MSEP 503 Defects, Diffusion and Transformation Slide ... · ,qwhuvwlwldo dwrp...
Transcript of MSEP 503 Defects, Diffusion and Transformation Slide ... · ,qwhuvwlwldo dwrp...
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Diffusion
Factors that Influence Diffusion Temperature - diffusion rate increases very rapidly with increasing temperature
Diffusion mechanism - interstitial is usually faster than vacancy
Diffusing and host species - Do, Qd is different for every solute, solvent pair
Microstructure - diffusion faster in polycrystalline vs. single crystal materialsbecause of the rapid diffusion along grain boundaries and dislocation cores.
Atomic diffusion is a process whereby the random thermally-activated hopping of atoms in a solid results in the net transport of atoms. For example, helium atoms inside a balloon can diffuse through the wall of the balloon and escape, resulting in the balloon slowly deflating. Other air molecules (e.g. oxygen, nitrogen) have lower mobilities and thus diffuse more slowly through the balloon wall. There is a concentration gradient in the balloon wall, because the balloon was initially filled with helium, and thus there is plenty of helium on the inside, but there is relatively little helium on the outside (helium is not a major component of air).
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General Note.
Diffusion is a flux of matter in which the atoms or molecules of a certain type move differently (rate, amount etc) with respect to the atoms/molecules of other type.
Please note the difference from gas or liquid flow, in this case ALL components move in the same way.
The definition: “Diffusion is the movement of molecules from a high concentration to a low concentration” is wrong because there are cases when diffusion process does just opposite.
Flux of matter can be caused not only by the difference in concentration of the atom that diffuses, but also the difference in concentration of other atoms and or gradient of physical parameters ( temperature, pressure, electric or magnetic field).
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Diffusion
Types of diffusion Diffusion paths:
HRTEM image of an interface between an aluminum (left) and a germanium grain. The black dots correspond to atom columns.
Surface diffusion
Bulk diffusion
Grainbaoundarydiffusion
In general: Dgp >Dsd >Dgb >>Db for hightemperatures and short diffusion times
Diffusion through the gas phase
Self diffusion: Motion of host lattice atoms. The diffusion coefficient for self diffusion depends on the diffusion mechanism:
Vacancy mechanism: Dself = [Cvac] Dvac
Interstitial mechanism: Dself = [Cint] Dint
Inter diffusion, multicomponent diffusion:Motion of host and foreign species. The fluxes and diffusion coefficient are correlated
Diffusion - Mass transport by atomic motion
Mechanisms• Gases & Liquids – random (Brownian) motion• Solids – self, vacancy, interstitial, or inter
diffusion
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Oxidation
Roles of Diffusion
Creep
AgingSintering
Doping Carburizing
Metals
Precipitates
SteelsSemiconductors
Many more…
Many mechanisms
Material Joining Diffusion bonding
Diffusion is relative flow of one material into another Mass flow process by which species change their position relative to their neighbours.
Diffusion of a species occurs from a region of high concentration to low concentration (usually). More accurately, diffusion occurs down the chemical potential (µ) gradient.
To comprehend many materials related phenomenon (as in the figure below) one must understand Diffusion.
The focus of the current chapter is solid state diffusion in crystalline materials. In the current context, diffusion should be differentiated with flow (of usually fluids and
sometime solids).
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Processing Using Diffusion
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.
• Doping silicon with phosphorus for n-type semiconductors:
• Process:
3. Result: Dopedsemiconductorregions.
silicon
magnified image of a computer chip
0.5 mm
light regions: Si atoms
light regions: Al atoms
2. Heat it.
1. Deposit P rich layers on surface.
silicon
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Ar H2
Movable piston with an orifice
H2 diffusion direction
Ar diffusion direction
Piston motion
Piston moves in thedirection of the slower
moving species
When a perfume bottle is opened at one end of a room, its smell reaches the other end via the diffusion of the molecules of the perfume.
If we consider an experimental setup as below (with Ar and H2 on different sides of a chamber separated by a movable piston), H2 will diffuse faster towards the left (as compared to Ar). As obvious, this will lead to the motion of movable piston in the direction of the slower moving species.
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A B
Inert Marker is basically a thin rod of a high melting material, which is insoluble in A & B
Kirkendall effect
Let us consider two materials A and B welded together with Inert marker and given
a diffusion anneal (i.e. heated for diffusion to take place).
Usually the lower melting component diffuses faster (say B). This will lead to the
shift in the marker position to the right.
Direction of marker motion
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Mass flow process by which species change their position relative to their neighbours. Diffusion is driven by thermal energy and a ‘gradient’ (usually in chemical potential).
Gradients in other physical quantities can also lead to diffusion (as in the figure below). In this chapter we will essentially restrict ourselves to concentration gradients.
