Improved Determination of Q Quality Factor and Resonance Frequency in Sensors Based on the Magnetoelastic Resonance Through the Fitting to Analytical Expressions

materials Article Improved Determination of Q Quality Factor and Resonance Frequency in Sensors Based on the Magnetoelastic Resonance Through the Fitting to Analytical Expressions Beatriz Sisniega 1,* , Jon Gutiérrez 1,2 , Virginia Muto 3and Alfredo García-Arribas 1,2 1Departamento de Electricidad y Electrónica, Universidad del País Vasco/Euskal Herriko Unibertsitatea (UPV/EHU), Barrio Sarriena s/n, 48940 Leioa, Spain; [email protected] (J.G.); alfredo.gar[email protected] (A.G.-A.) 2BCMaterials, Basque Center for Materials, Applications and Nanostructures, UPV/EHU Science Park, 48940 Leioa, Spain 3 Departamento de Matem á tica Aplicada y Estad í stica e Investigaci ó n Operativa, Universidad del Pa í s Vasco UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain; [email protected] *Correspondence: [email protected] Received: 17 September 2020; Accepted: 19 October 2020; Published: 22 October 2020   Abstract: The resonance quality factor Q is a key parameter that describes the performance of magnetoelastic sensors. Its value can be easily quantified from the width and the peak position of the resonance curve but, when the resonance signals are small, for instance when a lot of damping is present (low quality factor), this and other simple methods to determine this parameter are highly inaccurate. In these cases, numerical fittings of the resonance curves allow to accurately obtain the value of the quality factor. We present a study of the use of different expressions to numerically fit the resonance curves of a magnetoelastic sensor that is designed to monitor the precipitation reaction of calcium oxalate. The study compares the performance of both fittings and the equivalence of the parameters obtained in each of them. Through these numerical fittings, the evolution of the different parameters that define the resonance curve of these sensors is studied, and their accuracy in determining the quality factor is compared. Keywords: magnetoelastic resonance; quality factor; resonance curve fit 1. Introduction Magnetoelastic resonance sensors are usually made of amorphous, ribbon-shaped ferromagnetic alloys [ 1 ]. Magnetostriction in these materials provides a strong coupling of their magnetic and mechanical properties, so that when an external alternating magnetic field is applied to a magnetoelastic ribbon, elastic waves are induced in it. Conversely, the magnetic state of these materials is highly dependent on the application of external mechanical forces and loads, and therefore these mechanical variations generate changes in the magnetic flux that can be remotely detected by a detection coil placed on the ribbon. Matching the physical dimensions and elastic properties of the material used in the sensor, the phenomenon of magnetoelastic resonance occurs at specific frequencies that can be expressed as: fn=n 2LsE ρ(1) where L is the length of the ribbon, E is the Young’s modulus of the material, ρ its density, and n= 1 corresponds to the fundamental resonant frequency. This resonant behavior is highly sensitive to Materials 2020,13, 4708; doi:10.3390/ma13214708 www.mdpi.com/journal/materials Materials 2020,13, 4708 2 of 15 different external parameters, which has generated a great interest in the use of these materials as a main part of different sensing systems [ 2 ]. Magnetoelastic sensors have been used to measure several environmental parameters such as pressure, humidity or temperature [ 3 , 4 ], liquid viscosity and density [ 5 , 6 ], chemical agents and pH [ 7 , 8 ], and biological agents [ 9 – 11 ]. Specifically, and with respect to its use as a mass sensor, when the magnetoelastic ribbon is loaded with a uniform and small amount of mass ( δm ), a shift in its resonance frequency occurs, that can be evaluated from Equation (1) to be, to first order approximation: δf=−1 2 δm m0 f0(2) where f0is the resonance frequency of the bare material and m0the initial mass of the sensor. The performance of this type of mass sensor is mainly determined by two parameters: the sensitivity of the sensor ( S ) and the quality factor ( Q ) [ 12 ]. The sensitivity indicates the minimum detectable frequency change when the mass of the sensor increases: S=−δf δm(3) The quality factor Q is a dimensionless parameter that describes how sharp the resonance curve is, and decreases when the resonator is damped. Damping influences the shape of the magnetoelastic resonance curve and the resonance frequency itself, considered as the frequency of the maximum of the curve. A high Q value indicates a lower rate of energy loss, a sharp magnetoelastic resonance peak (Figure 1) and therefore, good resolution when determining the resonance frequency and frequency shifts. For developing high