DIFFUSION IN SOLIDSDIFFUSION IN SOLIDS FICK’S LAWS

KIRKENDALL EFFECT

ATOMIC MECHANISMS

Diffusion in SolidsP.G. Shewmon

McGraw-Hill, New York (1963)

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh Balani

Materials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016

Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh

AN INTRODUCTORY E-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

Oxidation

Roles of Diffusion

Creep

AgingSintering

Doping Carburizing

Metals

Precipitates

SteelsSemiconductors

Many more…

Some mechanisms

Material JoiningDiffusion bonding

To comprehend many materials related phenomenon one must understand Diffusion.

The focus of the current chapter is solid state diffusion in crystalline materials.

Ar H2

Movable piston with an orifice

H2 diffusion direction

Ar diffusion direction

Piston motion

Piston moves in thedirection of the slower

moving species

A B

Inert Marker – thin rod of a high melting material which is basically insoluble in A & B

Kirkendall effect

Materials A and B welded together with Inert marker and given a diffusion anneal Usually the lower melting component diffuses faster (say B)

Marker motion

Diffusion

Mass flow process by which species change their position relative to their neighbours.

Driven by thermal energy and a ‘gradient’ Thermal energy → thermal vibrations → Atomic jumps

Concentration / Chemical potential

ElectricGradient

Magnetic

Stress

Flux (J) (restricted definition) → Flow / area / time [Atoms / m2 / s]

1 dnJ

A dt

Flow direction

Fick’s I law

A

Flow direction

Assume that only B is moving into A

Assume steady state conditions → J f(x,t) (No accumulation of matter)

dx

dcDA

dt

dn

( ) / / Flux J atoms area time concentration gradient

dx

dcJ

dx

dcDJ

dx

dcD

dt

dn

AJ

1 Diffusivity (D) → f(Concentration of the components, T)

Fick’s first law

dc dJx x

dt dx

Continuity equation

(Truly speaking it is the chemical potential gradient!)

dx

dcDA

dt

dn

No. of atoms crossing area A

per unit time

Cross-sectional area

Concentration gradient

ve implies matter transport is down the concentration gradient

Diffusion coefficient/ Diffusivity

A

Flow direction

As a first approximation assume D f(t)

dx

dcDJ

Diffusion

Steady state J f(x,t)

Non-steady stateJ = f(x,t) D = f(c)

D = f(c)

D f(c)

D f(c)Steady and non-steady state diffusion

0x t

dc J

dt x

Under steady state conditions

0J c

Dx x

Substituting for flux from Fick’s first law

2

20

cD

x

If D is constant

Slope of c-x plot is constant under steady state conditions

constantc

Dx

If D is NOT constant

If D increases with concentration then slope (of c-x plot) decreases with ‘c’

If D decreases with ‘c’ then slope increases with ‘c’

Fick’s II law

Jx Jx+x

x

xxx JJonAccumulati

xx

JJJonAccumulati xx

xx

JJJx

t

cxx J

sm

Atomsm

sm

Atoms

23.

1

xx

Jx

t

c

x

cD

xt

cFick’s first law

x

cD

xt

c D f(x)2

2

x

cD

t

c

2

2

x

cD

t

c

RHS is the curvature of the c vs x curve

x →

c →

x →c

→

+ve curvature c ↑ as t ↑ ve curvature c ↓ as t ↑

LHS is the change is concentration with time

2

2

x

cD

t

c

Dt

xerfBAtxc

2 ),(

Solution to 2o de with 2 constantsdetermined from Boundary Conditions and Initial Condition

0

2exp2

duuErf

Erf () = 1 Erf ( ) = 1 Erf (0) = 0 Erf ( x) = Erf (x)

u →

Exp

( u

2 ) →

0

Area

Also For upto x~0.6 Erf(x) ~ x x 2, Erf(x) 1

A B

Example where the erf solution can be used

x →

Con

cent

rati

on →

Cavg

↑ t

t1 > 0 | c(x,t1)t2 > t1 | c(x,t1) t = 0 | c(x,0)

