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Diffusion in solids
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DIFFUSION IN SOLIDS
FICKS LAWS
KIRKENDALL EFFECT
ATOMIC MECHANISMS
Diffusion in SolidsP.G. Shewmon
McGraw-Hill, New York (1963)
Oxidation
Roles of Diffusion
Creep
AgingSintering
Doping Carburizing
Metals
Precipitates
SteelsSemiconductors
Many more
Some mechanisms
Material Joining
Diffusion bonding
To comprehend many materials related phenomenon one must understand Diffusion.
The focus of the current chapter is solid state diffusion in crystalline materials.
Ar H2
Movable piston
with an orifice
H2 diffusion direction
Ar diffusion direction
Piston motion
Piston moves in the
direction of the slower
moving species
A B
Inert Marker thin rod of a high melting material which is basically insoluble in A & B
Kirkendall effect
Materials A and B welded together with Inert marker and given a diffusion anneal Usually the lower melting component diffuses faster (say B)
Marker motion
Diffusion
Mass flow process by which species change their position relative to their neighbours.
Driven by thermal energy and a gradient
Thermal energy thermal vibrations Atomic jumps
Concentration / Chemical potential
ElectricGradient
Magnetic
Stress
Flux (J) (restricted definition) Flow / area / time [Atoms / m2 / s]
1 dnJ
A dt
Flow direction
Ficks I law
A
Flow direction
Assume that only B is moving into A
Assume steady state conditions J f(x,t) (No accumulation of matter)
dx
dcDA
dt
dn
( ) / / Flux J atoms area time concentration gradient
dx
dcJ
dx
dcDJ
dx
dcD
dt
dn
AJ
1 Diffusivity (D) f(Concentration of the components, T)
Ficks first law
dc dJx x
dt dx
Continuity equation
(Truly speaking it is the
chemical potential gradient!)
dx
dcDA
dt
dn
No. of atoms
crossing area A
per unit time
Cross-sectional area
Concentration gradient
ve implies matter transport is down the concentration gradient
Diffusion coefficient/ Diffusivity
A
Flow direction
As a first approximation assume D f(t)
dx
dcDJ
Diffusion
Steady state
J f(x,t)
Non-steady state
J = f(x,t)D = f(c)
D = f(c)
D f(c)
D f(c)Steady and non-steady state diffusion
0x t
dc J
dt x
Under steady state conditions
0J c
Dx x
Substituting for flux from Ficks first law
2
20
cD
x
If D is constant
Slope of c-x plot is constant under steady state conditions
constantc
Dx
If D is NOT constant
If D increases with concentration then slope (of c-x plot)
decreases with c
If D decreases with c then slope increases with c
Ficks II law
Jx Jx+x
x
xxx JJonAccumulati
x
x
JJJonAccumulati xx
x
x
JJJx
t
cxx J
sm
Atomsm
sm
Atoms
23
.1
xx
Jx
t
c
x
cD
xt
cFicks first law
x
cD
xt
c D f(x)2
2
x
cD
t
c
22
x
cD
t
c
RHS is the curvature of the c vs x curve
x
c
x c
+ve curvature c as t ve curvature c as t
LHS is the change is concentration with time
22
x
cD
t
c
Dt
xerfBAtxc
2 ),(
Solution to 2o de with 2 constants
determined from Boundary Conditions and Initial Condition
0
2exp2
duuErf
Erf () = 1
Erf ( ) = 1
Erf (0) = 0
Erf ( x) = Erf (x)
u
Ex
p(
u2)
0
Area
Also
For upto x~0.6 Erf(x) ~ x
x 2, Erf(x) 1
A B
Example where the erf solution can be used
x
Con
cen
trat
ion
Cavg
t
t1 > 0 | c(x,t1)t2 > t1 | c(x,t1) t = 0 | c(x,0)
A & B welded together and heated to high temperature (kept constant T0)
Flux
f(x)|t
f(t)|x
Non-steady
state
If D = f(c) c(+x,t) c(x,t)
i.e. asymmetry about y-axis
C(+x, 0) = C1 C(x, 0) = C2
C1
C2
A = (C1 + C2)/2
B = (C2 C1)/2
Dt
xerfBAtxc
2 ),(
AB = C1A+B = C2
1 2 2 1( , ) 2 2 2
C C C C xc x t erf
Dt
kTQ
eDD 0
Temperature dependence of diffusivity
Arrhenius type
Diffusivity depends exponentially on temperature.
