diffusion
TRANSCRIPT
Chapter 5-
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?
1
• How does diffusion depend on structure
and temperature?
CHAPTER 5:
DIFFUSION IN SOLIDS
Chapter 5- 2
• 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.
DIFFUSION DEMO
Chapter 5- 3
• Interdiffusion: In an alloy, atoms tend to migrate
from regions of large concentration.
Initially After some time
100%
Concentration Profiles0
Adapted
from Figs.
5.1 and 5.2,
Callister 6e.
Cu-Ni Diffusion Couple
Chapter 5- 3
The couple is heated at extended period at elevated
temperature
The concentrations of nickel and copper is a function of
position across the couple.
After some time
100%
Concentration Profiles0
Adapted from Figs. 5.1 and 5.2, Callister 6e.
Interdiffusion or Impurity Diffusion
The concentration profile can
be determined by elemental
analysis.
This indicates that Cu atoms
have diffused into the Ni and
that Ni atoms have diffused into
the Cu
Chapter 5- 4
Occurs for pure metals, but all atoms exchanging positions
are of the same type
Not normally subject to observation by noting composition
change
Initially
Note the labeled atomsAfter some time
Note the final positions
A
B
C
D
Self-diffusion
100%
Concentration Profiles0
Cu100%
Concentration Profiles0
Cu
Chapter 5- 5
Stepwise migration of atoms from lattice/interstitial
site to lattice/interstitial site
Required conditions
• there must be an empty adjacent site
*vacancy or interstitial
• sufficient energy to break bonds with its neighbor
atoms and then cause some lattice distortion during the
displacement
DIFFUSION MECHANISMS
Chapter 5- 5
VACANCY Diffusion:
• applies to both
interdiffusion (substitutional impurities) andself-diffusion
• atoms exchange with vacancies
• rate depends on:
--number of vacancies
--activation energy to exchange.
Metallic Diffusion Mechanisms
Chapter 5- 5
VACANCY Diffusion:
• applies to both
interdiffusion (substitutional impurities) andself-diffusion
• atoms exchange with vacancies
• rate depends on:
--number of vacancies
--activation energy to exchange.
Metallic Diffusion Mechanisms
Initially After some time
A
B
C
D
Chapter 5- 6
• Simulation of
interdiffusion
across an interface:
• Rate of substitutional
diffusion depends on:--vacancy concentration
--frequency of jumping.
(Courtesy P.M. Anderson)
DIFFUSION SIMULATION
Chapter 5- 7
(Courtesy P.M. Anderson)
Interstitial Diffusion
• Applies to interstitial impurities.
• In most metal alloys, interstitial
diffusion occurs more rapidly than
vacancy mode.
• There are more interstitials the vacancies
• Applications
Steel manufacturing (C and Fe)
Purification of gas using thin sheet of
metal
Metallic Diffusion Mechanisms
Chapter 5- 7
(Courtesy P.M. Anderson)
Interstitial 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.
Metallic Diffusion Mechanisms
Chapter 5- 7
SUBSTITUTIONAL vs INTERSTITIAL
Chapter 5-
• Case Hardening:--Diffuse carbon atoms
into the host iron atoms
at the surface.
--Example of interstitial
diffusion is a case
hardened gear.
• Result: The "Case" is--hard to deform: C atoms
"lock" planes from shearing.
--hard to crack: C atoms put
the surface in compression.
8
Fig. 5.0,
Callister 6e.
(Fig. 5.0 is
courtesy of
Surface
Division,
Midland-
Ross.)
PROCESSING USING DIFFUSION
Chapter 5-
• Doping Silicon with P for n-type semiconductors:
• Process:
9
1. Deposit P rich
layers on surface.
2. Heat it.
3. Result: Doped
semiconductor
regions.
silicon
silicon
Fig. 18.0,
Callister 6e.
PROCESSING USING DIFFUSION
Chapter 5-
Diffusion is a time dependent process.
Rate of mass transfer is frequently expressed as FLUX (J).
In differential form
Definition:
amount of M (mass or number of atoms)diffusing through and perpendicular to aa unit cross sectional area A of solid per unitof time t
10
DIFFUSION: FLUX
Chapter 5- 10
• Directional Quantity
• Flux can be measured for:--vacancies
--host (A) atoms
--impurity (B) atoms
DIFFUSION: FLUX
Unidirectional
Chapter 5-
• Concentration Profile, C(x): [kg/m3]
11
• Fick's First Law:
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Position, x
Cu flux Ni flux
• The steeper the concentration profile,
the greater the flux!
Adapted
from Fig.
5.2(c),
Callister 6e.
CONCENTRATION PROFILES & FLUX
Chapter 5-
• Steady State: the concentration profile doesn't
change with time.
12
• Apply Fick's First Law:
• Result: the slope, dC/dx, must be constant
(i.e., slope doesn't vary with position)!