performance sensors, the highest Sand Qare desirable. Materials 2020, 13, x FOR PEER REVIEW 2 of 15 to different external parameters, which has generated a great interest in the use of these materials as a main part of different sensing systems [2]. Magnetoelastic sensors have been used to measure several environmental parameters such as pressure, humidity or temperature [3,4], liquid viscosity and density [5,6], chemical agents and pH [7,8], and biological agents [9–11]. Specifically, and with respect to its use as a mass sensor, when the magnetoelastic ribbon is loaded with a uniform and small amount of mass (𝛿𝑚), a shift in its resonance frequency occurs, that can be evaluated from Equation (1) to be, to first order approximation: 𝛿𝑓=−12𝛿𝑚 𝑚 𝑓  (2) where 𝑓 is the resonance frequency of the bare material and 𝑚 the initial mass of the sensor. The performance of this type of mass sensor is mainly determined by two parameters: the sensitivity of the sensor (𝑆) and the quality factor (𝑄) [12]. The sensitivity indicates the minimum detectable frequency change when the mass of the sensor increases: 𝑆=−𝛿𝑓 𝛿𝑚 (3) The quality factor 𝑄 is a dimensionless parameter that describes how sharp the resonance curve is, and decreases when the resonator is damped. Damping influences the shape of the magnetoelastic resonance curve and the resonance frequency itself, considered as the frequency of the maximum of the curve. A high 𝑄 value indicates a lower rate of energy loss, a sharp magnetoelastic resonance peak (Figure 1) and therefore, good resolution when determining the resonance frequency and frequency shifts. For developing high performance sensors, the highest 𝑆 and 𝑄 are desirable. Figure 1. Two magnetoelastic resonance curves from the same resonator with different 𝑄 values, caused by different amount of mass loading. The peak corresponding to the higher 𝑄 (blue) is sharper and narrower, the resonance curve with a lower 𝑄 is wider (red). The quality factor can be estimated from the resonance curve as the experimentally measured resonance frequency 𝑓 (at which the amplitude is maximum, 𝐴), divided by the width of the signal ∆𝑓 (measured as the full width at half maximum, corresponding to amplitudes 𝐴/√2) [13]: 𝑄= 𝑓  ∆ 𝑓 (4) But the accuracy of this estimation method is very low (it can lead to errors up to 20% in the determination of 𝑄) [14], so finding alternative and robust methods to determine this parameter with greater accuracy is of great importance for the study of the performance of these sensors, especially in cases where the curves have considerable noise or damping, as it is the case when the Figure 1. Two magnetoelastic resonance curves from the same resonator with different Q values, caused by different amount of mass loading. The peak corresponding to the higher Q (blue) is sharper and narrower, the resonance curve with a lower Qis wider (red). The quality factor can be estimated from the resonance curve as the experimentally measured resonance frequency fmax (at which the amplitude is maximum, Amax ), divided by the width of the signal ∆f (measured as the full width at half maximum, corresponding to amplitudes Amax/√2 ) [ 13 ]: Q=fmax ∆f(4) But the accuracy of this estimation method is very low (it can lead to errors up to 20% in the determination of Q ) [ 14 ], so finding alternative and robust methods to determine this parameter with Materials 2020,13, 4708 3 of 15 greater accuracy is of great importance for the study of the performance of these sensors, especially in cases where the curves have considerable noise or damping, as it is the case when the magnetoelastic sensor operates immersed in a fluid. Some studies have been made on the different methods of determining this quality factor Q [ 13 , 15 ], and Lopes et al. [ 16 ] reported an extensive study of the determination of the quality factor of a magnetoelastic sensor using different strategies. In this last work, a numerical fitting of the experimental data to an expression of the magnetic susceptibility of the sample around the resonance is carried out and used as a suitable technique to calculate the value of Q . In the present work, the numerical fittings of the resonance curves obtained with a magnetoelastic sensor have been carried out using two different expressions: on the one hand, the expression used by Lopes et al. [ 16 ], that appears as Equation (6) below in this work and, on the other hand, a phenomenological approach that describes the frequency response of these magnetoelastic sensors and that has already been used to fit these resonance curves [ 17 , 18 ], appearing as Equation (9) within this work. The performance of each of the fitting expressions to determine the quality factor Q has been studied, and their comparison and equivalence carried out. The resonance curves used in the fit analysis we will show in the following, correspond to those obtained in the monitoring of the precipitation reaction of calcium oxalate salt crystals (inorganic salts resulting from various metabolic activities in humans) using a magnetoelastic sensor. This monitoring process was first described by Bouropoulos et al. [ 19 ], and the measurements obtained in this reaction sensing and used in the present work are explained in detail in a previous publication [ 20 ]. Through the numerical fitting of these curves, we have studied the evolution of the different parameters that define the resonance curve as the precipitation reaction progress, which provides a better understanding of the effect that the changes of mass in the sensor and damping have on the resonance signal obtained. 