Download Chapter 6 Diffusion

Chapter 1 Diffusion
in Solids
Diffusion - Introduction
• A phenomenon of material transport by atomic
migration
The mass transfer in macroscopic level is
implemented by the motion of atoms in
microscopic level
• Self-diffusion and interdiffusion (or impurity
diffusion)
• Topics: mechanisms of diffusion, mathematics
of diffusion, effects of temperature and
diffusing species on the rate of diffusion, and
diffusion of vacancy-solute complexes
Demonstration of diffusion
Before heat treatment
After heat treatment
Diffusion – Mechanisms (1)
Two mechanisms:
• Vacancy diffusion
• Interstitial diffusion
Diffusion – Mechanisms (2)
• Vacancy diffusion
In substitutional solid solutions, the diffusion (both
self-diffusion and interdiffusion) must involve
vacancies
For self-diffusion, the activation energy is vacancy formation
energy + vacancy migration energy.
Diffusion – Mechanisms (3)
• Interstitial diffusion
In interstitial solid solutions, the diffusion of interstitial
solute atoms is the migration of the atoms from
interstitial site to interstitial site
Position of interstitial
atom after diffusion
The activation energy is the migration energy of the interstitial atom.
Mathematics of Diffusion (1)
• Steady-state diffusion
– Time-dependent process, the rate of mass transfer
is expressed as a diffusion flux (J)
M
J
At
Mass transferred through a crosssectional area
Diffusion time
Area across which the diffusion occurs
In differential form
1 dM
J 
A dt
J = Mass transferred through a unit area per unit time (g/m2 s))
Mathematics of Diffusion (2)
Concentration profile does not change with time –
steady-state diffusion
e.g. the diffusion of atoms of a gas through a metal plate
concentration gradient = dC/dx
dC/dx
J
Mathematics of Diffusion (3)
For steady-state diffusion, the diffusion flux is
proportional to the concentration gradient
The mathematics of steady-state diffusion in one
dimension is given by
dC
J  D
dx
where D is the diffusion
coefficient (m2/s), showing the
rate of diffusion
Minus sign indicates the diffusion is down the
concentration gradient
Negative
For this case, the unit of C is in
mass per unit volume, e.g. g/m3
Fick’s first law
Mathematics of Diffusion (4)
• Nonsteady-state diffusion
The diffusion flux at a particular point varies with time.
Mathematics of Diffusion (5)
The diffusion equation is represented by
C

C

(D
)
t
x
x
Fick’s second law
C is a function of x and t
If D is independent of the composition, the above equation
changes to
C
 C
 D
2
t
x
2
Unit area
cross-section
C = mass per unit
volume (concentration)
Volume of
the box: 1dx
dC/dt = mass increase in the box
per unit volume per unit time
Mass decrease in the box per
unit volume per unit time
Mathematics of Diffusion (6)
• Solutions of diffusion equation
According to error function
solutions for diffusion
equation, the solution for
these profiles can be given by
C x  A  Berf (
x
2 Dt
)
A and B are constants and erf(z)
is the error function, defined as
erf ( z ) 
2


z
0
e
 y2
dy
FFF
Mathematics of Diffusion (7)
Boundary conditions:
For t>0, Cx=Cs at x=0
Cx=Co at x=
Therefore
C s  A  Berf (
Co  A  Berf (
0
2 Dt

)
A=Cs
Co=A+B
)
2 Dt
C x  Cs  (C s  Co )erf (
x
2 Dt
)
B=-(Cs-Co)
C x  Co
x
 1  erf (
)
Cs  Co
2 Dt
Mathematics of Diffusion (8)
Boundary conditions:
For t=0, Cx=C1 at x<0
Cx=C2 at x>0
Therefore
C1  A  Berf ()
C1=A-B
A= (C1+C2)/2
C2  A  Berf ()
C2=A+B
(C1  C2 ) (C1  C2 )
x
Cx 

erf (
)
2
2
2 Dt
B=-(C1-C2)/2
Mathematics of Diffusion (9)
x
C x  A  Berf (
xh
)  Cerf (
xh
)
2 Dt
2 Dt
Boundary conditions:
For t=0, Cx=0 at x<-h; Cx=Co at -h<x<h; Cx=0 at x>h
A-B-C=0
A  Berf ()  Cerf ()  0
A+B-C=Co
A  Berf ()  Cerf ()  Co
A  Berf ()  Cerf ()  0
A+B+C=0
Co
xh
xh
Cx 
[erf (
)  erf (
)]
2
2 Dt
2 Dt
A=0
B=Co/2
C=-Co/2
Mathematics of Diffusion (10)
C x  Co
x
 1  erf (
)
Cs  Co
2 Dt
When Cx reaches a certain value at a particular position at
different temperatures
C x  Co
 constant
Cs  C o
Therefore
x2
 constant
4 Dt
For example, if the same diffusion effect is obtained at
two different temperatures T1 and T2, there is
D1t1=D2t2
Mathematics of Diffusion (11)
Factors Affecting Diffusion (1)
(1) Diffusing species
Diffusing species affect diffusion coefficient. Different
diffusing species have different diffusion coefficients.
e.g. carbon diffusion in -Fe, Carbon diffusion is much
faster than Fe self-diffusion. At 500oC, DC=2.4E-12 m2/s
(interstitial diffusion) and DFe=3.0E-21 m2/s (vacancy
diffusion), DC/DFe = 8.0E8
(2) Temperature
 Qd 
D  Do exp  

