Diffusion in solid

7/28/2019 Diffusion in solid

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Diffusion


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Important Concepts

Applications of Diffusion

Activation Energy for Diffusion

Mechanisms for Diffusion

Rate of Diffusion (Ficks First Law)

Factors Affecting Diffusion

Composition Profile (Ficks Second Law)


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Diffusion

How does diffusion occur? Why is diffusion an important part

ofprocessing?

How can the rate of diffusion be

predicted for some simple cases?

How does diffusion depend onstructure and temperature?


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Interdiffusion (impurity diffusion): In an alloy,

atoms tend to migrate from regions of highconcentration to regions of low concentration.

Initially

Interdiffusion

After some time


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Self-diffusion: In an elemental solid,

atoms also migrate.

specific atom movement

Self-Diffusion

A

B

C

D

After some time

AB

C

D


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Diffusion Mechanisms

Atoms in solid materials are in constant motion, rapidly

changing positions. For an atom to move, 2 conditions must be met:

1. There must be an empty adjacent site, and

2. The atom must have sufficient (vibrational) energy to

break bonds with its neighboring atoms and thencause lattice distortion during the displacement.

At a specific temperature, only a small fraction of the

atoms is capable of motion by diffusion. This fraction

increases with rising temperature. There are 2 dominant models for metallic diffusion:

1. Vacancy Diffusion

2. Interstitial Diffusion


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Vacancy Diffusion

Vacancy Diffusion: atoms exchange with vacancies applies to substitutional impurity atoms rate depends on:

-- number of vacancies

-- activation energy to exchange.

increasing elapsed time


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Interstitial Diffusion

Interstitial diffusion smaller atoms (H,C, O, N) can diffuse between atoms.

More rapid than vacancy diffusion due to more

mobile small atoms and more empty interstitial sites.


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Diffusion

How do we quantify the rate of diffusion?

sm

kgor

scm

mol

timeareasurface

diffusingmass)(ormolesFlux

22J

J slope

dt

dM

AAt

MJ

1

M=mass

diffused

time

Measured empirically

Make thin film (membrane) of known surface area Impose concentration gradient

Measure how fast atoms or molecules diffuse through themembrane


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Rate of diffusion is independent of time; the diffusion flux does

not change with time.

The concentration profile shows the concentration (C) vs the position within

the solid (x); the slope at a particular point is the concentration gradient.

Steady-state diffusion across a thin plate


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Steady-State Diffusion

dx

dCDJ

Ficks first law of diffusionC1

C2

x

C1

C2

x1 x2

D diffusion coefficient

Flux proportional to concentration gradient =dxdC

12

12linearifxx

CC

x

C

dx

dC


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Diffusion and Temperature

Diffusion coefficient increases with increasing T.

DDoexp

Qd

RT

= pre-exponential [m2/s]

= diffusion coefficient [m2/s]

= activation energy [J/mol or eV/atom]

= gas constant [8.314 J/mol-K]

= absolute temperature [K]

D

Do

Qd

R

T

Activation energy - energy required to produce the movement

of 1 mole of atoms by diffusion.


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The diffusing species, host material and temperatureinfluence the diffusion coefficient.

For example, there is a significant difference in magnitudebetween self-diffusion and carbon interdiffusion in ironat 500 C.

Factors that influence diffusion


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Example 2:At 300C the diffusion coefficient and

activation energy for Cu in Si are:

D(300C) = 7.8 x 10-11

m2

/sQd= 41,500 J/mol

What is the diffusion coefficient at 350C?

350

0350

300

03001lnlnand1lnlnTR

QDDTR

QDD dd

300350300

350

300350

11lnlnln

TTR

Q

D

DDD d

transformdata

D

Temp = T

ln D

1/T

DDoexp Q

d

RT


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Example 2 (cont.)

K573

1

K623

1

K-J/mol314.8

J/mol500,41exp/s)m10x8.7( 2112D

12

12

11exp

TTR

QDD d

T1 = 273 + 300 = 573K

T2 = 273 + 350 = 623K

D2 = 15.7 x 10-11 m2/s


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Nonsteady State Diffusion

The concentration of diffusing species is afunction of both time and positionC= C(x,t).More likely scenario than steady state.

In this case, Ficks Second Law is used.

2

2

x

CD

t

C

Ficks Second Law


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10 hours at 600C gives C(x). How many hours would it take to get the same C(x) if processed at 500C?

Answer:

Processing Ex 6.3

Dtx

CCCtxCos

o

2erf1),(

Copper diffuses into a bar of aluminum.

pre-existing concentration Co of copper atoms

Surface concentration

C of Cu atoms bars

115.5 hrs


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Non-steady State Diffusion

Example 3: 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 ofcarbon 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. 6.5

Dt

x

CC

CtxC

os

o

2erf1

),(


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Example 3 Solution (1):

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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We must now determine from Table 6.1 the value ofzfor which the

error function is 0.8125. An interpolation is necessary as follows

z erf(z)

0.90 0.7970

z 0.8125

0.95 0.8209

7970.08209.0

7970.08125.0

90.095.0

90.0

z

z 0.93

Now solve forD

Dt

xz

2

tz

xD

2

2

4

/sm10x6.2s3600

h1

h)5.49()93.0()4(

m)10x4(

4

2112

23

2

2

tz

xD


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To solve for the temperature at

which D has the calculatedvalue, we use a rearranged

form of Equation (6.9a);)lnln(

o

d

DDR

QT

from Table 6.2, for diffusion of C in FCC Fe

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

/s)m10x3.2ln/sm10x6.2K)(ln-J/mol314.8(

J/mol000,148

25211 T


T= 1300 K = 1027C

DDoexp

Qd

RT


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Applications of Diffusion


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Furnace for heat treating steel using carburization.

Carburizing is the addition of carbon to the surface of low-carbon steels at temperatures ranging from 1560F to

1740F.

Hardening is achieved when a high carbon martensitic casewith good wear and fatigue resistance is superimposed on atough, low-carbon steel core.

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Case hardening orsurface

hardening is the process of

hardening the surface of a metal,

often a low carbon steel, by diffusing

elements into the material's surface,

forming a thin layer of a harder alloy. Carbon atoms diffuse into the iron

lattice atoms at the surface.

This is an example ofinterstitial

diffusion. The C atoms make iron (steel)

harder.

Case Hardening

Carbide band saw blade can cut

through case hardened materials.


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Thermal barrier coatings (TBC) with a ceramic topcoat are widely used for

protecting highly loaded gas turbine components against overheating.

For example, on internally cooled turbine blades the ceramic topcoat maintains a

high temperature difference between the outer surface and the underlying metallic

substrate.
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Doping by Diffusion

Integrated circuits (ICs), found in

numerous electronic deviceshave been fabricated using

doping techniques.

The base material for these ICs

is silicon that has been dopedwith other materials.

More precisely, controlled

concentrations ofimpurities have

been diffused into specific

regions of the device to change

the properties (improve electrical

conductivity).


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Doping silicon with phosphorus forn-type semiconductors:

Process:

3. Result: Doped

semiconductorregions.

silicon

Processing Using Diffusion

magnified image of a computer chip

0.5mm

light regions: Si atoms

light regions: Al atoms

2. Heat.

1. Deposit P rich

layers on surface.

silicon

Diffusion in solid

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