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

Abstract

The Sauerbrey equation is a useful empirical model in material science to represent the dynamics of frequency change denoted by Δ𝑓 in an area, denoted by 𝐴, of the electrode in terms of the increment of the mass, which is denoted by Δ𝑚, loaded on the surface of the crystal under a certain resonant frequency 𝑓₀. For the purpose of studying Δ𝑓 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 Δ𝑓 as a function expressed by 𝑥(𝑡,𝑓₀,𝐴,Δ𝑚), where 𝑡 is time. Usually, Δ𝑓 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 Δ𝑓. We will show that Δ𝑓 is long range dependent (LRD). Consequently, it is heavy tailed according to the Taqqu's theorem. The Hurst parameter (𝐻) of Δ𝑓 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 Δ𝑓 relating to the Sauerbrey equation in material science.

1. Introduction

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. [1].

In the aspect of ammonia sensing, an interesting phenomenon of the time-dependent frequency increment that is denoted by Δ𝑓, that is, frequency change, responding to the coated sensors working under certain humidity was observed in the experimental research by Wang et al. [2]. That phenomenon of the time-dependent Δ𝑓 appears a random pulse series. Nevertheless, it was only qualitatively described in [2]. 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 Δ𝑓 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 𝐻 of Δ𝑓 is approximately equal to one. Hence, Δ𝑓 relating to the Sauerbrey equation used in the experiments of [2] has strong long-range persistence, which may be served as a new explanation to describe the repeatability of the experiments done in [2].

The rest of the paper is organized as follows. We will give the preliminaries regarding the experiments on the time-dependent Δ𝑓 and propose modifications of the standard Sauerbrey equation in Section 2. The fractal behavior of the pulse phenomenon of Δ𝑓 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. [2], 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 [2].

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 [2], 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 byΔ𝑓=−2.26×10⁻⁶𝑓²⁰𝐴Δ𝑚,(2.1) where 𝑓(MHz) is the fundamental frequency of the unloaded piezoelectric crystal, 𝑓₀ is the resonant frequency (Hz), Δ𝑓 is the frequency change (Hz), Δ𝑚(g) is the mass change loading on the surface of the crystal, and 𝐴 (cm²) is the surface area of the electrode. Figure 1 indicates the flow chart of the experiments performed in [2]. The derivation from the standard Sauerbrey equation reported in [3] to (2.1) is given in the appendix.

Figure 1 

Flow chart of the experiments for obtaining Δ𝑓 for a gas sensing system.

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 [2] for the details of the experiments.

In (2.1), Δ𝑓 is in reality a function of time. To clarify this, we express (2.1) byΔ𝑓≈𝑑𝑓=𝑑𝑓𝑑𝑡𝑑𝑡≈−2.26×10⁻⁶𝑓²⁰𝐴𝑑𝑚=−2.26×10⁻⁶𝑓²⁰𝐴𝑑𝑑𝑚𝑡𝑑𝑡.(2.2) Thus, for a given 𝑛th round of experiment, we write (2.1) by𝑦_𝑛𝑡,𝑓₀,Δ𝑚,𝐴=−2.26×10⁻⁶𝑓²⁰𝐴Δ𝑚,𝑡⁰<𝑡<𝑡₀+𝑇,(2.3) where 𝑇 is the time duration of a round of experiment, 𝑡₀ is the starting time of the 𝑛th experiment, 𝑦_𝑛 means the frequency increment in the 𝑛th experiment, and 𝑛 is a positive integer. We call (2.3) the modified representation of the Sauerbrey equation of type I.

Considering uncertainty in experiments and measurements [4], the result of the 𝑛th experiment is generally not equal to that of the (𝑛+1)th’s. That is,𝑦_𝑛≠𝑦_𝑚for𝑚≠𝑛.(2.4) The above expression implies that 𝑦_𝑛 is a random variable. Therefore, we propose the representation of the Sauerbrey equation on the round-by-round basis by𝑥(𝑡)=𝑦_𝑛𝑡,𝑓₀,Δ𝑚,𝐴[𝑢(𝑡)−𝑢(𝑡−𝑛𝑇)],𝑛=1,2,…,(2.5) where 𝑢(𝑡) is the unit step function. We call (2.5) the modified representation of the Sauerbrey equation of type II. In what follows, Δ𝑓 as well as 𝑥(𝑡) are in the sense of (2.5). For convenience, we may write 𝑥(𝑡) as 𝑥(𝑡)=𝑥𝑡,𝑓₀,Δ𝑚,𝐴.(2.6)

