Non-Continuum Energy Transfer: Boltzmann Transport Equation

1. Non-Continuum Energy Transfer: Boltzmann Transport Equation

2. Phonons – What We’ve Learned • Phonons are quantized lattice vibrations • store and transport thermal energy • primary energy carriers in insulators and semi-conductors (computers!) • Phonons are characterized by their • energy • wavelength (wave vector) • polarization (direction) • branch (optical/acoustic)  acoustic phonons are the primary thermal energy carriers • Phonons have a statistical occupation (Bose-Einstein), quantized (discrete) energy, and only limited numbers at each energy level • we can derive the specific heat! • We can treat phonons as particles and therefore determine the thermal conductivity based on kinetic theory

3. Electrons – What We’ve Learned • Electrons are particles with quantized energy states • store and transport thermal and electrical energy • primary energy carriers in metals • usually approximate their behavior using the Free Electron Model • energy • wavelength (wave vector) • Electrons have a statistical occupation (Fermi-Dirac), quantized (discrete) energy, and only limited numbers at each energy level (density of states) • we can derive the specific heat! • We can treat electrons as particles and therefore determine the thermal conductivity based on kinetic theory • Wiedemann Franz relates thermal conductivity to electrical conductivity • In real materials, the free electron model is limited because it does not account for interactions with the lattice • energy band is not continuous • the filling of energy bands and band gaps determine whether a material is a conductor, insulator, or semi-conductor

4. Gases – What We’ve Learned • Gases can be treated as individual particles • store and transport thermal energy • primary energy carriers fluids  convection! • Gases have a statistical (Maxwell-Boltzmann) occupation, quantized (discrete) energy, and only limited numbers at each energy level • we can derive the specific heat, and many other gas properties using an equilibrium approach • We can use non-equilibrium kinetic theory to determine the thermal conductivity, viscosity, and diffusivity of gases • The tables in the back of the book come from somewhere!

5. BTE – Particle Approach • Phonons and electrons (and photons) possess wave-like characteristics • to track waves we need to know amplitude, phase, direction  very difficult! • We already treat gases as particles and we also like to treat phonons, electrons, and photons as particles as well  we’ve already applied kinetic theory to derive thermal conductivity! • can’t capture phase coherence effects (interference, diffraction, etc.) • can capture propagation, reflection, transmission, etc.

6. BTE – Transport Modeling • To understand energy transfer, we must be able to model the transport  scaling determines proper modeling approach • based on physical dimensions in space and time • compared to basic transport properties (wavelength, mean free path, mean free time, collision time usually ~ps-fs) • For very small time and length scales (on the order of a wavelength and collision time)  quantum approaches must be used & wave behavior is significant • Green’s Functions, molecular dynamics, density functional theory • For large time and length scales (greater than mean free path/time)  The macroscopic, continuum equations apply • as the time scale increases  time-averaged equilibrium • as the length scale increases  space-average equilibrium • The Boltzmann Transport Equation is a general transport equation that allows for non-equilibrium transport (on the order of the mean free path/time) • derived for gases but applicable to any particle systems (phonons, electrons and photons)

7. BTE – Equilibrium Distributions for a Gas At equilibrium, we can use Maxwell-Boltzmann statistics to determine the gas distribution for the relevant properties velocity (Gaussian) speed (Maxwellian) energy (Maxwellian)

8. BTE – Boltzmann Transport Equation • Consider a packet of particles with a distribution fin time t, space r and momentum p • typically we consider equilibrium distributions (Maxwell-Boltzmann, Bose-Einstein, Fermi-Dirac) but here we are considering how the distribution changesin time time, t Look at the terms recall

9. BTE – Boltzmann Transport Equation Plugging in and expanding, we can show Or in general mathematical terms … • Notes: • f here is a scalar (a distribution) and this equation describes the transport and time evolution of the distribution • we can multiply this equation (called taking moments) by other quantities to determine the transport of those quantities • for example – multiply by momentum to determine the transport of momentum (Navier-Stokes!) • the F term is an external force (such as an electric field for electrons!) What about collisions!

10. BTE - Collisions this implies that a small non-equilibrium distribution f will relax back to the equilibrium distribution f0 due to collisions in time τ • Thus far we’ve only described the transport of the distribution  collisions between particles will modify the distribution • in-scattering - a collision that increases the distribution of particles with momentum p from momentum p’ • out-scattering - a collision that increases the distribution of particles with momentum p’ from momentum p • Collisions act as sources and sinks of the distribution function! • Relaxation time approximation

11. BTE – Control Volume Perspective collision (disappear) x-direction in velocity space, vx dvx dx collision (appear) x-direction in physical space, x

12. BTE – Boltzmann Transport Equation The general Boltzmann Transport Equation • Notes: • f here is a scalar (a distribution) and this equation describes the transport of the distribution • we can multiply this equation (called taking moments) by other quantities to determine the transport of those quantities • for example – multiply by momentum to determine the transport of momentum (Navier-Stokes!) • the F term is an external force (such as an electric field for electrons!) • the BTE is a PDE in 7 dimensions (3 space, 3 momentum, and 1 space) and the collision term is often treated as an integral so the BTE is a 7-dimensional integro-differential equation!

13. BTE – Using the Distribution Function The BTE describes the transport of the statistical distribution function in time, space, and momentum We seek to determine macroscopic quantities (thermal conductivity, viscosity) and macroscopic conservation equations (mass, momentum, energy) from this distribution We can determine properties through simplifications of the general BTE and compare to other approaches We can derive macroscopic conservation equations by taking moments of the BTE

14. BTE – Thermal Conductivity Consider heat conduction, there is no external force … per se Using the relaxation time approximation, the BTE simplifies to Simplify further by assuming steady, 1-D transport Now assume that the distribution is near equilibrium such that called local equilibrium We now have an equation for the distribution function f in local equilibrium!

15. BTE – Thermal Conductivity Consider a general description of the heat flux  the flow of internal energy recall density of states energy occupation (distribution function) also recall that energy and momentum are related: But for steady heat conduction, we have a form of the distribution function Plugging into the general description of heat flux

16. BTE – Thermal Conductivity We can split this integral into independent expressions Consider each term separately f0 is the equilibrium distribution and at equilibrium an equal amount of heat flows left as does right  no net heat flow Fourier’s Law Thus if τ and v are constant

17. BTE – Conservation Equations We can derive macroscopic conservation equations by taking moments of the BTE multiplying by a scalar quantity (such as energy) and integrating over all states consider a general scalar quantity Φ where This can be written as 2 1 3 4

18. BTE – Conservation Equations We define the local average as Consider prior expression term-by-term (v is the bulk velocity) 1 2 3 4

19. BTE – Mass Conservation Equation We now have the following general scalar conservation equation Consider mass conservation whereΦ = mand ρ = nm plugging in … no external force affecting distribution of mass simplifying when reactions are not considered, these “source” & “sink” terms cancel out this is our traditional mass conservation equation!