Files Chapters ME215 Ch5 R

7/30/2019 Files Chapters ME215 Ch5 R

1/32

1

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:DIFFUSION IN SOLIDS

CHAPTER 5:DIFFUSION IN SOLIDS


2/32

2

Chapter 5: DIFFUSIONChapter 5: DIFFUSION

z Why study Diffusion?z Heat-treated to improve their properties.

z Heat-treatment almost always involveatomic diffusion.

z desired results depends on diffusionrate

z Heat-treatment temperature, time,and/or rate of heating/cooling can bepredicted by the mathematics of

diffusionz Steel gear Case hardened to improve

hardness and resistance to fatigue diffusing excess carbon or nitrogen into

outer surface layer.

z Why study Diffusion?z Heat-treated to improve their properties.

z Heat-treatment almost always involveatomic diffusion.

z desired results depends on diffusionrate

z Heat-treatment temperature, time,and/or rate of heating/cooling can bepredicted by the mathematics of

diffusionz Steel gear Case hardened to improve

hardness and resistance to fatigue diffusing excess carbon or nitrogen into

outer surface layer.


3/32

3

5.1 Introduction5.1 Introductionz 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:z Either with a specific solid ( at microscopic level )

z Or from a liquid, a gas, or another solid phase.

z This chapter covers:

zAtomic mechanism

z

Mathematics of diffusionz Influence of temperature and diffusing species of the

diffusion rate

z Diffusion: The phenomenon of material transport byatomic motion.

Many reactions and processes that are important in the material

treatment rely on the mass transfer:z Either with a specific solid ( at microscopic level )

z Or from a liquid, a gas, or another solid phase.

z This chapter covers:

zAtomic mechanism

z

Mathematics of diffusionz Influence of temperature and diffusing species of the

diffusion rate


4/32

4

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

z Phenomenon of diffusion

z Explained using diffusion couple,formed by joining bars of two

different materials having intimatecontact

z Copper and Nickel diffusion couple

z Figure 5.1 shows as formed

z Atom locations and concentration

z Heated for an extended period at an

elevated temperature ( but belowmelting temperature of both ) andcooled to room temperature.

z Phenomenon of diffusion

z Explained using diffusion couple,formed by joining bars of two

different materials having intimatecontact

z Copper and Nickel diffusion couple

z Figure 5.1 shows as formedz Atom locations and concentration

z Heated for an extended period at an

elevated temperature ( but belowmelting temperature of both ) andcooled to room temperature.


5/32

5

100%

Concentration Profiles

0

Cu Ni

3

Interdiffusion: In an alloy, atoms tend to migratefrom regions of large concentration.

Initially After some time

100%

Concentration Profiles

0

DIFFUSIONDIFFUSION


6/32

6

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

z

Chemical analysis revealszAlloy region

zVariation of concentration

zAtoms migrated or diffused into oneanother

z Interdiffusion or impurity diffusion

zAtoms of one metal diffuses into anotherz Net drift of atoms from high to lower

concentration

z Chemical analysis reveals

zAlloy region

zVariation of concentration

zAtoms migrated or diffused into oneanother

z Interdiffusion or impurity diffusion

z

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

concentration


7/32

74

Self-diffusion: In an elemental solid, atoms

also migrate. SelfSelf--diffusiondiffusion

All atoms exchanging positions are of same typeAll atoms exchanging positions are of same type

No compositional Diffusion in pure metalNo compositional Diffusion in pure metal

changeschanges

Label some atomsAfter some time

A

B

C

DA

B

C

D

DIFFUSIONDIFFUSION


8/32

8

5.2 Diffusion Mechanism5.2 Diffusion Mechanism

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

z Diffusion is just the stepwise migration of atoms from a lattice site

to other lattice site.z Two conditions for movement:

1. There must be an empty adjacent site

2. Atom must have sufficient energy to break bonds with neighboratoms

Atomic vibration (Section 4.7):

z Every atom is vibrating very rapidly about its lattice position within

the crystalz At any instant, not all vibrate with same frequency and amplitude.

