Thermoelectric Properties of Two Component Composites
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THERMOELECTRIC PROPERTIES OF TWO COMPONENT COMPOSITES
OHAD LEVY AND DAVID J. BERGMAN School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel.
ABSTRACT An explicit expression for the bulk effective thermoelectric coefficient a, of a two component composite is derived. This coefficient is found to depend only on the bulk effective electric conductivity oe and thermal conductivity 7Y, as well as on the moduli of the pure components. Using this expression and making standard scaling assumptions about the forms of oe and ,Ye, the scaling behavior of ce, and of the effective figure of merit Ze are investigated. This behavior depends strongly on the thermal and electric conductivities ratios in the pure components. Upper and lower bounds for a, are calculated from its analytical properties.
INTRODUCTION The electric current density that flows in a homogeneous and isotropic material due to electric and temperature fields under linear response assumptions is: Je = -oV4
- oaVT
(1)
where E is the electric potential, T is the temperature, c is the electric conductivity and a is the thermoelectric coefficient. The entropy current density under the same conditions is: J. = -oczV4b!VT (2)
T
where 7 is the thermal conductivity. The transport matrix representation is:
Ia/
V ((3)
ice keT
ee3 -ft
2
Z
Ic T
V (kT)
where e is the electron charge and k is the Boltzman constant. In this representation the different elements of each vector and those of the matrix S have the same dimentions. The figure of merit is defined as: , 0ca
_= -( T)_-
2
(4)
This quantity determi*nes the efficiency of a thermoelectric generator or refrigerator constructed from this material [1]. Each component material of an inhomogeneous composite shows such a behavior with its own transport matrix.
Mat. Res. Soc. Symp. Proc. Vol. 195. 01990 Materials Research Society
206
The material as a whole has a bulk effective transport matrix:
Se
=
e2
ke
ke
VcT
(5)
Note that although we assumed that the components are isotropic, we do not assume this for the composite: In the case of an anisotropic composite, our treatment assumes that both of the volume averaged fields < V4 >, < VT > point along the same direction, say along the x-axis, and that only the x-components of the volume averaged currents < Je >, < Js > are determined by the 2 x 2 matrix Se. The problem of the effective conductivities of two component composites when a =- 0 is treated extensively elsewhere [2,3]. It is formulated in terms of reduced variables and a characteristic geometric function. A single characteristic function, whose form depends only on the microgeometry of the composite, determines both bulk effective moduli cra and -y. One such possible representation is by the reduced variables s, = (1 - _)l and s, = (12')-, where A and B denote the two components. The characteristic function is then:
F(so) = 1- " _ c'B
n8
-- $
an-d F(s) =I--aY'B = E on Sl "'
(6)
0 < s, < 1; 0 < Fn. The s,, de
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