A simple model for the effect of nonuniform mass loading on the response of gravimetric chemical sensors

We reconsider the effects of nonuniform mass loading in gravimetric chemical or biological sensors by applying a mode-matching approach. The approach yields an eigenvalue equation which is solved numerically. The results reveal how an increasing concentration of the mass deposited on top of a microacoustic resonator leads to a frequency shift different from that predicted by the Sauerbrey equation valid for uniform mass loading. Depending on the mode type and the mass distribution, a nonuniform mass loading can lead to a frequency shift higher or lower than that of the uniform loading case. We then derive a point-mass solution which can be considered exact in all practical applications and compare it to available data in the literature. The point-mass solution is used to obtain a closed formula for the frequency shift produced by an arbitrary nonuniform mass loading, and some consequences of this formula are discussed. Altogether, the model presented is quite simple and yet allows one to judge the effects of nonuniform mass loading in gravimetric sensors very efficiently.