Jx D
dC
dx
dC
dx
left
dC
dx
right
• If Jx)left = Jx)right , then
STEADY STATE DIFFUSION
Chapter 5- 13Adapted from Fig. 5.4, Callister 6e.
STEADY STATE DIFFUSION
Linear Concentration Profile
BA
BA
xx
CC
x
C
dx
dC
gradientionConcentrat
Chapter 5-
PROBLEM: A plate of iron is exposed to a carburizing
(carbon-rich) atmosphere on one side and decarburizing
(carbon deficient) on the other side at 700°C. If a
condition of steady state is achieved, calculate the
diffusion flux of carbon through the plate.
Refer to the figure for
Additional information
about the system.
13
STEADY STATE DIFFUSION
Linear Concentration Profile
Chapter 5-
• Steel plate at
700C with
geometry
shown:
13
• Q: How much
carbon transfers
from the rich to
the deficient side?
Adapted
from Fig.
5.4,
Callister 6e.
STEADY STATE DIFFUSION
Linear Concentration Profile
Chapter 5- 13
STEADY STATE DIFFUSION
Purification of Hydrogen Gas
Mixture of
gases
LowConcentration of
H2
x1 x20
Steady State = straight line!
HighConcentration of
O2
H2O
H2
N2
H2
Palladium
Hydrogen gas selectively diffuses through the Palladium metal sheet
from the region of high to low concentration of hydrogen gas.
Chapter 5-
PROBLEM: Compute the kg of H2
that pass per hour through a 6
mm thick sheet of palladium
having an area of 0.25 m2 at
600°C. Assume the diffusion
coefficient of 1.7x10-8 m2/s, that
the concentrations at the high
and at the low pressure sides of
the plates are 2.0 and 0.4 kg
H2/m3 of palladium, and that
steady state conditions have
been attained.
13
STEADY STATE DIFFUSION
Purification of Hydrogen Gas
Chapter 5-
GIVEN:
A = 0.25 m2
D = 1.7x10-8 m2/s @ T = 600°C
C1 = 2.0 kg H2/m3 of Pd
C2 = 0.4 kg H2/m3 of Pd
Δx = 6 mm
Compute the kg of H2 that pass per hour or rate of transport
13
STEADY STATE DIFFUSION
Purification of Hydrogen Gas
PROBLEM: When -Fe is subjected to an atmosphere of
N2, the CN (in wt%) is a function of N2 pressure PN2 (in
MPa), and absolute temperature (T) according to
Furthermore, the values of D0 and
Qd for this diffusion system are
3.0x10-7 m2/s and 76.25 kJ/mol,
respectively. Consider a thin iron
Membrane 1.5 mm thick that is at
300°C. Compute the flux through this membrane if PN2 on
one side is 0.10 MPa and on the other side 5.0 MPa.13
STEADY STATE DIFFUSION
Linear Concentration Profile
RTPC mol
kJ
NN
6.37exp1090.4
2
3
Chapter 5-
• Diffusivity increases with T.
D has exp. dependence on T
Recall: Vacancy does also!
19
DIFFUSION AND TEMPERATURE
pre-exponential [m2/s]
activation energy
gas constant [8.31J/mol-K]
D Doexp Q
d
RT
diffusivity
[J/mol],[eV/mol]
(Absolute Temperature)
Chapter 5-
• Experimental Data: D has exp. dependence on T
Recall: Vacancy does also!
19
Dinterstitial >> Dsubstitutional
C in -FeC in -Fe Al in Al
Cu in Cu
Zn in Cu
Fe in -FeFe in -Fe
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.)
DIFFUSION AND TEMPERATURE
Chapter 5-
PROBLEM: Using the data below, compute the value of D
for the following:
a. magnesium in aluminum at 500°C
b. zinc in copper at 1000K
c. copper in nickel at 0°C
13
DIFFUSION and TEMPERATURE
Solute Host metal Do (m2/s) EA or QD, (kJ/mol)
Zn Cu 2.4x10-5 189
Cu Ni 2.7x10-5 256
Mg Al 1.2x10-4 131
D Doexp Q
d
RT
Chapter 5-
PROBLEM: Using the data below, compute the value of D
for the following:
a. magnesium in aluminum at 500°C (Ans:1.7x10-13 m2/s)
b. zinc in copper at 1000K (Ans:3.2x10-15 m2/s)
c. copper in nickel at 0°C (Ans:2.8x10-54 m2/s)
13
DIFFUSION and TEMPERATURE
Solute Host metal Do (m2/s) EA or QD, (kJ/mol)
Zn Cu 2.4x10-5 189
Cu Ni 2.7x10-5 256
Mg Al 1.2x10-4 131
D Doexp Q
d
RT
Chapter 5-
PROBLEM: The diffusion coefficient for carbon and
nickel are given at two temperatures
a. Determine the values of Do and QD.
b. What is the magnitude of D at 900°C?
