Topic5 Diffusion

8/9/2019 Topic5 Diffusion

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WHY STUDY DIFFUSION?

Materials often heat treated to improve properties

Atomic diffusion occurs during heat treatment

Depending on situation higher or lower diffusion ratesdesired

Heat treating temperatures and times, and heating or coolingrates can be determined using the mathematics/physics of diffusion

Example: steel gears are case-hardened bydiffusing C or N to outer surface

Topic 5:

DIFFUSION IN SOLIDS


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ISSUES TO ADDRESS...

Atomic mechanisms of diffusion Mathematics of diffusion

Influence of temperature and diffusing species onDiffusion rate

Topic 5:

DIFFUSION IN SOLIDS


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DIFFUSIONPhenomenon of material transport by atomic or particle

transport from region of high to low concentration

What forces the particles to go from left to right?Does each particle know its local concentration?

Every particle is equally likely to go left or right!At the interfaces in the above picture, there aremore particles going right than left this causes anaverage flux of particles to the right!Largely determined by probability & statistics


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Glass tube filled with water. At time t = 0, add some drops of ink to one end

of the tube. Measure the diffusion distance, x, over some time.

t ot 1

t 2t 3

xo x1 x2 x3time (s)

x (mm)

DIFFUSION DEMO


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100%

Concen trati on Pr of iles0

Cu Ni

Interdiffusion : In an alloy or diffusion couple, atoms tendto migrate from regions of large to lower concentration.Initially (diffusion couple) After some time

100%

Concen trati on Pr of iles0

Adapted fromFigs. 5.1 and5.2, Callister 6e .

DIFFUSION: THE PHENOMENA (1)


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Self-diffusion : In an elemental solid, atomsalso migrate.

Label some atoms After some time

A

B

C

DA

B

C

D

DIFFUSION: THE PHENOMENA (2)


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Conditions for diffusion: there must be an adjacent empty site

atom must have sufficient energy to break bonds with itsneighbors and migrate to adjacent site (activation energy)

DIFFUSION MECHANISMSDiffusion at the atomic level is a step-wise migration of atoms fromlattice site to lattice site

Higher the temperature, higher is the probability that an atom will havesufficient energy

hence, diffusion rates increase with temperature

Types of atomic diffusion mechanisms: substitutional (through vacancies)

interstitial


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Substitutional Diffusion: applies to substitutional impurities atoms exchange with vacancies rate depends on:

-- number of vacancies-- temperature-- activation energy to exchange.

incr ing l p d ti

DIFFUSION MECHANISMS


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ACTIVATION ENERGY FOR

DIFFUSION

Also called energy barrier for diffusion

Initial state Final stateIntermediate state

Energy Activation energy


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Simulation of interdiffusionacross an interface:

Rate of substitutionaldiffusion depends on:-- vacancy concentration-- activation energy (which is

related to frequency of jumping).

(Courtesy P.M. Anderson)

DIFFUSION SIMULATION


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(Courtesy P.M. Anderson)

Applies to interstitial impurities. More rapid than vacancy

diffusion (Why?).Interstitial atoms smaller and

more mobile; more number of interstitial sites than vacancies

INTERSTITIAL SIMULATION

Simulation:--shows the jumping of a

smaller atom (gray) fromone interstitial site to

another in a BCCstructure. Theinterstitial sitesconsidered here areat midpoints along theunit cell edges.


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Case Hardening :-- Example of interstitial

diffusion is a casehardened gear.

-- Diffuse carbon atoms

into the host iron atomsat the surface.

Result: The "Case" is--hard to deform: C atoms

"lock" planes from shearing .

Fig. 5.0,Callister 6e .(Fig. 5.0 iscourtesy of SurfaceDivision,Midland-Ross.)

PROCESSING USING DIFFUSION (1)

--hard to crack: C atoms putthe surface in compression.


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Doping Silicon with P for n-type semiconductors:

1. Deposit P richlayers on surface.

2. Heat it.3. Result: Doped

semiconductor regions.

silicon

siliconmagnifi ed imag e of a comp uter chip

0.5 mm

ligh t re gions: S i a t oms

ligh t re gions: Al a t oms

Fig. 18.0,Callister 6e .

PROCESSING USING DIFFUSION (2)

Process


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Flux : amount of material or atoms moving past a unit area in unit timeFlux, J = ( M/(A ( t)

J !