Usually, concentration gradients imply chemical potential gradients; but there are exceptions to this rule. Hence, sometimes diffusion occurs ‘uphill’ in concentration gradients, but downhill in chemical potential gradients.
Thermal energy leads to thermal vibrations of atoms, leading to atomic jumps. In the absence of a gradient, atoms will still randomly ‘jump about’, without any net flow of
matter.
Diffusion
Chemical potential
ElectricGradient
Magnetic
Stress
First we will consider a continuum picture of diffusion and later consider the atomic basis for the same in crystalline solids. The continuum picture is applicable to heat transfer (i.e., is closely related to mathematical equations of heat transfer). 8
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Diffusion mechanisms Vacancy diffusion Interstitial diffusion Impurities
Conditions necessary for diffusion An empty adjacent site Enough energy to break bonds and cause lattice distortions during
displacement
Mathematics of diffusion Steady-state diffusion (Fick’s first law) Nonsteady-State Diffusion (Fick’s second law)
Factors that influence diffusion Diffusing species Host solid Temperature Microstructure
Diffusion – How atoms move in solids
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What is diffusion?
Diffusion transport by atomic motion.
Inhomogeneous material can become homogeneous by diffusion. Temperature should be high enough to overcome energy barrier.
Diffusion
Part1. Constitutional effects
Diffusion is the phenomenon of spontaneous material transport byatomic motion.
Diffusion is classified according to
a) conditions: self-diffusion, diffusion from surface, interdiffusion, fastpath diffusion etc.
b) mechanism: interstitial, substitutional;
Part 2.
Non-constitutional effects. Kirkendall effect, Einstein equation, etc.11 12
Atom migration Vacancy migration
AfterBefore
Diffusion Mechanisms
Vacancy diffusion
To jump from lattice site to lattice site, atoms need energy to break bonds withneighbors, and to cause the necessary lattice distortions during jump.
Therefore, there is an energy barrier.
Energy comes from thermal energy of atomic vibrations (Eav ~ kT) Atom flow is opposite to vacancy flow direction.
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Interstitial atom before diffusion
Interstitial atom after diffusion
Diffusion Mechanisms
Interstitial diffusion
Generally faster than vacancy diffusion because bonding of interstitials tosurrounding atoms is normally weaker and there are more interstitial sites thanvacancy sites to jump to.
Smaller energy barrier Only small impurity atoms (e.g. C, H, O) fit into interstitial sites. The rate of interstitial diffusion is controlled only by the easiness with which a
diffusing atom can move into an interstice. 14
Self-diffusion: In an elemental solid, atoms also migrate.
A
B
C
D
After some time
AB
C
D
Vacancy Diffusion:• atoms exchange with vacancies• applies to substitutional impurities atoms • rate depends on:
-- number of vacancies-- activation energy to exchange.
increasing elapsed time
Probability of an atom jumping over the energy barrier:
𝑃 = 𝑒𝑥𝑝 −𝑄
𝑘𝑇
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Interstitial diffusion – smaller atoms can diffuse between atoms.
Interdiffusion: : In an alloy, atoms tend to migrate from regions of high conc. to regions of low conc.
More rapid than vacancy diffusion
Initially After some time
There is an energy barrier which must be overcome when an atom changes site.
Mechanisms of interdiffusion:
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Before After
(Heat)
Inter-diffusion vs. Self-diffusion
Self-diffusion: One-component material, atoms are of same type.
1 dnJ
A dtFlow direction
Area (A)
Concentration gradient. Concentration can be designated in many ways (e.g. moles per unit volume). Concentration gradient is the difference in concentration between two points (usually close by).
We can use a restricted definition of flux (J) as flow per unit area per unit time: → mass flow / area / time [Atoms / m2 / s].
Steady state. The properties at a single point in the system does not change with time. These properties in the case of fluid flow are pressure, temperature, velocity and mass flow rate. In the context of diffusion, steady state usually implies that, concentration of a given species at a given point in space, does not change with time.
Important terms
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mass atomsJ
area time m s
In diffusion problems, we would typically like to address one of the following problems.(i) What is the composition profile after a contain time (i.e. determine c(x,t))?
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Fick’s 1 law
Assume that only species ‘S’ is moving across an area ‘A’. Concentration gradient for species ‘S’ exists across the plane.
The concentration gradient (dc/dx) drives the flux (J) of atoms. Flux (J) is assumed to be proportional to concentration gradient. The constant of proportionality is the Diffusivity or Diffusion Coefficient (D).
‘D’ is assumed to be independent of the concentration gradient. Diffusivity is a material property. It is a function of the composition of the material and
the temperature. In crystals with cubic symmetry the diffusivity is isotropic (i.e. does not depend on direction).
