Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 758245, 9 pages
doi:10.1155/2011/758245
Research Article

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

1School of Information Science and Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China
2College of Computer Science, Zhejiang University of Technology, Hangzhou 310023, China

Received 10 November 2010; Accepted 24 November 2010

Academic Editor: Cristian Toma

Copyright © 2011 Xiaohua Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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