Download - CHAPTER 5: DIFFUSION IN SOLIDS
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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?
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• How does diffusion depend on structure and temperature?
CHAPTER 5:DIFFUSION IN SOLIDS
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Chapter 5: DIFFUSION Why study Diffusion? Heat-treated to improve their
properties.
Heat-treatment almost always involve atomic diffusion. desired results depends on
diffusion rate
Heat-treatment temperature, time, and/or rate of heating/cooling can be predicted by the mathematics of diffusion
Steel gear Case hardened to improve hardness and resistance to fatigue diffusing excess carbon or nitrogen into outer surface layer.
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5.1 Introduction Diffusion: The phenomenon of material transport by
atomic motion.
Many reactions and processes that are important in the material treatment rely on the mass transfer: Either with a specific solid ( at microscopic level ) Or from a liquid, a gas, or another solid phase.
This chapter covers: Atomic mechanism Mathematics of diffusion Influence of temperature and diffusing species of the
diffusion rate
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5.1 Introduction (Contd.) Phenomenon of diffusion
Explained using diffusion couple, formed by joining bars of two different materials having intimate contact
Copper and Nickel diffusion couple Atom locations and concentration
Heated for an extended period at an elevated temperature ( but below melting temperature of both ) and cooled to room temperature.
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100%
Concentration Profiles0
Cu Ni
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• Interdiffusion: In an alloy, atoms tend to migrate from regions of large concentration.
Initially After some time
100%
Concentration Profiles0
DIFFUSION
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5.1 Introduction (Contd.) Chemical analysis reveals
Alloy regionVariation of concentrationAtoms migrated or diffused into one
another Interdiffusion or impurity diffusion
Atoms of one metal diffuses into anotherNet drift of atoms from high to lower
concentration
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• Self-diffusion: In an elemental solid, atoms also migrate. All atoms exchanging positions are of same type
No compositional Diffusion in pure metalchanges
Label some atoms After some time
A
B
C
DA
B
C
D
DIFFUSION
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5.2 Diffusion Mechanism Atoms in solids are in constant motion rapidly changing positions.
Diffusion is just the stepwise migration of atoms from a lattice site to other lattice site.
Two conditions for movement:1. There must be an empty adjacent site2. Atom must have sufficient energy to break bonds with neighbor atoms
Atomic vibration (Section 4.7): Every atom is vibrating very rapidly about its lattice position within
the crystal At any instant, not all vibrate with same frequency and amplitude. Not all atoms have same energy Same atom may have different level of energy at different time Energy increases with temperature
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5.2 Diffusion Mechanism (Contd.) Several different models for atomic motion
Two dominate for metallic diffusion
VACANCY DIFFUSION Involves interchange of an atom from a normal lattice
position to an adjacent vacant lattice site or vacancyNecessitates presence of vacanciesDiffusing atoms and vacancies exchange positions
they move in opposite directionsBoth self- and inter-diffusion occurs by this mechanism
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Vacancy Diffusion:• applies to substitutional impurities• atoms exchange with vacancies• rate depends on: --number of vacancies --activation energy to exchange.
increasing elapsed time
DIFFUSION MECHANISMS
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5.2 Diffusion Mechanism (Contd.) INTERSTITIAL DIFFUSION
Atoms migrate from an interstitial position to a neighboring one that is empty
Found for interdiffusion of impuries such as hydrogen, carbon, nitrogen, and oxygen atoms small enough to fit into interstitial positions.
Host or substitutional impurity atoms rarely have insterstitial diffusion
Interstitial atoms are smaller and thus more mobile interstitial diffusion occurs much more rapidly then by vacancy mode
There are more empty interstitial positions than vacancies interstitial atomic movement have greater probability
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• Simulation of interdiffusion across an interface:
• Rate of substitutional diffusion depends on: --vacancy concentration --frequency of jumping.
(Courtesy P.M. Anderson)
Diffusion Simulation
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Diffusion Mechanisms Interstitial diffusion – smaller atoms
can diffuse between atoms.
More rapid than vacancy diffusionAdapted from Fig. 5.3 (b), Callister 7e.
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Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy ofSurface Division, Midland-Ross.)
• Case Hardening: --Diffuse carbon atoms into the host iron atoms at the surface. --Example of interstitial diffusion is a case hardened gear.
• Result: The presence of C atoms makes iron (steel) harder.
Processing Using Diffusion
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5.3 Steady-State Diffusion The quantity of an element that is transported within another is a
function of time diffusion is a time-dependent process.
Diffusion flux (J)
Rate of diffusion or mass transfer
Defined as “mass or number of atoms (M) diffusing through and perpendicular to a unit cross-sectional area of solid per unit time.
Mathematically, In differential form:
A: area across which diffusion is occuring t: elapsed diffusion time dt
dM
A
l
At
MJ
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Steady-State Diffusion
dx
dCDJ
Fick’s first law of diffusionC1
C2
x
C1
C2
x1 x2
D diffusion coefficient
Rate of diffusion independent of time
Flux proportional to concentration gradient =dx
dC
12
12 linear ifxx
CC
x
C
dx
dC
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5.3 Steady-State Diffusion (Contd.)
