“Eigen-periodic”-in-space surface heating in conduction with application to conductivity measurement of thin films

A two-dimensional heat conduction problem in Cartesian coordinates subject to a periodic-in-space boundary condition is analyzed by the Green’s functions approach. It is pointed out that when the frequency of the spatial periodic heating equates one of the natural frequencies (eigenvalues) of the system, the solution of the 2D heat conduction problem can be written down very simply as the product of the periodic surface condition (termed the “eigen-periodic”) by the solution of a 1D fin problem along the nonhomogeneous direction. This result suggests a novel and simple algebraic equation for determining the thermal conductivity of thin films placed on substrates under steady state conditions. High space frequencies of the sinusoidal heating, larger than the deviation frequency, are used to make negligible the thermal deviation effects due to the presence of the substrate.