z Not all atoms have same energy

z Same atom may have different level of energy at different time

z Energy increases with temperature

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

z Diffusion is just the stepwise migration of atoms from a lattice site

to other lattice site.z Two conditions for movement:

1. There must be an empty adjacent site

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

atoms

Atomic vibration (Section 4.7):

z Every atom is vibrating very rapidly about its lattice position within

the crystalz At any instant, not all vibrate with same frequency and amplitude.

z Not all atoms have same energy

z Same atom may have different level of energy at different time

z Energy increases with temperature


9/32

9

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

z Several different models for atomic motion

z Two dominate for metallic diffusion

zVACANCY DIFFUSION

z Involves interchange of an atom from a normal

lattice position to an adjacent vacant lattice site orvacancy

z Necessitates presence of vacancies

z Diffusing atoms and vacancies exchange positions they move in opposite directions

z Both self- and inter-diffusion occurs by this

mechanism

z Several different models for atomic motion

z Two dominate for metallic diffusion

zVACANCY DIFFUSION

z Involves interchange of an atom from a normal

lattice position to an adjacent vacant lattice site orvacancy

z Necessitates presence of vacancies

z Diffusing atoms and vacancies exchange positions they move in opposite directions

z Both self- and inter-diffusion occurs by this

mechanism


10/32

105

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


11/32

11



12/32

12


z INTERSTITIAL DIFFUSION

zAtoms migrate from an interstitial position to a neighboring

one that is emptyz Found for interdiffusion of impuries such as hydrogen,

carbon, nitrogen, and oxygen atoms small enough to fitinto interstitial positions.

z Host or substitutional impurity atoms rarely haveinsterstitial diffusion

z Interstitial atoms are smaller and thus more mobile interstitial diffusion occurs much more rapidly then byvacancy mode

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

z INTERSTITIAL DIFFUSION

zAtoms migrate from an interstitial position to a neighboringone that is empty

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

z Host or substitutional impurity atoms rarely haveinsterstitial diffusion

z Interstitial atoms are smaller and thus more mobile interstitial diffusion occurs much more rapidly then byvacancy mode

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


13/32

137

(Courtesy P.M. Anderson)

Applies to interstitial

impurities. More rapid than

vacancy diffusion.

Simulation:

--shows the jumping of asmaller atom (gray) from

one interstitial site to

another in a BCC

structure. Theinterstitial sites

considered here are

at midpoints along the

unit cell edges.

INTERSTITIAL DIFFUSIONINTERSTITIAL DIFFUSION


14/32

14

5.3 Steady-State Diffusion5.3 Steady-State Diffusion

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

z Diffusion flux (J)

z Rate of diffusion or mass transfer

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

z Mathematically, J = M / (At)z In differential form: J = (1/A)(dM/dt)

A: area across which diffusion is occurring

t: elapsed diffusion time

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

z Diffusion flux (J)

z Rate of diffusion or mass transferz Defined as mass or number of atoms (M) diffusing through and

perpendicular to a unit cross-sectional area of solid per unit time.

z

Mathematically, J = M / (At)z In differential form: J = (1/A)(dM/dt)

A: area across which diffusion is occurring

t: elapsed diffusion time


15/32

15

DiffusionDiffusionz How do we quantify the amount or rate of

diffusion?

z Measured empiricallyz Make thin film (membrane) of known surface area

z Impose concentration gradient

z Measure how fast atoms or molecules diffuse through themembrane

z How do we quantify the amount or rate ofdiffusion?