2. Materials and Methods 2.1. Magnetoelastic Material of the Sensor The material used as sensor platform is a magnetoelastic ribbon of composition Fe73Cr5Si10B12 provided by Vacuumschmelze GmbH & Co. KG, Hanau Germany. This material has an excellent corrosion resistance behavior due to the fact that it contains a small amount of chromium (5% at .), which forms a superficial passivation layer in the material. Magnetic, magnetoelastic and corrosion resistance properties of this material were studied by Sagasti et al. [ 21 ], and are summarized in Table 1. The strip was laser cut with dimensions 20 mm ×2 mm ×25 µm (length to width ratio R=10). Table 1. Magnetic, magnetoelastic, and corrosion behaviour parameters of the Fe73Cr5Si10B12 sample [ 21 ]. Ms is the spontaneous magnetization, λs the saturation magnetostriction, ∆E the change in the Young’s modulus with the applied magnetic field (or ∆E effect), k the magnetoelastic coupling coefficient and Ecorr the corrosion potential. Composition µ0Ms(T)λs(ppm) ∆E(%)kEcorr (mV) Corrosion Rate (µm/Year) Fe73Cr5Si10B12 1.12 14 17 0.41 47 0.035 2.2. Precipitation Reaction Measurements The measurements used in this work to make the numerical fittings correspond to the monitoring of the precipitation reaction of calcium oxalate crystals (CaC2O4): CaCl2(aq)+H2C2O4(aq)→CaC2O4(s)+2HCl(aq)(5) The formation of the salt crystals during the precipitation process was tracked by detecting changes in the sensor resonance frequency fmax , as fully described by Sisniega et al. [ 20 ]. The magnetoelastic sensor described in the previous subsection was placed in a vial with a mixture of equal parts of the Materials 2020,13, 4708 4 of 15 reagents ( CaCl2 and H2C2O4 ) at the same concentration (for three different concentrations: 30, 50, and 100 mM) and the changes in the resonance frequency of the sensor (determined by Equation (2)) were monitored along time, as the crystals precipitate and settle onto the sensor (increasing its mass). The experimental setup used to register the resonance curves consists of three coaxial solenoids: one to apply a constant bias field; a second one to produce the alternating field to magnetostrictively excite the sample; and the third one, consisting of a compensated pick-up coil, to detect the induced magnetization oscillations on the sensor. A spectrum analyzer (HP 3589A, Hewlett-Packard, Palo Alto, CA, USA) working in swept mode is used to produce the excitation and to receive the signal induced in the pick-up coil. After recording the resonance curves, the frequency of the maximum fmax and its amplitude are measured using the built-in analysis procedures of the analyzer and transmitted to a control computer. The sweep speed was set looking for a compromise between good curve resolution and the necessary speed to be able to record the whole resonance-antiresonance curve fast enough to follow the precipitation process. With this configuration, the experimental uncertainty in the frequency determination is 100 Hz, and the measurement noise can be up to 1 kHz when the resonance curves are wide and with low amplitude. Figure 2presents an example of the results obtained in the precipitation monitoring measurements. The curves in Figure 2a show the decrease of the sensor resonance frequency ( fmax ) as the precipitation takes place, for the three solutions with different concentrations. Figure 2b shows, for one particular solution (30 mM), the evolution of the complete resonance curve as the reaction progresses (curves taken at different reaction times). It can be seen that, in addition to the decrease in the resonance frequency, there is also a reduction of the quality factor Q that characterizes the resonance (or increase in damping), and a decrease in the amplitude of the sensor signal. The resonance curves displayed in Figure 2b were used in the present work to carry out the numerical fittings, together with analogous curves corresponding to the solutions of concentration 50 and 100 mM. Successive experiments in the same conditions yield similar results. Here we present the analysis in a single run for each concentration. Nine resonance curves corresponding to different