Even if steady state conditions do not exist (concentration at a point is changing with time, there is accumulation/depletion of matter), Fick’s 1-equation is still valid (but not easy to use).
dx
dcDA
dt
dn
dx
dcJ
dx
dcDJ
dx
dcD
dt
dn
AJ
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Fick’s first law (equation)
As we shall see the ‘law’ is actually an equation
Area
Flow directionThe negative sign implies that diffusion occurs down the
concentration gradient
* Adolf Fick in 1855
A material property
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dx
dcDA
dt
dn
No. of atoms crossing area A
per unit time
Cross-sectional area
Concentration gradient
ve sign implies matter transport is down the concentration gradient
Diffusion coefficient/Diffusivity
AFlow direction
As a first approximation assume D f(t)
Let us emphasize the terms in the equation
Note the strange unit of D: [m2/s]
2 3
1[ ]
dc number numberJ D D
dx m s m m
Let us look at the units of Diffusivity
2
[ ]m
Ds
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Diffusion
How do we quantify the amount or rate of diffusion?
sm
kgor
scm
mol
timearea surface
diffusing mass) (or molesFlux
22J
J slope
dt
dM
A
l
At
MJ
M = mass diffused
time
Measured empirically Make thin film (membrane) of known surface area Impose concentration gradient Measure how fast atoms or molecules diffuse through the
membrane
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Steady-State Diffusion
dx
dCDJ
Fick’s first law of diffusion
D diffusion coefficient
Rate of diffusion independent of time
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12 linear ifxx
CC
x
C
dx
dC
Flux proportional to concentration gradient =
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: C1 = 0.44 g/cm3
C2 = 0.02 g/cm3
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Example: Chemical Protective Clothing (CPC)
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12- xx
CCD
dx
dCDJ
Dtb 6
2
glove
C1
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:
scm
g 10 x 16.1
cm) 04.0(
)g/cm 44.0g/cm 02.0(/s)cm 10 x 110(
25-
3328-
J
Fick’s 2 law
The equation as below is often referred to as the Fick’s 2 law (though clearly this is an equation and not a law).
This equation is derived using Fick’s 1-equation and mass balance. The concentration of diffusing species is a function of both time and position C =
C(x,t) The equation is a second order PDE requiring one initial condition and two
boundary conditions to solve.
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2
x
cD
t
c
c cD
t x x
If ‘D’ is not a function of the position, then it can be ‘pulled out’.
Derivation of this equation will taken up next.
Another equation
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Let us consider a 1D diffusion problem. Let us consider a small element of width x in the
body. Let the volume of the element be the control
volume (V) = 1.1. x = x. (Unit height and thickness).
Let the concentration profile of a species ‘S’ be as in the figure.
The slope of the c-x curve is related to the flux via the Fick’s I-equation.
In the figure the flux is decreasing linearly. The flux entering the element is Jx and that leaving
the element is Jx+x. Since the flux at x1 is not equal to flux leaving
that leaving at x2 and since J(x1) > J(x2), there is an accumulation of species ‘S’ in the region x.
The increase in the matter (species ‘S’) in the control volume (V) = (c/t).V = (c/t). x.
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Jx Jx+x
x
xxx JJonAccumulati
xx
JJJonAccumulati xx
xx
JJJx
t
cxx J
sm
Atomsm
sm
Atoms
23.
1
xx
Jx
t
c
x
cD
xt
cFick’s first law
x
cD
xt
c D f(x)2
2
x
cD
t
c
A B
Calculation of units
If Jx is the flux arriving at plane A and Jx+x is the flux leaving plane B. Then the Accumulation of matter is given by: (Jx Jx+x).
cJ
t
In 3D
Arises from mass conservation (hence not valid for vacancies)
2cD c
t
In 3D26
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
2
2
x
cD
t
c
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Diffusion
Steady stateJ f(x,t)
Non-steady stateJ = f(x,t) D = f(c)
D = f(c)
D f(c)
D f(c)
Steady and non-steady state diffusion
Diffusion can occur under steady state or non-steady state (transient) conditions.
Under steady state conditions, the flux is not a function of the position within the material or time. Under non-steady state conditions this is not true.
This implies that under steady state the concentration profile does not change with time.
In each of these circumstances, diffusivity (D) may or may not be a function of concentration (c). The term concentration can also be replaced with composition.
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0x t
dc J
dt x
Under steady state conditions
0c
Dx x
Substituting for flux from Fick’s first law
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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’
x
cD
xt
c cJ
t
In 3DThe general form of the Fick’s 2-equation is:
The equation is a second order differential equation involving time and one spatial dimension. This equation can be simplified for various circumstances and solved, as we will consider one
by one. These include: (i) steady state conditions and (ii) non-steady state conditions.
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Cases:
Steady state
Zero accumulation
Unsteady state
• Flux in ≠ flux out
• Enrichment or depletion
Fick’s second law
Fick’s laws
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The first simplification we make for the non-steady state conditions is that ‘D’ is not a function of the position.
