Lecture 4

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 where u is a particle mobility, Average particle velocity

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 where u is a particle mobility, Average particle velocity 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 where u is a particle mobility, Average particle velocity 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 where u is a particle mobility, Average particle velocity 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 where u is a particle mobility, Average particle velocity 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 where u is a particle mobility, Average particle velocity 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