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الكلية كلية هندسة المواد
القسم قسم هندسة المعادن
المرحلة 2
أستاذ المادة جاسم محمد سلمان المرشدي
6/29/2011 7:20:19 PM
Diffusion in Solids p. 8.1
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
Diffusion is…
• The transport of atoms through matter
• The mechanism by which many important
processes occur in materials:
• Case hardening of steel
• Doping of semiconductors
• Oxidation of metals
• Solid-state formation of compounds from
individual components
• Sintering — the process by which an object
made from powders becomes dense and
strong
• Types of diffusion in solids
• Self-diffusion — movement of atoms through their
own lattice
• Interdiffusion (a.k.a. impurity diffusion) — e.g.,
movement of Ni through the lattice of Cu
• Mechanisms
• Vacancy diffusion
• Interstitial diffusion
Typically, interstitial diffusion is much faster than vacancy
diffusion
Diffusion in Solids p. 8.2
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
VACANCY DIFFUSION vs. INTERSTITIAL DIFFUSION
Vacancy diffusion: a host or substitutional
atom exchanges places with a vacancy
Vacancy
Before jump After jump
Callister, Fig. 5.3 — schematic illustration of diffusion
Interstitial diffusion: an interstitial atom jumps
into an adjacent unoccupied interstice
Before jump After jump
Diffusion in Solids p. 8.3
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
MACROSCOPIC DESCRIPTION OF DIFFUSION
• Fick’s first law (in one dimension)
J = –D
dC
dx
• J: flux, è ç
?
? ÷
? number
area time or è ç
?
? ÷
? mass
area time
• J =
1
A
dM
dt
• dC
dx : concentration gradient, è ç
?
? ÷
? number/volume
distance or è ç
?
? ÷
? mass/volume
distance
• “driving force” for diffusion
• D: diffusion coefficient for diffusing species in
solid, è ç
?
? ÷
? distance2
time
• Minus sign denotes flux is toward lower
concentrations, i.e. “down the concentration
gradient”
Diffusion in Solids p. 8.4
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
MACROSCOPIC DESCRIPTION OF DIFFUSION (cont.)
• Fick’s second law (in one dimension)
[rate of accumulation] = –[flux gradient]
¶C
¶t = – ¶J
¶x
J = –D
dC
dx ? ¶C
¶t = ¶
¶x è ç ç ç ?
? ÷ ÷ ÷ ?
D
¶C
¶x
• ¶C
¶t : rate of accumulation, è ç
?
? ÷
? number/volume
time or è ç
?
? ÷
? mass/volume
time
• ¶J
¶x : flux gradient;
“flux in - flux out”
distance
• Cases
• Steady state
• Flux out = flux in
è ç ç ç ?
? ÷ ÷ ÷ ?
U ¶J
¶x = 0
• U Zero accumulation
è ç ç ç ?
? ÷ ÷ ÷ ?
U ¶C
¶t = 0
• Unsteady state
• Flux in ¹ flux out
• Accumulation or depletion
• D ¹ D(x) ? ¶C
¶t = D
¶2C
¶x2
Diffusion in Solids p. 8.5
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
STEADY STATE DIFFUSION
Starting from
Fick s 2nd law in 1-D
with D ¹ D(x),
and assuming steady state,
? ?
D
¶2C
¶x2 = ¶C
¶t
= 0
we find:
¶2C
¶x2 = 0
i.e., the concentration profile is linear
(U only a line has a second derivative equal to zero)
steady state:
C(x)
x
conc n gradient
= slope
Jin Jout
=
= C/ x
Jin Jout
and the flux J can be calculated from Fick s 1st law simply as
Jin = Jout = –D
DC
Dx
Diffusion in Solids p. 8.6
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
SOLUTIONS TO FICK’S SECOND LAW (start)
¶C
¶t = D
¶2C
¶x2
[for D ¹ D(x); 1-D]
• General comment:
Many functions
C(x,t)
will satisfy this differential equation.