A & B welded together and heated to high temperature (kept constant → T0)

Flux

f(x)|t

f(t)|x

Non-steadystate

If D = f(c) c(+x,t) c(x,t)

i.e. asymmetry about y-axis

C(+x, 0) = C1

C(x, 0) = C2

C1

C2

A = (C1 + C2)/2

B = (C2 – C1)/2

Dt

xerfBAtxc

2 ),(

AB = C1

A+B = C2

1 2 2 1( , ) 2 2 2

C C C C xc x t erf

Dt

kT

Q

eDD 0

Temperature dependence of diffusivity

Arrhenius type

Diffusivity depends exponentially on temperature.

This dependence has important consequences with regard to material behaviour at elevated temperatures. Processes like precipitate coarsening, oxidation, creep etc. occur at very high rates at elevated temperatures.

ATOMIC MODELS OF DIFFUSION

1. Interstitial Mechanism

Usually the solubility of interstitial atoms (e.g. carbon in steel) is small. This implies that most of the interstitial sites are vacant. Hence, if an interstitial species wants to jump, ‘most likely’ the neighbouring site will be vacant and jump of the atomic species can take place.

Light interstitial atoms like hydrogen can diffuse very fast. For a correct description of diffusion of hydrogen anharmonic and quantum (under barrier) effects may be very important (especially at low temperatures).

The diffusion of two important types of species needs to be distinguished: (i) species ‘sitting’ in a lattice site(ii) species in a interstitial void

2. Vacancy Mechanism

For an atom in a lattice site (and often we are interested in substitutional atoms) jump to a neighbouring lattice site can take place if it is vacant. Hence, vacancy concentration plays an important role in the diffusion of species at lattice sites via the vacancy mechanism.

Vacancy clusters and defect complexes can alter this simple picture of diffusion involving vacancies

Interstitial Diffusion

1 2

1 2

Hm

At T > 0 K vibration of the atoms provides the energy to overcome the energybarrier Hm (enthalpy of motion)

→ frequency of vibrations, ’ → number of successful jumps / time

kT

Hm

e '

Substitutional Diffusion

Probability for a jump (probability that the site is vacant).(probability that the atom has sufficient energy)

Hm → enthalpy of motion of atom

’ → frequency of successful jumps

kT

HkT

Hmf

ee '

kT

HH mf

e '

kT

HH mf

eD 2Where, is the jump distance

Interstitial Diffusion

kT

Hm

eD 2

Substitutional Diffusion

kT

HH mf

eD 2

D (C in FCC Fe at 1000ºC) = 3 1011 m2/s

D (Ni in FCC Fe at 1000ºC) = 2 1016 m2/s

0 f mH H

kTD D e

of the form

0 mH

kTD D e

of the form

Diffusion Paths with Lesser Resistance

Qsurface < Qgrain boundary < Qpipe < Qlattice

Experimentally determined activation energies for diffusion

Core of dislocation lines offer paths of lower resistance → PIPE DIFFUSION

Lower activation energy automatically implies higher diffusivity

Diffusivity for a given path along with the available cross-section forthe path will determine the diffusion rate for that path

Comparison of Diffusivity for self-diffusion of Ag → single crystal vs polycrystal

1/T →

Log

(D

) →

Schematic

Polycrystal

Singlecrystal

← Increasing Temperature

Qgrain boundary = 110 kJ /mole

QLattice = 192 kJ /mole

Applications based on Fick’s II law Carburization of steel

Surface is often the most important part of the component, which is prone to degradation.

Surface hardening of steel components like gears is done by carburizing or nitriding.

Pack carburizing → solid carbon powder used as C source. Gas carburizing → Methane gas CH4 (g) → 2H2 (g) + C (diffuses into steel).

C(+x, 0) = C1

C(0, t) = CS

A = CS

B = CS – C1

SolvedExample

A 0.2% carbon steel needs to be surface carburized such that the concentration of carbon at 0.2mm depth is 1%. The carburizing medium imposes a surface concentration of carbon of 1.4% and the process is carried out at 900C (where, Fe is in FCC form).