z Measured empiricallyz Make thin film (membrane) of known surface areaz Impose concentration gradient

z Measure how fast atoms or molecules diffuse through themembrane

( )( ) smkgor

scmmol

timeareasurfacediffusingmass)(ormolesFlux

22=J

dt

dM

A

l

At

MJ ==

M=mass

diffused

time

J slope


16/32

16

Steady-State DiffusionSteady-State Diffusion

dx

dC

DJ =

Ficks 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

12

12linearifxx

CC

x

C

dx

dC

=


17/32

17

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

z If the diffusion flux does not change with time steady-statediffusion

z Example:z Diffusion of a gas through a plate of metal

z Concentration (or pressure) of diffusing species on both side areheld constant

z Concentration profile: Concentration versus positionz Assumed linear concentration profile as shown in figure (b)

z If the diffusion flux does not change with time steady-statediffusion

z Example:z Diffusion of a gas through a plate of metal

z Concentration (or pressure) of diffusing species on both side areheld constant

z Concentration profile: Concentration versus positionz Assumed linear concentration profile as shown in figure (b)


18/32

18


z Concentration gradient

z Slope at a particular point on the concentration profilecurve

z Concentration gradient = dC / dx

z For linear concentration shown in figure 5.4b:

Conc. Gradient = C/x = (CA CB) / (xA xB)

z Ficks first law: For steady-state diffusion, the flux is

proportional to the concentration gradientJ = -D(dC/dx)

D: diffusion coefficient (sq. m per second )

-ve sign: direction of diffusion from a high to a lowconcentration

z Concentration gradient

z Slope at a particular point on the concentration profilecurve

z Concentration gradient = dC / dx

z For linear concentration shown in figure 5.4b:

Conc. Gradient = C/x = (CA CB) / (xA xB)

z Ficks first law: For steady-state diffusion, the flux is

proportional to the concentration gradientJ = -D(dC/dx)

D: diffusion coefficient (sq. m per second )

-ve sign: direction of diffusion from a high to a lowconcentration


19/32

19



20/32

20

Example: Chemical Protective Clothing (CPC)Example: Chemical Protective Clothing (CPC)

z Methylene chloride is a common ingredient of paint

removers. Besides being an irritant, it also may be absorbedthrough skin. When using this paint remover, protectivegloves should be worn.

z If butyl rubber gloves (0.04 cm thick) are used, what is the

diffusive flux of methylene chloride through the glove?z Data:

z diffusion coefficient in butyl rubber:D= 110 x10-8 cm2/s

z surface concentrations:

z Methylene chloride is a common ingredient of paint

removers. Besides being an irritant, it also may be absorbedthrough skin. When using this paint remover, protectivegloves should be worn.

z If butyl rubber gloves (0.04 cm thick) are used, what is the

diffusive flux of methylene chloride through the glove?z Data:

z diffusion coefficient in butyl rubber:D= 110 x10-8 cm2/s

z surface concentrations:

C2 = 0.02 g/cm3

C1 = 0.44 g/cm3


21/32

21

scm

g10x16.1

cm)04.0(

)g/cm44.0g/cm02.0(/s)cm10x110(

25-

3328- =

=J

Example (cont).Example (cont).

12

12

- xx

CC

Ddx

dC

DJ

=

Dtb

6

2l

=

glove

C1

C2

skinpaintremover

x1

x2

Solution assuming linear conc. gradient

D = 110x10-8 cm2/s

C2 = 0.02 g/cm3

C1 = 0.44 g/cm3

x2 x1 = 0.04 cm

Data:


22/32

22

5.4 Nonsteady-State Diffusion5.4 Nonsteady-State Diffusion

z Most practical diffusion situations

are non-steady

z Non-steady

z Diffusion flux and the

concentration flux at someparticular point of solid varywith time

z

Net accumulation or depletionof the diffusing species

z Figure shown concentrationprofile at three different times

z Most practical diffusion situations

are non-steady

z Non-steady

z Diffusion flux and the

concentration flux at someparticular point of solid varywith time

z

Net accumulation or depletionof the diffusing species

z Figure shown concentrationprofile at three different times


23/32

23

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

dC

dt = D

d2C

dx 2

dx

=

dC

dt J=

D

dC

dx or

J(left)J(right)

