Post on 06-Mar-2018
Kwang Kim
Yonsei University
kbkim@yonsei.ac.kr
Foundations of
Materials Science and Engineering
Lecture Note 8 (Chapter 5 Diffusion)May 13, 2013
39
Y88.91
8
O16.00
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I126.9
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Se78.96
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N14.01
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