If the diffusion distance is short relative to dimensions of the initial inhomogeneity, we can use the error function (erf) solution with 2 arbitrary constants.
The constants can be solved for from Boundary Condition(s) and Initial Condition(s). (we will
take up examples to clarify this).
Under non-steady state conditions
x
cD
xt
c If D is not a function of position 2
2
x
cD
t
c
2c
D ct
In 3D
Dt
xerfBAtxc
2 ),(
Under other conditions other solutions can be applied. For example, if a fixed amount of material is deposited on the surface of an infinite body and diffusion is allowed to take place, the concentration profile can be determined from the function below.
2
( , ) exp4
M xc x t
DtDt
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0
2exp2
duuErf
Erf () = 1 Erf ( ) = 1 Erf (0) = 0 Erf ( x) = Erf (x)
u →
Exp
(u2 )
→
0
Area
Also For upto x~0.6 Erf(x) ~ x x 2, Erf(x) 1
The error function (erf()) is defined as below. The modulus of the function represents the area under the curve of the exp(u2) function between ‘0’ and (with ‘some’ constant scaling factor). Some properties of the error function are also listed below.
Properties of the error function
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An example where the error function (erf) solution can be used
Let two materials M1 & M2 be joined together and kept at a temperature (T0), where diffusion is appreciable. Let C1 be the concentration of a species in M1 and C2 in M2.
This is a 1D diffusion problem (i.e. the species diffuses along x-direction only). The initial concentration profile (at t = 0, c(x,0)) of a species is like a step function (blue
line). If M1 and M2 are pure materials, then C1 would be zero. We can define an average composition of the species as: (C1 + C2)/2.
M2 M1
x →
Con
cent
rati
on →
Cavg
C1
C2
C(+x, 0) = C1
C(x, 0) = C2
The initial conditions (at t = 0) can be written as:
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M2 M1
x →
Con
cent
rati
on →
Cavg
↑ t
t1 > 0 | c(x, t1)t2 > t1 | c(x, t1) t = 0 | c(x,0)
Flux
f(x)|t
f(t)|x
Non-steadystate
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 = C1
A+B = C2
1 2 2 1( , ) 2 2 2
C C C C xc x t erf
Dt
With increasing time the species ‘S’ diffuses into M1 leading to a depletion of S in the region close to the interface on the M2-side and enrichment on the M1-side.
This implies that we are dealing with non-steady state (transient) diffusion. From the initial conditions the arbitrary constants A & B can be determined and the
concentration profile as a function of time (t) and position (x) can be determined. Such a profile for two specific times (t1 and t2) are shown below.
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Examples of Solutions:
1. A fixed quantity of solute (B) is plated onto a semi-infinite bar
0
),(and0)0,( BdxtxCxCBoundary conditions:
Solution:
Dtx
Dt
BtxC
4exp),(
2
This case is realized if a thin film of diffusant is deposited on a surface.
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2. Interdiffusion of ONE component and diffusion from constant source.
0)0,(and),0( CtxCCtxC s Boundary conditions:
Solution:
Dtx
erfCCCtxC ss 4)(),(
2
0
x
duuxerf0
2 )exp(:Reminder
2BA
s
CCC
Notice that the surface concentration remains fixed.
In the case of interdiffusion of TWO components concentration profiles may be very different!
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In ideal case the point of constant concentration propagates with a rate of (4Dt)-½
If there is a way to trace a point of constant concentration then diffusion coefficient can be determined explicitly.
x2
t
The slope is 4D
0 2 4 6 8 10
0.5
1.0
Con
cent
ratio
n
Depth
x2/(4Dt)= 1 2 4 8 16 32 64
x
This method can be used to measure diffusion coefficient by measuring experimentally :
Dt
x 2
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constDt
xconsttxC
4),(
2
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Temperature dependence of diffusivity
Arrhenius type
Diffusion is an activated process and hence the Diffusivity depends exponentially on temperature (as in the Arrhenius type equation below).
‘Q’ is the activation energy for diffusion. ‘Q’ depends on the kind of atomic processes (i.e. mechanism) involved in diffusion (e.g. substitutional diffusion, interstitial diffusion, grain boundary diffusion, etc.).
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.
Diffusion coefficient increases with increasing T.