reaction times were fitted for each concentration. Materials 2020, 13, x FOR PEER REVIEW 4 of 15 the sensor (increasing its mass). The experimental setup used to register the resonance curves consists of three coaxial solenoids: one to apply a constant bias field; a second one to produce the alternating field to magnetostrictively excite the sample; and the third one, consisting of a compensated pick-up coil, to detect the induced magnetization oscillations on the sensor. A spectrum analyzer (HP 3589A, Hewlett-Packard, Palo Alto, CA, USA) working in swept mode is used to produce the excitation and to receive the signal induced in the pick-up coil. After recording the resonance curves, the frequency of the maximum 𝑓 and its amplitude are measured using the built-in analysis procedures of the analyzer and transmitted to a control computer. The sweep speed was set looking for a compromise between good curve resolution and the necessary speed to be able to record the whole resonanceantiresonance curve fast enough to follow the precipitation process. With this configuration, the experimental uncertainty in the frequency determination is 100 Hz, and the measurement noise can be up to 1 kHz when the resonance curves are wide and with low amplitude. Figure 2 presents an example of the results obtained in the precipitation monitoring measurements. The curves in Figure 2a show the decrease of the sensor resonance frequency (𝑓) as the precipitation takes place, for the three solutions with different concentrations. Figure 2b shows, for one particular solution (30 mM), the evolution of the complete resonance curve as the reaction progresses (curves taken at different reaction times). It can be seen that, in addition to the decrease in the resonance frequency, there is also a reduction of the quality factor 𝑄 that characterizes the resonance (or increase in damping), and a decrease in the amplitude of the sensor signal. The resonance curves displayed in Figure 2b were used in the present work to carry out the numerical fittings, together with analogous curves corresponding to the solutions of concentration 50 and 100 mM. Successive experiments in the same conditions yield similar results. Here we present the analysis in a single run for each concentration. Nine resonance curves corresponding to different reaction times were fitted for each concentration. (a) (b) Figure 2. (a) Temporal evolution of the magnetoelastic resonance frequency of the sensor during the reaction of precipitation for different concentrations (30 mM, 50 mM and 100 mM) and a control curve (sensor in a vial with distilled water). The value of the resonance frequency, 𝑓, is obtained as the frequency at the maximum amplitude of measured resonance curves (Figure 2b); (b) Measured magnetoelastic resonance curve signals of the sensor at different times during the precipitation process for the concentration 30 mM. Data taken from Ref. [20]. 2.3. Numerial Fitting of the Sensor Resonance Curves The quality factor 𝑄 of the resonance can be quantified from the width and the peak position of the resonance curve. However, when the resonance signals are heavily damped this and other simple methods are unable to determine this parameter or are highly inaccurate. We propose to use a frequency response fitting method to determine the quality factor and other parameters that Figure 2. ( a ) Temporal evolution of the magnetoelastic resonance frequency of the sensor during the reaction of precipitation for different concentrations (30 mM, 50 mM and 100 mM) and a control curve (sensor in a vial with distilled water). The value of the resonance frequency, fmax , is obtained as the frequency at the maximum amplitude of measured resonance curves (Figure 2b); ( b ) Measured magnetoelastic resonance curve signals of the sensor at different times during the precipitation process for the concentration 30 mM. Data taken from Ref. [20]. Materials 2020,13, 4708 5 of 15 2.3. Numerial Fitting of the Sensor Resonance Curves The quality factor Q of the resonance can be quantified from the width and the peak position of the resonance curve. However, when the resonance signals are heavily damped this and other simple methods are unable to determine this parameter or are highly inaccurate. We propose to use a frequency response fitting method to determine the quality factor and other parameters that characterize the resonance curves. In 1978, Savage et al. [ 22 ] derived an expression for the susceptibility of the magnetoelastic resonator around the magnetoelastic resonance given by χ(ω)=χ0 1−8k2 π2X n 1 n2×1 1−ω2 n ω2+jQ−1ωn ω (6) where k is the magnetoelastic coupling coefficient ( k=sπ2 8 1−fr fa2! , fr and fa being the resonance and anti-resonance frequencies, respectively), ωn= 2 πfn is the resonance frequency of the nth harmonic of the excited fundamental mode ( n= 1, f1=fr ), Q−1 is a phenomenological