1. Lecture 4
2006

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

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

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

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

7. 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.

8. 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

9. 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

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

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

13. 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.

14. 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’

15. 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.

17. f = 1 - 2/z

18. 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.

19. 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.

20. 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

21. 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

22. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient

23. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient F

24. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient F
Average particle velocity

25. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient
Diffusivity F
Average particle velocity
where u is a particle mobility,

26. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient
Diffusivity F
Average particle velocity
where u is a particle mobility,
Boltzmann’s constant
temperature

27. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient
Diffusivity F
Average particle velocity
where u is a particle mobility,
Boltzmann’s constant
temperature So

28. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient
Diffusivity F
Average particle velocity
where u is a particle mobility,
Boltzmann’s constant
temperature
Why does force, F result in ‘velocity’ and not acceleration? So

29. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient
Diffusivity F
Average particle velocity
where u is a particle mobility,
Boltzmann’s constant
temperature
Why does force, F result in ‘velocity’ and not acceleration? So
Mobility is related to hopping from site to site. F causes bias in direction of hopping only.

30. Diffusion in the Presence of a Potential Gradient
Diffusion will occur when a potential gradient exists which biases atomic mobility in a particular direction.
Force due to a potential (V) gradient
Diffusivity F
Average particle velocity
where u is a particle mobility,
Boltzmann’s constant
temperature
Why does force, F result in ‘velocity’ and not acceleration? So
Mobility is related to hopping from site to site. F causes bias in direction of hopping only.

32. Field x charge

33. Field x charge
For diffusion of charged particles in an electric field
<V> = velocity down potential (dv/dx)

34. Field x charge
For diffusion of charged particles in an electric field
<V> = velocity down potential (dv/dx)

35. Field x charge
For diffusion of charged particles in an electric field
<V> = velocity down potential (dv/dx)
Flux units: m2s-1

36. Field x charge
For diffusion of charged particles in an electric field
<V> = velocity down potential (dv/dx)
Flux units: m2s-1
Compare with Ohm’s law
(i = sE)

37. Field x charge
For diffusion of charged particles in an electric field
<V> = velocity down potential (dv/dx)
Flux units: m2s-1
Compare with Ohm’s law
(i = sE)

38. Field x charge
For diffusion of charged particles in an electric field
<V> = velocity down potential (dv/dx)
Flux units: m2s-1
Compare with Ohm’s law
(i = sE)
Nernst-Einstein equation:

39. Field x charge
For diffusion of charged particles in an electric field
<V> = velocity down potential (dv/dx)
Flux units: m2s-1
Compare with Ohm’s law
(i = sE)
Nernst-Einstein equation: relates conductivity to intrinsic mobility of charged ion (Ds)

40. Combination of flux due to potential gradient and concentration gradient is now
Fick’s 1st law
Substituting for J in Fick’s 2nd law

42. Solution for a thin finite source

43. Solution for a thin finite source

44. Solution for a thin finite source+ -
Potential gradient

45. Solution for a thin finite source
<v>t + -
Potential gradient

46. + - Solution for a thin finite source <v>t 2 x √2Dt
Potential gradient

47. + - Solution for a thin finite source <v>t 2 x √2Dt
Potential gradient
Displacement <v>t is governed by the electric field

48. + - Solution for a thin finite source <v>t 2 x √2Dt
Potential gradient
Displacement <v>t is governed by the electric field
Dispersion or width is determined by the self-diffusion

49. Comparing conductivity to tracer diffusion

50. Comparing conductivity to tracer diffusion
Correlation factor

52. Radioactive 22Na coated onto the surface of a single crystal of NaCl.

53. Radioactive 22Na coated onto the surface of a single crystal of NaCl.
DT was determined from analysis of concn at different depths for each temperature.

54. Radioactive 22Na coated onto the surface of a single crystal of NaCl.
DT was determined from analysis of concn at different depths for each temperature.
NaCl FCC lattice - correlation factor = 0.78

55. Radioactive 22Na coated onto the surface of a single crystal of NaCl.
DT was determined from analysis of concn at different depths for each temperature.
NaCl FCC lattice - correlation factor = 0.78
DT corrected to Ds and plotted as open circles vs 1/T

56. Radioactive 22Na coated onto the surface of a single crystal of NaCl.
DT was determined from analysis of concn at different depths for each temperature.
NaCl FCC lattice - correlation factor = 0.78
DT corrected to Ds and plotted as open circles vs 1/T
Filled circles are D determined from conductivity measurements

57. Na diffusion in NaCl: Conductivity vs tracer diffusion
Extrinsic + intrinsic vacancies
Extrinsic vacancies

58. Na diffusion in NaCl: Conductivity vs tracer diffusion
Extrinsic + intrinsic vacancies
Extrinsic vacancies
Notice deviation in extrinsic region below 550˚C
Difference due to ‘bound vacancies’

59. Na diffusion in NaCl: Conductivity vs tracer diffusion
Extrinsic + intrinsic vacancies
Extrinsic vacancies
Notice deviation in extrinsic region below 550˚C
Difference due to ‘bound vacancies’

60. Na diffusion in NaCl: Conductivity vs tracer diffusion
Extrinsic + intrinsic vacancies
Extrinsic vacancies
Notice deviation in extrinsic region below 550˚C
Difference due to ‘bound vacancies’
Vacancy bound to fixed 2+ impurity

61. Na diffusion in NaCl: Conductivity vs tracer diffusion
Extrinsic + intrinsic vacancies
Extrinsic vacancies
Notice deviation in extrinsic region below 550˚C
Difference due to ‘bound vacancies’
Vacancy bound to fixed 2+ impurity
Bound vacancies contribute to tracer diffusion but not to conductivity (through going transport)

62. Na diffusion in NaCl: Conductivity vs tracer diffusion
Extrinsic + intrinsic vacancies
Extrinsic vacancies
Notice deviation in extrinsic region below 550˚C
Difference due to ‘bound vacancies’
Vacancy bound to fixed 2+ impurity
Bound vacancies contribute to tracer diffusion but not to conductivity (through going transport)
Transport of charge requires an equal movement (flux) of vacancies in opposite direction.

63. Fast ionic diffusion -Silver Iodide (AgI)
Iodine ions
Octahedral sites (6)
Tetrahedral sites (12)
Trigonal sites (24)
Z=2, but 42 available sites for Ag+

64. First experiments on AgI fast ion conductor
AgI heated to above 147˚C
Cathode weighed before and after connection to circuit
Charge flow recorded on coulometer
Ag+ + e- -> Ag
Ag-> Ag+ + e-
Mass gained at cathode = current flow through coulometer

65. Phase transition
Activation energy similar to alkali halides
(Below a-phase)
Frenkel
schottky
increases by 2-3 orders of magnitude at PT
Activation energy is low above phase transition
At high T, s is 10 orders of magnitude higher than KCL (schottky/direct vacancy mechanims)