= pre-exponential [m2/s]
= diffusion coefficient [m2/s]
= activation energy [J/mol or eV/atom]
= gas constant [8.314 J/mol-K]
= absolute temperature [K]
D
Do
Q
R
T
𝐷 = 𝐷 𝑒𝑥𝑝 −𝑄
𝑅𝑇
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Non-Steady State Solution of Diffusion - Superposition Principle
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Non-Steady State Solution of Diffusion - Superposition Principle
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Non-Steady State Solution of Diffusion – Application of Superposition Principle
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Non-Steady State Solution of Diffusion – Leak Test & Error Function
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Non-Steady State Solution of Diffusion – Semi-Infinite Source
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Determination of Diffusivity – Grube method
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Determination of Diffusivity – Boltzmann-Matano
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Non-Steady State Solution of Diffusion – Separation of Variable
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Non-Steady State Solution of Diffusion – Separation of Variable
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Non-Steady State Solution of Diffusion – Separation of Variable
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Non-Steady State Solution of Diffusion – Separation of Variable
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Non-Steady State Solution of Diffusion – Separation of Variable
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Non-Steady State Solution of Diffusion – Separation of Variable
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Non-Steady State Solution of Diffusion – Separation of Variable
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Diffusion along High Diffusion Path – Grain Boundary Diffusion Model
dx
dCD
tL
mJ L
LL
2
dx
dCLDm LL
2
dx
dCD
Lt
mJ gb
gbgb
2
dx
dCLDm gbgb 2
LD
D
dx
dCLD
dx
dCLD
m
m
L
gb
L
gb
L
gb 22
2
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Phenomenological description does not give dependence of the diffusion coefficient on any physical parameters.
Consider two adjacent planes in the crystal one can get thatN sites with n1 atoms
N sites with n2 atoms
1 2
Q
Energy profile
a v is the number of jumps per second
Q is the energy barrier separating two sites
N is the number of atoms per plane
Microscopic Mechanisms of Diffusion
In ideal case diffusion coefficient exponentially depends on temperature and written as:
𝐷 = 𝑣
6𝑁𝑒𝑥𝑝 −
𝑄
𝑘𝑇
𝐷 = 𝐷 𝑒𝑥𝑝 −𝑄
𝑅𝑇62
Diffusion: A thermally activated process I
Energy of red atom= ER
Minimum energy for jump = EA
Probability that an atom has an energy >EA:
PEN EAexp
EA
kT
Diffusion coefficient
D D0 exp EA
kT
D0: Preexponential factor, a constant which is a function of jump frequency, jump distance and coordination number of vacancies
Numberof atoms
EnergyEA ER
Boltzmann distribution
T2T1
T1 < T2
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Diffusion: A thermally activated process II
The preexponential factor and the activation energy for a diffusion process can bedetermined from diffuson experiments done at different temperatures. The result are presented in an Arrhenius diagram.
ln D0
lnD
1/T
EA
k
ln D ln D0 EA
k
1
T
D D0 expEA
kT
In the Arrhenius diagram the slope is proportional to the activation energy and the intercept gives the preexponential factor.
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Diffusion and TemperatureD has exponential dependence on T
Dinterstitial >> Dsubstitutional
C in a-FeC in -Fe
Al in AlFe in a-FeFe in -Fe
1000K/T
D (m2/s)
0.5 1.0 1.510-20
10-14
10-8
T(C)
1500
1000
600
300
Tracer diffusion coefficients of 18O determined by SIMS profiling for various micro-and nanocrystalline oxides: coarse grained titania c-TiO2
(- - - -), nanocrystalline titania n-TiO2 (- - - -), microcrystalline zirconia m-ZrO2 (– – –), zirconia doped with yttrium or calcium (YSZ —· · —, CSZ — · —), bulk diffusion DV ( ) and interface diffusion DB (♦) in nanocrystalline ZrO2 (——), after Brossmann et al. 1999.
Diffusion coefficients I
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Diffusion coefficients II
Self diffusion coefficient for cations and oxygen in corundum, hematite and eskolaite. Despite having the same structure, the diffusion coefficient differ by several orders of magnitude.
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ExampleAt 300 ºC the diffusion coefficient and activation energy for Cu in Si are D(300 ºC) = 7.8 x 10-11 m2/s and Qd = 41.5 kJ/mol. What is the diffusion coefficient at 350ºC?
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1lnln and
1lnln
TR
QDD
TR
QDD dd
121
212
11lnlnln
TTR
Q
D
DDD d
transform dataD
Temp = T
ln D
1/T
1212
11exp
TTR
QDD d
T1 = 273 + 300 = 573 K
K 573
1
K 623
1
K-J/mol 314.8
J/mol 500,41exp /s)m 10 x 8.7( 211
2D
T2 = 273 + 350 = 623 K
D2 = 15.7 x 10-11 m2/s
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Non-steady State Diffusion - Example
B.C. at t = 0, C = Co for 0 x
at t > 0, C = CS for x = 0 (constant surface conc.)