To correctly describe the concentration profile C(x,t)
that occurs in a particular physical situation,
a function must also satisfy the pertinent
initial condition — what prevails at t=0 —
and boundary conditions that hold at some interface or
boundary in our system.
Diffusion in Solids p. 8.7
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
SOLUTIONS TO FICK’S SECOND LAW (cont.)
• Example — thin film solution (start)
• S g/cm2 of solute between two semi-infinite bars
two semi-infinite bars solute layer, S g/cm2
x
0
• i.c.: C(x,0) = 0
• b.c. #1: C(¥,t) = C(–¥,t) = 0
(composition at “ends” of bars — x > 10?``Dt — will
not change)
• b.c. #2: ?
–¥
¥
C dx = S — holds for all t
(constant total amount of solute)
• Solution:
C(x,t) = S
`?`4`p`Dt
exp
è ç ç ç ?
? ÷ ÷ ÷ ? –x2
4Dt
Diffusion in Solids p. 8.8
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
SOLUTIONS TO FICK’S SECOND LAW (cont.)
• Example — thin film solution (end)
C(x,t) = S
`?`4`p`Dt
exp
è ç ç ç ?
? ÷ ÷ ÷ ? –x2
4Dt
C
x
t 1
t 2 = 2t 1
t 3 = 4t 1
• Gaussian distribution (“bell-shaped curve”)
• Finite supply of diffusing species
? area under curves remains constant for all t
• Useful exercise:
• Where does ¶2C/¶x2 = 0? Why is that significant?
Diffusion in Solids p. 8.9
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
SOLUTIONS TO FICK’S SECOND LAW (cont.)
• Example — semi-infinite solid; constant surface
composition CS
• i.c.: C(x,0) = CO
• b.c. #1: C(¥,t) = CO
(composition at “end”of bars — x > 10?``Dt — will not
change)
• b.c. #2: C(0,t) = CS î
?
? surroundings maintain
constant surface conc’n
• Solution: (see Callister, Eq. 5.5 and Figs. 5.5-5.6)
C(x,t) – CO
CS – CO
= 1 – erf
è ç ç ç ç ?
? ÷ ÷ ÷ ÷ ? x
2 ?``Dt
erf(z)
1.0
0.5
–0.5
–1.0
–3 –2 –1 1 2 3
z
Diffusion in Solids p. 8.10
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
SOLUTIONS TO FICK’S SECOND LAW (end)
• “erf” solution also gives C(x,t) for two joined semi-infinite
solids:
Callister Figs. 5.1, 5.2
Diffusion in Solids p. 8.11
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
FACTORS INFLUENCING D
A diffusing atom needs…
1) … a new site to occupy
• Vacancy concentrations — and hence
substitutional and self-diffusion — depend
strongly on temperature: NV = Nexp(–DGV/RT)
• In contrast, interstitial sites are ~always
available (but only for dilute solutes) ? number
of available interstitial sites is ~T-independent
2) … energy to leave its current location
a. b. c.
a. b. c.
Free energy
DG
†
Distance, x
l
l
high solute conc’n
? higher energy
low solute conc’n
? lower energy
\ Diffusion requires an activation energy
? probability of a successful jump µ exp(–DG†/RT)
Diffusion in Solids p. 8.12
EMSE 201 — Introduction to Materials Science & Engineering © 2003 Mark R. De Guirerev. 2/10/03
FACTORS INFLUENCING D (cont.)
D = DO exp
è ç ç ç ?
? ÷ ÷ ÷ ?
–Qd
RT
• Plot of lnD vs.
1
T will be linear, with slope –Qd
R
• Qd, apparent activation energy for diffusion
• Increases with size of diffusing atom
• Is typically larger (DGV + DG† terms) for vacancy
diffusion than for interstitial diffusion (DG† term only)
Callister, Figure 5.7 — log D vs. 1/T for several metals
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