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Lecture 4 2006

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

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

No ‘memory effect’

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

No ‘memory effect’ Squared displacement

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

No ‘memory effect’ Squared displacement Diagonal and off-diagonal terms

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**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.

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

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

Origin of the problem is distinguishable and indistinguishable particles

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**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.

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**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’

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**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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f = 1 - 2/z

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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.

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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.

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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

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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

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**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

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**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

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**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

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**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,

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**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

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**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

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**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

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**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.

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**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.

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Field x charge

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

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

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

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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)

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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)

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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:

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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)

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**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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**Solution for a thin finite source**

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**Solution for a thin finite source**

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**Solution for a thin finite source**

+ - Potential gradient

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**Solution for a thin finite source**

<v>t + - Potential gradient

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**+ - Solution for a thin finite source <v>t 2 x √2Dt**

Potential gradient

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**+ - Solution for a thin finite source <v>t 2 x √2Dt**

Potential gradient Displacement <v>t is governed by the electric field

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**+ - 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

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**Comparing conductivity to tracer diffusion**

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**Comparing conductivity to tracer diffusion**

Correlation factor

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

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**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.

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**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

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**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

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**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

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**Na diffusion in NaCl: Conductivity vs tracer diffusion**

Extrinsic + intrinsic vacancies Extrinsic vacancies

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**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’

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**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’

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**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

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**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)

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**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.

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**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+

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**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

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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)

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Diffusion (continued)

Diffusion (continued)

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