lecture 3. microscopic dynamics and macroscopic d we can see that if we want to understand the...
Post on 31-Mar-2015
212 Views
Preview:
TRANSCRIPT
1
2
3
4
5
Ian Farnan
Michaelmas Term 2004
Natural Sciences Tripos Part IB
MINERAL SCIENCES
Module B: Transport Properties
Lecture 3
Microscopic dynamics and Macroscopic D
We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required.
Consider a particle moving on a 1-D lattice
For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x)
are equally probable.
€
Δx = δ i1
n
∑ = 0
Microscopic dynamics and Macroscopic D
We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required.
Consider a particle moving on a 1-D lattice
For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x)
are equally probable.
€
Δx = δ i1
n
∑ = 0
Large # hops, n
Microscopic dynamics and Macroscopic D
We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required.
Consider a particle moving on a 1-D lattice
For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x)
are equally probable.
€
Δx = δ i1
n
∑ = 0
Large # hops, n
with the same individual hop distance
Microscopic dynamics and Macroscopic D
We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required.
Consider a particle moving on a 1-D lattice
For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x)
are equally probable.
€
Δx = δ i1
n
∑ = 0
Large # hops, n
with the same individual hop distance
On average, distance moved is zero.
Microscopic dynamics and Macroscopic D
We can see that if we want to understand the diffusion constant measured in any material a knowledge of how the microscopic structure and dynamics of the material determine the diffusion coefficient is required.
Consider a particle moving on a 1-D lattice
For a large number (n ) of random hops of distance on a 1-D lattice the mean displacement, , will be zero because moves left or right (±x)
are equally probable.
€
Δx = δ i1
n
∑ = 0
Large # hops, n
with the same individual hop distance
On average, distance moved is zero.
However, any individual particle may have moved a long way.
€
Δx = δ i1
n
∑ = 0
The mean of the squared displacements, however, will not be zero
€
Δx2 = δ i1
n
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
= δ i2
1
n
∑ = nδ 2
€
i2 = δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( ). δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( )
€
11 δ12
δ21 δ22
δ33
δnn
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
€
Δx = δ i1
n
∑ = 0
The mean of the squared displacements, however, will not be zero
€
Δx2 = δ i1
n
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
= δ i2
1
n
∑ = nδ 2
Use this to quantify extent of diffusion/particle mobility
€
i2 = δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( ). δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( )
€
11 δ12
δ21 δ22
δ33
δnn
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
€
Δx = δ i1
n
∑ = 0
The mean of the squared displacements, however, will not be zero
€
Δx2 = δ i1
n
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
= δ i2
1
n
∑ = nδ 2
Use this to quantify extent of diffusion/particle mobility
€
i2 = δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( ). δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( )
Squared displacement
€
11 δ12
δ21 δ22
δ33
δnn
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
€
Δx = δ i1
n
∑ = 0
The mean of the squared displacements, however, will not be zero
€
Δx2 = δ i1
n
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
= δ i2
1
n
∑ = nδ 2
Use this to quantify extent of diffusion/particle mobility
€
i2 = δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( ). δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( )
Squared displacement
€
11 δ12
δ21 δ22
δ33
δnn
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
n hops results in an nxn matrix
€
Δx = δ i1
n
∑ = 0
The mean of the squared displacements, however, will not be zero
€
Δx2 = δ i1
n
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
= δ i2
1
n
∑ = nδ 2
Use this to quantify extent of diffusion/particle mobility
€
i2 = δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( ). δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( )
Squared displacement
€
11 δ12
δ21 δ22
δ33
δnn
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
n hops results in an nxn matrix
All diagonal elements are positive
€
Δx = δ i1
n
∑ = 0
The mean of the squared displacements, however, will not be zero
€
Δx2 = δ i1
n
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
= δ i2
1
n
∑ = nδ 2
Use this to quantify extent of diffusion/particle mobility
€
i2 = δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( ). δ1 + δ2 + δ3 + δ4 + δ5 + ....δn( )
Squared displacement
€
11 δ12
δ21 δ22
δ33
δnn
⎛
⎝
⎜ ⎜ ⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟ ⎟ ⎟
n hops results in an nxn matrix
All diagonal elements are positive
Off-diagonal elements can be positive or negative, on average sum to zero
This is true because when squaring i, the off-diagonal terms will sum
to zero for large n
€
iδ ki≠k
∑ = 0
We have then
€
Δx2 = δ n
If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
€
n =t
τ
This is true because when squaring i, the off-diagonal terms will sum
to zero for large n
€
iδ ki≠k
∑ = 0
We have then
€
Δx2 = δ n
If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
€
n =t
τ
€
ii2
ii
∑ = nδ 2Recall,
This is true because when squaring i, the off-diagonal terms will sum
to zero for large n
€
iδ ki≠k
∑ = 0
We have then
€
Δx2 = δ n
Average distance a particle has moved is given by:
If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
€
n =t
τ
€
ii2
ii
∑ = nδ 2Recall,
This is true because when squaring i, the off-diagonal terms will sum
to zero for large n
€
iδ ki≠k
∑ = 0
We have then
€
Δx2 = δ n
Average distance a particle has moved is given by:
Elementary hop distance x square root of the number of hops
If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
€
n =t
τ
€
ii2
ii
∑ = nδ 2Recall,
This is true because when squaring i, the off-diagonal terms will sum
to zero for large n
€
iδ ki≠k
∑ = 0
We have then
€
Δx2 = δ n
Average distance a particle has moved is given by:
Elementary hop distance x square root of the number of hops
If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
€
n =t
τ
Total diffusion time
€
ii2
ii
∑ = nδ 2Recall,
This is true because when squaring i, the off-diagonal terms will sum
to zero for large n
€
iδ ki≠k
∑ = 0
We have then
€
Δx2 = δ n
Average distance a particle has moved is given by:
Elementary hop distance x square root of the number of hops
If t is the time taken to acquire an mean square displacement, and is the time for a single elementary hop then:
€
n =t
τ
Total diffusion time
Individual hop time
€
ii2
ii
∑ = nδ 2Recall,
It follows that
Δ x
2
=
t
If we nowtake the result fro m t heroot me -ansquar ed displacemen tdetermined fro m the
mean displacement i na Gaussian cur ve(solution t oFick’s 2 ndLaw for diffusi on o f a thin
source) i.e.,
Δ x
2
= 2 Dt
we can relat ethe macroscopic diffus ion constant to an elementa ry hoppi ng distance ()and an elementary hoppi ngtime () so that
D =
2
2
This is the fundamental equation that relates microscopi c atomic dynamics t othe
macroscopicall ymeasur ed diffusi .on
NB the facto r 2 in the denominator is related t othe possible directions arando mhop
might take i na 1- D lattice - this becomes 4 i n a 2 D lattice 6and in 3a D lattice.
