vi . diffusion in solids - university of houstonlsun3/exemplarylectures/mece3345.pdfv. diffusion in...
TRANSCRIPT
V. Diffusion in Solids
1
MECE 3345 Materials Science
VI . Diffusion in Solids
copyright © 2008 by Li Sun
V. Diffusion in Solids
2
MECE 3345 Materials Science
Examples• Ink in water • Biology
blood
air
Transpiration
• Smell
V. Diffusion in Solids
3
MECE 3345 Materials Science
Definition
Diffusion: The dispersal of material by random atom or molecularmovement from regions of high concentration to those of lower concentration.
The result of diffusion is to decrease the concentration non-uniformity.
V. Diffusion in Solids
4
MECE 3345 Materials Science
Ni-CuNi Cu
Cu in Ni Ni in Cu NiCu alloyNi Cu
x = 0
conc
entra
tion
t = 0
0
100
t = ∞t = t1
V. Diffusion in Solids
5
MECE 3345 Materials Science
Diffusion MechanismVacancy diffusion
Interstitial diffusion
V. Diffusion in Solids
6
MECE 3345 Materials Science
Activation Energy
∆G
STHG ∆−∆=∆Vibration frequency: ν
Ener
gy
Nearest neighbor sites: z
: FrequencyJumpAtom position
mmm STHG ∆−∆=∆
mm STH ∆>>∆
Interstitial diffusion Vacancy diffusion
⎟⎠
⎞⎜⎝
⎛ ∆−=Γ
RTG
vzX mV exp
=Γ
⎟⎠⎞
⎜⎝⎛ ∆−=
RTHDD mexp0
V. Diffusion in Solids
7
MECE 3345 Materials Science
Vacancy Diffusion
• Applies to substitutional impurities• Atoms must hop to open vacancy• Rate depends on probability of overcoming migration activation
energy, Qm
pQ = Bexp(-Hm/kT)And• Probability of a vacancy
pv = Nv/N = exp(-Hv/kT)• Here is Diffusion Coefficient is:
D = D0 exp(-Hd/kT)where Hd = Hm + Hv
V. Diffusion in Solids
8
MECE 3345 Materials Science
Interstitial Diffusion
• Applies to interstitial impurities (tend to be low concentration)• Most interstitial sites are not occupied• More rapid than vacancy diffusion• Activation energy involves overcoming the migration activation
energy, Hm.• Rate is defined by the Diffusion Coefficient
D = D0exp(-Hd/kT)
where D0 is a pre-exponential factor (m2/s) and Hd = Hm
V. Diffusion in Solids
9
MECE 3345 Materials Science
DataInterstitial diffusion in Fe Self-diffusion data for metals
solute ∆Hm(kJmol-1) D0(10-6m2s-1)
C 84 2.0
N 76 0.3
H 13 0.1
metal Tm(K)
Q (kJmol-1)
D0 (10-6m2s-1)
D(Tm) (10-12m2s-1)
Al 933 142 170
31
190
49
Cu 1356 200
1.9
0.59
0.65Ni 1726 279
γ-Fe 1805 284 0.29
x = 0
isotope
V. Diffusion in Solids
10
MECE 3345 Materials Science
Other Diffusion Mechanisms
Direct exchange
Zener ring
V. Diffusion in Solids
11
MECE 3345 Materials Science
Fick’s First Law
n2
a
x
dxdnann += 12 FrequencyJumpv :
1221 →→ −= JJJ
dxdna
nnaJ
τ
ττ
61
61
61
2
21
−=
⎟⎠⎞
⎜⎝⎛ −=
τ61: directionxalongFrequencyJump
site lattice at thestay atome if tuneaverage : 1v
=τn1
V. Diffusion in Solids
12
MECE 3345 Materials Science
Geometric Factor and Diffusion Coefficient
dxdnDJ −= 1221 →→ −= JJJ
τα
2aD =
Diffusion constant Geometric factor
Average stay time at interstitial site
Lattice parameter
2161 λτ
=D distance jump :λEinstein formula
Simple cubic FCC
a22
=λ
121
=α
BCC
a23
=λ
81
=α
a=λ
61
=α
V. Diffusion in Solids
13
MECE 3345 Materials Science
Steady State Diffusion
Steady state: the state where the concentration profile does not change with time, which means at any time the concentration at certain spot in space is a constant.