AdMd

kg

-

-

Directional Quantity

Flux can be measured for:--vacancies--host (A) atoms

--impurity (B) atoms

J x

J y

J z x

y

z

x-d ir c t ion

Uni t ar a A thr ou hwh ic hat o s m ove .

MODELING DIFFUSION: FLUX


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Concentration Profile , C(x): [kg/m 3]

Fick's First Law:

C o e tr tioo f Cu [ /m 3 ]

C o e tr tioo f Ni [ /m 3 ]

ositio , x

Cu f lu x Ni f lu x

The steeper the concentration profile,the greater the flux!

Adapted fromFig. 5.2(c),Callister 6e .

J x

! D d Cd x

D iffusion coefficien t [m 2 /s]

concen trat ion gra d ien t [kg/m 4 ]

flu x in x-d ir.

[kg/m 2 -s]

CONCENTRATION PROFILES & FLUX


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Steady State : Steady rate of diffusion from one end to the other.Implies that the concentration profile doesn't change with time. Why?

Apply Fick's First Law:

Result: the slope, dC/dx , must be constant(i.e., slope doesn't vary with position)!

J (l f t) = J (right)

St dy St t :

C o tr tio , C , i th bo do s t h g w/tim .

J (right)J (l f t)

x ! D

dCdx

dCdx

left! dC

dx

right If J x)left = J x)right , the n

STEA DY STATE DIFFUSION


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Steel plate at700C withgeometryshown:

Q: How muchcarbon transfersfrom the rich tothe deficient side?

J ! DC C1x2 x1

! 2 .4 v 10 9g2s

Adapted fromFig. 5.4,Callister 6e .

C 1 = 1 . 2 k

g / m 3

C 2 = . 8

k g / m 3

C arb on r ic hga s

1 0 m

m

C arb on d eficien t

ga s

x1 x205 m

m

D =3x 10 -11 m 2 /s

S t e a d S tat e =s tra ight line!

EX: STEADY STATE DIFFUSION

Note: Steady state does not set in instantaneously.


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STEADY STATE DIFFUSION:

ANOTHER PERSPECTIVEHose connected to tap; tap turned onAt the instant tap is turned on, pressure is high at the tapend, and 1 atmosphere at the other endAfter steady state is reached, pressure linearly dropsfrom tap to other end, and will not change anymore

Tap end Flow end

PressureIncreasing time

Steady state


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Concentration profile,C(x), changesw/ time.

To conserve matter: Fick's First Law:

Governing Eqn.:

Co nc ntr ti o n,C , in th ox

J (right)J (l f t)

d x

d C

dt

Dd 2C

d x2

d x! d C

d t ! D

d C

d xor(le f t)(ri t)

d J

d x

! d C

d t

d J

d x

!D

d 2 C

d x2

(i f D d oesot v r

wit x)

eq ua te

NON STEADY STATE DIFFUSION

Ficks second law

d x! d C

d t J ! D

d C

d xor

J (le f t)J (ri t)

d J

d x

! d C

d t

d J

d x

!D

d 2 C

d x2

(i f D d oesot v a r

wit x)

eq ua te

d x! d C

d t J ! D

d C

d xor

J (le f t)J (ri t)

d J

d x

! d C

d t

d J

d x

!D

d 2 C

d x2

(i f D d oesot v a r

wit x)eq ua te


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Copper diffuses into a bar of aluminum.

Boundary conditions:For t = 0, C = C 0 at x > 0For t > 0, C = C s at x = 0

C = C 0 at x =

p re-existi o ., C o o f o pp er a toms

u r f ac e c o c .,C s o f Cu a toms b

a r

C o

C s

p ositio , x

C (x,t )

t ot 1

t 2t 3 Adapted fromFig. 5.5,

Callister 6e .

EX: NON STEADY STATE DIFFUSION

d Cd t

Dd 2C

d x 2


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Copper diffuses into a bar of aluminum.

General solution:

"error function"Values calibrated in Table 5.1, Callister 6e .

C( x, t ) C oC s C o

! 1 e r f x2 t

p re-existi c o c ., C o o f c o pp er a toms

u r f ac e c o c .,C s o f Cu a toms b

a r

C o

C s

p ositio , x

C (x,t )

t ot 1

t 2t 3 Adapted fromFig. 5.5,

Callister 6e .

