Phonon hydrodynamics for nanoscale heat transport at ordinary temperatures

The classical Fourier's law fails in extremely small and ultrafast heat conduction even at ordinary temperatures due to strong thermodynamic nonequilibrium effects. In this work, a macroscopic phonon hydrodynamic equation beyond Fourier's law with a relaxation term and nonlocal terms is derived through a perturbation expansion to the phonon Boltzmann equation around a four-moment nonequilibrium solution. The temperature jump and heat flux tangential retardant boundary conditions are developed based on the Maxwell model of the phonon-boundary interaction. Extensive steady-state and transient nanoscale heat transport cases are modeled by the phonon hydrodynamic model, which produces quantitative predictions in good agreement with available phonon Boltzmann equation solutions and experimental results. The phonon hydrodynamic model provides a simple and elegant mathematical description of non-Fourier heat conduction with a clear and intuitive physical picture. The present work will promote deeper understanding and macroscopic modeling of heat transport in extreme states.

Figure 1

Derivation of nonequilibrium boundary conditions for the phonon hydrodynamic equation: (a) heat flux tangential retardant (HFTR) boundary condition; (b) temperature jump boundary condition. The solid lines represent the heat flux and temperature profiles. $f^{-}$ and $f^{+}$ represent the distribution functions of incident phonons and reflecting phonons, respectively.

Figure 2

Schematic of phonon transport through a thin film with a thickness $d$: (a) in-plane transport; (b) cross-plane transport. $T_h$ and $T_c$ are the temperatures of hot and cold sources, respectively.

Figure 3

The dimensionless cross-sectional heat flux distributions for in-plane phonon transport through thin film at different spatial Knudsen numbers: (a) $\mathrm{K}{\mathrm{n}}_l=0.05$, (b) $\mathrm{K}{\mathrm{n}}_l=0.1$, (c) $\mathrm{K}{\mathrm{n}}_l=0.2$, (d) $\mathrm{K}{\mathrm{n}}_l=0.3$. The solid lines represent the analytical solution Eq. (64) based on the present phonon hydrodynamic model, whereas the symbols represent the analytical solution Eq. (59) (Fuchs-Sondheimer model) based on the phonon Boltzmann transport equation (BTE). Different specularity parameters are considered for the lateral surface of the thin film: $s=1$ (circle), $s=0.9$ (plus), $s=0.7$ (diamond), $s=0.4$ (triangle), $s=0$ (square). The spatial Knudsen number is defined as the ratio of phonon mean-free path to the thin-film thickness: $\mathrm{K}{\mathrm{n}}_l\equiv \mathrm{\Lambda }/d$.

Figure 4

The dimensionless effective in-plane thermal conductivity of the thin film. (a) Comparison to the theoretical solution. The solid lines represent the analytical solution Eq. (65) based on the present phonon hydrodynamic model, whereas the symbols represent the analytical solution Eq. (61) (Fuchs-Sondheimer model) based on the phonon Boltzmann transport equation (BTE). Different specularity parameters are considered for the lateral surface of the thin film: $s=1$ (circle), $s=0.9$ (plus), $s=0.7$ (diamond), $s=0.4$ (triangle), $s=0$ (square). The spatial Knudsen number is defined as the ratio of phonon mean-free path to the thin-film thickness: $\mathrm{K}{\mathrm{n}}_l\equiv \mathrm{\Lambda }/d$. (b) Comparison to experimental results of thin films with a diffuse lateral surface ($s=0$) at room temperature ($T=300$ K). The symbols with error bars are experimental data from the literature [16, 73, 74, 75, 76], whereas the dashed line and solid line represent respectively the solutions of the gray phonon BTE [16] and the present phonon hydrodynamic model with a median-thermal-conductivity phonon mean-free path ${\mathrm{\Lambda }}_m=441$ nm [72, 77]. The solid line with the cross symbol denotes the solution of the spectral phonon BTE [16].

Figure 5

Schematic of nonequilibrium effects and hydrodynamic modeling in the micro/nanoscale gas flow and phonon flow in the slip regime. (a) The gas flow. The light blue dashed line and purple dashed line represent the solutions of the Navier-Stokes equation (N-S Eq.) with the nonslip velocity boundary condition and velocity slip boundary condition, respectively. The red solid line denotes the real velocity profile. (b) The phonon flow. The light blue dashed line and purple dashed line represent the solutions of Fourier's law and the present phonon hydrodynamic equation (PH Eq.) with the heat flux tangential retardant (HFTR) boundary condition, respectively. The red solid line denotes the real heat flux profile. The Knudsen layer near the lateral boundary has a thickness of $1\sim 2$ mean-free paths ($\mathrm{\Lambda }$).

