CHAPTER 6:
DIFFUSION IN SOLIDS
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?
• How does diffusion depend on structure
and temperature?
1
DIFFUSION DEMO
• 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.
• Compare the results with theory.
2
Diffusion
• Introduction and Motivation
– Why do you care about diffusion?
• Many solids materials are often heat-treated to improve their
properties
• From an “atomistic” view, this thermal treatment leads to
diffusion – the phenomenon of material transport that occurs
by atomic motion
Diffusion
• Diffusion : an introduction
– Many reactions/other processes that are important in materials
engineering require the transfer of mass within a specific solid
(i.e. mixing/alloying), or from a liquid, gas, or another solid
– This is accomplished by diffusion – material transport by atomic
motion
– Chemical engineers know all about mass transfer
Diffusion
• Diffusion : an introduction
– Let’s use a picture to illustrate – diffusion couple
– Two big chunks of metal put in physical contact
• What happened?
• Upon heating, the copper
atoms moved into the
nickel and the nickel
moved into the copper
• So now you have, pure
copper, a Cu-Ni alloy, and
pure nickel
• Diffusion! (strictly
speaking interdiffusion or
impurity diffusion)
Before heating
After heating
Diffuse
• Diffusion mechanisms
– From the atomic view, diffusion is simply the stepwise migration
of atoms from lattice site to lattice site
– For atoms to move two conditions must be met
• There must be an empty site to move to
• The atom must have sufficient energy to break bonds with its
neighbors and then cause some lattice distortion during the
move (read about atomic vibrations, section 5.10)
– At a given temperature some fraction of the molecules will have
sufficient energy to undergo diffusive motion, by virtue of their
vibrational energy
• This fraction increases as T goes up
Diffusion
• Diffusion mechanisms
– Metals: two types of mechanisms dominate
• Vacancy diffusion
• Interstitial diffusion
Diffusion
• Diffusion mechanisms
– Metals: Vacancy diffusion
• Atom moves from normal lattice site to adjacent vacancy
• As you could imagine the efficiency of this process is strongly
dependent on the number of vacancies
• Typically more effective at higher temperatures
• Diffusion of atoms in one direction implies the “diffusion” of
vacancies in the opposite direction
• Self- and inter- diffusion occur by this mechanism
Diffusion
• Diffusion mechanisms
– Metals: Interstitial diffusion
• Atoms migrate from one interstitial position to a neighboring
one that is empty
• This mechanism is prevalent for atoms such as hydrogen,
carbon, nitrogen, oxygen (i.e. things small enough to easily fit
in the interstitial voids).
• Host or substitutional impurity atoms (i.e. atoms that fit in the
structure lattice) rarely form interstitials
• In metal alloys interstitial diffusion is more prevalent than
vacancy diffusion – why?
DIFFUSION: THE PHENOMENA
(1)
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of large concentration.
Initially
After some time
Adapted
from Figs.
5.1 and 5.2,
Callister 6e.
100%
0
Concentration Profiles
3
DIFFUSION: THE PHENOMENA
(2)
• Self-diffusion: In an elemental solid, atoms
also migrate.
Label some atoms
After some time
C
A
D
B
4
DIFFUSION MECHANISMS
Substitutional Diffusion:
• applies to substitutional impurities
• atoms exchange with vacancies
• rate depends on:
--number of vacancies
--activation energy to exchange.
5
DIFFUSION SIMULATION
• Simulation of
interdiffusion
across an interface:
• Rate of substitutional
diffusion depends on:
--vacancy concentration
--frequency of jumping.
Click on image to animate
(Courtesy P.M. Anderson)
6
INTERSTITIAL SIMULATION
• Applies to interstitial
impurities.
• More rapid than
vacancy diffusion.
• Simulation:
--shows the jumping of a
smaller atom (gray) from
one interstitial site to
another in a BCC
structure. The
interstitial sites
considered here are
at midpoints along the
unit cell edges.
Click on image to animate
(Courtesy P.M. Anderson)
7
Diffusion
• Mathematical description of diffusion
• Steady-state diffusion
• Diffusion is inherently a time-dependent process –
the quantity of an element that is transported within
another element as a function of time
• How fast does diffusion occur, or what is the rate of
mass transfer?
• This is expressed as the diffusive flux (J)
The flux J is defined as the mass M diffusing through and perpendicular to a unit area
(in this case of solid) per unit time
J
M
At
J
1 dM
A dt
J [=] mass/area-time
MODELING DIFFUSION: FLUX
• Flux:
• Directional Quantity
• Flux can be measured for:
--vacancies
--host (A) atoms
--impurity (B) atoms
10
• Mathematical description of diffusion
– Steady-state diffusion
• This is a limiting case: here the flux does not
change with time
• One example: consider a solid plate with gas
diffusing through it (see figure)
• The pressures on each side of the plate are
constant
CONCENTRATION PROFILES & FLUX
• Concentration Profile, C(x): [kg/m3]
Cu flux Ni flux
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Adapted
from Fig.
5.2(c),
Callister 6e.
Position, x
• Fick's First Law:
• The steeper the concentration profile,
the greater the flux!
11
STEADY STATE DIFFUSION
• Steady State: the concentration profile doesn't
change with time.
dC
• Apply Fick's First Law: J x  D
dx
dC 
dC 
  
