Kwang Kim

Yonsei University

kbkim@yonsei.ac.kr

Foundations of

Materials Science and Engineering

Lecture Note 8 (Chapter 5 Diffusion)May 13, 2013

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Introduction

Processing Using 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.

Introduction

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

Processing Using Diffusion• 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 interstitialdiffusion.

• The C atoms make iron (steel) harder.

Introduction

Processing Using Diffusion

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

Introduction

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 impuritieshave been diffused into specific regions of the device to change the properties (improve electrical conductivity).

Introduction

• Doping silicon with phosphorus for n-type semiconductors:

3. Result: Dopedsemiconductorregions.

silicon

magnified image of a computer chip

0.5 mm

light regions: Si atoms

light regions: Al atoms

2. Heat it.

1. Deposit P richlayers on surface.

silicon

Adapted from Figure 18.27, Callister & Rethwisch 8e.

Introduction

Diffusion - Phenomenon of material/matter transportby atomic motion

Often think of diffusion in a medium where atoms arerelatively free to move around such as liquids and gases…

• Materials processingMany materials are heat treated to achieve necessary properties: prefer to develop methods that will allow relativelyhigh diffusion rates to achieve an efficient/cost-effectivelyprocess (e.g. alloying, case hardening etc…)

• MechanismsGases & Liquids – random (Brownian) motionSolids – vacancy diffusion or interstitial diffusion

Introduction

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.

Introduction

• Glass tube filled with water.• At time t = 0, add some drops of ink to one end

of the tube.• Measure the diffusion distance, x, over some time.

to

t1

t2

t3

xo x1 x2 x3time (s)

x (mm)

DIFFUSION DEMO

Diffusion

Diffusion

• Interdiffusion: In an alloy, atoms tend to migrate fromregions of high conc. to regions of low conc.

Initially

Adapted from Figs. 5.1 and 5.2, Callister & Rethwisch 8e.

After some time

Diffusion

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

A

B

C

DA

B

C

D

Diffusion Mechanism

• 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

Diffusion Mechanism

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

-- number of vacancies-- activation energy to exchange.-- frequency of jumping

increasing elapsed time

Diffusion Mechanism

ACTIVATION ENERGY FOR DIFFUSION

• Also called energy barrier for diffusion

Initial state Final stateIntermediate state

Energy Activation energy

Diffusion Mechanism

Atom needs enough thermal energy to break bonds andsqueeze through its neighbors.Energy neededenergy barrierCalled the activation energy Em (like Q)

Diagram for Vacancy DiffusionDiffusion Thermally Activated Process

AtomVacancyEm

Distance

Ener

gy

Diffusion Mechanism

• 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.

Diffusion Mechanism

• Case Hardening:-- Example of interstitial

diffusion is a casehardened gear.

-- Diffuse carbon atomsinto the host iron atomsat the surface.

• Result: The "Case" is-- hard to deform: C atoms

"lock" planes from shearing.

Fig. 5.0, Callister 6e.(Fig. 5.0 iscourtesy ofSurface Division, Midland-Ross.)

PROCESSING USING DIFFUSION (1)

-- hard to crack: C atoms put the surface in compression.

Introduction

• Doping silicon with phosphorus for n-type semiconductors:

3. Result: Dopedsemiconductorregions.

silicon

magnified image of a computer chip

0.5 mm

light regions: Si atoms

light regions: Al atoms

2. Heat it.

1. Deposit P richlayers on surface.

silicon

Adapted from Figure 18.27, Callister & Rethwisch 8e.

Diffusion Flux

Diffusion Flux

Diffusion Flux

Diffusion Flux

How do we quantify the rate of diffusion?

smkgor

scmmol

timearea surfacediffusing mass) (or molesFlux 22J

J slopedtdM

Al

AtMJ

M =mass

diffusedtime

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

membrane

Steady State Diffusion

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

Steady State Diffusion

dxdCDJ

Fick’s first law of diffusionC1

C2

x

C1

C2

x1 x2

D diffusion coefficientD : independent of time

Flux proportional to concentration gradient =dxdC

12

12 linear ifxxCC

xC

dxdC

Steady State: the concentration profile doesn't change with time.

Steady State Diffusion

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

Steady State Diffusion

Example: Chemical Protective Clothing (CPC)

scmg 10 x 16.1

cm) 04.0()g/cm 44.0g/cm 02.0(/s)cm 10 x 110( 2

5-33

28-

J

12

12- xxCCD

dxdCDJ

D

tb 6

2

gloveC1

C2

skinpaintremove

rx1 x2

• Solution – assuming linear conc. gradient

D = 110 x 10-8 cm2/s

C2 = 0.02 g/cm3C1 = 0.44 g/cm3

x2 – x1 = 0.04 cm

Data:

Diffusion Coefficient

• 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

Diffusion Coefficient

• Diffusivity increases with T.

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

gas constant [8.31J/mol-K]

D Doexp QdRT

diffusivity

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

DIFFUSION AND TEMPERATURE

• Remember vacancy concentration: NV = N exp(-QV/kT)• QV is vacancy formation energy

(larger this energy, smaller the number of vacancies)• Qd is the activation energy

(larger this energy, smaller the diffusivity and lower the probability of atomic diffusion)

Diffusion Mechanism

ACTIVATION ENERGY FOR DIFFUSION

• Also called energy barrier for diffusion

Initial state Final stateIntermediate state

Energy Activation energy

Diffusion Coefficient

• 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

Diffusion Coefficient

• Diffusion coefficient increases with increasing T.D has exponential dependence on T

Dinterstitial >> D substitutional

C in -FeC in -Fe

Al in AlFe in -FeFe in -Fe

1000 K/T

D (m2/s)

0.5 1.0 1.510-20

10-14

10-8T(C)

1500

1000

600

300

Diffusion Coefficient

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

D(300ºC) = 7.8 x 10-11 m2/s, Qd = 41.5 kJ/molWhat is the diffusion coefficient at 350ºC?

101

202

1lnln and 1lnlnTR

QDDTR

QDD dd

121

212

11lnlnln TTR

QDDDD d

transform data

D

Temp = T

ln D

1/T

Diffusion Coefficient

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 = 573K

T2 = 273 + 350 = 623K

D2 = 15.7 x 10-11 m2/s

Non-steady State Diffusion

2

2

xCD

tC

Fick’s Second Law

• 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.

Non-steady State Diffusion

37

B.C. at t = 0, C = Co for 0 x

at t > 0, C = CS for x = 0 (constant surface conc.)

C = Co for x =

• Copper diffuses into a bar of aluminum.

pre-existing conc., Co of copper atoms

Surface conc., C of Cu atoms bars

Cs

Non-steady State Diffusion

C(x,t) = Conc. at point x at time t

erf (z) = error function

CS

Co

C(x,t)

Dtx

CCCt,xC

os

o

2 erf1

dye yz 2

0

2

• Diffusion depth given by:

xi Dti

Non-steady State Diffusion

Diffusion in ionic solids

Which do you expect to diffuse faster, cations or anions?

Smaller cations will usually diffuse faster. But analogous to charge neutrality required for defect formation, each ion will need counter charge to move with it (e.g. vacancy, impurity or free electrons or holes).

Diffusion in Solids

Diffusion FASTER for...

• open crystal structures

• lower melting T materials

• materials w/secondary

bonding

• smaller diffusing atoms

• lower density materials

Diffusion SLOWER for...

• close-packed structures

• higher melting T materials

• materials w/covalent

bonding

• larger diffusing atoms

• higher density materials