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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 758245, 9 pages
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.
The Sauerbrey equation is a useful empirical model in material science to represent the dynamics of frequency change denoted by in an area, denoted by , of the electrode in terms of the increment of the mass, which is denoted by , loaded on the surface of the crystal under a certain resonant frequency . For the purpose of studying 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 as a function expressed by where is time. Usually, 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 . We will show that is long range dependent (LRD). Consequently, it is heavy tailed according to the Taqqu's theorem. The Hurst parameter () of 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 relating to the Sauerbrey equation in material science.
Ammonia is a type of gas useful for synthesizing various materials in chemical engineering. On the other side, it is a gas harmful to human body. Therefore, the research regarding monitoring the ammonia in different concentrations is desired for atmospheric environmental measurements and control. The ammonia sensor may yet be a desirable device for this purpose; see for example, Wang et al. .
In the aspect of ammonia sensing, an interesting phenomenon of the time-dependent frequency increment that is denoted by , that is, frequency change, responding to the coated sensors working under certain humidity was observed in the experimental research by Wang et al. . That phenomenon of the time-dependent appears a random pulse series. Nevertheless, it was only qualitatively described in . Its statistical properties remain unknown. This paper aims at revealing the statistical properties of that pulse time series. The contribution points of this paper are in three aspects. First, we will give two types of the modified representations of the Sauerbrey equation towards investigating the dynamics of based on time series. Second, we will point out that it is LRD and accordingly it is heavy tailed according to the Taqqu’s theorem. Finally, we will show that the value of of is approximately equal to one. Hence, relating to the Sauerbrey equation used in the experiments of  has strong long-range persistence, which may be served as a new explanation to describe the repeatability of the experiments done in .
The rest of the paper is organized as follows. We will give the preliminaries regarding the experiments on the time-dependent and propose modifications of the standard Sauerbrey equation in Section 2. The fractal behavior of the pulse phenomenon of is explained in Section 3. Discussions are given in Section 4. Finally, Section 5 concludes the paper.
2. Modified Representations of the Sauerbrey Equation
In the experiments by Wang et al. , Pd2+ doped ZnO (zinc oxide) nanotetrapods were prepared and studied for the detection of ammonia. The investigated gas sensors were featured by the combination of a quartz crystal microbalance (QCM) as a transducer and Pd2+ doped ZnO nanotetrapods as a sensing element. The characteristics, including the sensitivity, stability, and reproducibility of the resulted sensors, were studied under different concentration of ammonia in .
Note that quartz crystal microbalance (QCM) is an extremely sensitive mass device. The sensing principle of QCM is to transform the mass change into frequency shifts. In the experiments described in , Pd2+ doped ZnO nanotetrapods were put on the QCM; if the resonance frequency of uncoated QCM was recorded, the mass of the Pd2+ doped ZnO nanotetrapods can be calculated according to the frequency shifts between the uncoated QCM and the coated one. Based on the same principle, if the coated QCM adsorbed ammonia, the mass of ammonia can also be calculated. The measured frequency shifts were used to evaluate the mass change based on the Sauerbrey equation expressed by where (MHz) is the fundamental frequency of the unloaded piezoelectric crystal, is the resonant frequency (Hz), is the frequency change (Hz), (g) is the mass change loading on the surface of the crystal, and (cm2) is the surface area of the electrode. Figure 1 indicates the flow chart of the experiments performed in . The derivation from the standard Sauerbrey equation reported in  to (2.1) is given in the appendix.
The experiment system in Figure 1 consists of sample gas inlets with valves, a mass flowmeter, a sensing chamber with QCM, a frequency meter, and a computer for data acquisition and analysis. The valve is a switch for opening and closing the gas tunnel. The mass flowmeter was used for testing the gas concentration. The frequency meter was to calculate the frequency shifts and send the result to the computer; refer to  for the details of the experiments.
In (2.1), is in reality a function of time. To clarify this, we express (2.1) by Thus, for a given th round of experiment, we write (2.1) by where is the time duration of a round of experiment, is the starting time of the th experiment, means the frequency increment in the th experiment, and is a positive integer. We call (2.3) the modified representation of the Sauerbrey equation of type I.
Considering uncertainty in experiments and measurements , the result of the th experiment is generally not equal to that of the th’s. That is, The above expression implies that is a random variable. Therefore, we propose the representation of the Sauerbrey equation on the round-by-round basis by where is the unit step function. We call (2.5) the modified representation of the Sauerbrey equation of type II. In what follows, as well as are in the sense of (2.5). For convenience, we may write as
3. Fractal Analysis of the Pulse Phenomenon of Frequency Response to Coated Sensors
Let be a second-order stationary random process or random function. Denote by the probability density function (PDF) of . Then, the probability is given by The mean and the autocorrelation function (ACF) of based on PDF is written by (3.2) and (3.3), respectively, Let be the variance of . Then,
Note 2. If is heavy tailed, in (3.3) is slowly decayed. By slowly decayed, we mean that
Note that (3.5) can be taken as a definition of LRD time series; see for example, Beran . As a matter of fact, according to the Taqqu’s theorem, see Abry et al. , LRD property of a random function is a consequence of the heavy-tailed (Li ).
