Microscale and Nanoscale Heat Transfer

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1 Sebastian Volz (Ed.) Microscale and Nanoscale Heat Transfer With 144 Figures and 7 Tables In Collaboration with Remi Carminati, Patrice Chantrenne, Stefan Dilhaire, Severine Gomez, Nathalie Trannoy, and Gilles Tessier ^Spri ringer

2 Contents Laws of Macroscopic Heat Transfer and Their Limits Jean-Jacques Greifet 1 1 Heat Conduction in Solids Macroscopic Approach Characteristic Length and Time Scales Short-Scale Transfer 5 2 Conduction in Fluids. Convection Macroscopic Approach Short-Scale Transfer. Ballistic Transport 7 3 Radiation Macroscopic Approach Characteristic Length and Time Scales 9 4 Conclusion 12 References 12 Index 12 Transport in Dilute Media Remi Carminati 15 1 Distribution Function and Flux Distribution Function Averages Conductive Flux 17 2 Thermodynamic Equilibrium Definition Equilibrium Distribution Function 18 3 Boltzmann Equation Dynamical Equation for the Distribution Function The Relaxation Time Model 20 4 Local Thermodynamic Equilibrium. Perturbation Method Dimensionless Boltzmann Equation Mean Free Path. Collision Time. Knudsen Number Local Thermodynamic Equilibrium Perturbation Method. Linear Response Fourier Law and Thermal Conductivity 24

3 X Contents 5 Example of a Non-LTE System. Short-Scale Conduction in a Gas Can One Speak of Temperature on Short Scales? Calculating the Conductive Flux in the Ballistic Regime Transitions Between Regimes 28 6 Conclusion 30 A Equilibrium Distribution Function 31 B Dynamical Evolution of the Distribution Function C for Free Particles 32 Calculating the Constants A and B for the Flux in the Ballistic Regime 33 References 34 Index 35 Electrons and Phonons Jean-Jacques Greffet 37 1 Electrons Free Electrons Electrons in a Periodic Potential Electrical Conduction Semi-Classical Approach Electrical Conductivity in the Collisional Regime Electrical Conduction in the Ballistic Regime 46 2 Phonons Vibrational Modes in a Lattice Phonon Energy Density of States. Optical and Acoustic Modes Calculating the Heat Flux Calculating the Thermal Conductivity 52 References 53 Index 53 Introductiori to Radiative Transfer Remi Carminati 55 1 Radiative Transfer Equation Specific Intensity, Flux, Energy Density Absorption, Scattering and Thermal Emission Establishing the RTE. Radiative Energy Balance Discussion 60 2 From the RTE to the Diffusion Approximation From the P\ Approximation to the Diffusion Equation Discussion Rosseland Approximation 65 3 Transport Regimes Static Transmission. Ohmic Conductance and Short-Scale Deviations 66

4 Contents XI 3.2 Transitions Between Regimes in the Dynamic Case Ballistic and Multiple Scattering Components in the RTE Electromagnetic Approach to Thermal Emission Intuitive View of the Thermal Emission Mechanism Principle Underlying the Calculation of Thermal Emission. Fluctuation-Dissipation Theorem 70 References 74 Index 75 Solution of the Boltzmann Equation for Phonon Transport Denis Lemonnier 77 1 Introduction 77 2 Theoretical Model Intensity. Internal Energy. Flux Transfer Equation Diffusive Regime 82 3 The Fi Method General Idea Boundary Conditions Numerical Solution Advantages and Disadvantages 88 4 Discrete Ordinate Method General Idea Choice of Quadratures Integrating the RTE over a Control Volume Integrating over a Control Volume Advantages and Disadvantages 100 References 105 Index 106 Radiative Transfer on Short Length Scales Karl Joulain Review of Electromagnetism Maxwell's Equations and Constitutive Relations Plane Wave Expansion Energy Conservation, Poynting Vector, and Energy Density Potentials Dipole Radiation Calculating Radiative Transfer on Short Length Scales Thermal Emission from a Nanoparticle Radiative Power Exchanged Between Two Spherical Nanoparticles Thermal Near-Field Emission from a Plane Surface Near-Field Radiative Transfer Between Two Planes 126