• If Jx)left = Jx)right , then  
dx left dx right
• Result: the slope, dC/dx, must be constant
(i.e., slope doesn't vary with position)!
12
EX: STEADY STATE DIFFUSION
• Steel plate at
700C with
geometry
shown:
Adapted
from Fig.
5.4,
Callister 6e.
• Q: How much
carbon transfers
from the rich to
the deficient side?
13
Diffusion
• Mathematical description of diffusion
• Steady-state diffusion
• This leads to a linear concentration profile of the
gas species through the plate
• So the concentration gradient is given as
C C A  C B

x
x A  xB
From this, one can obtain Fick’s 1st law:
J  D
dC
dx
You will often hear that the concentration gradient is the driving force for
diffusion … this is only partially true!
Diffusion
• Mathematical description of diffusion
– Steady-state diffusion
• The real driving force is the chemical potential gradient
• What is the chemical potential (mi) – you have seen it in
thermodynamics!
• It is the same thing as the partial molar Gibbs free energy for
species i in a mixture
• How do I get concentrations out?
• Thermodynamics!  Can relate partial molar Gibbs free
energy to fugacities, etc.
• More generally though  why is it the chemical potential
gradient and not the concentration gradient?
Diffusion
• Mathematical description of diffusion
– Nonsteady-state diffusion
• This is more common – the flux varies with time!
• For what follows we will assume the diffusivity is independent
of concentration (which is not always the case!)
• Write a shell balance
Consider a differential volume element of size Ax
AJ i
AJ i
x
x
x
x + x
x  x
{ Accumulation}  {Material in}  {Material out}
C
Ax i  A J i x  A J i x  x
t
Ci J i x  J i x  x

; take limit x  0
t
x
Ci
dJ i

t
dx
NON STEADY STATE DIFFUSION
• Concentration profile,
C(x), changes
w/ time.
• To conserve matter:
• Fick's First Law:
• Governing Eqn.:
14
Diffusion
• Mathematical description of diffusion
– Nonsteady-state diffusion
• Fick’s 2nd law is not as easy to solve … can solve it but need
boundary conditions (2) and an initial condition (1)
Ci
 2Ci
D 2
t
x
Consider the following example:
1. Before diffusing, the diffusing solute is uniformly distributed in
the solid with a concentration of C = Co
2. The value of C at the surface (x = 0) is constant (i.e. Cs)
3. The value of C “infinitely deep” into the solid is Co
Or in mathematical terms
t  0, C  C for 0  x  
o
t  0, C  Cs at x  0
C  Co at x  
Diffusion
• Mathematical description of diffusion
– Nonsteady-state diffusion
• The solution of the PDE using the BCs on the last slide is
C  x , t   Co
 x 
 1  erf 

