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The Effective Thermoelectric Properties of Composite Materials

Yang Yang and Jiangyu Li Department of Mechanical Engineering

University of Washington

3rd Thermoelectrics Applications Workshop 2012

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Thermoelectric Figure of Merit

Snyder and Toberer (2008)

phe

TZκκσα+

=2

HCCTE TTZT

ZT/111

++−+

=ηη

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Nanostructured Thermoelectrics

Li et al (2010)

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Intrisic Heterogeneity

Pichanusakorn and Prabhakar Bandaru (2010)

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Thermoelectric Composites

Can the effective thermoelectric figure of merit of a composite material be higher than all its constituents, excluding the effects of size and interface? and how can we take advantages of the size and interfacial effects of composite materials to further enhance their thermoelectric figure of merit?

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Effective Figure of Merit

Bergman et al (1991, 1999): The effective figure of merit of a composite can never exceed the largest value of figure of merit in any of its component, in the absence of size and interface effects.

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Current State of Art

Graeme Milton (2002): There is a lot of confusion, particularly in the composite materials community, as to appropriate form of the thermoelectric equations.

0

10

20

30

40

50

60

70

80

90

100

TE Composite

Homogenization

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Governing Equations

σ φ σα− = ∇ + ∇TJ2( )σα φ σα κ α κ= − ∇ − + ∇ = − ∇Q T T T T TJ J

φ= +U QJ J J

0∇⋅ =UJ0∇⋅ =J

φ∇ ⋅ = −∇ ⋅QJ J

Transport Equations

Field Equations

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Homogeneous Thermoelectric

2 2

2 σκ= −

d T Jdx

2 2

2

φ ασκ

=d Jdx

22

1 22JT x c x cσκ

= − + +

2 22

1 0 3[ ( ) ]2 2α αφ ασκ σκ σ

= − + + − +J J Jx T T x c

0 1 0 1( ) ( )ασ σ φ φ= − + −J T T

All the coefficients are assumed to be temperature independent

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One-dimensional Analysis

Bi2Te3

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Heterogeneous Thermoelectric

22

22

,02

, 12

σ κ

σ κ

− + + ≤ <= − + + < ≤

A AA A

B BB B

J x a x b x fT

J x a x b f x

22

22

( ) ,02

( ) , 12

α ασ κ σ

φα ασ κ σ

− + + ≤ <

= − + + < ≤

AA A A

A A A

BB B B

B B B

J Jx a x c x f

J Jx a x c f x

All the coefficients are assumed to be temperature independent

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One-dimensional Analysis

Bi2Te3-LAST

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Equivalency Principle

Under identical boundary conditions of temperature and electric potential, the composite and homogenized medium would have: (1) identical current density and (2) identical energy flux.

Yang et al (2012)

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Enhanced Effective Figure of Merit

Bi2Te3-LAST

Due to nonlinearity, the effective properties depend on boundary conditions

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Q 0 LT T>

Q

0T

LT LTlR

Q

p n

Conversion Efficiency Analysis

TE material

Q

Super conductor

0 LT T>

Q

0T

LT LTlR

Q κ α φ= − ∇ + +UJ T TJ J

0 0==∑U xE SJ

2

0

lL

I RHE

=

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Enhanced Conversion Efficiency

Z of constituent phases is fixed, but individual coefficients are allowed to vary

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Enhanced Conversion Efficiency

Snyder (2004) Bian and Shakouri (2006)

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Thermoelectric Composite

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One-dimensional Asymptotic Analysis

η

L

ξη

=x

ξ1

x

K

BK

AKx: slow or macroscopic variable

ξ: fast or microscopic variable

20 1 2( , ) ( , ) ( , ) ( , ) ...ξ ξ η ξ η ξ= + + +T x T x T x T x

20 1 2( , ) ( , ) ( , ) ( , ) ...φ ξ φ ξ ηφ ξ η φ ξ= + + +x x x x

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0( ) 0κξ ξ

∂∂=

∂ ∂T2η−

1η− 0 0 0φσ σαξ ξ

∂ ∂+ =

∂ ∂T

0 0 010( ) ( ) φκ α κ κ

ξ ξ ξ ξ∂ ∂ ∂∂∂ ∂

+ − + + =∂ ∂ ∂ ∂ ∂ ∂

T T TJT Jx x

0 01 1φ φσ σ σα σαξ ξ

∂ ∂∂ ∂− = + + +

∂ ∂ ∂ ∂T TJ

x x

0 1 1 20 1

0 1

( ) ( )

( )

α κ κ α κ κξ ξ ξ

φ φξ

∂ ∂ ∂ ∂∂ ∂− + + + − + +

∂ ∂ ∂ ∂ ∂ ∂∂ ∂

= +∂ ∂

T T T TJT JTx x x

Jx

0η

Asymptotic Analysis

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Homogenized Equations

20 1 2( , ) ( , ) ( , ) ( , ) ...ξ ξ η ξ η ξ= + + +T x T x T x T x 2

0 1 2( , ) ( , ) ( , ) ( , ) ...φ ξ φ ξ ηφ ξ η φ ξ= + + +x x x x

2 22 2 20

02

1 1 1( ) 0α ακ κ κ σ κ

− < > − < >< > + < >< > =d T J T Jdx

22

2

1 0σκ

+ =d T Jdx

2 22 10 0

2

23 1 2 2

0

1( )

1 1( ) 0

φ α ακ κ κ

α α α ακ κ κ κ σ κ

−

−

+ < > − < > < >

− < >< > − < > < > − < >< > =

d dTJdx dx

J T J

22

2 0φ ασκ

− =d Jdx

Homogenized

Homogeneous

Homogenized and homogeneous materials satisfy different type of governing equations!

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Homogenized Solution

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Concluding Remarks

Rigorous asymptotic analysis has been developed to predict the effective thermoelectric behaviors of composite materials, and preliminary results suggest that conversion efficiency higher than that of constituents is possible in composites