EX: NON STEADY STATE DIFFUSION


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Suppose we desire to achieve a specific concentration C1at a certain point in the sample at a certain time

PROCESS DESIGN EXAMPLE

!

Dt x

erf C C

C t xC

s 21

),(

0

0

!!

Dt x

erf C C C C

s 21constant

0

01

becomes

constant2

!

D t x


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The experiment: record combinations of t and x that kept C constant.

t ot 1

t 2

t 3x o x 1 x 2 x 3

Diffusion depth given by:

x i w Dt i

( i t i ! rf it i

= (c n tan t h r )

IFFUSION EMO: ANALYSIS


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Experimental result: x ~ t 0.58

Theory predicts x ~ t 0.50 Reasonable agreement!

00.5

11 .5

22 .5

33 .5

0 0.5 1 1 .5 2 2 .5 3

ln[t (m in)]

Lin ea r r e gr e ss io n f it t o d a t a :

ln[ x(mm )] ! 0.58ln[t (m in)] 2 .2R2 ! 0.999

DATA FROM DIFFUSION DEMO


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Copper diffuses into a bar of aluminum. 10 hours at 600C gives desired C(x). How many hours would it take to get the same C(x)

if we processed at 500C, given D 500 and D 600 ?

(Dt) C (Dt) 6 C

s

C ( x , t) CoC Co

= 1 e r f x

2Dt

Result: Dt should be held constant .

Answer:Note: valuesof D areprovided here.

Key point 1: C(x,t 500C ) = C(x,t 600C ).Key point 2: Both cases have the same C o and C s .

t !(Dt )6

D! 11 r

. x1 -1 m 2 /s

.3x1 -13 m 2 /s 1 rs

PROCESSING QUESTION


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Diffusivity increases with T.

pr e -e xp o n e nti a l [m2 /s ] (s ee Tab le 5. 2 , C a llis t e r e )a cti a tio n e n e rgy

g a s c o n s t a nt [8. 31 J/ mo l-K]

D ! Do e xpQd

RT

di ff s i ity[J/ mo l],[ eV /mo l](s ee Tab le 5. 2 , C a llis t er e )

DIFFUSION AND TEMPERATURE

Remember vacancy concentration: N V = N exp(-Q V/kT)QV is vacancy formation energy (larger this energy,

smaller the number of vacancies)Qd is the activation energy (larger this energy, smaller

the diffusivity and lower the probability of atomic diffusion)


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ACTIVATION ENERGY FORDIFFUSION

Also called energy barrier for diffusion

Initial state Final stateIntermediate state

Energy Activation energy


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Experimental Data:

1 /

D (m 2 /s) C i E - F e

C i K - F e

l i

l C

u i C

u

Z n i n C u F e

i n E

- F e

F e i n

K - F

e

. 1. 1. 2.1 -2

1 -1

1 -(C ) 1 1 6 3

D has exp. dependence on TRecall: Vacancy does also!

pr e -e xp o n e nti a l [m2 /s ] (s ee Tab le 5. 2 , C a llis t e r e )

a cti a tio n e n e rgy

g a s c o n s t a nt [8. 31 J/ mo l-K]

D ! Do e xpQdRT

di ff s i ity

[J/ mo l],[ eV /mo l](s ee Tab le 5. 2 , C a llis t er e )

Dint er s titi a l >> Dsu b s tit u tio n a l

C in E- eC in K- e Al in AlC u in C u

Zn in C u

Fe in E-FeFe in K-Fe

Adapted from Fig. 5.7, Callister 6e . (Date for Fig. 5.7 taken from E.A.

Brandes and G.B. Brook (Ed.) Smithells Metals Reference Book , 7thed., Butterworth-Heinemann, Oxford, 1992.)

DIFFUSION AND TEMPERATURE

NOTE: log(D) = log(D0) Qd/(RT)


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Diffusion FASTER for...

open crystal structures

lower melting T materials

materials w/secondarybonding

smaller diffusing atoms

lower density materials

Diffusion SLOWER for...

close-packed structures

higher melting T materials

materials w/covalentbonding

larger diffusing atoms

higher density materials

SUMMARY:STRUCTURE & DIFFUSION

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Topic5 Diffusion