Figure 6

The dimensionless temperature distributions in the cross-plane phonon transport through the thin film at different spatial Knudsen numbers: (a) $\mathrm{K}{\mathrm{n}}_l=0.05$, (b) $\mathrm{K}{\mathrm{n}}_l=0.1$, (c) $\mathrm{K}{\mathrm{n}}_l=0.2$, (d) $\mathrm{K}{\mathrm{n}}_l=0.3$. The solid lines represent the analytical solution Eq. (78) based on the present phonon hydrodynamic model, whereas the symbols represent the numerical solution of Eq. (71) based on the phonon Boltzmann transport equation (BTE). Different specularity parameters are considered for the surface of the isothermal heat sources: $s=0$ (circle), $s=0.3$ (diamond), $s=0.6$ (square), $s=0.9$ (triangle). The spatial Knudsen number is defined as the ratio of phonon mean-free path to the thin-film thickness: $\mathrm{K}{\mathrm{n}}_l\equiv \mathrm{\Lambda }/d$.

Figure 7

The dimensionless effective cross-plane thermal conductivity of the thin film. The solid lines represent the analytical solution Eq. (80) based on the present phonon hydrodynamic model, whereas the symbols represent the numerical solution Eq. (75) based on the phonon Boltzmann transport equation (BTE). Different specularity parameters are considered for the surface of isothermal heat sources: $s=0$ (circle), $s=0.3$ (diamond), $s=0.6$ (square), $s=0.9$ (triangle). The spatial Knudsen number is defined as the ratio of the phonon mean-free path to the thin-film thickness: $\mathrm{K}{\mathrm{n}}_l\equiv \mathrm{\Lambda }/d$.

Figure 8

Schematic of phonon transport through a nanowire with radius $R$.

Figure 9

The dimensionless cross-sectional heat flux distributions for phonon transport through a nanowire at different spatial Knudsen numbers: (a) $\mathrm{K}{\mathrm{n}}_l=0.05$, (b) $\mathrm{K}{\mathrm{n}}_l=0.1$, (c) $\mathrm{K}{\mathrm{n}}_l=0.2$, (d) $\mathrm{K}{\mathrm{n}}_l=0.3$. The solid lines represent the analytical solution Eq. (85) based on the present phonon hydrodynamic model, whereas the symbols represent the analytical solution Eq. (81) based on the phonon Boltzmann transport equation (BTE). Different specularity parameters are considered for the lateral surface of the nanowire: $s=1$ (circle), $s=0.9$ (plus), $s=0.7$ (diamond), $s=0.4$ (triangle), $s=0$ (square). The spatial Knudsen number is defined as the ratio of the phonon mean-free path to the nanowire radius: $\mathrm{K}{\mathrm{n}}_l\equiv \mathrm{\Lambda }/R$.

Figure 10

The dimensionless effective thermal conductivity of the nanowire. The solid lines represent the analytical solution Eq. (86) based on the present phonon hydrodynamic model, whereas the symbols represent the analytical solution Eq. (82) based on the phonon Boltzmann transport equation (BTE). Different specularity parameters are considered for the lateral surface of the nanowire: $s=1$ (circle), $s=0.9$ (plus), $s=0.7$ (diamond), $s=0.4$ (triangle), $s=0$ (square). The spatial Knudsen number is defined as the ratio of the phonon mean-free path to the nanowire radius: $\mathrm{K}{\mathrm{n}}_l\equiv \mathrm{\Lambda }/R$.

Figure 11

Schematic of transient nanoscale heat transport: (a) 1D transient phonon transport across the thin film; (b) high-frequency periodic heating of a semi-infinite surface; (c) heat conduction in the transient thermal grating (TTG) experiment.

Figure 12

The temporal evolutions of the dimensionless temperature distribution for the 1D transient phonon transport across the thin film at different spatial Knudsen numbers: (a) $\mathrm{K}{\mathrm{n}}_l=0.01$, (b) $\mathrm{K}{\mathrm{n}}_l=0.1$, (c) $\mathrm{K}{\mathrm{n}}_l=0.2$, (d) $\mathrm{K}{\mathrm{n}}_l=0.3$. The solid lines and dashed-dotted lines represent the analytical solutions based on the present phonon hydrodynamic model and C-V law, respectively, whereas the symbols represent the analytical solution based on Fourier's law [79] in (a) or the energy-based deviational Monte Carlo solution of the phonon Boltzmann transport equation (BTE) in (b)–(d). The dashed lines represent the analytical solution based on Fourier's law [79] in (b)–(d). The temporal Knudsen number is defined as the ratio of phonon relaxation time to the characteristic time (inverse of the dimensionless time), $\mathrm{K}{\mathrm{n}}_t={\tau }_R/t$, whereas the spatial Knudsen number is defined as the ratio of the phonon mean-free path to the thin-film thickness, $\mathrm{K}{\mathrm{n}}_l=\mathrm{\Lambda }/d$.

