Derivation for Mass–Frequency Relationship of a Quartz Crystal Microbalance Based on an Equivalent Circuit Network Analysis Method

Saurbrey equation has solved frequency dependence on mass change for a rigid film on a quartz crystal microbalance device. Does it hold true for any rigid film that is deposited on a quartz crystal microbalance? A new expression for the mass–frequency relationship of a quartz crystal microbalance device is derived based on equivalent circuit network analysis combined with physical properties of the quartz crystal. Compared to Sauerbrey equation, the proposed expression is derived without applying any approximations. Theoretical analysis is validated with finite-element analysis and experimental test. The results demonstrate that frequency shift of a quartz crystal microbalance is related not only to mass of the deposited addition but also density of the deposited addition. This work is significant to supplement and extend the theory of quartz crystal microbalance.