C = Co for x =
• Copper diffuses into a bar of aluminum.
pre-existing conc., Co of copper atoms
Surface conc., C of Cu atoms bar
s
Cs
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Solution:
C(x,t) = Conc. at point x at time t
erf (z) = error function
erf(z) values are given in Tables
CS
Co
C(x,t)
Dt
x
CC
Ct,xC
os
o
2 erf1
dye yz 2
0
2
71
• 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
Dt
x
CC
CtxC
os
o
2erf1
),(
Non-steady State Diffusion - Example
– t = 49.5 h x = 4 x 10-3 m
– Cx = 0.35 wt% Cs = 1.0 wt%
– Co = 0.20 wt%
)(erf12
erf120.00.1
20.035.0),(z
Dt
x
CC
CtxC
os
o
erf(z) = 0.812572
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
69 70
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For diffusion of C in FCC Fe
Do = 2.3 x 10-5 m2/s Qd = 148,000 J/mol
73
• To solve for the temperature at which Dhas the above value, we use a rearranged form of Equation (5.9a); )lnln( DDR
QT
o
d
/s)m 10x6.2 ln/sm 10x3.2 K)(ln-J/mol 314.8(
J/mol 000,14821125
T
Solution (cont.):
T = 1300 K = 1027ºC
74
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 breakthrough time (tb), i.e., how long could the gloves be used before methylene chloride reaches the hand?
Data
– diffusion coefficient in butyl rubber:
D = 110 x10-8 cm2/s
glove
C1
C2
skinpaintremover
x1 x2
Solution – assuming linear conc. gradient
Dtb 6
2
cm 0.04 12 xx
D = 110 x 10-8 cm2/s
Breakthrough time = tb
Time required for breakthrough ca. 4 min
min 4 s 240/s)cm 10 x 110)(6(
cm) 04.0(28-
2bt
Assumptions:
• These are 2 different metals in ratio 1:1
• They are joined by welding
• They are not completely miscible with each other
Let’s consider a chemical diffusion which occurs in presence of a contact between two metals.
Metal A Metal B
Diffusion in Multiphase Binary System
75
Diffusion Coefficient – Inter Diffusion
76
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Diffusion Coefficient – Inter Diffusion
77
Diffusion Coefficient – Self/Tracer Diffusion
78
Diffusion Coefficient – Intrinsic Diffusion Coefficient
79
Diffusion Coefficient – Inter Diffusion Coefficient
80
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B
BBAAB Nd
dDNDND
ln
ln1)(
~ **
Inter-diffusion Coefficient in a binary alloy linked to intrinsic diffusion by the Darken’s relation
Intrinsic diffusion Coefficient composed of mobility term (Tracer Diffusion) and
thermodynamic factor
B
BBB Nd
dDD
ln
ln1*
Tracer diffusion Coefficient – as a function of composition & temp.
)0(* BB ND
RTNQB
oBBB
BBeNDTND /)(* )(),(
: tracer impurity diffusion coefficient
: self-diffusion of A in the given structure)0(* BA ND
Diffusion Coefficient – Modeling
selfABB DND ** )0(
81
Diffusion Coefficient – Modeling
assuming composition independent D o
21
2
221
1
1
221
21
21
1
**/
221
21
21
1* nn
n
Nnn
n
N
RTQnn
nQ
nn
n
oBB DDeDN
nn
nN
nn
nD
Linear composition dependence of QB in a composition range N1 ~ N2
221
21
21
12
21
21
21
1 )()( Qnn
nQ
nn
nN
nn
nN
nn
nQNQ
Tracer diffusion coefficient at an intermediate composition is a geometrical mean of those at both ends – from experiments
the same for the D o term
RTNQNQRTNQNDBB
BBBoBBBB
oB eeeeTND /)()(/)()(ln* ),(
Both Q o & Q are modeled as a linear function of composition82
A hypothetical phase diagram A-B
A diffusion couple made by welding together pure A and pure B will result in layered structure containing α, β and γ.
83
Annealing at temperature T1 will produce a phase distribution and composition profile like that:
where:a, b, c and d – are solubility limits of the phases at T1.
84
81 82
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Concentration profile across the α/β interface and its associated movement assuming diffusion
control
})(~
)(~
{1
x
bBC
Dx
aBC
DaBCb
BCdt
dxv
a
dxaBCb
BC )(
dtx
aBC
Dx
bBC
D )})(~
())(~
{(
a
temerature absolute T constant gas R
energy activationQ~
constantoD
tcoefficien sioninterdiffu D~
)/~
exp(~
:where
RTQoDD
85
Optical micrograph of ion-nitrided ironshowing the multiplayer structure. The sample was ion nitrided at 605 °C for 2.5 h
Nitrogen concentration profile of ion-nitrided iron. The profile was measured
by electron probe microanalysis
Example
86
Nitrogen concentration profile of ion-nitrided iron. The profile was
measured by electron probe microanalysis
Example
87
Atomic Models of Diffusion
1) Interstitial Diffusion
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 (like carbon) wants to jump, this is ‘most likely’ possible as the the neighbouring site will be vacant.
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 in a interstitial void (interstitial diffusion)
(ii) species ‘sitting’ in a lattice site (substitutional diffusion).
