### 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%).

Original language | English |
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Journal | Nanoscale and Microscale Thermophysical Engineering |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Fingerprint

### Keywords

- ballistic thermal resistance
- ballistic–diffusive equations
- effective thermal conductivity
- Phonon mean free path
- phonon mean free path spectrum

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Materials Science(all)
- Condensed Matter Physics
- Mechanics of Materials

### Cite this

*Nanoscale and Microscale Thermophysical Engineering*. https://doi.org/10.1080/15567265.2019.1619885

**Modified ballistic–diffusive equations for obtaining phonon mean free path spectrum from ballistic thermal resistance : I. Introduction and validation of the equations.** / Kwon, Oh Myoung; Wehmeyer, Geoff; Dames, Chris.

Research output: Contribution to journal › Article

*Nanoscale and Microscale Thermophysical Engineering*. https://doi.org/10.1080/15567265.2019.1619885

}

TY - JOUR

T1 - Modified ballistic–diffusive equations for obtaining phonon mean free path spectrum from ballistic thermal resistance

T2 - I. Introduction and validation of the equations

AU - Kwon, Oh Myoung

AU - Wehmeyer, Geoff

AU - Dames, Chris

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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%).

AB - 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%).

KW - ballistic thermal resistance

KW - ballistic–diffusive equations

KW - effective thermal conductivity

KW - Phonon mean free path

KW - phonon mean free path spectrum

UR - http://www.scopus.com/inward/record.url?scp=85066617085&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066617085&partnerID=8YFLogxK

U2 - 10.1080/15567265.2019.1619885

DO - 10.1080/15567265.2019.1619885

M3 - Article

AN - SCOPUS:85066617085

JO - Nanoscale and Microscale Thermophysical Engineering

JF - Nanoscale and Microscale Thermophysical Engineering

SN - 1556-7265

ER -