3. Fractal Analysis of the Pulse Phenomenon of Frequency Response to Coated Sensors

Let 𝑋(𝑡) be a second-order stationary random process or random function. Denote by 𝑝(𝑋) the probability density function (PDF) of 𝑋(𝑡). Then, the probability is given by𝑃𝑋₂−𝑃𝑋₁=Prob𝑋₁<𝜉<𝑋₂=^𝑋2_𝑋1𝑝(𝜉)𝑑𝜉.(3.1) The mean and the autocorrelation function (ACF) of 𝑋(𝑡) based on PDF is written by (3.2) and (3.3), respectively,𝑅𝜇_𝑋=^∞−∞𝑋𝑝(𝑋)𝑑𝑋,(3.2)_𝑋(𝜏)=^∞−∞𝑋(𝑡)𝑋(𝑡+𝜏)𝑝(𝑋)𝑑𝑋.(3.3) Let 𝑉_𝑥 be the variance of 𝑋. Then,𝑉_𝑋=E𝑋(𝑡)−𝜇_𝑋²=^∞−∞𝑋−𝜇_𝑋²𝑝(𝑋)𝑑𝑋.(3.4)

Note 1. If the tail of 𝑝(𝑋) is so heavy such that the integrals of (3.2) for 𝜇_𝑋 and (3.4) for 𝑉_𝑋 are divergent, we say that 𝑝(𝑋) is heavy tailed (Adler et al. [5], Li [6]).

Note 2. If 𝑝(𝑋) is heavy tailed, 𝑅_𝑋(𝜏) in (3.3) is slowly decayed. By slowly decayed, we mean that ^∞−∞𝑅_𝑥(𝜏)𝑑𝜏=∞.(3.5)

Note that (3.5) can be taken as a definition of LRD time series; see for example, Beran [7]. As a matter of fact, according to the Taqqu’s theorem, see Abry et al. [8], LRD property of a random function 𝑋(𝑡) is a consequence of the heavy-tailed 𝑝(𝑋) (Li [9]).

In the Gaussian assumption of 𝑋(𝑡), we have𝑝(𝑋)=1√2𝜋𝑉_𝑋𝑒^{−(𝑋−𝜇𝑋)2/2𝑉𝑋}.(3.6) In this case, the heavy-tailed 𝑝(𝑋) implies that 𝑉_𝑋→∞. In the engineering sense, we do not need infinite variance but 𝑉_𝑋 is large enough, see Li [10].

Note that Δ𝑓 in the experiments was sampled in the discrete case. Therefore, without loss of generality, we denote Δ𝑓 byΔ𝑓=𝑥𝑡,𝑓₀,𝐴,Δ𝑚=𝑥𝑡_𝑖,(3.7) where 𝑥(𝑡_𝑖) represents the value of Δ𝑓 of the 𝑖th sample, and 𝑖 is the sample index. Figure 2 indicates the series of 𝑥(𝑡_𝑖), where the vertical coordinates is indicated by 𝑥(𝑡) and abscissa axis by 𝑡 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 𝑥(𝑡). 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 Δ𝑓 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 𝑥(𝑖)=𝑥(𝑡_𝑖), where 𝑖=1,…,400. Then, we replot Figure 2 by Figure 3.

Figure 2 

Measured Δ𝑓 of the Pd-doped ZnO nanotetrapods to the concentration of NH3 from 0 to 240 ppm in time.

Figure 3 

Measured Δ𝑓 of the Pd-doped ZnO nanotetrapods to the concentration of NH3 from 0 to 240 ppm in sampling index.

Taking into account the random pulse width, pulse amplitude, and pulse transition, we say that 𝑥(𝑖) is a random function. By numeric computations, we obtain the ACF of Δ𝑓 as indicated in Figure 4, which implies the remark below.

Figure 4 

ACF of 𝑥(𝑖).