It follows that
Δ x
2
=
t
If we nowtake the result fro m t heroot me -ansquar ed displacemen tdetermined fro m the
mean displacement i na Gaussian cur ve(solution t oFick’s 2 ndLaw for diffusi on o f a thin
source) i.e.,
Δ x
2
= 2 Dt
we can relat ethe macroscopic diffus ion constant to an elementa ry hoppi ng distance ()and an elementary hoppi ngtime () so that
D =
2
2
This is the fundamental equation that relates microscopi c atomic dynamics t othe
macroscopicall ymeasur ed diffusi .on
NB the facto r 2 in the denominator is related t othe possible directions arando mhop
might take i na 1- D lattice - this becomes 4 i n a 2 D lattice 6and in 3a D lattice.
Substituting for n
It follows that
Δ x
2
=
t
If we nowtake the result fro m t heroot me -ansquar ed displacemen tdetermined fro m the
mean displacement i na Gaussian cur ve(solution t oFick’s 2 ndLaw for diffusi on o f a thin
source) i.e.,
Δ x
2
= 2 Dt
we can relat ethe macroscopic diffus ion constant to an elementa ry hoppi ng distance ()and an elementary hoppi ngtime () so that
D =
2
2
This is the fundamental equation that relates microscopi c atomic dynamics t othe
macroscopicall ymeasur ed diffusi .on
NB the facto r 2 in the denominator is related t othe possible directions arando mhop
might take i na 1- D lattice - this becomes 4 i n a 2 D lattice 6and in 3a D lattice.
Substituting for n
Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance
It follows that
Δ x
2
=
t
If we nowtake the result fro m t heroot me -ansquar ed displacemen tdetermined fro m the
mean displacement i na Gaussian cur ve(solution t oFick’s 2 ndLaw for diffusi on o f a thin
source) i.e.,
Δ x
2
= 2 Dt
we can relat ethe macroscopic diffus ion constant to an elementa ry hoppi ng distance ()and an elementary hoppi ngtime () so that
D =
2
2
This is the fundamental equation that relates microscopi c atomic dynamics t othe
macroscopicall ymeasur ed diffusi .on
NB the facto r 2 in the denominator is related t othe possible directions arando mhop
might take i na 1- D lattice - this becomes 4 i n a 2 D lattice 6and in 3a D lattice.
Substituting for n
Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance
‘Einstein relationship’ : relates microscopic dynamics to macroscopically measured diffusion through a fundamental hopping distance and a fundamental hopping time.
It follows that
Δ x
2
=
t
If we nowtake the result fro m t heroot me -ansquar ed displacemen tdetermined fro m the
mean displacement i na Gaussian cur ve(solution t oFick’s 2 ndLaw for diffusi on o f a thin
source) i.e.,
Δ x
2
= 2 Dt
we can relat ethe macroscopic diffus ion constant to an elementa ry hoppi ng distance ()and an elementary hoppi ngtime () so that
D =
2
2
This is the fundamental equation that relates microscopi c atomic dynamics t othe
macroscopicall ymeasur ed diffusi .on
NB the facto r 2 in the denominator is related t othe possible directions arando mhop
might take i na 1- D lattice - this becomes 4 i n a 2 D lattice 6and in 3a D lattice.
Substituting for n
Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance
‘Einstein relationship’ : relates microscopic dynamics to macroscopically measured diffusion through a fundamental hopping distance and a fundamental hopping time.
The factor 2 represents the probability of hops (left or right) on a 1D lattice
Dimensionality of diffusion
1D
€
12
δ 2
τ
Dimensionality of diffusion
1D
2D
€
12
δ 2
τ
€
14
δ 2
τ
Dimensionality of diffusion
1D
2D
3D
€
12
δ 2
τ
€
14
δ 2
τ
€
16
δ 2
τ
Dimensionality of diffusion
1D
2D
3D
€
12
δ 2
τ
€
14
δ 2
τ
€
16
δ 2
τ
Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.
Dimensionality of diffusion
1D
2D
3D
€
12
δ 2
τ
€
14
δ 2
τ
€
16
δ 2
τ
Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.
NB 1/ is often replaced by a
frequency of hopping,
to give:
€
D = 1g νδ
2
Dimensionality of diffusion
1D
2D
3D
€
12
δ 2
τ
€
14
δ 2
τ
€
16
δ 2
τ
Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.
NB 1/ is often replaced by a
frequency of hopping,
to give:
€
D = 1g νδ
2
2, 4 or 6
Types of vacancySchottky defect
Frenkel Defect
Types of vacancySchottky defect
Frenkel Defect
Missing anion or cation in a lattice
Types of vacancySchottky defect
Frenkel Defect
Types of vacancySchottky defect
Frenkel Defect
Missing anion or cation in a lattice
Occur in pairs to maintain electrical neutrality not necessarily together.
Types of vacancySchottky defect
Frenkel Defect
Missing anion or cation in a lattice
Occur in pairs to maintain electrical neutrality not necessarily together.
Vacant lattice site created by an atom moving into an interstititial position
Types of vacancySchottky defect
Frenkel Defect
Missing anion or cation in a lattice
Occur in pairs to maintain electrical neutrality not necessarily together.
Vacant lattice site created by an atom moving into an interstititial position
These are intrinsic vacancies
Other vacancies may be created by trace amounts of impurities or variable oxidation states of some constituent ions e.g. in NaCl a 2+ impurity (Ca2+, say)will require a missing anion (Cl-) as charge balance.
Types of vacancySchottky defect
Frenkel Defect
Missing anion or cation in a lattice
Occur in pairs to maintain electrical neutrality not necessarily together.
Vacant lattice site created by an atom moving into an interstititial position
These are intrinsic vacancies
Other vacancies may be created by trace amounts of impurities or variable oxidation states of some constituent ions e.g. in NaCl a 2+ impurity (Ca2+, say)will require a missing anion (Cl-) as charge balance.
These are extrinsic vacancies
Mechanisms of diffusion(1) Direct interstitial mechanisms for light gaseous elements, e.g. H2, He, N2, O2
'dissolved' in solids. Do not take up rational lattice sites.
(2) Direct vacancy mechanism - atom on rational lattice site moves to adjacent,
vacant lattice site. Flux of atoms in one direction requires a flux of vacancies
in the other direction.
(3)+(4) Exchange of lattice sites by two atoms 'squeezing past each other (3) or through
a ring mechanism (4).
(5) Intersticialcy mechanism - an interstitial atom takes up a rational lattice site as
atom occupying rational position moves off into adjacent interstitial position.
1
2
3
4
5
2
3
5
4
11
Mechanisms of diffusion(1) Direct interstitial mechanisms for light gaseous elements, e.g. H2, He, N2, O2
'dissolved' in solids. Do not take up rational lattice sites.
