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ISSUES TO ADDRESS...

• How does diffusion occur?

• Why is it an important part of processing?

• How can the rate of diffusion be predicted for some simple cases?

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• How does diffusion depend on structure and temperature?

CHAPTER 5:DIFFUSION IN SOLIDS

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Chapter 5: DIFFUSION Why study Diffusion? Heat-treated to improve their

properties.

Heat-treatment almost always involve atomic diffusion. desired results depends on

diffusion rate

Heat-treatment temperature, time, and/or rate of heating/cooling can be predicted by the mathematics of diffusion

Steel gear Case hardened to improve hardness and resistance to fatigue diffusing excess carbon or nitrogen into outer surface layer.

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5.1 Introduction Diffusion: The phenomenon of material transport by

atomic motion.

Many reactions and processes that are important in the material treatment rely on the mass transfer: Either with a specific solid ( at microscopic level ) Or from a liquid, a gas, or another solid phase.

This chapter covers: Atomic mechanism Mathematics of diffusion Influence of temperature and diffusing species of the

diffusion rate

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5.1 Introduction (Contd.) Phenomenon of diffusion

Explained using diffusion couple, formed by joining bars of two different materials having intimate contact

Copper and Nickel diffusion couple Atom locations and concentration

Heated for an extended period at an elevated temperature ( but below melting temperature of both ) and cooled to room temperature.

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100%

Concentration Profiles0

Cu Ni

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• Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration.

Initially After some time

100%

Concentration Profiles0

DIFFUSION

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5.1 Introduction (Contd.) Chemical analysis reveals

Alloy regionVariation of concentrationAtoms migrated or diffused into one

another Interdiffusion or impurity diffusion

Atoms of one metal diffuses into anotherNet drift of atoms from high to lower

concentration

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• Self-diffusion: In an elemental solid, atoms also migrate. All atoms exchanging positions are of same type

No compositional Diffusion in pure metalchanges

Label some atoms After some time

A

B

C

DA

B

C

D

DIFFUSION

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5.2 Diffusion Mechanism Atoms in solids are in constant motion rapidly changing positions.

Diffusion is just the stepwise migration of atoms from a lattice site to other lattice site.

Two conditions for movement:1. There must be an empty adjacent site2. Atom must have sufficient energy to break bonds with neighbor atoms

Atomic vibration (Section 4.7): Every atom is vibrating very rapidly about its lattice position within

the crystal At any instant, not all vibrate with same frequency and amplitude. Not all atoms have same energy Same atom may have different level of energy at different time Energy increases with temperature

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5.2 Diffusion Mechanism (Contd.) Several different models for atomic motion

Two dominate for metallic diffusion

VACANCY DIFFUSION Involves interchange of an atom from a normal lattice

position to an adjacent vacant lattice site or vacancyNecessitates presence of vacanciesDiffusing atoms and vacancies exchange positions

they move in opposite directionsBoth self- and inter-diffusion occurs by this mechanism

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Vacancy Diffusion:• applies to substitutional impurities• atoms exchange with vacancies• rate depends on: --number of vacancies --activation energy to exchange.

increasing elapsed time

DIFFUSION MECHANISMS

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5.2 Diffusion Mechanism (Contd.) INTERSTITIAL DIFFUSION

Atoms migrate from an interstitial position to a neighboring one that is empty

Found for interdiffusion of impuries such as hydrogen, carbon, nitrogen, and oxygen atoms small enough to fit into interstitial positions.

Host or substitutional impurity atoms rarely have insterstitial diffusion

Interstitial atoms are smaller and thus more mobile interstitial diffusion occurs much more rapidly then by vacancy mode

There are more empty interstitial positions than vacancies interstitial atomic movement have greater probability

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• Simulation of interdiffusion across an interface:

• Rate of substitutional diffusion depends on: --vacancy concentration --frequency of jumping.

(Courtesy P.M. Anderson)

Diffusion Simulation

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Diffusion Mechanisms Interstitial diffusion – smaller atoms

can diffuse between atoms.

More rapid than vacancy diffusionAdapted from Fig. 5.3 (b), Callister 7e.

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Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy ofSurface Division, Midland-Ross.)

• Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear.

• Result: The presence of C atoms makes iron (steel) harder.

Processing Using Diffusion

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5.3 Steady-State Diffusion The quantity of an element that is transported within another is a

function of time diffusion is a time-dependent process.

Diffusion flux (J)

Rate of diffusion or mass transfer

Defined as “mass or number of atoms (M) diffusing through and perpendicular to a unit cross-sectional area of solid per unit time.

Mathematically, In differential form:

A: area across which diffusion is occuring t: elapsed diffusion time dt

dM

A

l

At

MJ

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Steady-State Diffusion

dx

dCDJ

Fick’s first law of diffusionC1

C2

x

C1

C2

x1 x2

D diffusion coefficient

Rate of diffusion independent of time

Flux proportional to concentration gradient =dx

dC

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12 linear ifxx

CC

x

C

dx

dC

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5.3 Steady-State Diffusion (Contd.)

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Example: Chemical Protective Clothing (CPC)

Methylene chloride is a common ingredient of paint removers. Besides being an irritant, it also may be absorbed through skin. When using this paint remover, protective gloves should be worn.

If butyl rubber gloves (0.04 cm thick) are used, what is the diffusive flux of methylene chloride through the glove?