In the Gaussian assumption of , we have In this case, the heavy-tailed implies that . In the engineering sense, we do not need infinite variance but is large enough, see Li .
Note that in the experiments was sampled in the discrete case. Therefore, without loss of generality, we denote by where represents the value of of the th sample, and is the sample index. Figure 2 indicates the series of , where the vertical coordinates is indicated by and abscissa axis by for short. Figure 2 is actually a curve of random pulse series representing the time-cycling responses of the Pd-doped ZnO nanotetrapods for the concentration of ammonia gas from 0 to 240 ppm. The ratio Pd2+/ZnO is 0.04 : 1 in Moore quality. In Figure 2, each pulse stands for the result of a round of experiment. For instance, the second pulse comes from the result of the second round of experiment for . From Figure 2, one sees that the sensor performed in a reproducible manner but random in nature. The experiment was repeated over 20 times and different dosage of Pd. The pulse series indicated in Figure 2 exhibits that is random in terms of pulse width, pulse amplitude, and transition time of pulse, as one can see by eye. The random behavior of the pulse series is the focus we work on in this paper. Without loss of generality, we let , where . Then, we replot Figure 2 by Figure 3.
Taking into account the random pulse width, pulse amplitude, and pulse transition, we say that is a random function. By numeric computations, we obtain the ACF of as indicated in Figure 4, which implies the remark below.
Remark 3.1. The random function , that is, , which we investigated, is LRD. In other words, the ACF is slowly decayed such that (3.5) holds.
According to the Taqqu’s theorem, therefore, comes the following remark.
Remark 3.2. The random function described by (2.1) is heavy tailed.
Remark 3.1 implies that the ACF of has the asymptotic expression given by where can be either a constant or a slowly varying function.
Denote by the power spectrum density (PSD) of , where is the angular frequency. Then, Since (3.5) holds, has to be considered in the domain of generalized functions. According to the Fourier transform in the domain of generalized functions (Kanwal , Li and Lim ), we immediately obtain where stands for the operator of the Fourier transform. Therefore, for the measured , we have the asymptotic expression of the PSD of below: Hence, we have the following remark. We note that in (3.9), (3.10), and (3.11) is the angular frequency, which is an argument in the Fourier transform of the ACF of , while stands for the frequency increment in the Sauerbrey equation. That is, the frequency in differs in meaning from that in the Sauerbrey equation.
Remark 3.3. The measured, that is, , is in the class of noise.
The previous discussions exhibit three properties of from a view of fractals. They are heavy-tailed PDF, slowly decayed ACF, and noise type PSD. Considering the Hurst parameter, we have which characterizes the LRD of from a view of fractals. By the least square fitting, for the curve in Figure 4, we have Therefore, the frequency change of the Pd-doped ZnO nanotetrapods to the concentration of NH3 from 0 to 240 ppm has strong LRD. This implies that the result from the th experiment is strongly correlated to that from the th experiment, even if is large. Hence, we hereby provide, from the point of view of fractals, a quantitative explanation of repeatability of the experiments that were qualitatively stated in .
We note that this paper does not say that obeys the widely used fractal time series model, we mean, fractional Gaussian noise (fGn). In addition, we did not claim anything about the fractal dimension of either. In the future, we will work on the accurate ACF and the fractal dimension of . To the best of our knowledge, the fractal properties described above may yet imply a considerable advance in the field regarding the frequency change of the Pd-doped ZnO nanotetrapods to the concentration of NH3 from 0 to 240 ppm. Further, we will investigate such a type of random pulses with the differential equations as reported in [13–17]. Finally, we note that appears deterministic but we did fractal analysis about it to provide an explanation of the uncertainty principle in measurements.
We have processed the real data of the frequency change of the Pd-doped ZnO nanotetrapods to the concentration of NH3 from 0 to 240 ppm. The present results exhibit that such a pulse series is heavy tailed and has strong LRD. We have also explained the repeatability of the experiments from a view of fractals.
Derivation of (2.1)
The theory of the quartz crystal microbalance (QCM) is the piezoelectric qualities of quartz crystals. The application of an electric field to the electrode of the quartz crystals causes a shear deformation (parallel to the electrode surface). The crystal can be made to resonate if an alternating electric field is applied at a particular frequency . Deposition of the working electrode layer dampens this resonant frequency. The Sauerbrey equation relates the dampening of frequency, , to the change in surface attached mass .
The Sauerbrey equation was first introduced by Sauerbrey in . Its standard form is given by where is the resonant frequency (Hz), is the frequency change (Hz), is the mass change (), is the piezoelectrically active crystal area (cm2) between electrodes, is the density of quartz, and is the shear modulus of quartz for AT-cut crystal. In the experiments described in , Replacing and in (A.1) with (A.2) produces , which is the expression (2.1).
This work was partly supported by the National Natural Science Foundation of China (NSFC) under the project Grant nos. 60573125, 60873264, 61070214, and 60870002, the 973 plan under the Project no. 2011CB302800/2011CB302802, NCET, and the Science and Technology Department of Zhejiang Province (2009C21008, 2010R10006, 2010C33095, and Y1090592).
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