11

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?

1

• How does diffusion depend on structure and temperature?

CHAPTER 5:CHAPTER 5:DIFFUSION IN SOLIDSDIFFUSION IN SOLIDS

22

Chapter 5: DIFFUSIONChapter 5: DIFFUSIONWhy study Diffusion?Why study Diffusion?Heat-treated to improve their Heat-treated to improve their properties.properties.

Heat-treatment almost always involve Heat-treatment almost always involve atomic diffusion.atomic diffusion.

desired results depends on diffusion desired results depends on diffusion raterate

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

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

33

5.1 Introduction5.1 IntroductionDiffusionDiffusion: The phenomenon of material transport by : The phenomenon of material transport by

atomic motion.atomic motion.

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

This chapter This chapter coverscovers:: Atomic mechanismAtomic mechanism Mathematics of diffusionMathematics of diffusion Influence of temperature and diffusing species of the Influence of temperature and diffusing species of the

diffusion ratediffusion rate

44

5.1 Introduction (Contd.)5.1 Introduction (Contd.)

Phenomenon of diffusionPhenomenon of diffusion Explained using Explained using diffusion couple, diffusion couple,

formed by joining bars of two formed by joining bars of two different materials having intimate different materials having intimate contactcontact

Copper and Nickel diffusion couple Copper and Nickel diffusion couple Figure 5.1 shows as formedFigure 5.1 shows as formed Atom locations and concentrationAtom locations and concentration

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

55

100%

Concentration Profiles0

Cu Ni

3

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

Initially After some time

100%

Concentration Profiles0

DIFFUSIONDIFFUSION

66

5.1 Introduction (Contd.)5.1 Introduction (Contd.)Chemical analysisChemical analysis reveals reveals Alloy regionAlloy region Variation of concentrationVariation of concentration Atoms migrated or diffused into one Atoms migrated or diffused into one

anotheranotherInterdiffusion or impurity diffusionInterdiffusion or impurity diffusion Atoms of one metal diffuses into anotherAtoms of one metal diffuses into another Net drift of atoms from high to lower Net drift of atoms from high to lower

concentrationconcentration

774

• Self-diffusion: In an elemental solid, atoms also migrate. Self-diffusionSelf-diffusion

All atoms exchanging positions are of same typeAll atoms exchanging positions are of same typeNo compositional Diffusion in pure metalNo compositional Diffusion in pure metalchangeschanges

Label some atoms After some time

A

B

C

DA

B

C

D

DIFFUSIONDIFFUSION

88

5.2 Diffusion Mechanism5.2 Diffusion Mechanism

Atoms in solids are in Atoms in solids are in constant motionconstant motion rapidly changing positions. rapidly changing positions.

Diffusion is just the Diffusion is just the stepwise migrationstepwise migration of atoms from of atoms from aa lattice site to lattice site to otherother lattice site. lattice site.

Two conditionsTwo conditions for movement: for movement:

1. There must be an empty adjacent site1. There must be an empty adjacent site

2. Atom must have sufficient energy to break bonds with neighbor 2. Atom must have sufficient energy to break bonds with neighbor atomsatoms

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

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

99

5.2 Diffusion Mechanism (Contd.)5.2 Diffusion Mechanism (Contd.)

Several different models for atomic motionSeveral different models for atomic motion Two dominate for metallic diffusionTwo dominate for metallic diffusion

VACANCY DIFFUSIONVACANCY DIFFUSION Involves interchange of an atom from a normal lattice Involves interchange of an atom from a normal lattice

position to an adjacent vacant lattice site or vacancyposition to an adjacent vacant lattice site or vacancy Necessitates presence of vacanciesNecessitates presence of vacancies Diffusing atoms and vacancies exchange positions Diffusing atoms and vacancies exchange positions

they move in opposite directionsthey move in opposite directions Both self- and inter-diffusion occurs by this mechanismBoth self- and inter-diffusion occurs by this mechanism

10105

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

increasing elapsed time

DIFFUSION MECHANISMSDIFFUSION MECHANISMS

1111

5.2 Diffusion Mechanism (Contd.)5.2 Diffusion Mechanism (Contd.)

1212

5.2 Diffusion Mechanism (Contd.)5.2 Diffusion Mechanism (Contd.)

INTERSTITIAL DIFFUSIONINTERSTITIAL DIFFUSION Atoms migrate from an interstitial position to a neighboring one Atoms migrate from an interstitial position to a neighboring one

that is emptythat is empty Found for interdiffusion of impuries such as hydrogen, carbon, Found for interdiffusion of impuries such as hydrogen, carbon,

nitrogen, and oxygen nitrogen, and oxygen atoms small enough to fit into interstitial atoms small enough to fit into interstitial positions.positions.