1 2
1 2
Hm
At T > 0 K vibration of the atoms provides the energy to overcome the energy barrier Hm (enthalpy of motion).
→ frequency of vibrations, ’ → number of successful jumps / time.
kT
H m
e ' 88
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2) Substitutional diffusion via Vacancy Mechanism
For an atom in a lattice site (or a large atom associated with the motif), a jump to a neighbouring site can take place only 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.
Probability for an atomic jump (probability that the site is vacant) (probability that the atom has sufficient energy)
'f m
H
kTH
kT e e
kT
HH mf
e '
Hm → enthalpy of motion of atom across the barrier.
’ → frequency of successful jumps.
89
kT
Hm
eD 2
For Substitutional Diffusion
kT
HH mf
eD 2
D (C in FCC Fe at 1000ºC) = 3 1011 m2/s
D (Ni in FCC Fe at 1000ºC) = 2 1016 m2/s
0 f mH H
kTD D e
0 mH
kTD D e
which is of the form
A comparison of the value of diffusivity for interstitial diffusion and substitutional diffusion is given below. The comparison is made for C in -Fe and Ni in -Fe (both at 1000C).
It is seen that Dinterstitial is orders of magnitude greater than Dsubstitutional.
This is because the “barrier” (in the exponent) for substitutional diffusion has two ‘opposing’ terms: Hf and Hm (as compared to interstitial diffusion, which has only one term).
For Substitutional Diffusion
which is of the form
2D
Hence, ’ is of the form:
( )EnthalpykT e
If is the jump distance then the diffusivity can be written as:
( )
2Enthalpy
kTD e
90
Important During self-diffusion there is no change of chemical potential. Realization of each of the mechanisms depends on Type of intrinsic defects that prevails in the solid Activation energy for each of the mechanisms, if more than one
may be realized. Presence of other defects (vacancies).
Realization of vacancy or kick-out diffusion is possible only at the temperatures with sufficient concentration of vacancies. Therefore, prevailing mechanism may change with temperature.
In general, EVERY component in solid undergoes self-diffusion, however, if a solid contains more than one component, the ratio between self-diffusion coefficient depends on the type of bonding: Solids with covalent bonding typically have very low self-
diffusion coefficients. Solids with ionic bonding may have very different self-diffusion
coefficients for anion and cation. Metals and metal alloys usually show fast self-diffusion. 91
Diffusion Paths with Lesser Resistance
Qsurface < Qgrain boundary < Qpipe < QlatticeExperimentally determined activation energies for diffusion
The diffusion considered so far (both substitutional and interstitial) is ‘through’ the lattice.
In a microstructure there are many features, which can provide ‘easier’ paths for diffusion. These paths have a lower activation barrier for atomic jumps.
The ‘features’ to be considered include grain boundaries, surfaces, dislocation cores, etc. Residual stress can also play a major role in diffusion.
The order for activation energies (Q) for various paths is as listed below. A lower activation energy implies a higher diffusivity.
However, the flux of matter will be determined not only by the diffusivity, but also by the cross-section available for the path.
The diffusion through the core of a dislocation (especially so for edge dislocations) is termed as Pipe Diffusion.
92
89 90
91 92
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Comparison of Diffusivity for self-diffusion of Ag → single crystal vs polycrystal
1/T →
Log
(D
) →
Schematic
Polycrystal
Single crystal
← Increasing Temperature
Qgrain boundary = 110 kJ /mole
QLattice = 192 kJ /mole
If the ‘true’ effect of the high diffusivity of a low cross-section path is to be observed, then we need to go to low temperatures. At low temperatures, the high activation energy (low diffusivity) path is practically frozen and the effect of low activation energy path can be observed.
93
Applications based on Fick’s 2 law Carburization of steelCarburization 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.
Carburizing is done in the -phase field, where in the solubility of carbon is higher that that in the a phase. The high temperature enhances the kinetics as well.
In pack carburizing, a solid carbon powder used as C source.
In gas carburizing Methane gas is used a carbon source using the following reaction.CH4 (g) → 2H2 (g) + C (the carbon relased diffuses into steel).
It is usually assumed that the carbon concentration on the surface (CS) is constant (i.e. the carburizing medium imposes a constant concentration on the surface). An uniform homogeneous carbon concentration (C0) is assumed in the material before the carburization. Transient diffusion conditions exist and C diffuses into the steel.
C(+x, 0) = C0
C(0, t) = CS
A = CS
B = CS – C0
( , ) 2
xC x t A B erf
Dt
94
Approximate formula for depth of penetration
12
x Dt
0
0
( , )1
2S
c x t C xerf
C C Dt
Let the distance at which [(C(x,t)C0)/(CSC0)] = ½ 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
( , )=
2S
S
C x t C xerf
C C Dt
Often we would like to work with approximate formulae, which tell us the ‘effective’ depth of penetration and the depth which remains un-penetrated.