(2) Direct vacancy mechanism - atom on rational lattice site moves to adjacent,
vacant lattice site. Flux of atoms in one direction requires a flux of vacancies
in the other direction.
(3)+(4) Exchange of lattice sites by two atoms 'squeezing past each other (3) or through
a ring mechanism (4).
(5) Intersticialcy mechanism - an interstitial atom takes up a rational lattice site as
atom occupying rational position moves off into adjacent interstitial position.
1
2
3
4
5
2
3
5
4
11
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high.
Mechanisms of diffusion(1) Direct interstitial mechanisms for light gaseous elements, e.g. H2, He, N2, O2
'dissolved' in solids. Do not take up rational lattice sites.
(2) Direct vacancy mechanism - atom on rational lattice site moves to adjacent,
vacant lattice site. Flux of atoms in one direction requires a flux of vacancies
in the other direction.
(3)+(4) Exchange of lattice sites by two atoms 'squeezing past each other (3) or through
a ring mechanism (4).
(5) Intersticialcy mechanism - an interstitial atom takes up a rational lattice site as
atom occupying rational position moves off into adjacent interstitial position.
1
2
3
4
5
2
3
5
4
11
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high.
2,3,4 all mechanisms proposed to move one atom onto the site of another. Elastic energy required by (3) was considered too great so a cooperative mechanism (4) was postulated.
}
Mechanisms of diffusion(1) Direct interstitial mechanisms for light gaseous elements, e.g. H2, He, N2, O2
'dissolved' in solids. Do not take up rational lattice sites.
(2) Direct vacancy mechanism - atom on rational lattice site moves to adjacent,
vacant lattice site. Flux of atoms in one direction requires a flux of vacancies
in the other direction.
(3)+(4) Exchange of lattice sites by two atoms 'squeezing past each other (3) or through
a ring mechanism (4).
(5) Intersticialcy mechanism - an interstitial atom takes up a rational lattice site as
atom occupying rational position moves off into adjacent interstitial position.
1
2
3
4
5
2
3
5
4
11
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high.
2,3,4 all mechanisms proposed to move one atom onto the site of another. Elastic energy required by (3) was considered too great so a cooperative mechanism (4) was postulated.
(2) allows direct hopping onto adjacent vacant site, explicitly requires vacancies, (3) + (4) do not.
}
Mechanisms of diffusion(1) Direct interstitial mechanisms for light gaseous elements, e.g. H2, He, N2, O2
'dissolved' in solids. Do not take up rational lattice sites.
(2) Direct vacancy mechanism - atom on rational lattice site moves to adjacent,
vacant lattice site. Flux of atoms in one direction requires a flux of vacancies
in the other direction.
(3)+(4) Exchange of lattice sites by two atoms 'squeezing past each other (3) or through
a ring mechanism (4).
(5) Intersticialcy mechanism - an interstitial atom takes up a rational lattice site as
atom occupying rational position moves off into adjacent interstitial position.
1
2
3
4
5
2
3
5
4
11
Small concentrations (up to a few percent) dissolved in lattice, taking up interstitial positions - diffusion often rapid as number of vacant interstitial sites is high.
2,3,4 all mechanisms proposed to move one atom onto the site of another. Elastic energy required by (3) was considered too great so a cooperative mechanism (4) was postulated.
(2) allows direct hopping onto adjacent vacant site, explicitly requires vacancies, (3) + (4) do not.
}
(5) Mechanism actually observed in some fast ion conductors (see later) combination of vacancy (2) and interstitial (1) mechanisms.
Kirkendall's ExperimentFine Molybdenum
Markers
Brass
Copper
Brass block was surrounded by fine (inert) molybdenum markers and then
copper. The whole arrangement was annealed at high temperature for increasing lengths
of time. The markers moved closer together for longer times.
Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out
and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move
nward.
This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).
The markers move inward because new lattice sites are being created on the outside and
here is a net vacancy flow inward.
First direct evidence of vacancy mechanisms and their importance in solid state diffusion.
Brass - alloy of Cu/Zn
Kirkendall's ExperimentFine Molybdenum
Markers
Brass
Copper
Brass block was surrounded by fine (inert) molybdenum markers and then
copper. The whole arrangement was annealed at high temperature for increasing lengths
of time. The markers moved closer together for longer times.
Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out
and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move
nward.
This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).
The markers move inward because new lattice sites are being created on the outside and
here is a net vacancy flow inward.
First direct evidence of vacancy mechanisms and their importance in solid state diffusion.
Brass - alloy of Cu/Zn
Concentration gradient exists between Cu and Cu/Zn -
Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.
Kirkendall's ExperimentFine Molybdenum
Markers
Brass
Copper
Brass block was surrounded by fine (inert) molybdenum markers and then
copper. The whole arrangement was annealed at high temperature for increasing lengths
of time. The markers moved closer together for longer times.
Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out
and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move
nward.
This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).
The markers move inward because new lattice sites are being created on the outside and
here is a net vacancy flow inward.
First direct evidence of vacancy mechanisms and their importance in solid state diffusion.
Brass - alloy of Cu/Zn
Concentration gradient exists between Cu and Cu/Zn -
Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.
What happens to the inert markers?
Kirkendall's ExperimentFine Molybdenum
Markers
Brass
Copper
Brass block was surrounded by fine (inert) molybdenum markers and then
copper. The whole arrangement was annealed at high temperature for increasing lengths
of time. The markers moved closer together for longer times.
Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out
and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move
nward.
This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).
The markers move inward because new lattice sites are being created on the outside and
here is a net vacancy flow inward.
First direct evidence of vacancy mechanisms and their importance in solid state diffusion.
Brass - alloy of Cu/Zn
Concentration gradient exists between Cu and Cu/Zn -
Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.
What happens to the inert markers?
They move closer together the longer time goes on.
Conclusion: Zn diffuses faster than Cu.
Kirkendall's ExperimentFine Molybdenum
Markers
Brass
Copper
Brass block was surrounded by fine (inert) molybdenum markers and then
copper. The whole arrangement was annealed at high temperature for increasing lengths
of time. The markers moved closer together for longer times.
Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out
and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move
nward.
This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).
The markers move inward because new lattice sites are being created on the outside and
here is a net vacancy flow inward.
First direct evidence of vacancy mechanisms and their importance in solid state diffusion.
Brass - alloy of Cu/Zn
Concentration gradient exists between Cu and Cu/Zn -
Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.
What happens to the inert markers?
They move closer together the longer time goes on.
Conclusion: Zn diffuses faster than Cu.
Must be vacancy mechanism, because if direct (3) or cooperative (4) then Dcu = DZn
Kirkendall's ExperimentFine Molybdenum
Markers
Brass
Copper
Brass block was surrounded by fine (inert) molybdenum markers and then
copper. The whole arrangement was annealed at high temperature for increasing lengths
of time. The markers moved closer together for longer times.
Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out
and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move
nward.
This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).
The markers move inward because new lattice sites are being created on the outside and
here is a net vacancy flow inward.
First direct evidence of vacancy mechanisms and their importance in solid state diffusion.
Brass - alloy of Cu/Zn
Concentration gradient exists between Cu and Cu/Zn -
Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.
What happens to the inert markers?
They move closer together the longer time goes on.
Conclusion: Zn diffuses faster than Cu.
Must be vacancy mechanism, because if direct (3) or cooperative (4) then Dcu = DZn
If markers move in then new sites are being created beyond markers with vacancy flow inwards
Kirkendall's ExperimentFine Molybdenum
Markers
Brass
Copper
Brass block was surrounded by fine (inert) molybdenum markers and then
copper. The whole arrangement was annealed at high temperature for increasing lengths
of time. The markers moved closer together for longer times.
Brass is an alloy of Copper and zinc. The concentration gradient causes Zn to diffuse out
and Cu to diffuse in. However, Zn diffuses faster than Cu causing the markers to move
nward.
This experiment rules out direct exchange (mech 3) and cyclic exchange (mech 4) asdiffusion mechanisms since they require DCu (in) = DZn (out).
The markers move inward because new lattice sites are being created on the outside and
here is a net vacancy flow inward.
First direct evidence of vacancy mechanisms and their importance in solid state diffusion.
Brass - alloy of Cu/Zn
Concentration gradient exists between Cu and Cu/Zn -
Zn will diffuse out, ‘down’ the gradient’ as sample is heated for long periods of time.
What happens to the inert markers?
They move closer together the longer time goes on.
Conclusion: Zn diffuses faster than Cu.
Must be vacancy mechanism, because if direct (3) or cooperative (4) then Dcu = DZn
If markers move in then new sites are being created beyond markers with vacancy flow inwards
Direct vacancy mechanism is the predominant mechanism in solid state diffusion.
Microscopic diffusion by atoms and vacancies
Earlier, we have seen that the diffusion constant D is related to microscopic (atomistic)
processes by the elementary hop distance, r, and a time between hops, , which we now
express a s a hopping frequenc , y = 1/.
For diffusi on of a vacancy on a cubic lattice t he diffusi onconstant becomes
Dv
=
1
6
r
2
v
v
≡ vacancy hoppi ngfrequency
≡ atomic hopping frequency
Similarly for diffusi on o f interstitials
DI
=
1
6
r
2
I
However, for atomic diffusion vi a a vacancy mechanism an ato mrequires an adjacent
vacancy in orde r t omove (this is not requir edin interstitial or vacancy diffusion)
Ds
=
1
6
r
2
v
cv
cv
≡
n
N
, the concentrati on of vacancies
Ds i sknown as the se -lf diffusion constan t andrefers t othe itineran t motion of the
constitue nt atoms withi n a material which occurs wit h or withou t aconcentration (or
other potentia )l gradien .t In t heabsence of a gradient this atomic motion still occurs but
ther e isno tnet atomic movemen .t
Microscopic diffusion by atoms and vacancies
Earlier, we have seen that the diffusion constant D is related to microscopic (atomistic)
processes by the elementary hop distance, r, and a time between hops, , which we now
express a s a hopping frequenc , y = 1/.
For diffusi on of a vacancy on a cubic lattice t he diffusi onconstant becomes
Dv
=
1
6
r
2
v
v
≡ vacancy hoppi ngfrequency
≡ atomic hopping frequency
Similarly for diffusi on o f interstitials
DI
=
1
6
r
2
I
However, for atomic diffusion vi a a vacancy mechanism an ato mrequires an adjacent
vacancy in orde r t omove (this is not requir edin interstitial or vacancy diffusion)
Ds
=
1
6
r
2
v
cv
cv
≡
n
N
, the concentrati on of vacancies
Ds i sknown as the se -lf diffusion constan t andrefers t othe itineran t motion of the
constitue nt atoms withi n a material which occurs wit h or withou t aconcentration (or
other potentia )l gradien .t In t heabsence of a gradient this atomic motion still occurs but
ther e isno tnet atomic movemen .t
V
Microscopic diffusion by atoms and vacancies
Earlier, we have seen that the diffusion constant D is related to microscopic (atomistic)
processes by the elementary hop distance, r, and a time between hops, , which we now
express a s a hopping frequenc , y = 1/.
For diffusi on of a vacancy on a cubic lattice t he diffusi onconstant becomes
Dv
=
1
6
r
2
v
v
≡ vacancy hoppi ngfrequency
≡ atomic hopping frequency
Similarly for diffusi on o f interstitials
DI
=
1
6
r
2
I
However, for atomic diffusion vi a a vacancy mechanism an ato mrequires an adjacent
vacancy in orde r t omove (this is not requir edin interstitial or vacancy diffusion)
Ds
=
1
6
r
2
v
cv
cv
≡
n
N
, the concentrati on of vacancies
Ds i sknown as the se -lf diffusion constan t andrefers t othe itineran t motion of the
constitue nt atoms withi n a material which occurs wit h or withou t aconcentration (or
other potentia )l gradien .t In t heabsence of a gradient this atomic motion still occurs but
ther e isno tnet atomic movemen .t
V
(r = )
Microscopic diffusion by atoms and vacancies
Earlier, we have seen that the diffusion constant D is related to microscopic (atomistic)
processes by the elementary hop distance, r, and a time between hops, , which we now
express a s a hopping frequenc , y = 1/.
For diffusi on of a vacancy on a cubic lattice t he diffusi onconstant becomes
Dv
=
1
6
r
2
v
v
≡ vacancy hoppi ngfrequency
≡ atomic hopping frequency
Similarly for diffusi on o f interstitials
DI
=
1
6
r
2
I
However, for atomic diffusion vi a a vacancy mechanism an ato mrequires an adjacent
vacancy in orde r t omove (this is not requir edin interstitial or vacancy diffusion)
Ds
=
1
6
r
2
v
cv
cv
≡
n
N
, the concentrati on of vacancies
Ds i sknown as the se -lf diffusion constan t andrefers t othe itineran t motion of the
constitue nt atoms withi n a material which occurs wit h or withou t aconcentration (or
other potentia )l gradien .t In t heabsence of a gradient this atomic motion still occurs but
ther e isno tnet atomic movemen .t
V
(r = )
Microscopic diffusion by atoms and vacancies
Earlier, we have seen that the diffusion constant D is related to microscopic (atomistic)
processes by the elementary hop distance, r, and a time between hops, , which we now
express a s a hopping frequenc , y = 1/.