damping coefficient, χ0 is the magnetic susceptibility measured at a frequency far below the resonance and j =√−1 . Due to the experimental procedure used to measure the resonance curves [ 20 ], the voltage induced in the pick-up coil (which is displayed in Figures 1and 2b) is proportional to the magnetic susceptibility. The parameters to be fitted in Equation (6) are: ωr , ωa , χ0 , and Q . The numerical fitting of the experimental data to the modulus or magnitude of the Equation (6) for its first resonant mode ( n= 1) was performed using Mathematica © software. The parameters ωr , ωa , and χ0 , were allowed to vary around their experimental corresponding values (extracted from the experimental data), until the value of Q that minimizes the norm between the fit and the experimental data was found. The norm (or residual) is defined as: R=1 NX i=1,N χexp,i−χfit,i χmax !2 (7) where χmax =maxχmax,exp,χmax,fit , N is the number of experimental points ( N= 401, in our measurements), and 0 ≤ R ≤ 1, meaning values close to 0 that the fitting is good. Additionally, a phenomenological expression using the formalism of linear systems was developed to describe the frequency response of a magnetoelastic material [ 17 ]. The magnetoelastic resonance is characterized by maximum amplitude at the resonance frequency fr , and a null minimum at the anti-resonance frequency fa . The analytical expression of the transference function ( G ) that describes a system with such frequency response is composed of a couple of complex conjugated poles in the denominator (describing the resonance) and a couple of complex conjugated zeros in the numerator (describing the anti-resonance): G(s)=ω2 r ω2 a·s2+2δaωas+ω2 a s2+2δrωrs+ω2 r (8) where ωr= 2 πfr is the resonance frequency, ωa= 2 πfa the anti-resonance frequency, and δr and δa are damping parameters. The transfer function is expressed as a Laplace operator with s=jω , where j=√−1 and ωis the frequency. The amplitude of the frequency response of the system, when submitted to harmonic excitation, based on the frequency response described on the transference function is: V(ω)=A·ω2 r ω2 a ω2−2jδaωaω−ω2 a ω2−2jδrωrω−ω2 r +aω+b(9) Materials 2020,13, 4708 6 of 15 where A accounts for the amplitude and a and b for the background signal of the experiment (assumed to be linear). The parameters to be fitted in the expression are: ωr , ωa , δr , δa , A , a and b . Note that in this expression, the quality factor Q does not appear explicitly. Instead, the damping parameters δr and δa carry the relevant information. The numerical fitting of Equation (9) to the experimental data was carried out using MATLAB ® software and a nonlinear least-squares fitting. The parameters ωr , ωa , and Ato be fitted are initially taken as their values obtained from the experimental curve. All parameters are fitted until those that best fit the experimental curve are found, in this case those that correspond to a local minimum of a function that is a sum of squared residuals (being the residual the difference between the experimental value of the dependent variable and the value predicted by the fitting model). 3. Results and Discussion 3.1. Numerical Fitting of the Resonance Curves and Resonance Frequency Numerical fittings were performed to the resonance curves measured at different times in the precipitation reaction, for solution with different concentrations (30, 50 and 100 mM). Figure 3shows an example of the numerical fittings made with Equation (6), while Figure 4shows the same fitting but using Equation (9). It can be seen that both models fit the experimental data reasonably well, resulting in a low value of the corresponding residuals. Materials 2020, 13, x FOR PEER REVIEW 6 of 15 3. Results and Discussion 3.1. Numerical Fitting of the Resonance Curves and Resonance Frequency Numerical fittings were performed to the resonance curves measured at different times in the precipitation reaction, for solution with different concentrations (30, 50 and 100 mM). Figure 3 shows an example of the numerical fittings made with Equation (6), while Figure 4 shows the same fitting but using Equation (9). It can be seen that both models fit the experimental data reasonably well, resulting in a low value of the corresponding residuals. (a) (b) Figure 3. Fit to Equation (6). (a) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (6); (b) Measured magnetoelastic resonance curve, fitting to Equation (6) (in dashed line), and corresponding error (in red) of the sensor at time t = 125 s for reactants of concentration 30 mM. (a) (b) Figure 4. Fit to Equation (9). (a) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (9); (b) Measured magnetoelastic resonance curve, fitting to Equation (9) (in dashed line), and corresponding error (in red) of the sensor at time t = 125 s for reactants of concentration 30 mM. Figure 3. Fit to Equation (6). ( a ) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (6); ( b ) Measured magnetoelastic resonance curve, fitting to Equation (6) (in dashed line), and corresponding error (in red) of the sensor at time t=125 s for reactants of concentration 30 mM. However, when the deposited mass is high (at high reaction times or higher concentration of reagents), the fitting to Equation (9) behaves slightly better (with lower error between the experimental data and the values obtained through the model) than the one using the Equation (6). Materials 2020,13, 4708 7 of 15 Materials 2020, 13, x FOR PEER REVIEW 6 of 15 3. Results and Discussion 3.1. Numerical Fitting of the Resonance Curves and Resonance Frequency Numerical fittings were performed to the resonance curves measured at different times in the precipitation reaction, for solution with different concentrations (30, 50 and 100 mM). Figure 3 shows an example of the numerical fittings made with Equation (6), while Figure 4 shows the same fitting but using Equation (9). It can be seen that both models fit the experimental data reasonably well, resulting in a low value of the corresponding residuals. (a) (b) Figure 3. Fit to Equation (6). (a) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (6); (b) Measured magnetoelastic resonance curve, fitting to Equation (6) (in dashed line), and corresponding error (in red) of the sensor at time t = 125 s for reactants of concentration 30 mM. (a) (b) Figure 4. Fit to Equation (9). (a) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (9); (b) Measured magnetoelastic resonance curve, fitting to Equation (9) (in dashed line), and corresponding error (in red) of the sensor at time t = 125 s for reactants of concentration 30 mM. Figure 4. Fit to Equation (9). ( a ) Measured magnetoelastic resonance curves of the sensor at different times during the precipitation process for reactants of concentration 30 mM, and corresponding numerical fittings (in dashed lines) to Equation (9); ( b ) Measured magnetoelastic resonance curve, fitting to Equation (9) (in dashed line), and corresponding error (in red) of the sensor at time t=125 s for reactants of concentration 30 mM. If we analyze the evolution of the parameters obtained by the fitting procedure, we can see that the two damping parameters in Equation (9) ( δr and δa ) increase progressively as the mass deposited in the sensor increases (Figure 5). As the reaction advances, the crystals are formed in the solution of the vial and are deposited on the surface of the sensor, increasing its mass, and this increases the damping suffered by this sensor. In the same manner, as the concentration of the reagents increases (30, 50, 100 mM), the mass of calcium oxalate crystals is greater and, therefore, so is the damping. The numerical fittings show this direct relationship between the deposited mass and the damping in the resonance curves obtained. Similarly, it can be seen that the Q parameter obtained by fitting the data to Equation (6) decreases as the precipitation reaction progresses and the mass is deposited on the sensor (Figure 6). Materials 2020, 13, x FOR PEER REVIEW 7 of 15 However, when the deposited mass is high (at high reaction times or higher concentration of reagents), the fitting to Equation (9) behaves slightly better (with lower error between the experimental data and the values obtained through the model) than the one using the Equation (6). If we analyze the evolution of the parameters obtained by the fitting procedure, we can see that the two damping parameters in Equation (9) ( 𝛿 and 𝛿) increase progressively as the mass deposited in the sensor increases (Figure 5). As the reaction advances, the crystals are formed in the solution of the vial and are deposited on the surface of the sensor, increasing its mass, and this increases the damping suffered by this sensor. In the same manner, as the concentration of the reagents increases (30, 50, 100 mM), the mass of calcium oxalate crystals is greater and, therefore, so is the damping. The numerical fittings show this direct relationship between the deposited mass and the damping in the resonance curves obtained. Similarly, it can be seen that the 𝑄 parameter obtained by fitting the data to Equation (6) decreases as the precipitation reaction progresses and the mass is deposited on the sensor (Figure 6). (a) (b) Figure 5. Evolution of the damping parameters obtained through the numerical fitting to Equation (9) of the resonance curves during the precipitation time for different concentration of reagents (30, 50, and 100 mM). (a) Damping parameter 𝛿; (b) Damping parameter 𝛿. Figure 6. Evolution of the quality factor 𝑄 of the resonator during the precipitation process for different reagents concentrations (30, 50, and 100 mM) obtained through the numerical fittings to Equation (6) of the resonance curves. Figure 5. Evolution of the damping parameters