1. Lecture 3

2. 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.

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
Large # hops, n
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.

4. 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
Large # hops, n
with the same individual hop distance
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.

5. 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
Large # hops, n
with the same individual hop distance
On average, distance moved is zero.
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.

6. 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
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.
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.

7. The mean of the squared displacements, however, will not be zero

8. The mean of the squared displacements, however, will not be zero
Use this to quantify extent of diffusion/particle mobility

9. The mean of the squared displacements, however, will not be zero
Use this to quantify extent of diffusion/particle mobility
Squared displacement

10. The mean of the squared displacements, however, will not be zero
Use this to quantify extent of diffusion/particle mobility
Squared displacement
n hops results in an nxn matrix

11. The mean of the squared displacements, however, will not be zero
Use this to quantify extent of diffusion/particle mobility
Squared displacement
n hops results in an nxn matrix
All diagonal elements are positive

12. The mean of the squared displacements, however, will not be zero
Use this to quantify extent of diffusion/particle mobility
Squared displacement
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

13. This is true because when squaring i, the off-diagonal terms will sum to zero for large n
We have then
If t is the time taken to acquire an mean square displacement, and  is the time for a single elementary hop then:

14. This is true because when squaring i, the off-diagonal terms will sum to zero for large n
Recall,
We have then
If t is the time taken to acquire an mean square displacement, and  is the time for a single elementary hop then:

15. This is true because when squaring i, the off-diagonal terms will sum to zero for large n
Recall,
We have then
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:

16. This is true because when squaring i, the off-diagonal terms will sum to zero for large n
Recall,
We have then
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:

17. This is true because when squaring i, the off-diagonal terms will sum to zero for large n
Recall,
We have then
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:
Total diffusion time

18. This is true because when squaring i, the off-diagonal terms will sum to zero for large n
Recall,
We have then
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:
Total diffusion time
Individual hop time

20. Substituting for n

21. Substituting for n
Equate the average distance from analysis of macroscopic diffusion profile for thin source with microscopic ‘rms’ distance

22. 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.

23. 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

24. Dimensionality of diffusion

25. Dimensionality of diffusion

26. Dimensionality of diffusion

27. Dimensionality of diffusion
Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.

28. NB 1/t is often replaced by a frequency of hopping, n
to give:
Dimensionality of diffusion 1D 2D 3D
Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.

29. NB 1/t is often replaced by a frequency of hopping, n
to give:
Dimensionality of diffusion 1D
2, 4 or 6 2D 3D
Although we deal with real 3D materials, the dimensionality of the diffusive process may well be lower.

31. Missing anion or cation in a lattice

33. Missing anion or cation in a lattice
Occur in pairs to maintain electrical neutrality not necessarily together.

34. 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

35. 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.

36. 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

38. 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.

39. 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.

40. 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.

41. 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.

42. Brass - alloy of Cu/Zn

43. 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.

44. 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?

45. 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.

46. 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

47. 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

48. 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.

50. V

51. V
(r = d)

52. V
(r = d)

53. V
(r = d) V V

54. V
(r = d) 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

56. Migration energy DEm

57. Migration energy DEm
Elastic energy is required to distort lattice and allow atom to pass from site A through to an adjacent vacant site C.

58. Migration energy DEm
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, DEm . Or the ‘saddle point energy’.

59. Migration energy DEm
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, DEm . Or the ‘saddle point energy’.
Hopping frequency [v=voexp(- DEm / kT)] increases with temperature as the atom occupies vibrational states nearer the top of the well.

60. Migration energy DEm
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, DEm . Or the ‘saddle point energy’.
Hopping frequency [v=voexp(- DEm / 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:

61. Migration energy DEm
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, DEm . Or the ‘saddle point energy’.
Hopping frequency [v=voexp(- DEm / 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(- DEm / kT)

62. Migration energy DEm
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, DEm . Or the ‘saddle point energy’.
Hopping frequency [v=voexp(- DEm / 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(- DEm / kT)
interstitials, vacancies

64. For interstitial diffusion:

65. For interstitial diffusion:
measure D as a function of temperature

66. For interstitial diffusion:
measure D as a function of temperature
Plot loge D vs 1/T

67. For interstitial diffusion:
measure D as a function of temperature
Plot loge D vs 1/T
gradient

68. For interstitial diffusion:
measure D as a function of temperature
Plot loge D vs 1/T
gradient
What about self-diffusion?

69. 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?

71. Temperature dependence of vacancy concn

72. Temperature dependence of vacancy concn
Large entropic TDS factor in DG promotes vacancy formation even though Ev may be large.

73. Temperature dependence of vacancy concn
Large entropic TDS factor in DG 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

75. Two energetic factors controlling the temperature dependence of diffusion can often be separated.
DEm + Ev

76. Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd

77. Two energetic factors controlling the temperature dependence of diffusion can often be separated.
Example: NaCl doped with Cd
Cd2+ replaces Na+ creating Na+ vacancies.

78. 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.

79. 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 DEm

80. 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 DEm
At high temperatures, thermally created vacancies become important

81. 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 DEm
DEm + Ev
At high temperatures, thermally created vacancies become important
Activation energy is then DEm + Ev

82. 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 DEm
At high temperatures, thermally created vacancies beome important
Activation energy is then DEm + Ev
D increases much more rapidly as new vacancies are created.

84. Creation of a vacancy is a highly energetic process - breaking of all bonds and removal to the surface.

85. 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.

86. 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

87. 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…….

89. Very careful x-ray diffraction and dilatation experiments showed a difference between Da/a and Dl/l for aluminium

90. Very careful x-ray diffraction and dilatation experiments showed a difference between Da/a and Dl/l for aluminium
Extra volume is created by vacancies in the material

91. Very careful x-ray diffraction and dilatation experiments showed a difference between Da/a and Dl/l for aluminium
Extra volume is created by vacancies in the material
The nearer the melting point the greater the number of vacancies.

93. Random walk - > each hop is independent of the previous hop

94. Random walk - > each hop is independent of the previous hop
No ‘memory effect’

95. Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Squared displacement

96. Random walk - > each hop is independent of the previous hop
No ‘memory effect’
Squared displacement
Diagonal and off-diagonal terms

97. 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.

98. 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

99. 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

101. Tracer diffusion is correlated (non-random) - why?

102. Tracer diffusion is correlated (non-random) - why?
Origin of the problem is distinguishable and indistinguishable particles

103. 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.

104. 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’

105. 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.

107. f = 1 - 2/z

108. f = 1 - 2/z
Total displacement for n jumps (recall, d√n) for a tracer is less than for a true random walk because jumps are wasted back and forth on a site.

109. f = 1 - 2/z
Total displacement for n jumps (recall, d√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.

110. f = 1 - 2/z
Total displacement for n jumps (recall, d√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

111. f = 1 - 2/z
Total displacement for n jumps (recall, d√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