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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
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scm
g 10 x 16.1
cm) 04.0(
)g/cm 44.0g/cm 02.0(/s)cm 10 x 110(
25-
3328-
J
Example (cont).
12
12- xx
CCD
dx
dCDJ
Dtb 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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5.4 Nonsteady-State Diffusion Most practical diffusion situations
are non-steady Non-steady
Diffusion flux and the concentration flux at some particular point of solid vary with time
Net accumulation or depletion of the diffusing species
Figure shown concentration profile at three different times
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• Concentration profile, C(x), changes w/ time.
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• To conserve matter: • Fick's First Law:
• Governing Eqn.:
Concentration, C, in the box
J (right)J (left)
dx
dCdt
=Dd2C
dx2
dx
dC
dtJ D
dC
dxor
J (left)J (right)
dJ
dx
dC
dt
dJ
dx D
d2C
dx2
(if D does not vary with x)
equate
NON STEADY STATE DIFFUSION
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Solution for Semi-infinite Solid with constant surface concentration Assumptions
Initial concentration C0
X = 0 at the surface and increases with distance into the solid Initial time = 0
Boundary conditions For t = 0, C = Co at 0 x For t > 0, C = Cs (Constant surface concentration) at x=0
C = C0 at x = Solution
erf ( ) : Gaussian error function Values given in Table 5.1
Dt
xerf
CC
CC
s
x
21
0
0
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• Copper diffuses into a bar of aluminum.
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• General solution:
"error function"Values calibrated in Table 5.1, Callister 6e.
C(x,t) Co
Cs Co
1 erfx
2 Dt
pre-existing conc., Co of copper atoms
Surface conc., Cs of Cu atoms bar
Co
Cs
position, x
C(x,t)
tot1
t2t3
NON STEADY STATE DIFFUSION
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Non-steady State Diffusion Sample Problem: 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. 5.5
Dt
x
CC
CtxC
os
o
2erf1
),(
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Solution (cont.):
t = 49.5 h x = 4 x 10-3 m Cx = 0.35 wt% Cs = 1.0 wt%
Co = 0.20 wt%
Dt
x
CC
C)t,x(C
os
o
2erf1
)(erf12
erf120.00.1
20.035.0),(z
Dt
x
CC
CtxC
os
o
erf(z) = 0.8125
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Solution (cont.):We must now determine from Table 5.1 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
7970.08209.0
7970.08125.0
90.095.0
90.0
z
z 0.93
Now solve for D
Dt
xz
2
tz
xD
2
2
4
/sm 10 x 6.2s 3600
h 1
h) 5.49()93.0()4(
m)10 x 4(
4
2112
23
2
2
tz
xD
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To solve for the temperature at which D has above value, we use a rearranged form of Equation (5.9a);
)lnln( DDR
QT
o
d
from Table 5.2, for diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol
/s)m 10x6.2 ln/sm 10x3.2 K)(ln-J/mol 314.8(
J/mol 000,14821125
T
Solution (cont.):
T = 1300 K = 1027°C
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• 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?
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(Dt)500ºC =(Dt)600ºCs
C(x,t) CoC Co
=1 erfx
2Dt
• Dt should be held constant.
• Answer:Note: valuesof D areGiven here.
Key point 1: C(x,t500C) = C(x,t600C).
Key point 2: Both cases have the same Co and Cs.
t500(Dt)600
D500
110hr
4.8x10-14m2/s
5.3x10-13m2/s 10hrs
EXAMPLE PROBLEM
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Factors That Influence Diffusion
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Factors That Influence Diffusion (Contd.) DIFFUSING SPECIES
Magnitude of diffusion coefficient (D) indicative of the rate at which atoms diffuse
D depends on both the diffusing species as well as the host atomic structure
Self-diffusion Fe in -Fe 3.0E(-21) m2/sVacancy Diffusion
Inter-diffusion C in -Fe 2.4E(-12) m2/s Interstitial Diffusion
Interstitial is faster than vacancy diffusion
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Factors That Influence Diffusion (Contd.)TEMPERATURE
Temperature has a most profound influence on the coefficients and diffusion rate
Example: Fe in -Fe (Table 5.2)500oC D=3.0E(-21) m2/s900oC D=1.8E(-15) m2/s approximately six orders
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• Diffusivity increases with T.
• Experimental Data:
1000K/T
D (m2/s)
0.5 1.0 1.5 2.010-20
10-14
10-8T(C)1
50
0
10
00
60
0
30
0
D has exp. dependence on TRecall: Vacancy does also!
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pre-exponential [m2/s] (see Table 5.2, Callister 6e)activation energy
gas constant [8.31J/mol-K]
DDoexp QdRT
diffusivity
[J/mol],[eV/mol] (see Table 5.2, Callister 6e)
Dinterstitial >> DsubstitutionalC in -FeC in -Fe Al in Al
Cu in Cu
Zn in Cu
Fe in -FeFe in -Fe
DIFFUSION AND TEMPERATURE
TR
QDD d 1
lnln 0
TR
QDD d 1
3.2loglog 0