Host or substitutional impurity atoms rarely have insterstitial Host or substitutional impurity atoms rarely have insterstitial diffusiondiffusion

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

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

13137

(Courtesy P.M. Anderson)

• Applies to interstitial impurities.• More rapid than vacancy diffusion.• Simulation: --shows the jumping of a smaller atom (gray) from one interstitial site to another in a BCC structure. The interstitial sites considered here are at midpoints along the unit cell edges.

INTERSTITIAL DIFFUSIONINTERSTITIAL DIFFUSION

1414

5.3 Steady-State Diffusion5.3 Steady-State Diffusion

The quantity of an element that is transported within another is a The quantity of an element that is transported within another is a function of time function of time diffusion is a time-dependent process. diffusion is a time-dependent process.

Diffusion flux (J)Diffusion flux (J)

Rate of diffusion or mass transferRate of diffusion or mass transfer

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

Mathematically, Mathematically, J = M / (J = M / (AtAt))

In differential form: In differential form: J = (1/A)(dM/dt)J = (1/A)(dM/dt)

A: area across which diffusion is occuringA: area across which diffusion is occuring t: elapsed diffusion timet: elapsed diffusion time

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5.3 Steady-State Diffusion (Contd.)5.3 Steady-State Diffusion (Contd.)

If the diffusion flux does not change with time If the diffusion flux does not change with time steady-state diffusionsteady-state diffusionExampleExample: :

Diffusion of a gas through a plate of metalDiffusion of a gas through a plate of metal Concentration (or pressure) of diffusing species on both side are held Concentration (or pressure) of diffusing species on both side are held

constantconstant Concentration profileConcentration profile: Concentration versus position: Concentration versus position Assumed linear concentration profile as shown in figure (b)Assumed linear concentration profile as shown in figure (b)

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5.3 Steady-State Diffusion (Contd.)5.3 Steady-State Diffusion (Contd.)Concentration gradientConcentration gradient Slope at a particular point on the concentration profile curveSlope at a particular point on the concentration profile curve Concentration gradient = dC / dxConcentration gradient = dC / dx

For linear concentration shown in figure 5.4b:For linear concentration shown in figure 5.4b:

Conc. Gradient = Conc. Gradient = C/C/x = (Cx = (CAA – C – CBB) / (x) / (xAA – x – xBB))

Fick’s first lawFick’s first law: For steady-state diffusion, the flux is proportional to the : For steady-state diffusion, the flux is proportional to the concentration gradientconcentration gradient

J = -D(dC/dx)J = -D(dC/dx)

D: diffusion coefficient (sq. m per second )D: diffusion coefficient (sq. m per second )

-ve sign: direction of diffusion from a high to a low concentration-ve sign: direction of diffusion from a high to a low concentration

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5.3 Steady-State Diffusion (Contd.)5.3 Steady-State Diffusion (Contd.)

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5.4 Nonsteady-State Diffusion5.4 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 three different times

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

14

• 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 DIFFUSIONNON STEADY STATE DIFFUSION

2020

Solution for Semi-infinite Solid with constant surface concentrationSolution for Semi-infinite Solid with constant surface concentration

AssumptionsAssumptions Initial concentration CInitial concentration C00

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

Boundary conditionsBoundary conditions For t = 0, For t = 0, C = CC = Coo at 0 at 0 x x For t > 0, For t > 0, C = CC = Css (Constant surface concentration) at x=0 (Constant surface concentration) at x=0

C = CC = C00 at x = at x =

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

Dt

xerf

CC

CC

s

x

21

0

0

2121

• 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 DIFFUSIONNON STEADY STATE DIFFUSION

2222

2323

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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?

16

(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 PROBLEMEXAMPLE PROBLEM

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Factors That Influence DiffusionFactors That Influence Diffusion

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Factors That Influence Diffusion (Contd.)Factors That Influence Diffusion (Contd.)

DIFFUSING SPECIESDIFFUSING SPECIES

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

DD depends on both the depends on both the diffusing speciesdiffusing species as well as the as well as the hosthost atomic atomic structurestructure

Self-diffusionSelf-diffusion Fe in Fe in -Fe-Fe 3.0E(-3.0E(-2121) m) m22/s/s Vacancy Vacancy DiffusionDiffusion

Inter-diffusionInter-diffusion C in C in -Fe-Fe 2.4E(-2.4E(-1212) m) m22/s/s Interstitial Interstitial DiffusionDiffusion

Interstitial is Interstitial is fasterfaster than vacancy diffusion than vacancy diffusion

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Factors That Influence Diffusion (Contd.)Factors That Influence Diffusion (Contd.)

TEMPERATURETEMPERATURE

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

ExampleExample: Fe in : Fe in -Fe (Table 5.2)-Fe (Table 5.2)500500ooCC D=3.0E(-D=3.0E(-2121) m) m22/s/s900900ooCC D=1.8E(-D=1.8E(-1515) m) m22/s/s approximately approximately six orderssix 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!

19

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 TEMPERATUREDIFFUSION AND TEMPERATURE

TR

QDD d 1

lnln 0

TR

QDD d 1

3.2loglog 0