Data: diffusion coefficient in butyl rubber:

D = 110 x10-8 cm2/s surface concentrations:

C2 = 0.02 g/cm3

C1 = 0.44 g/cm3

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scm

g 10 x 16.1

cm) 04.0(

)g/cm 44.0g/cm 02.0(/s)cm 10 x 110(

25-

3328-

J

Example (cont).

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12- xx

CCD

dx

dCDJ

Dtb 6

2

gloveC1

C2

skinpaintremover

x1 x2

• Solution – assuming linear conc. gradient

D = 110 x 10-8 cm2/s

C2 = 0.02 g/cm3

C1 = 0.44 g/cm3

x2 – x1 = 0.04 cm

Data:

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5.4 Nonsteady-State Diffusion Most practical diffusion situations

are non-steady Non-steady

Diffusion flux and the concentration flux at some particular point of solid vary with time

Net accumulation or depletion of the diffusing species

Figure shown concentration profile at three different times

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• Concentration profile, C(x), changes w/ time.

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• To conserve matter: • Fick's First Law:

• Governing Eqn.:

Concentration, C, in the box

J (right)J (left)

dx

dCdt

=Dd2C

dx2

dx

dC

dtJ D

dC

dxor

J (left)J (right)

dJ

dx

dC

dt

dJ

dx D

d2C

dx2

(if D does not vary with x)

equate

NON STEADY STATE DIFFUSION

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Solution for Semi-infinite Solid with constant surface concentration Assumptions

Initial concentration C0

X = 0 at the surface and increases with distance into the solid Initial time = 0

Boundary conditions For t = 0, C = Co at 0 x For t > 0, C = Cs (Constant surface concentration) at x=0

C = C0 at x = Solution

erf ( ) : Gaussian error function Values given in Table 5.1

Dt

xerf

CC

CC

s

x

21

0

0

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• Copper diffuses into a bar of aluminum.

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• General solution:

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

C(x,t) Co

Cs Co

1 erfx

2 Dt

pre-existing conc., Co of copper atoms

Surface conc., Cs of Cu atoms bar

Co

Cs

position, x

C(x,t)

tot1

t2t3

NON STEADY STATE DIFFUSION

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Non-steady State Diffusion Sample Problem: An FCC iron-carbon alloy initially

containing 0.20 wt% C is carburized at an elevated temperature and in an atmosphere that gives a surface carbon concentration constant at 1.0 wt%. If after 49.5 h the concentration of carbon is 0.35 wt% at a position 4.0 mm below the surface, determine the temperature at which the treatment was carried out.

Solution: use Eqn. 5.5

Dt

x

CC

CtxC

os

o

2erf1

),(

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Solution (cont.):

t = 49.5 h x = 4 x 10-3 m Cx = 0.35 wt% Cs = 1.0 wt%

Co = 0.20 wt%

Dt

x

CC

C)t,x(C

os

o

2erf1

)(erf12

erf120.00.1

20.035.0),(z

Dt

x

CC

CtxC

os

o

erf(z) = 0.8125

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Solution (cont.):We must now determine from Table 5.1 the value of z for which the error function is 0.8125. An interpolation is necessary as follows

z erf(z)

0.90 0.7970z 0.81250.95 0.8209

7970.08209.0

7970.08125.0

90.095.0

90.0

z

z 0.93

Now solve for D

Dt

xz

2

tz

xD

2

2

4

/sm 10 x 6.2s 3600

h 1

h) 5.49()93.0()4(

m)10 x 4(

4

2112

23

2

2

tz

xD

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To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a);

)lnln( DDR

QT

o

d

from Table 5.2, for diffusion of C in FCC Fe

Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol

/s)m 10x6.2 ln/sm 10x3.2 K)(ln-J/mol 314.8(

J/mol 000,14821125

T

Solution (cont.):

T = 1300 K = 1027°C

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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?

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(Dt)500ºC =(Dt)600ºCs

C(x,t) CoC Co

=1 erfx

2Dt

• Dt should be held constant.

• Answer:Note: valuesof D areGiven 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

EXAMPLE PROBLEM

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Factors That Influence Diffusion

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Factors That Influence Diffusion (Contd.) DIFFUSING SPECIES

Magnitude of diffusion coefficient (D) indicative of the rate at which atoms diffuse

D depends on both the diffusing species as well as the host atomic structure

Self-diffusion Fe in -Fe 3.0E(-21) m2/sVacancy Diffusion

Inter-diffusion C in -Fe 2.4E(-12) m2/s Interstitial Diffusion

Interstitial is faster than vacancy diffusion

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Factors That Influence Diffusion (Contd.)TEMPERATURE

Temperature has a most profound influence on the coefficients and diffusion rate

Example: Fe in -Fe (Table 5.2)500oC D=3.0E(-21) m2/s900oC D=1.8E(-15) m2/s approximately six orders

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• Diffusivity increases with T.

• Experimental Data:

1000K/T

D (m2/s)

0.5 1.0 1.5 2.010-20

10-14

10-8T(C)1

50

0

10

00

60

0

30

0

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

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

gas constant [8.31J/mol-K]

DDoexp QdRT

diffusivity

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

Dinterstitial >> DsubstitutionalC in -FeC in -Fe Al in Al

Cu in Cu

Zn in Cu

Fe in -FeFe in -Fe

DIFFUSION AND TEMPERATURE

TR

QDD d 1

lnln 0

TR

QDD d 1

3.2loglog 0