DIFFUSION IN SOLIDS

FICKS LAWS

KIRKENDALL EFFECT

ATOMIC MECHANISMS

Diffusion in SolidsP.G. Shewmon

McGraw-Hill, New York (1963)

Oxidation

Roles of Diffusion

Creep

AgingSintering

Doping Carburizing

Metals

Precipitates

SteelsSemiconductors

Many more

Some mechanisms

Material Joining

Diffusion 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 the

direction 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

Ficks 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)

Ficks 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 state

J = 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 Ficks 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

Ficks II law

Jx Jx+x

x

xxx JJonAccumulati

x

x

JJJonAccumulati xx

x

x

JJJx

t

cxx J

sm

Atomsm

sm

Atoms

23

.1

xx

Jx

t

c

x

cD

xt

cFicks first law

x

cD

xt

c D f(x)2

2

x

cD

t

c

22

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

22

x

cD

t

c

Dt

xerfBAtxc

2 ),(

Solution to 2o de with 2 constants

determined from Boundary Conditions and Initial Condition

0

2exp2

duuErf

Erf () = 1

Erf ( ) = 1

Erf (0) = 0

Erf ( x) = Erf (x)

u

Ex

p(

u2)

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

cen

trat

ion

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-steady

state

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 = C1A+B = C2

1 2 2 1( , ) 2 2 2

C C C C xc x t erf

Dt

kTQ

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

kTHm

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

kTH

kT

Hmf

ee '

kT

HH mf

e '

kT

HH mf

eD 2

Where, is the jump distance

Interstitial Diffusion

kTHm

eD 2

Substitutional Diffusion

kT

HH mf

eD 2

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

D (Ni in FCC Fe at 1000C) = 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

Lo

g (

D)

Schematic

Polycrystal

Single

crystal

Increasing Temperature

Qgrain boundary = 110 kJ /mole

QLattice = 192 kJ /mole

Applications based on Ficks 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

Solved

Example

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

Given: T = 900 C, C0 = C(x, 0) = C(, t) = 0.2 % C,

Cf = C(0.2 mm, t1) = 1% C (at x = 0.2 mm), Cs = C(0, t) = 1.4% C

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

( , )=

2

S

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 = 1000s

t = 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 = 1000s

t = 7000s

t

0.4 0.6 0.8 1.0 1.2 1.4

The following points are to be noted:

The mechanism of C diffusion is interstitial diffusion

The diffusivity D has to be evaluated at 900C using: 0 exp

QD D

RT

0 expQ

D DRT

34 157 10(0.7 10 )exp

8.314 1173

2127.14 10

m

s

-41

12

1

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)] = 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

( , )=

2

S

S

C x t C xerf

C C Dt

End

Another solution to the Ficks 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

24

2 '

'

)/(

dxdc

JD

kTHm

eD 2

2

0 D

kTQ

eDD 0On comparison

with

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

centr

atio

n

4. Direct Interchange and Ring