An Exact Closed-Form Solution for the 2$ω$ Method for an Arbitrarily Anisotropic Substrate Available to Purchase

The 2$ω$ method, in which the temperature sensing line is separate from the heater line, and the closely related 3$ω$ method, with a heater line that is its own temperature sensor, are popular electrothermal techniques for measuring the thermal conductivity. For the 2$ω$ method, Ordonez-Miranda et al. (2023, “Analytical Integration of the Heater and Sensor 3$ω$ Signals of Anisotropic Bulk Materials and Thin Films,” J. Appl. Phys., 133(20), p. 205104) recently obtained an analytical solution for its thermal model for a substrate with differing in-plane and out-of-plane thermal conductivities. Here, we further generalize the $2ω$ thermal model by deriving an exact closed-form solution for a substrate of arbitrarily aligned thermal conductivity. The derivation builds on a Green's function from Mishra et al. (2015, “A 3 Omega Method to Measure an Arbitrary Anisotropic Thermal Conductivity Tensor,” Rev. Sci. Instrum., 86(5), p. 054902), and the resulting $2ω$ expression is shown to maintain a similar Meijer G-function form as the known analytical solution of the 3$ω$ method.