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### Size effect on thermal conductivity of thin films

Guihua Tang, Yue Zhao, Guangxin Zhai, Zengyao Li, Wenquan Tao

School of Energy & Power Engineering,

Xi’an Jiaotong University, China

Background

2

Local mean free path method

3.1

3

3.2

4

4

Outline

Results 1: Local thermal conductivity distribution

Results 2:Overall thermal conductivity

Conclusions

- Boundary or interface scattering becomes important when the characteristic length (film thickness, wire diameter) is comparable with the mean free path.
- The thermal conductivity (as well as other transport coefficients, viscosity) becomes size dependent.
- Numerous important applications of nanoscale thermal conduction (electronic devices cooling, thermal insulator, thermalelectric conversion, etc.)

y

The Phonon Gas

X

is the lattice volumetric specific heat;

is the average speed of phonons;

is the phonon mean free path.

- Specific heat of solid: Lattice vibration in solids.

Harmonic oscillator model of an atom

- Conduction in insulatorsis dominated by lattice waves or phonons.
- Simple expression of thermal conductivity based on the kinetic theory

Apply the Matthiessen’s rule

- Classical size effect based ongeometric consideration (1)

- In the ballistic transport limit, L<<Lb, the MFP is L
- L>>Lb, the MFP is the bulk mean free path Lb
- Intermediate region:

A thin film for paths originated from the boundary surface

(m≈3)

Simple interpolation between the two expressions

- Classical size effect based ongeometric consideration (2)

- When L<<Lb, assuming that all the energy carriers originate from the boundary surface

- L>>Lb, the MFP is the bulk mean free path Lb

(m≈3)

A thin film for paths originated from the centre

Interpolation

- Classical size effect based ongeometric consideration (3)

- The direction of transport was not considered and the anisotropic feature cannot be captured
- Filk and Tien employed a weighted average of the mean free path components in the parallel and normal directions of a thin film

Interpolation

(m≈4/3)

- Classical size effect based ongeometric consideration (4)

A thin circular wire for paths originated from the centre

Film

Thin

Wire

p is the probability of specular scattering on the boundary

- Classical size effect based onBoltzmann Transport Equation (BTE)

- The relaxation time approximation was adopted.
- The distribution function was assumed to be not too far away from equilibrium.

2. Local mean free path method

- For an unbounded phonon gas, the probability of a phonon gas can travel between two consecutive collisions with other phonons at location x and x+dx would be of the form:

The probability of a phonon gas having a free path between x and x+dx

- When the gas is bounded, a number of phonons will be terminated by the boundary, thus effective MFP < Lb

Local thermal conductivity distribution in a semi-infinitefilm

Local thermal conductivity distribution in a thin film

- An equation to calculate the size-dependent film thermal conductivity has been derived. No Matthiessen’s rule; No interpolation
- Local thermal conductivity distribution in the thin film has been obtained.
- The present solution seems to overpredicts reduction in thermal conductivity compared to the data in references when Knudsen number is larger than 1.
- More cases are needed for further validation and extension to complicated geometric structures.

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