5. Diffusion in solids

Two views of the diffusion processTwo views of the diffusion process1) Microscopic - Random walk 2) Macroscopic – Continuous spreading

Fick’s Laws and the Diffusion CoefficientC = concentration of a species in a mixture(atoms/cm3)

J = Flux of this speciesUnit plane

J sec2 cmatoms

Concentration of mass of speciesdx Unit area

J dxddJJ

x

C(1×dx) atoms

Time rate of change of atoms of species in dxg p= net influx of species + creation of species in dx by source.

QdxdxddJJJCdx

)() ( Q

dxt)()(

QxJ

tCor

Generally, J is due to diffusion :

CD J Fick’s first law (cm2/sec)

xD

J

For constant D,

QxCD

tC

2

2

Fick’s second law

Other coordination : QCDtC

2

Cartesian

x

z

2

2

2

2

2

22 CCCC

xy

222 zyxC

CylindricalCylindrical

222 1)(1

CCCrC

z

222)(

rzr

rrr

Crθ

Spherical

rθ

2

22

22 )(sin

sin1)(1

C

rrCr

rrC

Ф 2

2

22 sin1

C

r

Applicable to solids, liquids, gases

Atomic picture of diffusion in solids

ε*=diffusion activation energy or “barrier”

* ф фε*= ф(saddle point) – ф(equilibrium site)

ν = vibration frequency of atom in equilibrium site211 d ν = 1-2/1

2

2

sec , ])(1[21

eqdxd

m

Jump frequency : Г f f j i ( di ti )Jump frequency : Г = frequency of jumping(any direction)

ω = frequency of jumping(one direction)

Г =βω Where β = No of neigh neighboring sitesatom can jump to.

ω = ν e-ε*/RTω ν e

Attempt frequency Fraction of jumps(attempts) which havep q y(Einstein or Debye frequency)

Fraction of jumps(attempts) which have energy sufficient to overcome barrier

Fig 7.4 Eight random jumps of equal length λ. – frequencyⅠ

Compute from random walk theory2r

In time t, atom makes Гt jumps for one particular sequence of n jumps:

nr 1 (vector condition)

)()(2 rrr )()( 11 nnrrr

dot product

1

1 11

2 2n

i

n

ijji

n

iiir

1

1 1

2

1

2 cos2n

i

n

ijij

n

i

nλ2 random between -1 and +1

Average over many sequences averages to zeroijcos

222r n Гt

2r by Macroscopic Diffusion Analysis

- Spreading of N atoms initially placed at r=0

cDC 21 I.C. C(r,0)=0 for r≠0

rr

rrD

t2

2I.C. C(r,0) 0 for r 0B.C. C(∞,t)=0

also :

24 drtrCrN also : 0,4 drtrCrN

Solution : = Probability of finding atom i i l d

2/3

4/

)4(),(

2

Dte

NtrC Dtr

in unit volume at r and t

2/3)4( DtN

space

2

0 2/3

4/2322 64

)4(]),([

2

all

Dtr

DtdrrDt

errdN

trCrr

C

t 0t=0

t>0

rt=∞

Equate from random walk theory and diffusion analysis :2r

61 2D Einstein formulaГ

Applications of Einstein Formula

① Vacancy mechanism of diffusion in FCC lattice① Vacancy mechanism of diffusion in FCC lattice(self diffusion in metals)

barrier (half way along jump)

There are 12 equivalent sites around atom in FCC structure. And, the probability that anyone is empty is the vacancy fraction xv

∴β = 12xv a0

also λ=jump distance =x

av

* *2

/ 2 /01 12 RT RTaD x e a x e

0012

6 2f v vD x e a x e

for thermodynamic equilibrium vacancy fraction, RTv

rex /*( )/RT ( )/

0v RTD a e

Pre-exponential (3ⅹ10-8cm)2 (1013sec-1)=10-2cm2/sec

Diffusion of vacancy ( not atoms)

