A novel equation and some surprising results in tuning fork mass sensors: Critique, validation, and comparison

The use of Quartz Tuning Forks (QTF) as mass sensors in thin-film coating relies on a frequency shift formula derived from the Euler-Bernoulli equation, commonly cited in the literature. This study theoretically and experimentally evaluates the accuracy of this formula in film mass detection. Experiments were conducted using tuning forks (TF) larger than typical QTFs, and the calculated masses were compared to precision balance measurements. Results revealed that the conventional formula produced errors of up to 300%. Further investigation identified two primary error sources: the inability to account for the film's Young's modulus contribution and the assumption of longitudinally homogeneous coating. To address these issues, we developed a new mathematical model and validated it through extensive experiments. The proposed model predicts film mass and Young's modulus contributions with an error of approximately 1.5%. Conventional formulas predict a consistent frequency decrease with coating. However, our study demonstrates—both mathematically and experimentally—that frequency may increase or remain unchanged depending on the film’s Young’s modulus and coating uniformity. The proposed model effectively explains these unexpected results, and will very likely guide future studies involving QTF sensors by providing a more reliable framework for QTF-based mass sensing in thin-film applications.