 RT 
where Do = pre-exponential constant
Qd = activation energy for diffusion (J/mol)
R = gas constant (8.3144 J/mol-K
T = absolute temperature (K) (oC + 273.15)
Factors Affecting Diffusion (2)
Factors Affecting Diffusion (3)
Qd 1
ln D  ln Do 
( )
R T
logD has a linear relationship with
reciprocal temperature
Qd 1
log D  log Do 
( )
2.3R T
Qd
slope  
2. 3 R
Qd  2.3R(slope)
logDo is the intercept on
the vertical axis
Diffusion coefficients
are usually determined
by measuring these
straight lines
10-3 (1/K)
Example Problem - 1
For a steel, it has been determined that a carburizing heat treatment of
10 h duration will raise the carbon concentration to 0.45 wt% at a
point 2.5 mm from the surface. Estimate the time required to achieve
the same concentration at a 5.0-mm position for the same steel and at
the same carburizing temperature.
Solution
C x  Co
x
 1  erf (
)
Cs  Co
2 Dt
Since both cases reach the same carbon concentration at the same
temperature, the left hand side of equation is the same. Therefore
x1
x2

2 Dt1
2 Dt 2
t2 
x22
t
2 1
x1

52
2.5
2
10  40 h
Example Problem - 2
Example Problem – 2
Solution
C1 = 5 wt%; C2 = 2 wt%
Cx = 2.5 wt%
x = 50m = 510-5 m
Do = 8.510-5 m2/s
Qd = 202100 J/mol
Cx 
erf (
(C1  C2 ) (C1  C2 )
x

erf (
)
2
2
2 Dt
x
2 Dt
x
)(
T = 1023 K
(C1  C2 )
(C1  C2 )
 Cx )
2
2
2
erf (
)   0.6667
3
2 Dt
x
2 Dt
 0.70
BBB
Example Problem – 2
5
( x 1.4)
(5 10 1.4)
1.275 10
t


D
D
D
2
2
9
Qd
202100
5
D  Do exp( 
)  8.5  10 exp( 
)
RT
8.314  1023
 4.072  10
15
2
m /s
Finally
t = 1.27510-9/4.072 10-15 = 313113 s = 87 h
Other Diffusion Paths
“Short-circuit” diffusion paths: dislocations,
grain boundaries, external surfaces, Along these
paths, the diffusion is much faster than the bulk
diffusion
In normal cases, the short-circuit contribution to the
overall diffusion is insignificant because the path
cross-section or the total boundary area is very
small
In special cases, e.g., the diffusion in nanomaterials,
the contribution is significant because the total
boundary area is very large
Diffusion of vacancy-solute complexes (1)
There is a binding energy between vacancy and
solute atom
Vacancy + solute complexes
Non-equilibrium grain boundary segregation can be
produced by complex diffusion from the grain interior to
the boundary
Diffusion of vacancy-solute complexes (2)
1. Vacancy-substitutional
solute complexes in bcc
crystals
C
Migration energy of
complex  (Vacancy-solute
binding energy + vacancy
migration energy) or
vacancy-solute atom
interchange energy
A
B
(a)
(b)
Diffusion of vacancy-solute complexes (3)
2. Vacancy-substitutional
solute complexes in fcc
crystals
(a)
First mechanism needs partial
dissociation
Second mechanism does not
need partial dissociation
A
C
B
(b)
C
D
A
Thus, the first mechanism needs
more energy, i.e., the second
mechanism is more possible
Vacancy
Solute atom
Matrix atom
(c)
Diffusion of vacancy-solute complexes (4)
3. Vacancy-interstitial solute
complexes in bcc crystals
(1) A  B, then C  D
Migration energy of complex 
(solute atom migration energy +
vacancy-solute binding energy)
or vacancy migration energy
C
D
A
(a)
(2) C  D, then A  B
Migration energy of complex 
(vacancy migration energy +
vacancy-solute binding energy)
or solute atom migration energy
(b)
B
Diffusion of vacancy-solute complexes (5)
3. Vacancy-interstitial solute
complexes in fcc crystals
Complex migration
does not need its
partial dissociation
A
C
B
D
Migration energy of
complex  vacancy
migration energy or
solute atom migration
energy
(a)
A
C
(c)
(b)
E
B
(d)
Diffusion in Ionic Materials (1)
Diffusion occurs by vacancy mechanism and involves two types
of ions with opposite charges
Formation of vacancies:
• Schottky defects
Diffusion in Ionic Materials (2)
• Nonstoichiometry
FeO
Diffusion in Ionic Materials (3)
• Substitutional impurities
e.g. in NaCl-Ca solid solution
Replacement of a sodium ion by a calcium ion creates an
extra positive charge. To maintain the electroneutrality, the
formation of a sodium vacancy is required
Diffusion processes:
The diffusive migration of a single ion is the transport of electrical
charge. To maintain the electroneutrality near the moving ion,
another species with equal and opposite charges is required to
accompany this ion’s diffusive motion
Possible charged species: vacancy or electronic carrier like free
electron or hole
Diffusion in Ionic Materials (4)
Concepts: free electron, hole
Semiconductors – n-type semiconductor and p-type
semiconductor
n-type semiconductor: electron conduction (an extra valence
electron – excitation of the electron – a free electron)
p-type semiconductor: hole conduction, a deficiency of one
valence electron (hole)
FeO
Assignments
Problems 6.27, 6.29, and 6.30