*2 /RTD a e No Xv , since vacancy defined to be present, or Xv = 1

0vD a e② Impurity Diffusion in BCC structure (carbon, hydrogen in iron)

Impurity ATOM 42/0a

1 2 waD 4

261 0

*1 RTeaD /20

*

61

Where is settle point?a0

p

General RTEeDD /0

Thermal Diffusion

Irreversible thermodynamics: coupling of heat and mass fluxes by chemical potential and temperature gradientsp p g

11 12TJ L L

T

LL21 22

Tq L LT

1221 LL

Or, in conventional notation*Q TJ D C D C

*Q = Heat of transport( + or - )J D C D CkT T

Fick’s lawSoret effect

Q*Q

Heat of transport( + or )

= Thermal diffusion factor*q x T Q D C

Fourie’s Law

kT Thermal diffusion factor

Fourie s Law

Measurement of Q*Q

Mixture of A + B in tube-initially uniform

Apply temperature gradientApply temperature gradient

When final steady state reached:

TT

When final steady state reached:

*C0 or C

Q TJkT T

T

Conc.Of A

C /

kT Tbut C C L

init

Of A

*

T /

C

T L

Q T

L

C

QkT T

Typically 1~10

Thermal Diffusion in Solids (impurity atoms in host)( p y )

T(z)

One-dimensional random

J

Wwalk - ζ direction only

JzImpurity atoms/cm2

W

C Cn

Atomic plane

Con c = C =

nRnL atoms/cm2

Atomic planes

W J f f ld l t h t l 1W+ = Jump frequency from cold plane to hot plane, sec-1

W- = Jump frequency from hot plane to cold plane, sec-1

J- = nL w- = flux from hot plane to cold planeJ+ = nR w+= flux from cold plane to hot plane

Impose condition of zero net flux : J- = J+

or

ww

nn

R

L

wnR

In isothermal case : w+ = w = 1/*

RTe ГIn isothermal case : w+ w- 2

e

O di i

Г

One dimension,half jump left & half right

Assume : ε* = εi + εm + εf

εi = energy imparted to atom to start jumpε energy to cross barrier between planesεm = energy to cross barrier between planesεf = energy to prepare the site in the final plane for receiving

the jumping atom(e.g., relaxation of neighbors)

/w i kTe e

[ ]1( )2

m

k T T

e[ ]

( )f

k T T

[ ]1( )2

m

k T T

e w /f kTe [ ]

( )i

k T T

e

H

C

n ex p { } ex p { } ( ) n

i f f iww kT k T T

Note : If all energy required for barrier crossing(ε*= εm), there is no thermal diffusion.

Let Q*=εi – εf = heat of transport, because

- Q * /k T -Q * /k TL

R

n * / Q * T e e x p { } e e x p { (1 - ) } n k T T1

Q k TT

T

T

Q * T- k T

2

* e 1 Q Tk T

2k T

Expand

dnnn dz

nn RL

dTT dz

T

dTQdn2

*111 dzRTdzn 2

B t /λBut c=n/λ

dTcQdc *dzdT

RTcQ

dzdc

2

*

Surface diffusion (2 dimensional)

INITIAL DEPOSIST OF IMPURITY

SUBSTRATESURFACE

(Js)cons c = -Ds▽Cs

J

Surface flux

)sec

( 2 cmatoms Surface con C

)( atomsJsJs

seccm )( 2cm

Surface diff. coef(cm2/sec)

SURFACE AFTER SPREADING

( / )

Ds = 2

41

s Гs

s*21

RT

sss e2

41

ε*s < heat of vap.

Fig. 7.8 Surface diffusion of impurity species on a crystal surface

Surface thermal self diffusionSurface thermal self diffusion

Migration of lattice a forms along its own surface due to

a temperature gradient.

▽Cs=0(because the migrating species same as lattice)

T

Only soret effect left

T*Q surfacealong)

TT()C

RTQ(D)(J s

ssths