Transport Zuoan Li NorFERM-2008

Outline Diffusion basis ▲ Diffusion mechanism ▲ Mathematics of diffusion ▲ D vs. T Diffusion in electric gradient (conductivity) Transport in electrochemical gradient

Diffusion Cu-Ni diffusion couple Before heat treatment After

Diffusion Mechanism Vacancy diffusion

Interstitial diffusion

Interstitialcy diffusion Collinear jump

Free transport mechanism (Grotthuss mechanism) Proton diffusion Vehicle mechanism Free transport mechanism (Grotthuss mechanism)

BaCeO3 Kreur, Annu. Rev. Mater. Res. 2003

Mathematics of diffusion J: flux of particles across plane with area A Fick’s 1st law c x A J: flux of particles across plane with area A

Fick’s 2nd law

Solution to Fick’s 2nd law Thin film Thin layer of radioactive isotopes is located at x=0 of a semi-infinite sample (self-exhausting source). Boundary conditions: Concentration after time t:

Deposition Annealing Cutting

Error function The concentration of tracer at x=0 is kept constant after diffusion (non-exhausting source). Boundary conditions: Concentration after time t:

O18 depth profile

Activation energy of diffusion coefficient Random diffusion: frequency of successful jumps : probability that jump can overcome energy barrier : probability of site being ready for jump

DHm

: probability of site for jump Vacancy Interstitial Defects Interstitial solute Constituents T, defect structure, and pO2

ci Di Defects Constituents small & variable large & constant Constituents

Constant or frozen vacancy Vacancy diffusion in elemental solids or Constant or frozen vacancy

Predominant oxygen vacancy (MaOb-d) Constant oxygen vacancy Vacancy diffusion in oxides Predominant oxygen vacancy (MaOb-d) Constant oxygen vacancy

Jump depends on oxygen-oxygen distance Protons Rotation is easy Jump depends on oxygen-oxygen distance Large soft lattices

H/D isotope effects Classical effects: Non-classical effects:

Electrical potential gradient Transport Electrical potential gradient

Nernst-Einstein relationship Activation energy

Conductivity Solid ionic conductor

Electronic conductivity Intrinsic semiconductor Intrinsic ionization

Extrinsic semiconductor (n-type) Low temperature Log n 1/T, K Intermediate temperature High temperature intrinsic

Non-stoichiometric semiconductor Defect equilibrium, T, P

un & up Magnitude Non-polar solids Polar oxides

Electrochemical potential gradient Transport Electrochemical potential gradient

Voltage over a sample Neutral form itot= 0 → transport number (EMF) itot≠ 0 → fuel cell

Flux of a specific species Steady state

Some diffusion terms Self diffusion Tracer diffusion Defect diffusion Chemical diffusion Ambipolar diffusion

Nonstoichiometric (defects) Defect Chemistry Summary Intrinsic (n=p) Extrinsic (n≠p) Nonstoichiometric (defects)

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