C s  Co
 2 Dt 
erf is the so called “error function” (it is a tabulated function)
erf  y  
2

z
y
exp
dy

2
0
Okay, enough math…physically this will tell you the concentration
profile inside the solid as a function of time
Or, how long you need to heat treat something to alloy the metal!
EX: NON STEADY STATE
DIFFUSION
• Copper diffuses into a bar of aluminum.
Cs
C(x,t)
Co to
t1
t3
t2
Adapted from
Fig. 5.5,
Callister 6e.
position, x
• General solution:
"error function"
Values calibrated in Table 5.1, Callister 6e.
15
.5
0.5
0.38
C -- wt%
C( x .001)
C( x .1)
0.25
C( x 1)
0.13
.01
0
0
0.5
1
1.5
2
2.5
x
x, mm
3
3.5
4
4.5
5
5
0.489
0.5
0.4
C -- wt%
C( .1  t )
0.3
C( 1  t )
C( 5  t )
0.2
0.1
0.01
0
0
0.01
1
2
3
4
5
t
t, s
6
7
8
9
10
10
Diffusion
• Factors influencing diffusion
– Diffusing species
• Diffusion coefficients, not surprisingly, depend on the identity
of the diffusing species
• Example – carbon diffuses through iron much more quickly
than iron self-diffuses (i.e. steel)
– Why? Carbon diffuses via an interstitial mechanism, iron
diffuses via a vacancy mechanism
– Temperature – big effect
• Why? Can described diffusion in solids as an activated
process
  Qd 
D  Do exp 

RT


Do – preexponential (what are the units?), Qd – Activation energy for diffusion
DIFFUSION AND TEMPERATURE
• Diffusivity increases with T.
• Experimental Data:
D has exp. dependence on T
Recall: Vacancy does also!
Dinterstitial >> Dsubstitutional
Cu in Cu
C in -Fe
Al in Al
C in -Fe
Fe in -Fe
Fe in -Fe
Zn in Cu
Adapted from Fig. 5.7, Callister 6e. (Date for Fig. 5.7 taken from
E.A. Brandes and G.B. Brook (Ed.) Smithells Metals Reference
Book, 7th ed., Butterworth-Heinemann, Oxford, 1992.)
19
Diffusion
• Factors influencing diffusion
– Temperature
• In other words the diffusion in solids is described as an
Arrhenius type process
• Qd can be thought of as the energy required to produce
diffusive motion of one mole of atoms
– Can determine the activation energy from a plot of D
versus 1/T
Example: Given the following data, determine Do and Qd
T, K
500
600
700
800
900
1000
D, m2/s
4.46E-21
1.78E-18
1.30E-16
3.53E-15
4.09E-14
3.18E-13
Diffusion
Example: Given the following data, determine Do and Qd
Step 1: calculate 1/T and plot 1/T versus D
D versus
1/T1/T
D versus
1.0E-08
1.0E-08
  Qd
D  Do exp 
 RT
Q
ln D  ln Do  d
R
1.0E-12
1.0E-12
D (m 2 /s)
D (m 2/s)
1.0E-10
1.0E-10
1.0E-14
1.0E-14
1.0E-16
1.0E-16
y = -18085x - 10.71



