Analysis of 3ω method for thermal conductivity measurement

The 3ω method for measurement of thermal conductivity of a solid usually consists of a strip heater above a substrate, and the method is based on an approximate solution of the heat conduction problem in which the heater thickness is neglected. We establish a two-dimensional heat conduction model for the heater-on-substrate 3ω method which takes into account the finite thickness of the heater. Analytical solutions for the finiteheater problem and the vanishing heater-thickness problem are presented utilizing the method of separation of variables. It is found that neglecting the heater thickness is not accurate at low or high frequencies, but works well in between. This frequency range becomes narrower when the heater thickness increases. Nomenclature a, b heater thickness and width defined in Fig. 1(a) [m] c, d dimensions of substrate defined in Fig. 1(a) [m] A, C, D non-dimensional distances H(·) Heaviside function I0 alternating current amplitude [A] k thermal conductivity [W/m K] K thermal conductivity ratio between substrate and heater L coefficient vector M coefficient matrix P power generated per unit length [W/m] q heat flux [W/m] R coefficient vector t non-dimensional time t dimensional time [s] T non-dimensional temperature T ∗ dimensional temperature [K] T ∗ 0 reference temperature [K] (x, y) non-dimensional Cartesian coordinates (x, y) dimensional Cartesian coordinates [(m,m)]