Chapter 6:

Diffusion

Important Concepts

Applications of Diffusion Activation Energy for Diffusion Mechanisms for Diffusion Rate of Diffusion (Fick’s First Law) Factors Affecting Diffusion Composition Profile (Fick’s Second Law)

3

Diffusion• How does diffusion occur?• Why is diffusion an important part

of processing? • How can the rate of diffusion be

predicted for some simple cases? • How does diffusion depend on

structure and temperature?

Applications of Diffusion

• Furnace for heat treating steel using carburization. • Carburizing is the addition of carbon to the surface of low-carbon steels at temperatures ranging from 1560°F to 1740°F. • Hardening is achieved when a high carbon martensitic case with good wear and fatigue resistance is superimposed on a tough, low-carbon steel core.

http://www.americanmetaltreatinginc.com/carburizing.htm

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• Case hardening or surface 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 of interstitial diffusion.

• The C atoms make iron (steel) harder.

Case Hardening

“Carbide band saw blade can cut through case hardened materials.”

Schematic of the microstructure of the Co-Pt-Ta-Cr film after annealing. Most of the chromium diffuses from the grains to the grain boundaries after the annealing process. This helps improve the magnetic properties of the hard disk.

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• Hot-dip galvanizing is a form of galvanization. It is the process of coating iron, steel, or aluminum with a thin zinc layer, by passing the metal through a molten bath of zinc at a temperature of around 860 °F (460 °C). • When exposed to the atmosphere, the pure zinc (Zn) reacts with oxygen (O2) to form zinc oxide (ZnO), which further reacts with carbon dioxide (CO2) to form zinc carbonate (ZnCO3), a dull grey, fairly strong material. • In many environments, the steel below the coating will be protected from further corrosion. •Galvanized steel is widely used in applications where rust resistance is needed. A hot-dip galvanizing 'kettle' with fume hood

Galvanized steel and coils popular for applications in industrial goods, automobile components, precision tubes, consumer durable and many more.

Galvanized i-beams.

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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.

Doping by Diffusion• Integrated circuits (ICs), found in

numerous electronic devices have been fabricated using doping techniques.

• The base material for these ICs is silicon that has been “doped” with other materials.

• More precisely, controlled concentrations of impurities have been diffused into specific regions of the device to change the properties (improve electrical conductivity).

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• Doping silicon with phosphorus for n-type semiconductors:• Process:

3. Result: Doped semiconductor regions.

silicon

Processing Using Diffusion

magnified image of a computer chip

0.5 mm

light regions: Si atoms

light regions: Al atoms

2. Heat.

1. Deposit P rich layers on surface.

silicon

12

DiffusionDiffusion - Mass transport by atomic motion.

Diffusion is a consequence of the constant thermal motion of atoms, molecules and particles that results in material moving from areas of high to low concentration.

Mechanisms• Brownian motion is the seemingly random

movement of particles suspended in a liquid or gas.

• Solids – vacancy diffusion or interstitial diffusion.

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• Interdiffusion (impurity diffusion): In an alloy, atoms tend to migrate from regions of high concentration 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

A

B

C

D

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, and2. The atom must have sufficient (vibrational) energy to

break bonds with its neighboring atoms and then cause 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 Diffusion2. 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

18

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?

smkgor

scmmol

timearea surfacediffusing mass) (or molesFlux 22J

J slope

dtdM

AAtMJ 1

M =

massdiffused

time

• Measured empirically– Make thin film (membrane) of known surface area– Impose concentration gradient– Measure how fast atoms or molecules diffuse through the

membrane

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

dxdCDJ

Fick’s first law of diffusionC1

C2

x

C1

C2

x1 x2

D diffusion coefficient

Flux proportional to concentration gradient =dxdC

12

12 linear ifxxCC

xC

dxdC

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Example 1: 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 are worn.

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

• Data:– diffusion coefficient for 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 16.1cm) 04.0(

)g/cm 44.0g/cm 02.0(/s)cm 10 x 110( 25-

3328-

J

Example 1 (cont).

12

12- xxCCD

dxdCDJ

D

tb 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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Diffusion and Temperature

• Diffusion coefficient increases with increasing T.

D Do exp

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]

DDo

Qd

RT

Activation energy - energy required to produce the movement of 1 mole of atoms by diffusion.

• The diffusing species, host material and temperature influence the diffusion coefficient.

• For example, there is a significant difference in magnitude between self-diffusion and carbon interdiffusion in α iron at 500 °C.

Factors that influence diffusion

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Example 2: At 300ºC the diffusion coefficient and activation energy for Cu in Si are:

D(300ºC) = 7.8 x 10-11 m2/sQd = 41,500 J/mol

What is the diffusion coefficient at 350ºC?

3500350

3000300

1lnln and 1lnlnTR

QDDTR

QDD dd

300350300

350300350

11lnlnln TTR

QDDDD d

transform data

D

Temp = T

ln D

1/T

DDoexp

Qd

RT

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Example 2 (cont.)

K 5731

K 6231

K-J/mol 314.8J/mol 500,41exp /s)m 10 x 8.7( 211

2D

1212

11exp TTR

QDD d

T1 = 273 + 300 = 573 K

T2 = 273 + 350 = 623 K

D2 = 15.7 x 10-11 m2/s

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Nonsteady State Diffusion• The concentration of diffusing species is a

function of both time and position C = C(x,t). More likely scenario than steady state.

• In this case, Fick’s Second Law is used.

2

2

xCD

tC

Fick’s Second Law

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

(Dt)500ºC =(Dt)600ºC

• Answer:

Processing – Ex 6.3

Dtx

CCCtxC

os

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 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. 6.5

Dt

xCC

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%

Dtx

CCC)t,x(C

os

o

2erf1

)(erf12

erf120.00.120.035.0),( z

Dtx

CCCtxC

os

o

erf(z) = 0.8125

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Example 3 Solution (2):

We must now determine from Table 6.1 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

7970.08209.07970.08125.0

90.095.090.0

z

z 0.93

Now solve for D

Dtxz

2

tzxD 2

2

4

/sm 10 x 6.2s 3600

h 1

h) 5.49()93.0()4(

m)10 x 4(

4211

2

23

2

2

tzxD

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• To solve for the temperature at which D has the calculated value, we use a rearranged form of Equation (6.9a);

)lnln( o

d

DDRQT

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)m 10x3.2ln /sm 10x6.2K)(ln -J/mol 314.8(J/mol 000,148

25211 T

Example 4 Solution (3):

T = 1300 K = 1027°C

DDoexp

Qd

RT