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 Delta f in an area, denoted by A, of the electrode in terms of the increment of the mass, which is denoted by Delta m, loaded on the surface of the crystal under a certain resonant frequency f(0). For the purpose of studying Delta f 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 Delta f as a function expressed by x(t, f(0), A, Delta m), where t is time. Usually, Delta f 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 Delta f. We will show that Delta f is long range dependent (LRD). Consequently, it is heavy tailed according to the Taqqu's theorem. The Hurst parameter (H) of Delta f 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 Delta f relating to the Sauerbrey equation in material science.