# Lecture 4 2006.

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Lecture 4 2006

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

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

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

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

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.

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

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

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

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

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.

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’

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.

f = 1 - 2/z

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.

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.

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

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

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

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

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

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,

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

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

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

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.

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.

Field x charge

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

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

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

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)

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)

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:

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)

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

Solution for a thin finite source

Solution for a thin finite source

Solution for a thin finite source

Solution for a thin finite source

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

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

+ - 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

Comparing conductivity to tracer diffusion

Comparing conductivity to tracer diffusion
Correlation factor

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

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.

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

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

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

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

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’

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’

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

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)

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.

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+

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

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)