Role of Al₄C₃ on the stability of the thermal conductivity of Al/diamond composites subjected to constant or oscillating temperature in a humid environment

Abstract

Present technologies allow fabrication of aluminum/diamond particles composites with excellent thermal properties, in particular showing the by far highest Thermal Conductivity of any of the materials being evaluated for thermal management. Although there is a widespread consensus concerning the essential role that the interface plays, it is not yet fully clear how the aluminum carbide formed at the interface affects thermal properties. In particular, how it affects the stability of the thermal properties of composites subjected to thermal treatments under wetting conditions. This is precisely the objective of the present work. To this end, samples were fabricated by means of gas pressure infiltration of liquid Al into preforms of packed diamond particles. Infiltration was carried out at two temperatures and three contact times. Thermal fatigue with cooling phase in water and performance in moisture environments at temperatures close to 100 °C were evaluated. The results show that those samples having low amounts of carbide at the interface (shorter contact times) are more prone to a decrease of the thermal conductivity.

Appendix

Interfacial thermal conductance

The ITC in the Al/diamond interface was calculated taking into account the presence of Al₄C₃ at the interface. Both the AMM and the DMM will be used.

Assuming that heat transport is mostly due to phonons, a general expression for the interfacial thermal conductance in going from medium 1 to medium 2 is [29]

$$ h_{12} = \frac{1}{4}C_{p,1} \eta_{12} c_{{{\text{D}},1}} , $$

(3)

where C _p,1 is the specific heat per volume of medium 1, η ₁₂ is the transmission coefficient of phonons across the interface from phase 1 to phase 2, and c _D,1 is the Debye (or effective) velocity of medium 1 given by [30]

$$ c_{{{\text{D}},1}} = \left( {\frac{1}{3}\left[ {\frac{2}{{\left( {c_{{{\text{t}},1}} } \right)^{2} }} + \frac{1}{{\left( {c_{{{\text{l}},1}} } \right)^{2} }}} \right]} \right)^{ - 1/2} , $$

(4)

where c ¹ _t and c ¹ _l are the transversal and longitudinal sound velocities in medium 1 that in turn can be written in terms of the elastic constant of the material [31],

$$ c_{{{\text{l}},1}} = \left( {\frac{{B_{1} + \frac{4}{3}G_{1} }}{{\rho_{1} }}} \right)^{{\frac{1}{2}}} , $$

(5)

$$ c_{{{\text{t}},1}} = \left( {\frac{{G_{1} }}{{\rho_{1} }}} \right)^{{\frac{1}{2}}} . $$

(6)

Before proceeding, it should be noted that the two models here used only differ in the transmission coefficient across the interface. In particular, in the AMM this magnitude is written as

$$ \eta_{12} = pq, $$

(7)

where p is the transmission probability of phonons incident with an angle less than a critical angle θ _c and q is the fraction of the total number of incident phonons that actually fulfill that condition [29]. p can be written as

$$ p = \frac{{4\rho_{1} c_{{{\text{D}},1}} \rho_{2} c_{{{\text{D}},2}} }}{{\left( {\rho_{1} c_{{{\text{D}},1}} + \rho_{2} c_{{{\text{D}},2}} } \right)^{2} }}, $$

(8)

while for c _D,1 ≪ c _D,2, q in turn can be approximated by

$$ q = \frac{1}{2}\sin^{2} \theta_{c} \approx \frac{1}{2}\left( {\frac{{c_{{{\text{D}},1}} }}{{c_{{{\text{D}},2}} }}} \right)^{2} . $$

(9)

Thus, the final AMM expression for the interfacial thermal conductance is

$$ h_{12} = \frac{1}{2}C_{p,1} \frac{{\rho_{1} c_{{{\text{D}},1}} \rho_{2} c_{{{\text{D}},2}} }}{{\left( {\rho_{1} c_{{{\text{D}},1}} + \rho_{2} c_{{{\text{D}},2}} } \right)^{2} }}\left( {\frac{{c_{{{\text{D}},1}} }}{{c_{{{\text{D}},2}} }}} \right)^{2} c_{{{\text{D}},1}}. $$

(10)

The DMM leads to the following transmission coefficient (from medium1 to medium 2)

$$ \eta_{12} = \frac{{\mathop \sum \nolimits_{i} c_{i,2}^{ - 2} }}{{\mathop \sum \nolimits_{i} c_{i,1}^{ - 2} + \mathop \sum \nolimits_{i} c_{i,2}^{ - 2} }} = \frac{{c_{{{\text{D}},1}}^{2} }}{{c_{{{\text{D}},1}}^{2} + c_{{{\text{D}},2}}^{2} }}. $$

(11)

Inserting Eq. (11) into (3) h _1,2 can be easily calculated. The ITC was calculated by using the material properties gathered in Table 4. The results for the three h involved in the present work, obtained with the two methods described in this Appendix, are reported in Table 2.

Table 4 Reports several properties of the three materials involved in the resent work and required for the calculation of the interfacial conductances