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Lecture 4 2006 Slide 2 Slide 3 Random walk - > each hop is independent of the previous hop Slide 4 No memory effect Slide 5 Random walk - > each hop is independent of the previous hop No memory effect Squared displacement Slide 6 Random walk - > each hop is independent of the previous hop No memory effect Squared displacement Diagonal and off-diagonal terms Slide 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. Slide 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 Slide 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 Slide 10 Slide 11 Tracer diffusion is correlated (non-random) - why? Slide 12 Origin of the problem is distinguishable and indistinguishable particles Slide 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. Slide 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 Slide 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 selfdiffusing atom by a factor, f. Slide 16 Slide 17 f = 1 - 2/z Slide 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. Slide 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. Slide 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. Selfdiffusion constant, D s = D T / f Slide 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. Selfdiffusion constant, D s = D T / f Tracer diffusion Slide 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 Slide 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 Slide 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 Slide 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 Slide 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 Boltzmanns constant temperature Slide 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 Boltzmanns constant temperature So Slide 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 Boltzmanns constant temperature So Why does force, F result in velocity and not acceleration? Slide 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 Boltzmanns 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. Slide 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 Boltzmanns 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. Slide 31 Slide 32 Field x charge Slide 33 For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Slide 34 Field x charge For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Slide 35 Field x charge Flux units: m 2 s -1 For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Slide 36 Field x charge Flux units: m 2 s -1 Compare with Ohms law (i = E) For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Slide 37 Field x charge Flux units: m 2 s -1 Compare with Ohms law (i = E) For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Slide 38 Field x charge Flux units: m 2 s -1 Compare with Ohms law (i = E) For diffusion of charged particles in an electric field = velocity down potential (dv/dx) Nernst-Einstein equation: Slide 39 Field x charge Flux units: m 2 s -1 Compare with Ohms 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 ) Slide 40 Combination of flux due to potential gradient and concentration gradient is now Ficks 1st law Substituting for J in Ficks 2nd law Slide 41 Slide 42 Solution for a thin finite source Slide 43 Slide 44 + - Potential gradient Solution for a thin finite source Slide 45 t + - Potential gradient Solution for a thin finite source Slide 46 t + - Potential gradient 2 x 2Dt Solution for a thin finite source Slide 47 t + - Potential gradient 2 x 2Dt Solution for a thin finite source Displacement t is governed by the electric field Slide 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 Slide 49 Comparing conductivity to tracer diffusion Slide 50 Correlation factor Slide 51 Slide 52 Radioactive 22 Na coated onto the surface of a single crystal of NaCl. Slide 53 D T was determined from analysis of conc n at different depths for each temperature. Slide 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 Slide 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 Slide 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 Slide 57 Na diffusion in NaCl: Conductivity vs tracer diffusion Extrinsic + intrinsic vacancies Extrinsic vacancies Slide 58 Na diffusion in NaCl: Conductivity vs tracer diffusion Extrinsic + intrinsic vacancies Extrinsic vacancies Notice deviation in extrinsic region below 550C Difference due to bound vacancies Slide 59 Na diffusion in NaCl: Conductivity vs tracer diffusion Extrinsic + intrinsic vacancies Extrinsic vacancies Notice deviation in extrinsic region below 550C Difference due to bound vacancies Slide 60 Na diffusion in NaCl: Conductivity vs tracer diffusion Extrinsic + intrinsic vacancies Extrinsic vacancies Notice deviation in extrinsic region below 550C Difference due to bound vacancies Vacancy bound to fixed 2+ impurity Slide 61 Na diffusion in NaCl: Conductivity vs tracer diffusion Extrinsic + intrinsic vacancies Extrinsic vacancies Notice deviation in extrinsic region below 550C Difference due to bound vacancies Bound vacancies contribute to tracer diffusion but not to conductivity (through going transport) Vacancy bound to fixed 2+ impurity Slide 62 Na diffusion in NaCl: Conductivity vs tracer diffusion Extrinsic + intrinsic vacancies Extrinsic vacancies Notice deviation in extrinsic region below 550C 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 Slide 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 + Slide 64 First experiments on AgI fast ion conductor Ag-> Ag + + e - Ag + + e - -> Ag AgI heated to above 147C Cathode weighed before and after connection to circuit Charge flow recorded on coulometer Mass gained at cathode = current flow through coulometer Slide 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) Slide 66