dJ

dx

= dC

dt

dJ

dx

= Dd2 C

dx

2

(if D doesnot varywith x)

equate

NON STEADY STATE DIFFUSIONNON STEADY STATE DIFFUSION


24/32

24

z Solution for Semi-infinite Solid with constant surface

concentrationz Assumptions

z Initial concentration C0

z X = 0 at the surface and increases with distance into thesolid

z Initial time = 0

z Boundary conditionsz For t = 0, C = Co at 0 x

z For t > 0, C = Cs (Constant surface concentration) at

x=0C = C0 at x =

z Solution

z

erf ( ) : Gaussian error functionzValues given in Table 5.1

z Solution for Semi-infinite Solid with constant surface

concentrationz Assumptions

z Initial concentration C0

z X = 0 at the surface and increases with distance into thesolid

z Initial time = 0

z Boundary conditionsz For t = 0, C = Co at 0 x

z For t > 0, C = Cs (Constant surface concentration) at

x=0C = C0 at x =

z Solution

z

erf ( ) : Gaussian error functionzValues given in Table 5.1

=

Dt

xerfCC

CC

s

x

21

0

0


25/32

25

Copper diffuses into a bar of aluminum.

15

General solution:

"error function"

Values calibrated in Table 5.1, Callister 6e.

C(x, t) CoCs Co= 1 erf

x

2 Dt

pre-existing conc., C o of copper atoms

Surface conc.,Cs of Cu atoms

bar

Co

Cs

position, x

C(x,t)

to t1t2

t3

NON STEADY STATE DIFFUSIONNON STEADY STATE DIFFUSION


26/32

26


27/32

27


28/32

28

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) 500C =(Dt) 600C

s

C(x,t) CoC C

o

= 1 erfx

2Dt

Dt should be held constant.

Answer:Note: values

of D are

Given 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

= 110 hr

4.8x10

-14

m

2

/s

5.3x10 -13m2/s 10hrs

EXAMPLE PROBLEMEXAMPLE PROBLEM


29/32

29

Factors That Influence DiffusionFactors That Influence Diffusion


30/32

30

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

z DIFFUSING SPECIES

z

Magnitude of diffusion coefficient (D)

indicative of therate at which atoms diffuse

z D depends on both the diffusing species as well as the host

atomic structure

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

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

z Interstitial is faster than vacancy diffusion

z DIFFUSING SPECIES

z Magnitude of diffusion coefficient (D) indicative of therate at which atoms diffuse

z D depends on both the diffusing species as well as the host

atomic structure

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

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

z Interstitial is faster than vacancy diffusion


31/32

31

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

TEMPERATURE

z Temperature has a most profound influenceon the coefficients and diffusion rate

z Example: Fe in -Fe (Table 5.2)500oC D=3.0E(-21) m2/s

900oC D=1.8E(-15) m2/s approximately

six orders

TEMPERATURE

z Temperature has a most profound influenceon the coefficients and diffusion rate

z Example: Fe in -Fe (Table 5.2)500oC D=3.0E(-21) m2/s

900oC D=1.8E(-15) m2/s approximately

six orders


32/32

32

Diffusivity increases with T.

Experimental Data:

1000K/T

D (m 2/s) Cin-Fe

Cin-F

e

AlinAl

Cuin

Cu

Zn

inCu

Fein

-F

e

Fein

-F

e

0.5 1.0 1.5 2.010 -20

10-14

10 -8

T(C)1500

1000

600

300

D has exp. dependence on T

Recall: Vacancy does also!

19

pre-exponential [m 2/s] (see Table 5.2, Callister 6e )activation energy

gas constant [8.31J/mol-K]

D = Do

exp

Qd

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

DIFFUSION AND TEMPERATUREDIFFUSION AND TEMPERATURE

=TR

QDD d

1lnln 0

= TRQDD d 13.2

loglog 0

Documents

Files Chapters ME215 Ch5 R