Position x
c
Fick’s first law:
conc
entra
tion
A
B
dxdnDJ −= n: number density
Position x
J
V. Diffusion in Solids
14
MECE 3345 Materials Science
Fick’s Second Law
n2n1
Position x
Position x
c
t1A
B
conc
entra
tion
xc
t2A
B
21→J 12→J
conc
entra
tion
A
x
∆x
J
V. Diffusion in Solids
15
MECE 3345 Materials Science
Diffusion into a Semi-infinite Solid
t ≤ 0, before diffusion startn(x) = n0, 0≤ x≤ ∞
A B
x = 0
n0
x = ∞0
ns
t > 0, diffusion start n (x=0) = ns , n (x = ∞) = n0
x = 0
n0
x = ∞
A B
0
ns
Solution to the Fick’s second law
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
−
−
Dtxerf
nnnn
s
tx
21
0
0)(, erf: Gaussian error function
Position
conc
entra
tion
x
n
0
t=0
t increases
n0
ns
Fixed surface concentration.
V. Diffusion in Solids
16
MECE 3345 Materials Science
Error function ( ) ξπ
ξ dexerfx∫ −=0
22
V. Diffusion in Solids
17
MECE 3345 Materials Science
Examples
Steady state diffusion : Gas purification
Hydrogen gas purification can be achieved by pass mixed gas through palladium sheet.
Pump Pump Assume the there is a hydrogen purification system with cross-sectional area of 0.2m2, the palladium film thickness is 10mm hydrogen concentration on the two sides are 3kg/m3 and 0.5kg/m3. If the room temperature hydrogen diffusion coefficient in Pd is 1.0×10-15m2/s and the activation energy is 27.8kJ/mol. Calculate the pure hydrogen generation speed (kg/hour) when the system works at 500oC.
V. Diffusion in Solids
18
MECE 3345 Materials Science
xA0
C1
C2
diffusion direction
C1 C2
xB
What we know:C1= 3kg/m3, C2 = 0.5kg/m3
Palladium film thicknessd=10mm = 0.01mDiffusion area A=0.2m2
D @ 300K = 1.0×10-11m2/sQd = 27.8kJ/mol
Calculate how much hydrogen can diffuse through in 1hour at 500oC.
I.
V. Diffusion in Solids
19
MECE 3345 Materials Science
Diffusion into a Semi-infinite SolidII. Calculate the carburizing time to achieve 0.45 wt% C at a position of 2mm into a 0.2 weight percent steel. Sample was maintained at 1000oC with 1.3 wt% carbon atmosphere.
Cs C0
c
0.2 wt %1.3 wt %
Position
conc
entra
tion
x0
t=0C0
Cs
t2 > t1t = t1
0.0 0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
erf(z
)
V. Diffusion in Solids
20
MECE 3345 Materials Science
Diffusion Coefficient and Thermally Activated Process
Thermally activated processes: The processes have exponentially increasing rates versus temperature.
Vacancy formation, electrical conductivity, creep and diffusion are thermally activated processes.
⎟⎠⎞
⎜⎝⎛−=
RTQDD dexp0
D0: temperature independent parameter (m2/s).Qd: activation energy for diffusion (J/mol).R: gas constant 8.31J/mol⋅KT: absolute temperature
V. Diffusion in Solids
21
MECE 3345 Materials Science
Arrhenius Plot
⎟⎠⎞
⎜⎝⎛−=
RTQCrate exp
Arrhenius Plot: log (rate) vs. T1
TRQCrate 1)log()log( −=
From the slope the thermal activation energy can be calculated
(1/K)
10-8
D (m
2 /s)
10-12
10-15
10-18
10-21
10-24
0 0.4 0.8 1.2 1.6 2.0 2.4
T1000
V. Diffusion in Solids
22
MECE 3345 Materials Science
Diffusion and Temperature
D has exponential dependence on T
Dinterstitial >> Dsubstitutional
C in α-FeC in γ-Fe
Al in AlFe in a-FeFe in g-Fe
1000K/T
D (m2/s) C in α-Fe
C in γ-Fe
Al in Al
Fe in α-Fe
Fe in γ-Fe
0.5 1.0 1.510-20
10-14
10-8T(°C)15
00
1000
600
300