation by T.

Kwok, P. S. Ho, and S. Yip. Initial atomic positions

are shown by the circles, trajectories of atoms are

shown by lines. We can see the difference between

atomic mobility in the bulk crystal and in the grain

boundary region.

Mean square displacement:

2.3.1 Substitutional Diffusion: Self-Diffusion

RT

Gvz v

mexp


RT

H

R

S vve

v

expexp

ionconcentratvacancy e

v

2

6

1D

RT

HH

R

SSzvD vmvm expexp

6

1 2

RT

QDD SD

exp0 vmSD HHQ

metals fccin 12z

e

vv DD /

Note:

(111) plane


Dtr 4.2


Dt

x

Dt

MC

4exp

2

2


Substitutional and interstitial alloys

Different atomic mechanisms of alloy formation, showing pure metal, substitutional,

interstitial, and a combination of the two.

2.3.3 Diffusion in Binary Substitutional Alloys (I)

x

CDJ B

BB

x

CDJ A

AA

:lattice the torelativeDiffusion

BA CCC 0x

C

x

C BA

x

CDJ A

BB

x

CDJ A

AA

:Rewrite

Interdiffusion and vacancy flow. (a) Composition profile after interdiffusion of A and B. (b) The corresponding fluxes of atoms and vacancies as a function of position x. (c) The rate at which the

vacancy concentration created or destroyed by dislocation climb.

BA DD

|||| BA JJ

BvA JJJ

The jumping of atoms in one direction

can be considered as the jumping of

vacancies in the other direction.

(a) before, (b) after: a vacancy is absorbed

at a jog on an edge dislocation (positive

climb), (b) before, (a) after: a vacancy is

created by negative climb of an edge

dislocation, (c) Perspective drawing of a

jogged edge dislocation.

2.3.3 Diffusion in Binary Substitutional Alloys (II)


The jumping of atoms in one direction

can be considered as the jumping of

vacancies in the other direction.

:fluxnet Vacancy

BAv JJJ x

CDDJ A

BAv

)(

A flux of vacancies causes the atomic planes to move through the specimen.

To keep Cv constant, vacancies should be created

on B-rich side and destroyed on A-rich side.

x

J

t

C vv

new plane

2.3.3 Diffusion in Binary Substitutional Alloys (III)

BvA JJJ

: vplanes lattice ofmovement ofVelocity

0v CtAtAJv

# atoms is removed

by vacancies

crossing plane A

# atoms in the

volume, which is

swept by plane A

during time dt

0v CJv

x

CDDJ A

BAv

)(

0/ CCAA

xDD A

BA

)(v

new plane

A flux of vacancies causes the atomic

planes to move through the specimen.

2.3.3 Diffusion in Binary Substitutional Alloys (IV)

But is does not tell us how fast A and B

actually mix.

x

J

t

C AA

Fick’s first law for interdiffusion

The total flux of atoms A:

AA

AA Cx

CDJ v

Flux due to

lattice

velocity

Standard

diffusive

flux

xDD A

BA

)(v

x

CDDJ A

BAABA

)(

BAAB DDD ~

x

CDJ A

A

~

x

CDJ B

B

~

BA JJ

2.3.3 Diffusion in Binary Substitutional Alloys (V)

0/ CCAA

Interdiffusion coefficient:

*

2.3.3 Diffusion in Binary Substitutional Alloys (VI)

Fick’s second law for interdiffusion

x

J

t

C AA

x

CDJ A

A

~

+

=

x

CD

xt

C AA ~

BAAB DDD ~

RT

QDD

exp

~~0

RT

QDD B

BB

exp0

RT

QDD A

AA

exp0

Darken’s equations

2.3.3 Diffusion in Binary Substitutional Alloys (VII)

An experimental arrangement to show

the Kirkendall effect. 1947. inert wires.

BAAB DDD ~

Zn30wt%Cu

diffusion coefficients in the Cu-Ni system at 1000°C

CuZn DD

W decreases

2.7


2.4 Atomic Mobility

Drift velocity is related to the diffusive flux:

BBB CvJ

32 m

atoms

s

m

sm

atoms J Cv

Reasonable to guess that:

xMv B

BB

Some

constant

Mobility

‘Chemical’

force per

atom

Atoms

drift down

gradient

Combine both equations:

xCMJ B

BBB

x

CDJ B

BB


2.7 High Diffusivity Paths

RT

QDD l

ll

exp0

RT

QDD b

bb

exp0

RT

QDD S

SS

exp0

lbS DDD

Found experimentally that:

The effect of grain boundary diffusion combined with volume diffusion.

In general

But the size of the

highway matters!


Combined lattice and boundary fluxes during steady-

state diffusion through a thin slab of material.

x

CDJ ll

x

CDJ bb

The total flux:

x

C

d

dDD

ddJJJ

lb

lb

w/)ww(

x

C

The apparent diffusion coefficient:

dDDD blapp /

w


Combined lattice and boundary fluxes during steady-

state diffusion through a thin slab of material.

x

C

The apparent diffusion coefficient:

dD

D

D

D

l

b

l

app 1

Grain boundary diffusion is important:

dDD lb

dDDD blapp /

nm 5.0~

m 10001~ d

Grain boundary

Grain size

w

2.7 High Diffusivity Paths: Temperature effects

Diffusion in a polycrystalline metal

RT

QDD l

ll

exp0

RT

QDD b

bb

exp0

Note that:

T allat lb DD

dDDD blapp /

metals fccin 5.0 lb QQ

Low T

High T


2.7 High Diffusivity Paths: Along Dislocations

Dislocations act as a high conductivity path through the lattice.

l

p

l

app

D

Dg

D

D1

710gIn a well annealed material:

-25 mmper nsdislocatio 10

atoms 10

-213 mm atoms 10

High T: small l

p

D

Dg Low T: only dislocations

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Lecture: Diffusion in Metals and Alloysnikolai.priezjev/papers/Lecture_5_Ch_2_diffusion.pdfDiffusion Flux The flux of diffusing atoms, J, is used to quantify how fast diffusion occurs