Diffusion Callister

8/13/2019 Diffusion Callister

1/29

WHY STUDY DIFFUSION?

Materials often heat treated to improve properties

Atomic diffusion occurs during heat treatment

Depending on situation higher or lower diffusion rates

desired

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

diffusion

Example: steel gears are case-hardened by

diffusing C or N to outer surface

Topic 5:

DIFFUSION IN SOLIDS

AMI E

N BCA F

E

w w w w

. amie. nbca

f e.in/phpbb


2/29

ISSUES TO ADDRESS...

Atomic mechanisms of diffusion

Mathematics of diffusion

Influence of temperature and diffusing species on

Diffusion rate

Topic 5:

DIFFUSION IN SOLIDS


3/29

DIFFUSION

Phenomenon of material transport by atomic or particletransport 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 are

more particles going right than left this causes

an average flux of particles to the right! Largely determined by probability & statistics


4/29

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.

to

t1

t2

t3

xo x1 x2 x3time (s)

x (mm)

DIFFUSION DEMO


5/29

100%

Concentration Profiles0

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%

Concentration Profiles0

Adapted from

Figs. 5.1 and

5.2, Callister

6e.

DIFFUSION: THE PHENOMENA (1)


6/29

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)


7/29

Conditions for diffusion:

there must be an adjacent empty site

atom must have sufficient energy to break bonds with its

neighbors and migrate to adjacent site (activation energy)

DIFFUSION MECHANISMSDiffusion at the atomic level is a step-wise migration of atoms from

lattice site to lattice site

Higher the temperature, higher is the probability that an atom will have

sufficient energy

hence, diffusion rates increase with temperature

Types of atomic diffusion mechanisms:

substitutional (through vacancies) interstitial


8/29

Substitutional Diffusion:

applies to substitutional impurities

atoms exchange with vacancies

rate depends on:

-- number of vacancies

-- temperature

-- activation energy to exchange.

increasing elapsed time

DIFFUSION MECHANISMS


9/29

ACTIVATION ENERGY FOR

DIFFUSION

Also called energy barrier for diffusion

Initial state Final stateIntermediate state

nergy Activation energy


10/29

Simulation of

interdiffusion

across an interface:

Rate of substitutional

diffusion depends on:

-- vacancy concentration-- activation energy (which is

related to frequency of jumping).

(Courtesy P.M. Anderson)

DIFFUSION SIMULATION


11/29

(Courtesy P.M. Anderson)

Applies to interstitial impurities.

More rapid than vacancy

diffusion (Why?).

Interstitial atoms smaller andmore mobile; more number of

interstitial sites than vacancies

INTERSTITIAL SIMULATION

Simulation:--shows the jumping of a

smaller atom (gray) from

one interstitial site to

another in a BCCstructure. The

interstitial sites

considered here are

at midpoints along theunit cell edges.


12/29

Case Hardening:-- Example of interstitial

diffusion is a case

hardened gear.-- Diffuse carbon atoms

into the host iron atoms

at the surface.

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

"lock" planes from shearing.

Fig. 5.0,

Callister 6e.

(Fig. 5.0 is

courtesy of

SurfaceDivision,

Midland-

Ross.)

PROCESSING USING DIFFUSION (1)

--hard to crack: C atoms put

the surface in compression.


13/29

Doping Silicon with P for n-type semiconductors:

1. Deposit P rich

layers on surface.

2. Heat it.

3. Result: Doped

semiconductorregions.

silicon

silicon

magnified image of a computer chip

0.5mm

light regions: Si atoms

light regions: Al atoms

Fig. 18.0,

Callister 6e.

PROCESSING USING DIFFUSION (2)

Process


14/29

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

J =

1

A

dM

dt

kg

m2s

oratoms

m2s

Directional Quantity

Flux can be measured for:

--vacancies--host (A) atoms

--impurity (B) atoms

Jx

Jy

Jz x

y

z

x-direction

Unit area A

throughwhichatoms

move.

MODELING DIFFUSION: FLUX


15/29

Concentration Profile, C(x): [kg/m3]

Fick's First Law:

Concentrationof Cu [kg/m3]

Concentrationof Ni [kg/m3]

Position, x

Cu flux Ni flux

The steeper the concentration profile,

the greater the flux!

Adapted from

Fig. 5.2(c),

Callister 6e.

Jx = D dCdx

Diffusion coefficient [m2/s]

concentration

gradient [kg/m4]

flux in x-dir.

[kg/m2-s]

CONCENTRATION PROFILES & FLUX


16/29

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)!

Jx(left)=Jx(right)

Steady State:

Concentration, C, in the box doesnt change w/time.

Jx(right)Jx(left)

x

Jx = D

dC

dx

dCdx

left

= dCdx

right

If Jx)left = Jx)right , then

STEADY STATE DIFFUSION


17/29

Steel plate at

700C withgeometry

shown:

Q: How much

carbon transfers

from the rich to

the deficient side?J = D

C2 C1x2 x1

= 2.4 109kg

m2s

Adapted from

Fig. 5.4,

Callister 6e.

C1=1.2kg

/m3

C2=0

.8kg/m

3

Carbon

richgas

10mm

Carbondeficient

gas

x1 x205m

m

D=3x10-11m2/s

Steady State =

straight line!