For diffusi on of a vacancy on a cubic lattice t he diffusi onconstant becomes
Dv
=
1
6
r
2
v
v
≡ vacancy hoppi ngfrequency
≡ atomic hopping frequency
Similarly for diffusi on o f interstitials
DI
=
1
6
r
2
I
However, for atomic diffusion vi a a vacancy mechanism an ato mrequires an adjacent
vacancy in orde r t omove (this is not requir edin interstitial or vacancy diffusion)
Ds
=
1
6
r
2
v
cv
cv
≡
n
N
, the concentrati on of vacancies
Ds i sknown as the se -lf diffusion constan t andrefers t othe itineran t motion of the
constitue nt atoms withi n a material which occurs wit h or withou t aconcentration (or
other potentia )l gradien .t In t heabsence of a gradient this atomic motion still occurs but
ther e isno tnet atomic movemen .t
V
(r = )
V
V
Microscopic diffusion by atoms and vacancies
Earlier, we have seen that the diffusion constant D is related to microscopic (atomistic)
processes by the elementary hop distance, r, and a time between hops, , which we now
express a s a hopping frequenc , y = 1/.
For diffusi on of a vacancy on a cubic lattice t he diffusi onconstant becomes
Dv
=
1
6
r
2
v
v
≡ vacancy hoppi ngfrequency
≡ atomic hopping frequency
Similarly for diffusi on o f interstitials
DI
=
1
6
r
2
I
However, for atomic diffusion vi a a vacancy mechanism an ato mrequires an adjacent
vacancy in orde r t omove (this is not requir edin interstitial or vacancy diffusion)
Ds
=
1
6
r
2
v
cv
cv
≡
n
N
, the concentrati on of vacancies
Ds i sknown as the se -lf diffusion constan t andrefers t othe itineran t motion of the
constitue nt atoms withi n a material which occurs wit h or withou t aconcentration (or
other potentia )l gradien .t In t heabsence of a gradient this atomic motion still occurs but
ther e isno tnet atomic movemen .t
V
(r = )
V
V
Ds can be thought of as an average mobility of indistinguishable particles.Increased by increasing the hopping freqeuncy or the concn of vacancies
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Migration energy ΔEm
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.
Migration energy ΔEm
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.
The energy vs distance profile is a maximum at B, this is the migration
energy, ΔEm . Or the ‘saddle point energy’.
Migration energy ΔEm
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.
The energy vs distance profile is a maximum at B, this is the migration
energy, ΔEm . Or the ‘saddle point energy’.
Migration energy ΔEm
Hopping frequency [v=voexp(- ΔEm / kT)] increases with temperature as the atom occupies vibrational states nearer the top of the well.
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.
The energy vs distance profile is a maximum at B, this is the migration
energy, ΔEm . Or the ‘saddle point energy’.
Migration energy ΔEm
Hopping frequency [v=voexp(- ΔEm / kT)] increases with temperature as the atom occupies vibrational states nearer the top of the well.
Macroscopically this leads to an Arrhenian temperature dependence for D:
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.
The energy vs distance profile is a maximum at B, this is the migration
energy, ΔEm . Or the ‘saddle point energy’.
Migration energy ΔEm
Hopping frequency [v=voexp(- ΔEm / kT)] increases with temperature as the atom occupies vibrational states nearer the top of the well.
Macroscopically this leads to an Arrhenian temperature dependence for D:
DI,v = Do exp(- ΔEm / kT)
Temperature dependence of D
Temperature will have a profound effect on the diffusion constant. Increasing
temperature will increase the atomic hopping frequency, . i s related t othe vibrational
frequency o f atom sin a materia l and as temperature increas esmore of the atoms will
vibrat e neare r t hetop of their loca l potenti alenergy well and thus bemore likel y t o hop
over into t he nex tpotential wel .l
E
Δ E
A B C
A
B
C
r
I nthe case whe re a vacancyis available or fo r interstitia l diffusion, for the atom
t omove fron A to C it has t o distor t t helattice as it pass esthrough positi onB. Thisrequire s an additiona l ener gyΔE t o get t othe 'saddl e point' B - known as the saddle point
energy. At low T the jum pwill be infrequent a s the ato mwill oscillate ar oundits
equilibriu mpositi onat the botto mof the potential we .ll As T increases the probabilty
that the ato mwill be in higher ene rgy states closer to the top of t he well increase s and
hence the likelihood that it can make the jum .p This can be written
υ = υ
0
exp
− Δ E
kT
⎛
⎝
⎞
⎠
Where ΔE is t hesaddle point energy.
Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.
The energy vs distance profile is a maximum at B, this is the migration
energy, ΔEm . Or the ‘saddle point energy’.
Migration energy ΔEm
Hopping frequency [v=voexp(- ΔEm / kT)] increases with temperature as the atom occupies vibrational states nearer the top of the well.
Macroscopically this leads to an Arrhenian temperature dependence for D:
DI,v = Do exp(- ΔEm / kT)
interstitials, vacancies
Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,
T, can be written
DI , v
= Do
exp −
Δ Em
kT
⎛
⎝
⎜
⎞
⎠
⎟
where Δ Em
is knowna st hemigrati on energy andis related t othe micoscopic energy
barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D
over a range of temperatures and making an Arrhenius plot of loge D vs 1/T
loge
D = loge
Do
−
Δ Em
R
.
1
T
T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.
For interstitial diffusion:Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,
T, can be written
DI , v
= Do
exp −
Δ Em
kT
⎛
⎝
⎜
⎞
⎠
⎟
where Δ Em
is knowna st hemigrati on energy andis related t othe micoscopic energy
barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D
over a range of temperatures and making an Arrhenius plot of loge D vs 1/T
loge
D = loge
Do
−
Δ Em
R
.
1
T
T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.
For interstitial diffusion:
measure D as a function of temperature
Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,
T, can be written
DI , v
= Do
exp −
Δ Em
kT
⎛
⎝
⎜
⎞
⎠
⎟
where Δ Em
is knowna st hemigrati on energy andis related t othe micoscopic energy
barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D
over a range of temperatures and making an Arrhenius plot of loge D vs 1/T
loge
D = loge
Do
−
Δ Em
R
.
1
T
T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.
For interstitial diffusion:
measure D as a function of temperature
Plot loge D vs 1/T
Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,
T, can be written
DI , v
= Do
exp −
Δ Em
kT
⎛
⎝
⎜
⎞
⎠
⎟
where Δ Em
is knowna st hemigrati on energy andis related t othe micoscopic energy
barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D
over a range of temperatures and making an Arrhenius plot of loge D vs 1/T
loge
D = loge
Do
−
Δ Em
R
.
1
T
T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.