Figure 13

Temporal evolutions of dimensionless heat flux distribution for the 1D transient phonon transport across the thin film at different spatial Knudsen numbers: (a) $\mathrm{K}{\mathrm{n}}_l=0.01$, (b) $\mathrm{K}{\mathrm{n}}_l=0.1$, (c) $\mathrm{K}{\mathrm{n}}_l=0.2$, (d) $\mathrm{K}{\mathrm{n}}_l=0.3$. The solid lines and dashed-dotted lines denote the analytical solution based on the present phonon hydrodynamic model and C-V law, respectively, whereas the symbols represent the analytical solution based on Fourier's law [79] in (a) or the energy-based deviational Monte Carlo solution of the phonon Boltzmann transport equation (BTE) in (b)–(d). The dashed lines represent the analytical solution based on Fourier's law [79] in (b)–(d). The temporal Knudsen number is defined as the ratio of the phonon relaxation time to the characteristic time (inverse of the dimensionless time), $\mathrm{K}{\mathrm{n}}_t={\tau }_{\mathrm{R}}/t$, whereas the spatial Knudsen number is defined as the ratio of the phonon mean-free path to the thin-film thickness, $\mathrm{K}{\mathrm{n}}_l=\mathrm{\Lambda }/d$.

Figure 14

Spatial distribution of the amplitude of the dimensionless temperature oscillation for the high-frequency periodic heating of the semi-infinite surface at different frequencies. Circles represent the phonon Boltzmann transport equation (BTE) solution from Eq. (91), whereas the solid lines and dashed lines represent the analytical solution of the phonon hydrodynamic model and Fourier's law, respectively. Five different frequencies are considered: $\mathrm{K}{\mathrm{n}}_t=10^{-4}, \mathrm{K}{\mathrm{n}}_t=10^{-3}, \mathrm{K}{\mathrm{n}}_t=10^{-2}, \mathrm{K}{\mathrm{n}}_t=0.1, \mathrm{K}{\mathrm{n}}_t=0.3$. The temporal Knudsen number is defined as the ratio of the phonon relaxation time to the period of external cosine temperature oscillation (inverse of its frequency): $\mathrm{K}{\mathrm{n}}_t={\omega }_H{\tau }_{\mathrm{R}}$.

Figure 15

Heating-frequency dependence of the amplitude of the dimensionless surface temperature oscillation for the high-frequency periodic heating of the semi-infinite surface. The squares represent the phonon Boltzmann transport equation (BTE) solution from Eq. (91), whereas the solid line and dashed line represent the analytical solution of the phonon hydrodynamic model and Fourier's law, respectively. The temporal Knudsen number is defined as the ratio of the phonon relaxation time to the period of external cosine temperature oscillation (inverse of its frequency): $\mathrm{K}{\mathrm{n}}_t={\omega }_H{\tau }_{\mathrm{R}}$.

Figure 16

Heating-frequency dependence of the amplitude of the dimensionless surface heat flux oscillation for the high-frequency periodic heating of the semi-infinite surface. The squares represent the phonon Boltzmann transport equation (BTE) solution from Eq. (92), whereas the solid line and dashed line represent the analytical solution of the phonon hydrodynamic model and Fourier's law, respectively. The temporal Knudsen number is defined as the ratio of the phonon relaxation time to the period of external cosine temperature oscillation (inverse of its frequency): $\mathrm{K}{\mathrm{n}}_t={\omega }_H{\tau }_{\mathrm{R}}$.

Figure 17

Nondimensional effective in-plane thermal conductivity of the silicon thin film in the transient thermal grating (TTG) experiment at room temperature ($T=300$ K). (a) and (b) Comparison to theoretical solutions. The symbols represent the solutions of the phonon Boltzmann transport equation (BTE) [84] whereas the solid line and dashed line denote the analytical solution of the phonon hydrodynamic model and Fourier's law, respectively. The spatial Knudsen number and inverse nondimensional grating period are defined respectively as $\mathrm{K}{\mathrm{n}}_l=\mathrm{\Lambda }/d$ and $\eta =2\pi \mathrm{\Lambda }/L$. (c) Comparison to experimental results. The symbols represent the experimental data of two thin films with thickness 400 nm from the literature [72] whereas the solid line represents the results of the present phonon hydrodynamic model with a median-thermal-conductivity phonon MFP ${\mathrm{\Lambda }}_m=441$ nm [72, 77]. The dashed line and solid line with the cross symbol denote the gray and spectral Monte Carlo (MC) solutions of the phonon BTE [84], respectively.