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Predicting Thermal Conductivity from Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering

Predicting Thermal Conductivity from Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering Carnegie Mellon University, Pittsburgh PA, USA 2 nd International Conference on Phononics and Thermal Energy Science Tongji University, Shanghai, China May 26, 2014.

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Predicting Thermal Conductivity from Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering

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  1. Predicting Thermal Conductivity from Molecular Dynamics Simulations Alan McGaughey Department of Mechanical Engineering Carnegie Mellon University, Pittsburgh PA, USA 2nd International Conference on Phononics and Thermal Energy Science Tongji University, Shanghai, China May 26, 2014
  2. Pittsburgh and Carnegie Mellon
  3. Acknowledgements Students Dr. Eric Landry (→ UTRC) Dr. John Thomas (→ JHU APL) Dr. Joe Turney (w/ C. Amon) (→ UTRC) Dr. Dan Sellan (Toronto → UT Austin) Dr. Jason Larkin (→ SpiralGen) Kevin Parrish (CMU Ph.D.) Sam Huberman (Toronto→ MIT) Ryan Iutzi (Waterloo → MIT) Collaborators Cristina Amon (Toronto) Massoud Kaviany (Michigan) Pawel Keblinski (RPI) Funding Department of Energy (U. Michigan) Air Force Office of Scientific Research YIP National Science Foundation (Landry and Thomas fellowships)
  4. Outline Thermal Conductivity MD Simulation Green-Kubo (equilibrium) Direct Method (non-equilibrium) Summary
  5. What is Thermal Conductivity? heat flux = ̶ thermal conductivity × temperature gradient Defined empirically Any thermal conductivity prediction invokes the Fourier law Combines the effects of all energy carriers Provides no information about carrier-level transport
  6. Experimental Thermal Conductivity Data Phillpot and McGaughey, Materials Today,June 2005, 18-20.
  7. Atomic-Level Consideration McGaughey, Ph.D. thesis, U. Michigan (2004).
  8. Size-Dependence in Nanostructures Silicon thin films, in-plane direction Silicon nanowires Jain et al., PRB87 (2013) 195301. Li et al., APL83 (2003) 2934. Superlattices, carbon nanotubes, graphene, nanostructured alloys, …
  9. Why Simulations and Calculations? Can thermal conductivity be tailored by nanostructuring? Experiments can be time- and resource-intensive Nanoscale phenomena are challenging to observe experimentally Small length (10-10m) and time (10-15 s) scales Difficult to measure properties and elucidate physics
  10. What Does the Non-Modeler Want? Thermal conductivity values not available in handbooks or textbooks New bulk materials and nanostructures Extreme conditions (geophysics) Fundamental understanding Why does one material have a higher thermal conductivity than another? How to design materials with tailored thermal properties?
  11. Thermal Transport by Phonons Quantized lattice vibration with energyħω Phonons are the primary carriers of thermal energy in semiconductors and dielectrics (e.g., Si, GaN, graphene, quartz). Specific discussion in tomorrow’s lecture
  12. Atomistic Approaches How are atomic trajectories & dynamics related to thermal conductivity? Phonon → Abstraction Atom → Real
  13. Outline Thermal Conductivity MD Simulation Green-Kubo (equilibrium) Direct Method (non-equilibrium) Summary
  14. What is Molecular Dynamics Simulations? Real-space technique (atoms, positions, velocities) Generate trajectories using Newton’s laws of motion & potential Water flow in a CNT A. McGaughey, CMU Bending of a carbon nanotube J. Li, MIT Assembly of tethered nanoparticles S. Glotzer, UMichigan
  15. MD is Like a Mass-Spring System 1. Equations of Motion: 2. Convert to1storder equations: 3. Solve numerically
  16. Simulation Setup Integration scheme for equations of motion 2nd order typically sufficient (e.g., Verlet) Time step: small enough to resolve atomic vibrations Typically ~ 1 fs → Simulations for O(106) Dt System size: big enough so that behavior is size independent Depends on material and interest (102 -106 atoms) Interatomic potential and cutoff scheme Boundary conditions Periodic, free, fixed Initial conditions Positions from structure, randomize velocities
  17. Movies
  18. Advantages of MD Simulation Access huge systems (>106 atoms) Contains full anharmonicity of atomic interactions High temperatures Model different phases of matter Crystal, alloy, amorphous, fluid Naturally include disorder Potentials now being fit to phonon properties, DFT calculations CNT and graphene: Lindsay & Broido, PRB81, 205441 (2010) Silicon: Esfarjani et al., PRB84, 085204 (2011)
  19. Disadvantages of MD Simulation Newton’s laws are classical -> no quantum statistics Not valid at low temperatures High quality potentials do not exist for many materials Typically fit to structural and mechanical properties Broido et al., PRB72, 014308(2005).
