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# Size effect on thermal conductivity of thin films

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```									   Xi’an Jiaotong University

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
Outline
1     Background

2     Local mean free path method

3.1
3
Results 1: Local thermal conductivity
distribution

3.2
4    Results 2: Overall thermal conductivity

4     Conclusions
1. Background
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.)
Specific heat of solid: Lattice vibration in solids.
Z

y

X                The Phonon Gas

Harmonic oscillator model of an atom

Conduction in insulators is dominated by lattice waves or phonons.
Simple expression of thermal conductivity based on the kinetic
theory
 cv is the lattice volumetric specific heat;
1                      a is the average speed of phonons;
   cv a  ph
3                      ph is the phonon mean free path.
Classical size effect based on geometric consideration (1)

 b is bulk mean free path

In the ballistic transport limit, L<<b, the MFP is L
L>>b, the MFP is the bulk mean free path b
Intermediate region:

Apply the Matthiessen’s rule

1    1 1
                            eff eff   1            b
 eff L  b                 Eq.1:                  Kn 
b     b 1  Kn          L
Classical size effect based on geometric consideration (2)

When L<<b, assuming that all
the energy carriers originate from
the boundary surface
L
b                         ln  Kn   1
                b
Kn
A thin film for paths
originated from the             L>>b, the MFP is the bulk mean
boundary surface               free path b
1
 eff      Kn      
Kn  5    Eq.2:       1          
 b  ln  Kn   1 
            
 eff  Kn 1
Kn  1             1
b   

m      (m≈3)

1  Kn  5        Simple interpolation between the two expressions
Classical size effect based on geometric 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
 eff , z      Kn 
1

Kn  5     Eq.3:           1     1 
b         2  Kn 
L
 eff  Kn 1
Kn  1             1       (m≈3)
b         m 
A thin film for paths
originated from the centre
1  Kn  5            Interpolation
Classical size effect based on geometric consideration (4)

1
 eff         Kn       
Kn  5    Eq.4:       1              
b     1   4 Kn 1 


 eff  Kn 1
L                     Kn  1             1        (m≈4/3)
b         m 

A thin circular wire for paths
originated from the centre       1  Kn  5            Interpolation
Classical size effect based on Boltzmann Transport Equation (BTE)
The relaxation time approximation was adopted.
The distribution function was assumed to be not too far away from equilibrium.

Thin              eff      3 1  p  Kn   1 1  1  exp(t / Kn)
Film             b
 1
2      1  t 3  t 5  1  p exp  t / Kn  dt
           

Thin
 eff , w      12 1  p 
2

 m t  t  1
2

 mp 
1
1                                  dtd
m 1
 1                                            2
exp      
Wire            b                             m 1
0              1
 Kn  t
4

p is the probability of specular scattering on the boundary
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:

 x   x 
p  x   exp     d    
The probability of a phonon gas

 b   b 
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 < b

For a thin film:

  z   b 1    1 exp      2  t 1 exp  t  dt 

                                                

  z b ,              z0
Semi-infinite film:
  b ,                z

b    2    1 exp      2  t 1 exp  t  dt 
 z                                       
2    
                                                   

z
  z b ,           z0
Thin film:
   L  z  b ,   zL

z
3. Results

1.0
(2.9, 0.99)

0.9

0.8                                            z
 /b

0.7

o
0.6

0.5
0   1   2    3          4        5   6
z/b

Local thermal conductivity
distribution in a semi-infinite film
Local thermal conductivity
distribution in a thin film
Kn=0.01
1.0
Kn=0.1             Kn=0.2              Kn=0.05
Kn=0.3
0.8
Kn=0.5
z
0.6
b

Kn=1
L
0.4
Kn=2                           o
0.2                          Kn=3
Kn=5
Kn=10                                   b
Kn 
0.0
0.0        0.2      0.4           0.6   0.8        1.0
z/L                                    L
 eff            1
Kn / 2  
 1  exp( )  Kn   d  x 1 exp( x)dx
b               Kn      0      
Overall thermal conductivity
in a thin film VS Kn
 eff   1
1.0                                              Eq.1:       
 b 1  Kn
0.8                                                                                1
 eff  eff      Kn       
Eq.2:            1            , Kn  5
b      b  ln  Kn   1 
             
0.6
 eff           1
eff/b

Eq.1                                  Kn 
 1           , Kn  5
 b  2  Kn1 
Eq.3:
0.4                        Eq.2, m=3                                 
Eq.3, m=3
Eq.4, m =4/3                                            1
0.2                        Eq.5, p=0                    eff      Kn       
Eq.4:       1           
Present                      b  1   4 Kn 1 
             
0.0
100   10    1           0.1         0.01               eff  Kn 1
 1    , Kn  1
Kn =b/L                                   b      m 

L                               eff      3 1  p  Kn   1 1  1  exp(t / Kn)
Eq.5:
b
 1
2      1  t 3  t 5  1  p exp  t / Kn  dt
           
4. Conclusions
 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.
09/07/2010

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