This dependence has important consequences with regard to material behaviour at elevated temperatures. Processes like precipitate coarsening, oxidation, creep
etc. occur at very high rates at elevated temperatures.
ATOMIC MODELS OF DIFFUSION
1. Interstitial Mechanism
Usually the solubility of interstitial atoms (e.g. carbon in steel) is small. This implies that
most of the interstitial sites are vacant. Hence, if an interstitial species wants to jump, most
likely the neighbouring site will be vacant and jump of the atomic species can take place.
Light interstitial atoms like hydrogen can diffuse very fast. For a correct description of
diffusion of hydrogen anharmonic and quantum (under barrier) effects may be very
important (especially at low temperatures).
The diffusion of two important types of species needs to be distinguished:
(i) species sitting in a lattice site
(ii) species in a interstitial void
2. Vacancy Mechanism
For an atom in a lattice site (and often we are interested in substitutional atoms) jump to a
neighbouring lattice site can take place if it is vacant. Hence, vacancy concentration plays an
important role in the diffusion of species at lattice sites via the vacancy mechanism.
Vacancy clusters and defect complexes can alter this simple picture of diffusion involving
vacancies
Interstitial Diffusion
1 2
1 2
Hm
At T > 0 K vibration of the atoms provides the energy to overcome the energybarrier Hm (enthalpy of motion)
frequency of vibrations, number of successful jumps / time
kTHm
e '
Substitutional Diffusion
Probability for a jump
(probability that the site is vacant).(probability that the atom has sufficient energy)
Hm enthalpy of motion of atom
frequency of successful jumps
kTH
kT
Hmf
ee '
kT
HH mf
e '
kT
HH mf
eD 2
Where, is the jump distance
Interstitial Diffusion
kTHm
eD 2
Substitutional Diffusion
kT
HH mf
eD 2
D (C in FCC Fe at 1000C) = 3 1011 m2/s
D (Ni in FCC Fe at 1000C) = 2 1016 m2/s
0
f mH H
kTD D e
of the form
0 mH
kTD D e
of the form
Diffusion Paths with Lesser Resistance
Qsurface < Qgrain boundary < Qpipe < Qlattice
Experimentally determined activation energies for diffusion
Core of dislocation lines offer paths of lower resistance PIPE DIFFUSION
Lower activation energy automatically implies higher diffusivity
Diffusivity for a given path along with the available cross-section forthe path will determine the diffusion rate for that path
Comparison of Diffusivity for self-diffusion of Ag
single crystal vs polycrystal
1/T
Lo
g (
D)
Schematic
Polycrystal
Single
crystal
Increasing Temperature
Qgrain boundary = 110 kJ /mole
QLattice = 192 kJ /mole
Applications based on Ficks II law Carburization of steel
Surface is often the most important part of the component, which is prone to degradation.
Surface hardening of steel components like gears is done by carburizing or nitriding.
Pack carburizing solid carbon powder used as C source.
Gas carburizing Methane gas CH4 (g) 2H2 (g) + C (diffuses into steel).
C(+x, 0) = C1 C(0, t) = CS
A = CS B = CS C1
Solved
Example
A 0.2% carbon steel needs to be surface carburized such that the concentration
of carbon at 0.2mm depth is 1%. The carburizing medium imposes a surface
concentration of carbon of 1.4% and the process is carried out at 900C
(where, Fe is in FCC form).