For interstitial diffusion:
measure D as a function of temperature
Plot loge D vs 1/T
gradient
Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,
T, can be written
DI , v
= Do
exp −
Δ Em
kT
⎛
⎝
⎜
⎞
⎠
⎟
where Δ Em
is knowna st hemigrati on energy andis related t othe micoscopic energy
barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D
over a range of temperatures and making an Arrhenius plot of loge D vs 1/T
loge
D = loge
Do
−
Δ Em
R
.
1
T
T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.
For interstitial diffusion:
measure D as a function of temperature
Plot loge D vs 1/T
gradient
What about self-diffusion?
Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,
T, can be written
DI , v
= Do
exp −
Δ Em
kT
⎛
⎝
⎜
⎞
⎠
⎟
where Δ Em
is knowna st hemigrati on energy andis related t othe micoscopic energy
barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D
over a range of temperatures and making an Arrhenius plot of loge D vs 1/T
loge
D = loge
Do
−
Δ Em
R
.
1
T
T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.
For interstitial diffusion:
measure D as a function of temperature
Plot loge D vs 1/T
gradient
What about self-diffusion?
Vacancy concentration vs temperature?
Thus the macroscopic diffusion constant for interstitials or vacancies at any temperature,
T, can be written
DI , v
= Do
exp −
Δ Em
kT
⎛
⎝
⎜
⎞
⎠
⎟
where Δ Em
is knowna st hemigrati on energy andis related t othe micoscopic energy
barrier represented bythe saddle-point energy.We canmeas ure an activati on ene rgy fo r diffusion, by maki ngmeasurements of D
over a range of temperatures and making an Arrhenius plot of loge D vs 1/T
loge
D = loge
Do
−
Δ Em
R
.
1
T
T hesl ope of the li ne being ΔE/ (R for ane nergyi n kJ mol-1 ) and the y intercep t loge Do.For atomic diffusion, the activati on ene rgy fo r diffusion by a vacancy mechanism w ill bedetermi ned by t heactivati on ene rgy oft hejump from the lattice site to an adjacentunoccupied lattice site and by t he probabilit y of an adjacent vacancy be ing available.
Thermally Created Vacancies
Given that the free energy of formation of a vacancy is ΔGv then even for a 'pure'
material a tany temperature T above absolute zer ,o ther e w ill be a number of vacancies inequilibriu mwit hthe structur .e T hefracti on o f vacancies a s a functi on of temperaturedepends ont heenthal py of vac ancy formati ,on Ev.
n
N
= exp
− Ev
RT
⎛
⎝
⎜
⎞
⎠
⎟
ΔEv i s fairl y large, since it involve s brea king bonds and removing anato m fro mthe
structur ethen n/N will have a ve ry steep temperatur e dependence. If one looks a t the
equilibriu ,m thermall y generat ed vacancy concentrations at room temperatur ethey arever ysmall. I nfac t t heya re ver ymuch less than t he norma l level of impurities i n a
material .i .e, trace impurity atoms or slight off-stoichiometry, so tha tat low temperatures
the impurit y concentration is constan tbecaus e the number o f the seextrinsic defects willnot c . hange This is anadvantage as it allows ust omeasur ethe activation ener gyΔE at
lower T (fro m l n D vs 1/T plot )s and when the number of thermally generat edintrinsic
vacancies becomes important there will be a change of slope dueto the much higher
activation ener .gy T helowe r temperature slope is t heactivati on ene rgy for atomic
Thermally Created Vacancies
Given that the free energy of formation of a vacancy is ΔGv then even for a 'pure'
material a tany temperature T above absolute zer ,o ther e w ill be a number of vacancies inequilibriu mwit hthe structur .e T hefracti on o f vacancies a s a functi on of temperaturedepends ont heenthal py of vac ancy formati ,on Ev.
n
N
= exp
− Ev
RT
⎛
⎝
⎜
⎞
⎠
⎟
ΔEv i s fairl y large, since it involve s brea king bonds and removing anato m fro mthe
structur ethen n/N will have a ve ry steep temperatur e dependence. If one looks a t the
equilibriu ,m thermall y generat ed vacancy concentrations at room temperatur ethey arever ysmall. I nfac t t heya re ver ymuch less than t he norma l level of impurities i n a
material .i .e, trace impurity atoms or slight off-stoichiometry, so tha tat low temperatures
the impurit y concentration is constan tbecaus e the number o f the seextrinsic defects willnot c . hange This is anadvantage as it allows ust omeasur ethe activation ener gyΔE at
lower T (fro m l n D vs 1/T plot )s and when the number of thermally generat edintrinsic
vacancies becomes important there will be a change of slope dueto the much higher
activation ener .gy T helowe r temperature slope is t heactivati on ene rgy for atomic
Temperature dependence of vacancy concn
Thermally Created Vacancies
Given that the free energy of formation of a vacancy is ΔGv then even for a 'pure'
material a tany temperature T above absolute zer ,o ther e w ill be a number of vacancies inequilibriu mwit hthe structur .e T hefracti on o f vacancies a s a functi on of temperaturedepends ont heenthal py of vac ancy formati ,on Ev.
n
N
= exp
− Ev
RT
⎛
⎝
⎜
⎞
⎠
⎟
ΔEv i s fairl y large, since it involve s brea king bonds and removing anato m fro mthe
structur ethen n/N will have a ve ry steep temperatur e dependence. If one looks a t the
equilibriu ,m thermall y generat ed vacancy concentrations at room temperatur ethey arever ysmall. I nfac t t heya re ver ymuch less than t he norma l level of impurities i n a
material .i .e, trace impurity atoms or slight off-stoichiometry, so tha tat low temperatures
the impurit y concentration is constan tbecaus e the number o f the seextrinsic defects willnot c . hange This is anadvantage as it allows ust omeasur ethe activation ener gyΔE at
lower T (fro m l n D vs 1/T plot )s and when the number of thermally generat edintrinsic
vacancies becomes important there will be a change of slope dueto the much higher
activation ener .gy T helowe r temperature slope is t heactivati on ene rgy for atomic
Temperature dependence of vacancy concn
Large entropic TΔS factor in
ΔG promotes vacancy formation even though Ev may be large.
Thermally Created Vacancies
Given that the free energy of formation of a vacancy is ΔGv then even for a 'pure'
material a tany temperature T above absolute zer ,o ther e w ill be a number of vacancies inequilibriu mwit hthe structur .e T hefracti on o f vacancies a s a functi on of temperaturedepends ont heenthal py of vac ancy formati ,on Ev.
n
N
= exp
− Ev
RT
⎛
⎝
⎜
⎞
⎠
⎟
ΔEv i s fairl y large, since it involve s brea king bonds and removing anato m fro mthe
structur ethen n/N will have a ve ry steep temperatur e dependence. If one looks a t the
equilibriu ,m thermall y generat ed vacancy concentrations at room temperatur ethey arever ysmall. I nfac t t heya re ver ymuch less than t he norma l level of impurities i n a
material .i .e, trace impurity atoms or slight off-stoichiometry, so tha tat low temperatures
the impurit y concentration is constan tbecaus e the number o f the seextrinsic defects willnot c . hange This is anadvantage as it allows ust omeasur ethe activation ener gyΔE at
lower T (fro m l n D vs 1/T plot )s and when the number of thermally generat edintrinsic
vacancies becomes important there will be a change of slope dueto the much higher
activation ener .gy T helowe r temperature slope is t heactivati on ene rgy for atomic
Temperature dependence of vacancy concn
Large entropic TΔS factor in
ΔG promotes vacancy formation even though Ev may be large.