13
DIFFUSION and TEMPERATURE
Temperature, °C D (m2/s)
600 5.5x10-14
700 3.9x10-13
Chapter 5-
PROBLEM: The diffusion coefficient for carbon and
nickel are given at two temperatures
a. Determine the values of Do and QD.
13
DIFFUSION and TEMPERATURE
Temperature, °C D (m2/s)
600 5.5x10-14
700 3.9x10-13
TR
EDD A 1
3.2loglog 0
Chapter 5-
SOLUTION:
13
DIFFUSION and TEMPERATURE
Temperature, °C D (m2/s)
600 5.5x10-14
700 3.9x10-13
KR
EDx
KR
EDx
A
A
973
1
3.2log)109.3log()2(
873
1
3.2log)105.5log()1(
0
13
0
14
Chapter 5-
SOLUTION:
13
DIFFUSION and TEMPERATURE
Temperature, °C D (m2/s)
600 5.5x10-14
700 3.9x10-13
KR
EDx
KR
EDx
A
A
973
1
3.2log)109.3log()2(
873
1
3.2log)105.5log()1(
0
13
0
14
Chapter 5-
SOLUTION:
Linear Regression:
13
DIFFUSION and TEMPERATURE
Temperature, °C D (m2/s)
600 5.5x10-14
700 3.9x10-13
b
AA
DDbercepty
RslopeER
Eslope
xinT
vsyinDplot
10logint
)3.2(3.2
)(1
)(log
00
Chapter 5-
PROBLEM: The diffusion coefficient for carbon and
nickel are given at two temperatures
a. Determine the values of Do and QD.
Ans: Do = 1.04x10-5 m2/s and
Ans: QD = 138 kJ/mol
13
DIFFUSION and TEMPERATURE
Temperature, °C D (m2/s)
600 5.5x10-14
700 3.9x10-13
TR
EDD A 1
3.2loglog 0
Chapter 5-
• Concentration profile,
C(x), changes
w/ time.
14
• To conserve matter: • Fick's First Law:
• Governing Eqn.:
NON STEADY STATE DIFFUSION
∂C
∂t= D
∂2C
∂x2
Chapter 5- 15
"error function"
Values calibrated in Table 5.1, Callister 6e.
Co
Cs
position, x
C(x,t)
tot1
t2t3
Adapted from
Fig. 5.5,
Callister 6e.
NON STEADY STATE DIFFUSION
Diffusion of Cu in Al
x1
Chapter 5- 15
NON STEADY STATE DIFFUSION
Diffusion of Cu in Al
For semi-infinite solid (none of the diffusing atoms reaches the bar end during the time over which diffusion takes place),
at t<0 (before diffusion takes place)
C = C0 = uniform concentration of the solute
at t = or > 0 (during diffusion)
C x = 0 = Cs = constant surface concentration (Cs>C0)
C x > 0 = C(x,t) = unsteady concentration profile
pre-existing conc., Co of copper atomsbar
Chapter 5- 15
"error function”
ERROR FUNCTION
General solution:
Chapter 5-
PROBLEM: Consider a carbon iron alloy that initially has a
uniform carbon concentration of 0.25 wt% and is to be
treated at 950C. If the carbon concentration at the
surface is suddenly brought to and maintained at 1.20
wt%, how long will it take to achieve a carbon content of
0.80 wt% at the position 0.5 mm below the surface?
Assume D at this temperature is 1.6x10-11 m2/s and the
alloy is semi-infinite.
13
NON STEADY STATE DIFFUSION
Carburizing
During diffusion, C is a function of x and t
Surface conc., Cs of C atoms bar
Before diffusion, C0 = conc of C atoms
bar
Chapter 5-
• 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?
16
• Result: Dt should be held constant.
• Answer:Note: values
of D are
provided here.
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
PROCESSING QUESTION
Chapter 5- 17
• The experiment: we recorded combinations of
t and x that kept C constant.
• Diffusion depth given by:
C(xi, t i )Co
Cs Co 1 erf
xi
2 Dt i
= (constant here)
DIFFUSION DEMO: ANALYSIS
Chapter 5-
• Experimental result: x ~ t0.58
• Theory predicts x ~ t0.50
• Reasonable agreement!
18
DATA FROM DIFFUSION DEMO
Chapter 5- 20
Diffusion FASTER for...
• open crystal structures
• lower melting T materials
• materials w/secondary
bonding
• smaller diffusing atoms
• cations
• lower density materials
Diffusion SLOWER for...
• close-packed structures
• higher melting T materials
• materials w/covalent
bonding
• larger diffusing atoms
• anions
• higher density materials
SUMMARY:
STRUCTURE & DIFFUSION