1. Simulation of Nanoscale Thermal Transport Laurent Pilon UCLA Mechanical and Aerospace Engineering Department Copyright© 2003

2. Motivations Macroscopic laws break down when:Macroscopic laws break down when: –The length of the system is comparable to the mean free path of the heat carrier –The time scale of the physical system is smaller than the relaxation time of the heat carriers Experimentation at submicron scale poses great challengesExperimentation at submicron scale poses great challenges  Numerical Simulations have become critical for basic understanding and design purposes for basic understanding and design purposes

3. Equation for Phonon Radiative TransferEquation for Phonon Radiative Transfer –The transport of phonons is governed by the Boltzmann transport equation (BTE) –The so-called Equation for Phonon Radiative Transfer (EPRT) is an alternative formulation of the BTE Phonon Transport Equation (EPRT) advectionscattering T1T1T1T1 T2T2T2T2Dielectric thin film L ℓ ℓ

4. Method of Characteristics Along the heat carrier pathline: the EPRT simplifies Phonon pathline  Transform a PDE into an ODE solved along the phonon pathlines The temperature is recovered from:

5. + Very accurate + Multidimensional problems + Compatible with other methods (x i+1,y j+1,z k ) x y z (x i+1,y j,z k+1 ) t (x n,y n,z n ) (x i,y j,z k ) (x i+1,y j,z k ) (x i,y j+1,z k+1 ) pathline (x i,y j,z k+1 ) t+  t + Pre-specified grid + Backward in time + Transient and steady state (x i+1,y j+1,z k+1 ) Numerical Method

6. Transient Ballistic Transport T1T1T1T1 T2T2T2T2 Dielectric thin film 1m1m1m1m q z 0 Black surfaces

7. Steady-State Ballistic Transport Discontinuity T 1 =20 K T 2 =10 K Dielectric thin film 1m1m1m1m q z 0 Black surfaces

8. Conduction Across a Diamond Thin-Film Fourier’s law 100 nm 1  m 10  m T 1 =301 K T 2 =300 K Diamond thin film 1m1m1m1m q Black surfaces z 0

9. SiO 2 Rod with Specularly Reflecting Boundaries T1T1 T2T2T2T2 Reflecting Boundaries L 1  m   Fourier’s law (2D problem)