1
 
T 
Intercept Do, slope –Qd/R
1.0E-20
1.0E-20
1.0E-22
1.0E-22
0.00000 0.00050
0.00050
0.00000
T, K
500
600
700
800
900
1000
0.00100
0.00100
0.00150
0.00150
(1/K)
1/T1/T
(1/K)
D, m2/s
4.46E-21
1.78E-18
1.30E-16
3.53E-15
4.09E-14
3.18E-13
Note: plot is a straight line
Get parameters from linear regression
1.0E-06
1.0E-06
1.0E-18
1.0E-18
1/T
0.00200
0.00167
0.00143
0.00125
0.00111
0.00100
0.00200
0.00200
0.00250
0.00250
Do = 2.23 x 10-5 m2/s
Qd = 150,400 J/mol
Diffusion
• Other diffusion paths
– How else can migration/diffusion of atoms occur?
• Along dislocations, grain boundaries, surfaces
• Referred to as “short-circuit” diffusion paths since rates are
much faster than bulk diffusion
• The contribution of these paths to the overall diffusive flux is
minor however (why?)
Diffusion
•
Diffusion in ionic/polymeric materials
–
Ionic materials
•
More complex than in metals – why? -- have to consider
motion of two types of ions with different charge
• Diffusion is typically via a vacancy mechanism
1. Ion vacancies occur in pairs
2. They form in nonstoichiometric compounds
3. They are created by substitutional impurity compounds of
different charge state (an impurity atom, or a vacancy) than
the host ions
Diffusion
•
Diffusion in ionic/polymeric materials
–
Ionic materials – few other points
•
•
Transference of charge is associated with diffusion of ions
Charge neutrality – need an ion with opposite charge (and
same valence) to move with it
–
•
•
Candidates: vacancy, impurity atom, electron carrier
Diffusion of slowest moving species dictates overall
diffusion
Finally, apply an electric field  charges migrate, give rise
to an electric current
Diffusion
•
Diffusion in ionic/polymeric materials
–
Polymers
•
•
•
We will actually be more interested in the diffusion of small
species (H2O, CO2) occluded in the polymer
In other words we will be interested in the polymer
permeability (why do you think?)
Diffusion through amorphous regions is faster than that
through crystalline regions
–
Mechanism analogous to interstitial diffusion in metals
PROCESSING USING DIFFUSION
(1)
• Case Hardening:
--Diffuse carbon atoms
into the host iron atoms
at the surface.
--Example of interstitial
diffusion is a case
hardened gear.
Fig. 5.0,
Callister 6e.
(Fig. 5.0 is
courtesy of
Surface
Division,
MidlandRoss.)
• Result: The "Case" is
--hard to deform: C atoms
"lock" planes from shearing.
--hard to crack: C atoms put
the surface in compression.
8
PROCESSING USING DIFFUSION
(2)
• Doping Silicon with P for n-type semiconductors:
• Process:
1. Deposit P rich
layers on surface.
silicon
2. Heat it.
3. Result: Doped
semiconductor
regions.
Fig. 18.0,
Callister 6e.
silicon
Doped Si is the semiconductor material; Al wires the circuit elements together9
PROCESSING QUESTION
• 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?
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
• Result: Dt should be held constant.
• Answer:
Note: values
of D are
provided here.
16
DIFFUSION DEMO: ANALYSIS
• The experiment: we recorded combinations of
t and x that kept C constant.


C(x i , t i )  Co
x
i 
 1 erf 
= (constant here)


Cs  Co
2 Dt i 
• Diffusion depth given by:
17
DATA FROM DIFFUSION DEMO
• Experimental result: x ~ t0.58
• Theory predicts x ~ t0.50
• Reasonable agreement!
18
SUMMARY:
STRUCTURE & DIFFUSION
Diffusion FASTER for...
Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• lower melting T materials
• higher melting T materials
• materials w/secondary
bonding
• materials w/covalent
bonding
• smaller diffusing atoms
• larger diffusing atoms
• cations
• anions
• lower density materials
• higher density materials
20
ANNOUNCEMENTS
Reading: Chapter 6
HW # 5, Due Monday, February 26:
6.3; 6.4; 6.6; 6.7; 6.12; 6.15; 6.16; 6.18
6.21; 6.24; 6.D2
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