There are very many ways to arrange a small number of vacancies over a very large number of lattice sites - See BH48
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
ΔEm + Ev
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
Cd2+ replaces Na+ creating Na+
vacancies.
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
Cd2+ replaces Na+ creating Na+
vacancies.
At low temperatures this doping creates extrinsic vacancies.
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
Cd2+ replaces Na+ creating Na+
vacancies.
At low temperatures this doping creates extrinsic vacancies.
Number of thermally created vacancies is far less than extrinsic vacancies at low temperature ->
activation energy is simply ΔEm
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
Cd2+ replaces Na+ creating Na+
vacancies.
At low temperatures this doping creates extrinsic vacancies.
Number of thermally created vacancies is far less than extrinsic vacancies at low temperature ->
activation energy is simply ΔEm
At high temperatures, thermally created vacancies become important
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
Cd2+ replaces Na+ creating Na+
vacancies.
At low temperatures this doping creates extrinsic vacancies.
Number of thermally created vacancies is far less than extrinsic vacancies at low temperature ->
activation energy is simply ΔEm
At high temperatures, thermally created vacancies become important
Activation energy is then ΔEm + Ev
ΔEm + Ev
migration (hopping, ΔEm) and the higher T slope ΔEv+ΔEm. I n which case
D = Do
exp
− Δ Em
+ Δ Ev
( )
RT
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
An example of such data is shown below for NaCl substituted wit h a sma ll amount ofCdCl2.
Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
Cd2+ replaces Na+ creating Na+
vacancies.
At low temperatures this doping creates extrinsic vacancies.
Number of thermally created vacancies is far less than extrinsic vacancies at low temperature ->
activation energy is simply ΔEm
At high temperatures, thermally created vacancies beome important
Activation energy is then ΔEm + Ev
D increases much more rapidly as new vacancies are created.
Temperature dependence of vacancy concentration
Temperature dependence of vacancy concentration
Creation of a vacancy is a highly energetic process - breaking of all bonds and removal to the surface.
Temperature dependence of vacancy concentration
Creation of a vacancy is a highly energetic process - breaking of all bonds and removal to the surface.
In addition there is an associated volume expansion beyond that expected from the x-ray determined volume.
Temperature dependence of vacancy concentration
Creation of a vacancy is a highly energetic process - breaking of all bonds and removal to the surface.
In addition there is an associated volume expansion beyond that expected from the x-ray determined volume.
Nearly all atoms remain in register and there is some increase in the lattice spacing due to thermal expansion. The ‘ideal’ volume at any temperature can be determined from the lattice parameter at the same temperature
Temperature dependence of vacancy concentration
Creation of a vacancy is a highly energetic process - breaking of all bonds and removal to the surface.
In addition there is an associated volume expansion beyond that expected from the x-ray determined volume.
Nearly all atoms remain in register and there is some increase in the lattice spacing due to thermal expansion. The ‘ideal’ volume at any temperature can be determined from the lattice parameter at the same temperature
The real, macroscopic volume of a sample can also be measured…….
Macroscopic expansion vs ‘x-ray expansion’
Macroscopic expansion vs ‘x-ray expansion’
Very careful x-ray diffraction and dilatation experiments showed a
difference between Δa/a and Δl/l for aluminium
Macroscopic expansion vs ‘x-ray expansion’
Very careful x-ray diffraction and dilatation experiments showed a
difference between Δa/a and Δl/l for aluminium
Extra volume is created by vacancies in the material
Macroscopic expansion vs ‘x-ray expansion’
Very careful x-ray diffraction and dilatation experiments showed a
difference between Δa/a and Δl/l for aluminium
Extra volume is created by vacancies in the material
The nearer the melting point the greater the number of vacancies.
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk - > each hop is independent of the previous hop
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Squared displacement
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Squared displacement
Diagonal and off-diagonal terms
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Squared displacement
Diagonal and off-diagonal terms
If motion is not random then the off-diagonal terms no longer sum to zero for a large number of hops.
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Squared displacement
Diagonal and off-diagonal terms
If motion is not random then the off-diagonal terms no longer sum to zero for a large number of hops.
They are correlated by a factor, f
Random walk, correlation factors, tracer diffusion
Random Walk
ri
R
We have already seen that for a particle executing a number of hops per unit time(n) of elementary distance, ri, there will be a total displacement, R, in unit time.
R = r
i
i = 1
n
∑
Squari ng gives
R
2
= r
i
i , j
n
∑ . r
j
separatingR
2
= ri
2
i = 1
n
∑ + r
i
i ≠ j
n
∑ . r
j
If all hops are random and therefore do no t depend on t he precedi ng hop then the second
term in the right hand expressi on will be zero. If for som ereason the probability of a hop
is not random, and depends on where a particl e has hopped fro ,m t henthis term willintroduce acorrelati onfactor, f, into the mean squa re displacement. This i s aproces s we
need t o understand in orde r tointerpre t tracer diffusi .on T helabelling of an atom by the
substitution of a trac erisotope makes it inequivalent t o othe r atoms surrounding it and
bias est he hops itmigh t m akewhen a vacancy isa next neighbou.r
Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Squared displacement
Diagonal and off-diagonal terms
If motion is not random then the off-diagonal terms no longer sum to zero for a large number of hops.
They are correlated by a factor, f
€
f = r ii≠ j
n
∑ .r j
A tracer atom has a higher probability of moving back to the vacancy it has just left.
Jumps of a tracer are thus correlated because they depend on the direction of the previous
jump.
The correlation factor takes account of the fact that the total displacement acquired by a
tracer atom is less than that acquired in a true random walk because of these ‘wasted’
jumps back and forth on the same two sites. In general, a full matrix calculation of the
correlation term is required. However, a simple approximation can produce some
reasonable values of f.
f = 1 −
2
z
where z is the coordination number.
i.e., 2 jumps a re wast edif the tracer jumps back int othe adjacen t vacancy and theprobability of this happening i s1/z, the number of neighbouring sites.
A tracer atom has a higher probability of moving back to the vacancy it has just left.
Jumps of a tracer are thus correlated because they depend on the direction of the previous
jump.