Remark 3.1. The random function  𝑥(𝑖), that is, Δ𝑓, 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 Δ𝑓 described by (2.1) is heavy tailed.

Remark 3.1 implies that the ACF of 𝑥(𝑖) has the asymptotic expression given by𝑅(𝑘)∼𝑐𝑘^−𝑏(𝑘⟶∞),𝑏∈(0,1),(3.8) where 𝑐>0 can be either a constant or a slowly varying function.

Denote by 𝑆_𝑥(𝜔) the power spectrum density (PSD) of 𝑥(𝑖), where 𝜔 is the angular frequency. Then,𝑆_𝑥(𝜔)=^∞−∞𝑅_𝑥(𝜏)𝑒^{−𝑗𝜔𝜏}𝑑𝜏.(3.9) Since (3.5) holds, 𝑆_𝑥(𝜔) has to be considered in the domain of generalized functions. According to the Fourier transform in the domain of generalized functions (Kanwal [11], Li and Lim [12]), we immediately obtain F||𝑘||^−𝑏=2sin𝜋𝑏2Γ(1−𝑏)|𝜔|^𝑏−1,(3.10) where F stands for the operator of the Fourier transform. Therefore, for the measured Δ𝑓, we have the asymptotic expression of the PSD of 𝑥(𝑖) below:F[𝑅(𝑘)]∼|𝜔|^𝑏−1for𝜔⟶0.(3.11) Hence, we have the following remark. We note that 𝜔 in (3.9), (3.10), and (3.11) is the angular frequency, which is an argument in the Fourier transform of the ACF of 𝑥(𝑖), while 𝑥(𝑖) stands for the frequency increment in the Sauerbrey equation. That is, the frequency in 𝑆_𝑥(𝜔) differs in meaning from that in the Sauerbrey equation.

Remark 3.3. The measuredΔ𝑓, that is, 𝑥(𝑖), is in the class of 1/𝑓 noise.

4. Discussions

The previous discussions exhibit three properties of 𝑥(𝑖) from a view of fractals. They are heavy-tailed PDF, slowly decayed ACF, and 1/𝑓 noise type PSD. Considering the Hurst parameter, we have 𝐻=1−𝑏2.(4.1) which characterizes the LRD of 𝑥(𝑖) from a view of fractals. By the least square fitting, for the curve in Figure 4, we have 𝐻≈1.(4.2) 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 𝑛th experiment is strongly correlated to that from the (𝑛+𝑘)th experiment, even if 𝑘 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 [2].

We note that this paper does not say that 𝑥(𝑖) 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 𝑥(𝑖) either. In the future, we will work on the accurate ACF and the fractal dimension of 𝑥(𝑖). 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 𝑥(𝑖) appears deterministic but we did fractal analysis about it to provide an explanation of the uncertainty principle in measurements.

5. Conclusions

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.

Appendix

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 𝑓₀. Deposition of the working electrode layer dampens this resonant frequency. The Sauerbrey equation relates the dampening of frequency, Δ𝑓₀, to the change in surface attached mass Δ𝑚.

The Sauerbrey equation was first introduced by Sauerbrey in [3]. Its standard form is given byΔ𝑓=−2Δ𝑚𝑓²⁰𝐴√𝜌_𝑞𝜇_𝑞=−2𝑓²⁰𝐴√𝜌_𝑞𝜇_𝑞Δ𝑚,(A.1) where 𝑓₀ is the resonant frequency (Hz), Δ𝑓 is the frequency change (Hz), Δ𝑚 is the mass change (𝑔), 𝐴 is the piezoelectrically active crystal area (cm²) between electrodes, 𝜌_𝑞 is the density of quartz, and 𝜇_𝑞 is the shear modulus of quartz for AT-cut crystal. In the experiments described in [2], 𝜇𝜌_𝑞𝑞=2.648g/cm³,=2.947×10¹¹g/cms².(A.2) Replacing 𝜌_𝑞 and 𝜇_𝑞 in (A.1) with (A.2) produces Δ𝑓=−2.26×10⁻⁶(𝑓²⁰/𝐴)Δ𝑚, which is the expression (2.1).

Acknowledgments

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).