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Lecture 4 2006. Random walk - > each hop is independent of the previous hop.

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Presentation on theme: "Lecture 4 2006. Random walk - > each hop is independent of the previous hop."— Presentation transcript:

1 Lecture

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

4 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

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

12 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.

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17 f = 1 - 2/z

18 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.

19 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.

20 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, D s = D T / f

21 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, D s = D T / 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 F Average particle velocity where u is a particle mobility, Diffusivity

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 F Average particle velocity where u is a particle mobility, Diffusivity 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 F Average particle velocity where u is a particle mobility, Diffusivity 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 F Average particle velocity where u is a particle mobility, Diffusivity Boltzmann’s constant temperature So Why does force, F result in ‘velocity’ and not acceleration?

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 F Average particle velocity where u is a particle mobility, Diffusivity Boltzmann’s constant temperature So Why does force, F result in ‘velocity’ and not acceleration? 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 F Average particle velocity where u is a particle mobility, Diffusivity Boltzmann’s constant temperature So Why does force, F result in ‘velocity’ and not acceleration? Mobility is related to hopping from site to site. F causes bias in direction of hopping only.

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32 Field x charge

33 For diffusion of charged particles in an electric field = velocity down potential (dv/dx)

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

35 Field x charge Flux units: m 2 s -1 For diffusion of charged particles in an electric field = velocity down potential (dv/dx)

36 Field x charge Flux units: m 2 s -1 Compare with Ohm’s law (i =  E) For diffusion of charged particles in an electric field = velocity down potential (dv/dx)

37 Field x charge Flux units: m 2 s -1 Compare with Ohm’s law (i =  E) For diffusion of charged particles in an electric field = velocity down potential (dv/dx)

38 Field x charge Flux units: m 2 s -1 Compare with Ohm’s law (i =  E) For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Nernst-Einstein equation:

39 Field x charge Flux units: m 2 s -1 Compare with Ohm’s law (i =  E) For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Nernst-Einstein equation: relates conductivity to intrinsic mobility of charged ion (D s )

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

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42 Solution for a thin finite source

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44 + - Potential gradient Solution for a thin finite source

45 t + - Potential gradient Solution for a thin finite source

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

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

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

49 Comparing conductivity to tracer diffusion

50 Correlation factor

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

53 D T was determined from analysis of conc n at different depths for each temperature.

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

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

56 Radioactive 22 Na coated onto the surface of a single crystal of NaCl. D T was determined from analysis of conc n at different depths for each temperature. NaCl FCC lattice - correlation factor = 0.78 D T corrected to D s 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’ Bound vacancies contribute to tracer diffusion but not to conductivity (through going transport) Vacancy bound to fixed 2+ impurity

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’ 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. Vacancy bound to fixed 2+ impurity

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

65 Phase transition schottky Frenkel Activation energy similar to alkali halides (Below  -phase)  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)

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