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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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