5 XII Contents 5 Conclusion 129 References 129 Index 130 Monte Carlo Method Sebastian Volz Introduction Aims Heat Flux and Energy Carriers Calculating the Heat Flux with the Monte Carlo Method Basic Idea Sampling Random Walks Calculating the Statistical Error and Average Ballistic and Quasi-Ballistic Transport in Gases Molecules and Heat Flux Random Walk Distributions Collision Distributions Transfer Between a Hot Tip and a Surface Ballistic and Quasi-Ballistic Transport in Insulating Crystals Phonons, Temperature and Heat Flux Isothermal Cell Technique Modelling Random Walks Conduction in a Thin Film Conclusion 152 References 152 Index 153 Molecular Dynamics Patrice Chantrenne Principles of Molecular Dynamics Definitions and Notation Integrating Newton's Equation Interaction Potentials Implementing the Simulation Energy Distribution Thermal Conductivity Calculation Equilibrium Molecular Dynamics Non-Homogeneous Non-Equilibrium Molecular Dynamics Homogeneous Non-Equilibrium Molecular Dynamics Determining Vibrational Properties Heat Transfer by Phonons Determining Vibrational Properties 175 References 177 Index 179

6 Contents XIII Scanning Thermal Microscopy Bernard Cretin, Severine Gomes, Nathalie Trannoy, Pascal Vairac Introduction to Near-Field Microscopy Basic Principles of Near-Field Microscopy Historical Perspective: From Conventional Microscopy to Near-Field Microscopy Scanning Probe Microscopes Development of Scanning Thermal Microscopy Near-Field Microscopy and Heat Transfer Thermal Probes SThM with the Micrometric Thermoresistive Wire Probe Goals Method Thermal Image Contrast Controlling and Optimising SThM Functions Analysing Measurements in Constant Temperature Mode Analysing Measurements in Constant Current Mode Conclusion 230 References 230 Index 236 Optical Techniques for Local Measurement Stefan Dilhaire, Daniele Fournier, Gilles Tessier Generating Thermal and Thermoelastic Waves Generating Waves by Thermoelectric Effects Optical Generation Detecting Thermal and Thermoelastic Waves Reflectometry Interferometric Probes Applications Temperature and Displacement Fields. Orders of Magnitude Locating Hot Spots and Mapping Temperature Measuring Thermophysical Properties 278 References 281 Index 284 Hybrid Techniques and Multipurpose Microscopes Bernard Cretin, Pascal Vairac Physics of Microscopes Combining Thermal and Thermoelastic Effects Microscopes and Their Resolutions D Model with Cylindrical Symmetry Combined Photothermoelastic Microscopy 295

7 XIV Contents 3.1 Microscopes Based on a Thermoelectric Probe Microscopes Based on Detection of Expansion Prospects 301 References 303 Index 306 Energy Exchange at Short Time Scales: Electron-Phonon Interactions in Metals and Metallic Nanostructures Fabrice Vallee Introduction Electronic and Vibrational Structures in Metallic Systems Electronic Structure of Noble Metals Lattice Vibrations Optical Properties of Metals Optical Response at Equilibrium Femtosecond Pump-Probe Method Electron-Lattice Interactions. Energy Exchange Kinetic Model. Boltzmann Equation Electron-Phonon Interaction. Bulk Metals Energy Exchange in the Thermal Regime. Two-Temperature Model Electron-Lattice Interactions in Metallic Nanoparticles Acoustic Vibrational Modes of Nanospheres Vibrational Modes Time-Resolved Studies 328 References 330 Index 331 Investigation of Short-Time Heat Transfer Effects by an Optical Pump Probe Method Bernard Perrin Acoustic and Thermal Generation by Ultrashort Laser Pulse Acoustic Generation in the Absence of Heat Diffusion Taking Heat Diffusion into Account Optical Detection of Thermal and Acoustic Transients Experimental Setups Interferometric Detection Cumulative Effects Due to the Pump Pulse Train Conclusion 356 References 356 Index 358 Index 361

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