obtained through the numerical fitting to Equation (9) of the resonance curves during the precipitation time for different concentration of reagents (30, 50, and 100 mM). (a) Damping parameter δr; (b) Damping parameter δa. Materials 2020,13, 4708 8 of 15 Materials 2020, 13, x FOR PEER REVIEW 7 of 15 However, when the deposited mass is high (at high reaction times or higher concentration of reagents), the fitting to Equation (9) behaves slightly better (with lower error between the experimental data and the values obtained through the model) than the one using the Equation (6). If we analyze the evolution of the parameters obtained by the fitting procedure, we can see that the two damping parameters in Equation (9) ( 𝛿 and 𝛿) increase progressively as the mass deposited in the sensor increases (Figure 5). As the reaction advances, the crystals are formed in the solution of the vial and are deposited on the surface of the sensor, increasing its mass, and this increases the damping suffered by this sensor. In the same manner, as the concentration of the reagents increases (30, 50, 100 mM), the mass of calcium oxalate crystals is greater and, therefore, so is the damping. The numerical fittings show this direct relationship between the deposited mass and the damping in the resonance curves obtained. Similarly, it can be seen that the 𝑄 parameter obtained by fitting the data to Equation (6) decreases as the precipitation reaction progresses and the mass is deposited on the sensor (Figure 6). (a) (b) Figure 5. Evolution of the damping parameters obtained through the numerical fitting to Equation (9) of the resonance curves during the precipitation time for different concentration of reagents (30, 50, and 100 mM). (a) Damping parameter 𝛿; (b) Damping parameter 𝛿. Figure 6. Evolution of the quality factor 𝑄 of the resonator during the precipitation process for different reagents concentrations (30, 50, and 100 mM) obtained through the numerical fittings to Equation (6) of the resonance curves. Figure 6. Evolution of the quality factor Q of the resonator during the precipitation process for different reagents concentrations (30, 50, and 100 mM) obtained through the numerical fittings to Equation (6) of the resonance curves. Figure 6shows the inverse relation between the quality factor Q and the damping parameters ( δr and δa ). All these parameters show a clear and marked tendency as the precipitation reaction occurs. The evolution of resonance parameters, preferentially the resonance frequency, can be used to monitor the reaction by the changes caused by the mass deposition on the sensor signal. Numerical fittings improve the information obtained through the changes observed in the measured resonance frequency, since the result comes from the whole resonance curve, not only from the point of maximum amplitude and the width at half maximum. In addition, the fitting procedure is useful to reduce the noise, especially when the signal is low or highly damped, as evidenced in Figure 7a. This plot displays the frequency of the maxima (that is, the resonance frequency fmax ) of all the resonance curves measured during the precipitation process of the solution with concentration 50 mM. The data obtained directly from the measurements present a considerable ripple with an amplitude close to 1 kHz, which severely limits the resolution of the sensor. It is mainly caused by the limited resolution with which frequency differences are discriminated by the measuring system. The evolution of the frequency of the maxima obtained from the fitting of the resonance curves displays a smooth monotonous tendency, which facilitates the calibration and use of the sensor. One important consequence of the fitting procedure is that the values of the fr parameter obtained in the fittings do not correspond to fmax , the position of the maxima of the resonance curves. Figure 7b compares the fitted values from both fitting expressions with the experimental resonance frequency (fmax, taken as the maximum of the resonance curve). This tells us that what we usually take as the resonance frequency, that is the frequency of the maximum of the resonance curve, is not really the physical resonance frequency of the sensor, fr which can be obtained from the fitting to the analytical expressions. The frequency of the maximum of the measured curves, fmax , is systematically lower than the resonance frequency, fr , since it is affected by the damping of the curve. The same applies to the frequency of anti-resonance (minimum of the curve), which, when affected by damping due to the mass deposition, increases with respect to the value, fa , which is obtained from the fits. Figure 8a illustrates this discrepancy, showing the frequency response of the system, for ideal resonance and anti-resonance separately, and for the full curve (sum of resonance and anti-resonance) affected by