ME 381R Lecture 8

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ME 381R Lecture 8 9 Boltzmann Transport
Equation Thermal Conductivity Model
Dr. Li Shi Department of Mechanical Engineering
The University of Texas at Austin Austin, TX
78712 www.me.utexas.edu/lishi lishi_at_mail.utexas.
edu
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Drawbacks of Kinetic Theory

• Assumes single particle velocity and single mean
  free
• path or mean free time.
• Breaks down when, vg(w) or t(w)
• Assumes local thermodynamics equilibrium f
  f(T)
• Breaks down when L ? ? t ? t
• Cannot handle non-equilibrium problems
• Short pulse laser interactions
• High electric field transport in devices
• Cannot handle wave effects
• Interference, diffraction, tunneling

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Boltzmann Transport Equation for Particle
Transport
Distribution Function of Particles f
f(r,p,t) --probability of particle occupation of
momentum p at location r and time t
Equilibrium Distribution f0, i.e. Fermi-Dirac
for electrons, Bose-Einstein for phonons, Plank
for photons, etc.
Non-equilibrium, e.g. in a high electric field or
temperature gradient
Relaxation Time Approximation
t
Relaxation time
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Energy Flux
q
v
Energy flux in terms of particle flux carrying
energy
dk
q
k
f
Vector
Integrate over all the solid angle
Scalar
Integrate over energy instead of momentum
Density of States of phonon modes between e
and e de
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Quasi-equilibrium Condition
BTE Solution
Quasi-equilibrium
Direction x is chosen to in the direction of q
Energy Flux
Fourier Law of Heat Conduction
t(e) can be treated using Callaway (Phys. Rev.
113, 1046) or Holland model (Phys. Rev., 134,
A471-A480)
If v and t are independent of particle energy, e,
then ?
Kinetic theory