  20. Model System: LJ Argon Computationally fast, so good for methodology development Will consider crystal and liquid phases as a case study. Melting temperature ~ 90 K. http://www.cmbi.ru.nl/redock/
  21. Outline Thermal Conductivity MD Simulation Green-Kubo (equilibrium) Direct Method (non-equilibrium) Summary
  22. Top-Down: Green Kubo Method Fluctuation-Dissipation Theorem Track thermal center of mass, calculate autocorrelation. Numerical Integration Auto-correlation Heat Current
  23. Heat Current For a 2-body potential: For a 3-body potential:
  24. Heat Current Autocorrelation Function Correlation Time Time Origins Specify: Maximum correlation time (> longest lifetime) Total simulation time (>> longest lifetime) HCACF for multiple initial conditions (5-10), average. Ensures good sampling of phase space. Can perform autocorrelation in frequency domain using FFT.
  25. HCACF Averaging LJ Crystal, T = 20 K, 10 seeds
  26. HCACF for LJ Crystal McGaughey and Kaviany, IJHMT47, 1783(2004)
  27. HCACF for LJ Liquid McGaughey and Kaviany, IJHMT47, 1783(2004)
  28. Integration and Averaging Integrate numerically, trapezoidal rule fine due to small time step. LJ Crystal, T = 20 K, 10 seeds
  29. Integration for LJ Crystal and Liquid McGaughey and Kaviany, IJHMT47, 1783(2004)
  30. Specifying the Thermal Conductivity The major challenge. Remove human bias. 1. Identify averaging region by eye. 2. Fit functional form to HCACF. 3. First Dip 4. First Avalanche Track signal to noise in HCACF. Chen et al., Phys. Lett. A. 374 (2010) 2392. Sellanet al., PRB81, 214305(2010)
  31. Green Kubo Size Effects Typically easy to handle, but may need large simulation cells. LJ Argon Tersoff silicon, T = 300 K McGaughey and Kaviany, PRB71, 184305(2004) He et al., PCCP14, 16209(2012).
  32. LJ Argon Thermal Conductivities McGaughey, Ph.D. thesis, U. Michigan (2004).
  33. LJ Superlattices Landry et al., PRB77, 184302(2008)
  34. Silica Structures McGaughey and Kaviany, IJHMT47, 1799(2004)
  35. Specifying the Thermal Conductivity McGaughey and Kaviany, IJHMT47, 1799(2004)
  36. Outline Thermal Conductivity MD Simulation Green-Kubo (equilibrium) Direct Method (non-equilibrium) Summary
  37. Top-Down: Direct Method Fourier Law in a non-equilibrium MD simulation Curve Fitting Predict thermal conductance of interfaces, thin films, … Thomas et al., PRB 81 (2010) 045413.
  38. Setting up the Simulation 1. Fixed or periodic boundaries? 2. Set temperature difference or heat flux? 3. When at steady state? How long to average? 4. Cross section, reservoir size, heat flux (or DT) so that k is converged. 5. Data analysis is straightforward (one seed is typically enough) For CNTs: Salaway and Zhigilei., IJHMT 70(2014) 954.
  39. Size Effects: The Main Challenge 1. Increase length until k converges (does not work for most materials). 2. Use size-dependent data to plot 1/k vs. l/L, fit line. Dangerous! LJ Crystal SW Silicon Sellanet al., PRB81, 214305(2010)
  40. 1/k vs. l/L may not be linear! Use phonon properties from lattice dynamics calculations to test. LJ Crystal SW Silicon Sellanet al., PRB81, 214305(2010)
  41. Need Large Systems to Extrapolate May not be accessible in MD. Potential problems with large aspect ratio systemsHu et al., JAP110 (2011) 113511 Sellanet al., PRB81, 214305(2010)
  42. Outline Thermal Conductivity MD Simulation Green-Kubo (equilibrium) Direct Method (non-equilibrium) Summary
  43. Green-Kubo vs. Direct Method
  44. Comparing Green-Kubo and Direct Method LJ Crystal LJ Superlattices Turney et al., PRB79, 075316(2009) Landry et al., PRB77, 184302(2008)
  45. Recommendations LAMMPS has built in capabilities for both Green-Kubo and direct methods lammps.sandia.gov Don’t use quantum corrections! Turney et al. PRB79 (2009) 224305 Use Green-Kubo if possible First avalanche method for specifying integral[Chen et al., Phys. Lett. A. 374 (2010) 2392] Direct method has subtle size effects Sellan et al., PRB81, 214305 (2010)
  46. Suggested Reading A. J. H. McGaughey and M. Kaviany, "Phonon transport in molecular dynamics simulations: Formulation and thermal conductivity prediction."  Advances in Heat Transfer, Volume 39, 169-255 (Elsevier, 2006). D. P. Sellan, E. S. Landry, J. E. Turney, A. J. H. McGaughey, and C. H. Amon, "Size effects in molecular dynamics thermal conductivity prediction." Physical Review B81 (2010) 214305. P. C. Howell, “Thermal Conductivity Calculation with the Molecular Dynamics Direct Method I: More Robust Simulations of Solid Materials.” Journal of Computational and Theoretical Nanoscience8 (2011) 2129-2143.
  47. Tomorrow: Phonon Properties Alan McGaughey Department of Mechanical EngineeringCarnegie Mellon University, Pittsburgh PA mcgaughey@cmu.edu ntpl.me.cmu.edu
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