A statistical model for effective thermal conductivity of composite materials

• Paradox in circuit network approach for effective thermal conductivity was illustrated.
• A statistical approach is then adopted with ensemble containing microstates.
• A generalized interface model for microstate is constructed.
• The effective thermal conductivity at mesoscale is determined.
• The new statistical model fits excellently to experimental data.

Thermal interface materials composed of polymers and solid particles of high thermal conductivity have been used widely in electronic cooling industries. However, theoretical prediction of effective thermal conductivity of the composites remains as a crucial research topic. Theoretical modelings of the effective thermal conductivity of composites were based mainly on the analog between electric and thermal fields that satisfy Laplace equation under steady condition. Two approaches were employed, either by solving the Laplace equation or by equating the composite to a circuit network of conductors. In this study, the existing models obtained from these two approaches are first briefly reviewed. However, a close examination of the second approach reveals that there exists a paradox in this approach. To resolve this paradox, a statistical approach is then adopted. To this end, a mesoscopic ensemble that contains microscopic states is first constructed. The thermal conductivities of the microstates are identified by the principle of least action. A statistical parameter for each microstate is identified to characterize the effect of interface resistance on heat flow, as well as the connection between microstates. Effective thermal conductivity of the ensemble was then obtained from the variation principle that minimizes the standard deviation with optimal distribution of the parameter. The predictions by the present statistical model fit to experimental data with excellent agreement.