EX: STEADY STATE DIFFUSION

Note: Steady state does not set in instantaneously.

STEADY STATE DIFFUSION


18/29

STEADY STATE DIFFUSION:

ANOTHER PERSPECTIVE Hose connected to tap; tap turned on

At the instant tap is turned on, pressure is high at the tap

end, and 1 atmosphere at the other end After steady state is reached, pressure linearly drops

from tap to other end, and will not change anymore

Tap end Flow end

PressureIncreasing time

Steady state

NON STEADY STATE DIFFUSION


19/29

Concentration profile,C(x), changes

w/ time.

To conserve matter: Fick's First Law:

Governing Eqn.:

Concentration,C, in the box

J(right)J(left)

dx

dCdt

= Dd2

Cdx2

dx

=

dC

dt

J(left)J(right)

NON STEADY STATE DIFFUSION

Ficks second law

dJ

dx

= dC

dt

dJ

dx

= Dd2C

dx2

(if D doesnot vary

with x)

equate

J = DdC

dx

EX NON STEADY STATE DIFFUSION


20/29

Copper diffuses into a bar of aluminum.

Boundary conditions:For t = 0, C = C0 at x > 0

For t > 0, C = Cs at x = 0

C = C0 at x =

pre-existing conc., Coof copper atoms

Surface conc.,Csof Cu atoms

bar

Co

Cs

position, x

C(x,t)

to t1t2

t3 Adapted fromFig. 5.5,

Callister 6e.

EX: NON STEADY STATE DIFFUSION

dC

dt

= Dd2C

dx2

EX NON STEADY STATE DIFFUSION


21/29


General solution:

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

C(x,t) CoCs Co

= 1 erf x2 Dt

pre-existing conc., Coof copper atoms

Surface conc.,Csof Cu atoms

bar

Co

Cs

position, x

C(x,t)

to t1t2

t3 Adapted fromFig. 5.5,

Callister 6e.

EX: NON STEADY STATE DIFFUSION

PROCESS DESIGN EXAMPLE


22/29

Suppose we desire to achieve a specific concentration C1

at a certain point in the sample at a certain time

PROCESS DESIGN EXAMPLE

=

Dt

x

erfCC

CtxC

s 21

),(

0

0

==

Dt

xerf

CC

CC

s 21constant

0

01

becomes

constant2

=Dt

x

DIFFUSION DEMO: ANALYSIS


23/29

The experiment: record combinations oft and x that kept C constant.to

t1

t2

t3

x o x 1x

2 x 3

Diffusion depth given by:

xi

Dti

C(xi, t i ) CoCs Co

= 1 erfxi

2 Dt i

= (constant here)

DIFFUSION DEMO: ANALYSIS

DATA FROM DIFFUSION DEMO


24/29

Experimental result: x ~ t0.58

Theory predicts x ~ t0.50

Reasonable agreement!

BBBBBBBBBBBB

B

B

0

0.5

1

1.5

22.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3

ln[t(min)]

Linear regression fit to data:ln[x(mm)] = 0.58ln[t(min)]+ 2.2

R2 = 0.999

DATA FROM DIFFUSION DEMO

PROCESSING QUESTION


25/29


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 D500 and D600?

(Dt)500C=(Dt)600C

s

C(x,t)CoC C

o

= 1erfx

2Dt

Result: Dt should be held constant.

Answer:Note: values

of D are

provided here.

Key point 1: C(x,t500C) = C(x,t600C).Key point 2: Both cases have the same Co and Cs.

t500

=(Dt)

600

D500

= 110hr

4.8x10-14m2/s

5.3x10-13m2/s 10hrs

PROCESSING QUESTION

DIFFUSION AND TEMPERATURE


26/29

Diffusivity increases with T.

pre-exponential [m2/s] (see Table 5.2, Callister 6eactivation energy

gas constant [8.31J/mol-K]

D= Doexp Q

d

RT

diffusivity

[J/mol],[eV/mol](see Table 5.2, Callister 6e)


Remember vacancy concentration: NV = N exp(-QV/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)



27/29


DIFFUSION

Also called energy barrier for diffusion

Initial state Final stateIntermediate state

nergy Activation energy



28/29

Experimental Data:

1000K/T

D (m2/s) Cin-Fe

Cin-Fe

Alin

Al

Cuin

C

u

ZninCu

Fein

-F

e

Fein

-F

e

0.5 1.0 1.5 2.010-20

10-14

10-8T(C)

1500

1000

600

300

D has exp. dependence on T

Recall: Vacancy does also!

pre-exponential [m2/s] (see Table 5.2, Callister 6e

activation energy

gas constant [8.31J/mol-K]

D= Doexp Q

d

RT

diffusivity

[J/mol],[eV/mol](see Table 5.2, Callister 6e)

Dinterstitial >> Dsubstitutional

C in -FeC in -Fe Al in Al

Cu in Cu

Zn in Cu

Fe in -FeFe in -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, 7th


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

SUMMARY:


29/29

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

STRUCTURE & DIFFUSION

Diffusion Callister

Documents

Transcript of Diffusion Callister