Data: 4 20D (C in -Fe) 0.7 10 m / s 157 /Q kJ mole
Given: T = 900 C, C0 = C(x, 0) = C(, t) = 0.2 % C,
Cf = C(0.2 mm, t1) = 1% C (at x = 0.2 mm), Cs = C(0, t) = 1.4% C
The solution to the Fick second law: ( , ) 2
xC x t A B erf
Dt
The constants A & B are determined from boundary and initial conditions:
(0, ) 0.014SC t A C , 0( , ) 0.002C t A B C or 0( ,0) 0.002C x A B C
S 0B C C 0.012 , ( , ) 0.014 0.012 2
xC x t erf
Dt
-44
1
1
2 10(2 10 , ) 0.01 0.014 0.012
2C m t erf
Dt
S S 0( , ) C (C -C ) 2
xC x t erf
Dt
0
( , )=
2
S
S
C x t C xerf
C C Dt
(2)
(1)
-4
1
1 2 10
3 2erf
Dt
x (in mm from surface)
% C
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.2
t =0
t = t1 = 14580s
t = 1000s
t = 7000s
t
0.4 0.6 0.8 1.0 1.2 1.4
x (in mm from surface)
% C
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.2
t =0
t = t1 = 14580s
t = 1000s
t = 7000s
t
0.4 0.6 0.8 1.0 1.2 1.4
The following points are to be noted:
The mechanism of C diffusion is interstitial diffusion
The diffusivity D has to be evaluated at 900C using: 0 exp
QD D
RT
0 expQ
D DRT
34 157 10(0.7 10 )exp
8.314 1173
2127.14 10
m
s
-41
12
1
2 10(0.3333) 0.309
2 7.14 10erf
t
24
1 12
1 1014580
0.337.14 10t s
From equation (2)
-4
1
1 2 10
3 2erf
Dt
Approximate formula for depth of penetrationDtx
0
0
( , )1
2S
c x t C xerf
C C Dt
Let the distance at which
[(C(x,t)C0)/(CSC0)] = be called x1/2(which is an effective penetration depth)
1 211
2 2
xerf
Dt
1 2 1
22
xerf
Dt
1 1
2 2erf
1 2 1
22
x
Dt
penetrationx Dt
The depth at which C(x) is nearly C0 is (i.e. the distance beyond which is un-penetrated):
0 12
xerf
Dt
Erf(u) ~ 1 when u ~ 2
22
x
Dt
4x Dt
0
( , )=
2
S
S
C x t C xerf
C C Dt
End
Another solution to the Ficks II law
A thin film of material (fixed quantity of mass M) is deposited on the surface of another
material (e.g. dopant on the surface of a semi-conductor). The system is heated to allow
diffusion of the film material into the substrate.
For these boundary conditions we get a exponential solution.
2
( , ) exp4
M xc x t
DtDt
Boundary and Initial conditions
C(+x, 0) = 0
0cdx M
Ionic materials are not close packed
Ionic crystals may contain connected void pathways for rapid diffusion
These pathways could include ions in a sublattice (which could get disordered) and hence the transport is very selective
alumina compounds show cationic conduction Fluorite like oxides are anionic conductors
Due to high diffusivity of ions in these materials they are called superionic conductors. They are characterized by:
High value of D along with small temperature dependence of D Small values of D0
Order disorder transition in conducting sublattice has been cited as one of the mechanisms for this behaviour
Diffusion in ionic materials
Element Hf Hm Hf + Hm Q
Au 97 80 177 174
Ag 95 79 174 184
Calculated and experimental activation energies for vacancy Diffusion
1 2
Vacant site
c = atoms / volume
c = 1 / 3
concentration gradient dc/dx = (1 / 3)/ = 1 / 4
Flux = No of atoms / area / time = / area = / 2
24
2 '
'
)/(
dxdc
JD
kTHm
eD 2
2
0 D
kTQ
eDD 0On comparison
with
3. Interstitialcy Mechanism
Exchange of interstitial atom with a regular host atom (ejected from its regular site and occupies an interstitial site)
Requires comparatively low activation energies and can provide a pathway for fast diffusion
Interstitial halogen centres in alkali halides and silver interstitials in silver halides
D = f(c)
D f(c)C1
C2
Steady state diffusion
x
Con
centr
atio
n
4. Direct Interchange and Ring