Data: 4 20D (C in -Fe) 0.7 10 m / s 157 /Q kJ mole

The solution to the Fick’ second law: ( , ) 2

xC x t A B erf

Dt

The constants A & B are determined from boundary and initial conditions:

(0, ) 0.014SC t A C , 0( , ) 0.002C t A B C or 0( ,0) 0.002C x A B C

S 0B C C 0.012 , ( , ) 0.014 0.012 2

xC x t erf

Dt

-44

1

1

2 10(2 10 , ) 0.01 0.014 0.012

2C m t erf

Dt

S S 0( , ) C (C -C ) 2

xC x t erf

Dt

0

( , )=

2S

S

C x t C xerf

C C Dt

(2)

(1)

-4

1

1 2 10

3 2erf

Dt

x (in mm from surface)

% C

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2

t =0

t = t1 = 14580s

t = 1000st = 7000s

t

0.4 0.6 0.8 1.0 1.2 1.4

x (in mm from surface)

% C

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.2

t =0

t = t1 = 14580s

t = 1000st = 7000s

t

0.4 0.6 0.8 1.0 1.2 1.4

-4

1

121

2 10(0.3333) 0.309

2 7.14 10erf

t

24

1 12

1 1014580

0.337.14 10t s

From equation (2)

-4

1

1 2 10

3 2erf

Dt

Approximate formula for depth of penetrationDtx

0

0

( , )1

2S

c x t C xerf

C C Dt

Let the distance at which [(C(x,t)C0)/(CSC0)] = ½ be called x1/2 (which is an ‘effective penetration depth’)

1 211

2 2

xerf

Dt

1 2 1

22

xerf

Dt

1 1

2 2erf

1 2 1

22

x

Dt

penetrationx Dt

The depth at which C(x) is nearly C0 is (i.e. the distance beyond which is ‘un’-penetrated):

0 12

xerf

Dt

Erf(u) ~ 1 when u ~ 2

22

x

Dt

4x Dt

0

( , )=

2S

S

C x t C xerf

C C Dt

End

Another solution to the Fick’s II law

A thin film of material (fixed quantity of mass M) is deposited on the surface of another material (e.g. dopant on the surface of a semi-conductor). The system is heated to allow diffusion of the film material into the substrate.

For these boundary conditions we get a exponential solution.

2

( , ) exp4

M xc x t

DtDt

Boundary and Initial conditions

C(+x, 0) = 0

0cdx M

Ionic materials are not close packed Ionic crystals may contain connected void pathways for rapid diffusion These pathways could include ions in a sublattice (which could get disordered)

and hence the transport is very selective alumina compounds show cationic conduction Fluorite like oxides are anionic conductors

Due to high diffusivity of ions in these materials they are called superionic conductors. They are characterized by: High value of D along with small temperature dependence of D Small values of D0

Order disorder transition in conducting sublattice has been cited as one of the mechanisms for this behaviour

Diffusion in ionic materials

Element Hf Hm Hf + Hm Q

Au 97 80 177 174

Ag 95 79 174 184

Calculated and experimental activation energies for vacancy Diffusion

1 2

Vacant site

c = atoms / volume c = 1 / 3

concentration gradient dc/dx = (1 / 3)/ = 1 / 4 Flux = No of atoms / area / time = ’ / area = ’ / 2

242

''

)/(

dxdc

JD

kT

Hm

eD 2

20 D

kT

Q

eDD 0

On comparisonwith

3. Interstitialcy Mechanism

Exchange of interstitial atom with a regular host atom (ejected from its regular site and occupies an interstitial site)

Requires comparatively low activation energies and can provide a pathway for fast diffusion

Interstitial halogen centres in alkali halides and silver interstitials in silver halides

D = f(c)

D f(c)C1

C2

Steady state diffusion

x →

Con

cent

rati

on →

4. Direct Interchange and Ring