Long Memory from Sauerbrey Equation: A Case in Coated Quartz Crystal Microbalance in terms of Ammonia

The Sauerbrey equation is a useful empirical model in material science to represent the dynamics of frequency change denoted by Δ ð ‘“ in an area, denoted by ð ´ , of the electrode in terms of the increment of the mass, which is denoted by Δ ð ‘š , loaded on the surface of the crystal under a certain resonant frequency ð ‘“ 0 . For the purpose of studying Δ ð ‘“ from the point of view of time series, we first propose two types of the modified representations of the Sauerbrey equation by taking time as an argument to represent Δ ð ‘“ as a function expressed by ð ‘¥ ( ð ‘¡ , ð ‘“ 0 , ð ´ , Δ ð ‘š ) , where ð ‘¡ is time. Usually, Δ ð ‘“ is studied experimentally for the performance evaluation of the tested quartz used in ammonia sensors. Its properties in time series, however, are rarely reported. This paper presents the fractal properties of Δ ð ‘“ . We will show that Δ ð ‘“ is long range dependent (LRD). Consequently, it is heavy tailed according to the Taqqu's theorem. The Hurst parameter ( ð » ) of Δ ð ‘“ approaches one, implying its strong long memory, providing a new explanation of the repeatability of the experiments and novel point of view of the dynamics of Δ ð ‘“ relating to the Sauerbrey equation in material science.