The correlation factor takes account of the fact that the total displacement acquired by a
tracer atom is less than that acquired in a true random walk because of these ‘wasted’
jumps back and forth on the same two sites. In general, a full matrix calculation of the
correlation term is required. However, a simple approximation can produce some
reasonable values of f.
f = 1 −
2
z
where z is the coordination number.
i.e., 2 jumps a re wast edif the tracer jumps back int othe adjacen t vacancy and theprobability of this happening i s1/z, the number of neighbouring sites.
Tracer diffusion is correlated (non-random) - why?
A tracer atom has a higher probability of moving back to the vacancy it has just left.
Jumps of a tracer are thus correlated because they depend on the direction of the previous
jump.
The correlation factor takes account of the fact that the total displacement acquired by a
tracer atom is less than that acquired in a true random walk because of these ‘wasted’
jumps back and forth on the same two sites. In general, a full matrix calculation of the
correlation term is required. However, a simple approximation can produce some
reasonable values of f.
f = 1 −
2
z
where z is the coordination number.
i.e., 2 jumps a re wast edif the tracer jumps back int othe adjacen t vacancy and theprobability of this happening i s1/z, the number of neighbouring sites.
Tracer diffusion is correlated (non-random) - why?
Origin of the problem is distinguishable and indistinguishable particles
A tracer atom has a higher probability of moving back to the vacancy it has just left.
Jumps of a tracer are thus correlated because they depend on the direction of the previous
jump.
The correlation factor takes account of the fact that the total displacement acquired by a
tracer atom is less than that acquired in a true random walk because of these ‘wasted’
jumps back and forth on the same two sites. In general, a full matrix calculation of the
correlation term is required. However, a simple approximation can produce some
reasonable values of f.
f = 1 −
2
z
where z is the coordination number.
i.e., 2 jumps a re wast edif the tracer jumps back int othe adjacen t vacancy and theprobability of this happening i s1/z, the number of neighbouring sites.
Tracer diffusion is correlated (non-random) - why?
Origin of the problem is distinguishable and indistinguishable particles
tracer atom has a higher probability of hopping back into a site it has just left because it is distinguishable.
A tracer atom has a higher probability of moving back to the vacancy it has just left.
Jumps of a tracer are thus correlated because they depend on the direction of the previous
jump.
The correlation factor takes account of the fact that the total displacement acquired by a
tracer atom is less than that acquired in a true random walk because of these ‘wasted’
jumps back and forth on the same two sites. In general, a full matrix calculation of the
correlation term is required. However, a simple approximation can produce some
reasonable values of f.
f = 1 −
2
z
where z is the coordination number.
i.e., 2 jumps a re wast edif the tracer jumps back int othe adjacen t vacancy and theprobability of this happening i s1/z, the number of neighbouring sites.
Tracer diffusion is correlated (non-random) - why?
Origin of the problem is distinguishable and indistinguishable particles
tracer atom has a higher probability of hopping back into a site it has just left because it is distinguishable.
We call this a ‘correlation’ or a ‘memory effect’
A tracer atom has a higher probability of moving back to the vacancy it has just left.
Jumps of a tracer are thus correlated because they depend on the direction of the previous
jump.
The correlation factor takes account of the fact that the total displacement acquired by a
tracer atom is less than that acquired in a true random walk because of these ‘wasted’
jumps back and forth on the same two sites. In general, a full matrix calculation of the
correlation term is required. However, a simple approximation can produce some
reasonable values of f.
f = 1 −
2
z
where z is the coordination number.
i.e., 2 jumps a re wast edif the tracer jumps back int othe adjacen t vacancy and theprobability of this happening i s1/z, the number of neighbouring sites.
Tracer diffusion is correlated (non-random) - why?
Origin of the problem is distinguishable and indistinguishable particles
tracer atom has a higher probability of hopping back into a site it has just left because it is distinguishable.
We call this a ‘correlation’ or a ‘memory effect’
Random walk of a tracer will be less than that of a self–diffusing atom by a factor, f.
Approximate and actual values of f for different lattices
lattice
2D square
2D hexagonal
diamond
simple cubic
BCC
FCC
z_
4
6
4
6
8
12
approx.f (1-2/z)
1/2
2/3
1/2
2/3
3/4
5/6 (0.833)
calculated f
0.467
0.560
0.5
0.655
0.72
0.78
Approximate and actual values of f for different lattices
lattice
2D square
2D hexagonal
diamond
simple cubic
BCC
FCC
z_
4
6
4
6
8
12
approx.f (1-2/z)
1/2
2/3
1/2
2/3
3/4
5/6 (0.833)
calculated f
0.467
0.560
0.5
0.655
0.72
0.78
f = 1 - 2/z
Approximate and actual values of f for different lattices
lattice
2D square
2D hexagonal
diamond
simple cubic
BCC
FCC
z_
4
6
4
6
8
12
approx.f (1-2/z)
1/2
2/3
1/2
2/3
3/4
5/6 (0.833)
calculated f
0.467
0.560
0.5
0.655
0.72
0.78
f = 1 - 2/z
Total displacement for n jumps (recall, √n) for a tracer is less than for a true random walk because jumps are wasted back and forth on a site.
Approximate and actual values of f for different lattices
lattice
2D square
2D hexagonal
diamond
simple cubic
BCC
FCC
z_
4
6
4
6
8
12
approx.f (1-2/z)
1/2
2/3
1/2
2/3
3/4
5/6 (0.833)
calculated f
0.467
0.560
0.5
0.655
0.72
0.78
f = 1 - 2/z
Total displacement for n jumps (recall, √n) for a tracer is less than for a true random walk because jumps are wasted back and forth on a site.
These hops do not contribute to the total displacement.
Approximate and actual values of f for different lattices
lattice
2D square
2D hexagonal
diamond
simple cubic
BCC
FCC
z_
4
6
4
6
8
12
approx.f (1-2/z)
1/2
2/3
1/2
2/3
3/4
5/6 (0.833)
calculated f
0.467
0.560
0.5
0.655
0.72
0.78
f = 1 - 2/z
Total displacement for n jumps (recall, √n) for a tracer is less than for a true random walk because jumps are wasted back and forth on a site.
These hops do not contribute to the total displacement.
Self–diffusion constant, Ds = DT / f
Approximate and actual values of f for different lattices
lattice
2D square
2D hexagonal
diamond
simple cubic
BCC
FCC
z_
4
6
4
6
8
12
approx.f (1-2/z)
1/2
2/3
1/2
2/3
3/4
5/6 (0.833)
calculated f
0.467
0.560
0.5
0.655
0.72
0.78
f = 1 - 2/z
Total displacement for n jumps (recall, √n) for a tracer is less than for a true random walk because jumps are wasted back and forth on a site.
These hops do not contribute to the total displacement.
Self–diffusion constant, Ds = DT / f
Tracer diffusion
top related