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ABSTRACT
Phonon mean free path (MFP) spectra are essential for the accurate prediction and utilization of the classical size effect. Rebuilding an MFP spectrum from experimental data remains challenging. It requires solving the thermal transport phenomenon of a heat source of a given shape across the entire size range. Herein, to do this for a heat source embedded in an infinite medium, we derive a new set of modified ballistic–diffusive equations by analyzing the cause of the erroneous results observed in a steady-state solution of the original ballistic-diffusive equations. We demonstrate their ease and accuracy by obtaining the effective thermal conductivity for a spherical nanoparticle embedded in an infinite medium in an explicit closed-form and comparing it with that obtained by the Boltzmann transport equation (differences estimated as <3%).
Acknowledgments
This research was supported by the Nano-material Technology Development Program (No.2011–0030146) and Basic Science Research Program (NRF-2018R1A2B2002837) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology.
Nomenclature
| C | = | volumetric specific heat (J/m3∙K) |
| D | = | characteristic length of a heater, density of states |
| En | = | nth-order exponential integral function |
| f | = | distribution function |
| ħ | = | reduced Planck constant |
| j | = | polarization index |
| k | = | thermal conductivity (W/m∙K) |
| n | = | the unit vector normal to the boundary |
| q | = | heat flux (W/m2) |
| = | nondimensional heat flux | |
| q | = | heat flux vector (W/m2) |
| r | = | coordinate in r direction |
| = | nondimensional coordinate in r direction | |
| r | = | position vector |
| s | = | distance along the propagation direction |
| = | nondimensional distance along the propagation direction | |
| S | = | suppression function |
| t | = | time |
| T | = | temperature (K) |
| u | = | specific internal energy (J/m3) |
| v | = | group velocity (m/s) |
| v | = | group velocity vector (m/s) |
| x | = | coordinate in x direction |
Greek symbols
| χ | = | a parameter defined as Λ/D |
| ϕ | = | mean free path spectrum of a bulk medium |
| Λ | = | mean free path |
| μ | = | cosine of an angle θ |
| θ | = | angle |
| τ | = | relaxation time |
| τ1 | = | size parameter |
| ω | = | angular frequency |
| Ω | = | solid angle |
| = | direction vector |
Subscripts
| 0 | = | uniform equilibrium value at steady state |
| b | = | ballistic |
| bm | = | ballistic component for modified BDEs |
| e | = | emitting |
| eff | = | effective value |
| i | = | incident |
| m | = | diffusive |
| max | = | maximum value |
| mm | = | diffusive component for modified BDEs |
| nf | = | net flux |
| w | = | wall or boundary |
| ω | = | spectral property in terms of angular frequency |