A simple numerical method for evaluating heat dissipation from curved wires with periodic applied heating

In many situations, the dual-purpose heater/thermometer wires used in the three-omega method—one of the most precise and sensitive techniques for measuring the thermal conductivity of thin films and interfaces—must include bends and curves to avoid obstructions on the surface of a sample. Although the three-omega analysis assumes that the heating wire is infinitely long and straight, recent experimental work has demonstrated that, in some cases, curved-wire geometries can be used without introducing detectable systematic error. We describe a general numerical method that can be used to calculate the temperature of three-omega heating wires with arbitrary wire geometries. This method provides experimentalists with a simple quantitative procedure for calculating how large the systematic error caused by a particular wire asymmetry will be. We show calculations of two useful cases: a straight wire with a single bend of arbitrary angle and a wire that forms a circle. We find that the amplitude of the in-phase temperature oscillations near a wire that forms a circle differs from the prediction using the analytic straight-line source solution by