12
4x x 95
Another solution to the Fick’s 2 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 can use an exponential solution.
2
( , ) exp4
M xc x t
DtDt
Boundary and Initial conditions
C(+x, 0) = 0
0cdx M
Initially the species is only on the surface
The total mass of the species remains
constatant
The exponential solution
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Ionic materials are not close packed (like CCP or HCP metals).
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
97
There are materials where structural properties allow ultra-fast ion movement: superionics. In these materials ( for example AgRb3I4 one of the ions is much smaller than the available sites and there are far more available sites than ions.
Diffusion in polymers and glasses can be described by “randomly opening path” theory. Temperature dependence of diffusion coefficient in these materials is very complicated and time to time activation energy may become negative=> Diffusion coefficient may decrease with temperature.
Diffusion coefficient in anisotropic solids is a strong function of direction. Example: diffusion coefficient of Li and other alkaline metals in graphite along and across the layers may differ by 4 orders of magnitude.
Diffusion in Other Materials
98
Element Hf Hm Hf + Hm Q
Au 97 80 177 174
Ag 95 79 174 184
Calculated and experimental activation energies for vacancy Diffusion
99
SolvedExample
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
( , )=
2S
S
C x t C xerf
C C Dt
(2)
(1)
4
1
1 2 10
3 2erf
Dt
100
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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 = 1000st = 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 = 1000st = 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 e x pQ
D DR T
0 expQ
D DRT
34 157 10
(0.7 10 )exp8.314 1173
2127.14 10
m
s
-4
1
121
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
101
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
242
''
)/(
dxdc
JD
kT
H m
eD 2
20 D
kT
Q
eDD 0
On comparisonwith
102
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
cent
rati
on →
103
Diffusion of impurities.
a) Interstitial b) Vacancy b) Kick-out
Important. The diffusion mechanism of an impurity depends on many factors:
type of the solution: interstitial or substitutional; size of the diffusant and size of the host sites; temperature; presence of other impurities; electronic structure of the host: metal, dielectric or
semiconductor.104
101 102
103 104
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Ionic and charged in impurities in solids can drift in electric field. As a first approximation one can assume that the flux of ions is proportional to electric field and concentration, i.e., one can use a concept of mobility. In this idealized case the flux of ions is given:
ECgJ iii
Diffusion in the Presence of Electric Field: Electromigration
where gi is mobility of ions, Ci is the concentration of ions and E is electric field.
E
Electromigration Diffusion
Diffusion coefficient and mobility are linked
Thus mobility is kT
Dezgi
(Nernst) Einstein equation
105
Limitations
Electrical mobility and diffusion coefficient are linked to eachother.
Nernst-Einstein equation is valid whenever the followingconditions are met:
The system is not far from equilibrium, i.e., gradient ofpotential and concentration are small
The diffusion species follow Boltzmann statistics, i.e., theydo not interact with the host and with each other.
Important: Nernst-Einstein equation is applicable to electrons in somesemiconductors.
Nernst-Einstein equation is not valid for systems with stronginteractions.
106
Nernst-Einstein equation is a low electric field approximation! Itimplies that the energy acquired by ion during one jump is mushsmaller than the activation energy. =>
Systems with very low activation energy do not obey Nernst –Einstein equation.
Application of a sufficiently high electric field maysignificantly increase mobility. This electric field is, in fact,comparable with crystal field, the electric field between ionsin crystal.
Materials with fast-path diffusion may have different electricfields for each path at which non-linear dependence betweenmobility and diffusion coefficient becomes noticeable.
Comments
107
If a homogeneous alloy is placed in a temperature gradient, one of the elements willdiffuse under the influence of the temperature difference. This is known as the Soréteffect, and again shows an example of diffusion occurring without a compositiongradient. In the presence of temperature gradients we cannot use Gibbs freeenergies to define equilibrium conditions, so chemical potential arguments can notbe used.
Thermal Diffusion
T1 < T21 ≠ 2
In practice thermal diffusion (also calledthermomigration) can occur both down andup the temperature gradient.
Carbon in austenite thermomigrates up a temperature gradient, because theactivation energy in this case is required mainly for preparing the destination site.As the carbon moves, two Fe atoms have to separate to create room for the Catom. This occurs more easily at a higher temperature, so the carbon movespreferentially to the hotter region.
Thermal diffusion in ionic solids with onlyone atom mobile leads to thermo-electricvoltage, similar to Seebeck effect inelectronic conductors.
108
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107 108
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The presence of strain in the material can have a significant effect on thechemical potential of a solute. For example, in the case of an interstitialsolute such as carbon in iron, a tensile strain will increase the space availablefor the interstitial and so reduce the chemical potential.
Strain Field Induced Diffusion
The impurities that expand the lattice drift toward dilated regions andimpurities that cause contraction of the lattice drift towards compressedregions.
109
109