damping. In Figure 8b, the difference between the frequency of the maximum of the resonance fmax and the resonance frequency fr is shown as a function of the damping, and a linear relation between them is observed. Materials 2020,13, 4708 9 of 15 Materials 2020, 13, x FOR PEER REVIEW 8 of 15 Figure 6 shows the inverse relation between the quality factor 𝑄 and the damping parameters (𝛿 and 𝛿). All these parameters show a clear and marked tendency as the precipitation reaction occurs. The evolution of resonance parameters, preferentially the resonance frequency, can be used to monitor the reaction by the changes caused by the mass deposition on the sensor signal. Numerical fittings improve the information obtained through the changes observed in the measured resonance frequency, since the result comes from the whole resonance curve, not only from the point of maximum amplitude and the width at half maximum. In addition, the fitting procedure is useful to reduce the noise, especially when the signal is low or highly damped, as evidenced in Figure 7a. This plot displays the frequency of the maxima (that is, the resonance frequency 𝑓) of all the resonance curves measured during the precipitation process of the solution with concentration 50 mM. The data obtained directly from the measurements present a considerable ripple with an amplitude close to 1 kHz, which severely limits the resolution of the sensor. It is mainly caused by the limited resolution with which frequency differences are discriminated by the measuring system. The evolution of the frequency of the maxima obtained from the fitting of the resonance curves displays a smooth monotonous tendency, which facilitates the calibration and use of the sensor. One important consequence of the fitting procedure is that the values of the 𝑓 parameter obtained in the fittings do not correspond to 𝑓, the position of the maxima of the resonance curves. Figure 7b compares the fitted values from both fitting expressions with the experimental resonance frequency (𝑓, taken as the maximum of the resonance curve). (a) (b) Figure 7. (a) Evolution of the frequency corresponding to the maximum of the resonance curves resulting from the fitting to Equation (9) of the resonance curves measured during the precipitation, when the concentration of the reactants is 50 mM, compared to experimental data (𝑓𝑚𝑎𝑥); (b) Evolution of the 𝑓 obtained in the numerical fitting to Equations (6) and (9) of the measured resonance curves, compared to experimental data (𝑓𝑚𝑎𝑥). This tells us that what we usually take as the resonance frequency, that is the frequency of the maximum of the resonance curve, is not really the physical resonance frequency of the sensor, 𝑓 which can be obtained from the fitting to the analytical expressions. The frequency of the maximum of the measured curves, 𝑓, is systematically lower than the resonance frequency, 𝑓, since it is affected by the damping of the curve. The same applies to the frequency of anti-resonance (minimum of the curve), which, when affected by damping due to the mass deposition, increases with respect to the value, 𝑓, which is obtained from the fits. Figure 8a illustrates this discrepancy, showing the frequency response of the system, for ideal resonance and anti-resonance separately, and for the full curve (sum of resonance and anti-resonance) affected by damping. In Figure 8b, the difference Figure 7. ( a ) Evolution of the frequency corresponding to the maximum of the resonance curves resulting from the fitting to Equation (9) of the resonance curves measured during the precipitation, when the concentration of the reactants is 50 mM, compared to experimental data ( fmax ); ( b ) Evolution of the fr obtained in the numerical fitting to Equations (6) and (9) of the measured resonance curves, compared to experimental data (fmax ). Materials 2020, 13, x FOR PEER REVIEW 9 of 15 between the frequency of the maximum of the resonance 𝑓 and the resonance frequency 𝑓 is shown as a function of the damping, and a linear relation between them is observed. (a) (b) Figure 8. (a) Frequency response of the system for resonance and anti-resonance behavior, and the total response affected by the damping. Corresponding resonance and anti-resonance frequencies are indicated; (b) Evolution of the difference between the experimental resonance frequency 𝑓  and the resonance frequency obtained through the fitting to Equation (9) as the damping parameter 𝛿  increases, for the case of concentration 50 mM. The shift in the resonance frequency becomes greater as the damping increases. The curves in Figure 8a were obtained with the magnitude of the frequency response corresponding to the transfer functions representing systems of: a pure resonance (𝐺), a pure antiresonance (𝐺), and the combination of both (𝐺), which represents the observed resonanceantiresonance behaviour. 𝐺(𝑠)= 𝜔 𝑠+2𝛿𝜔𝑠+𝜔 (10) 𝐺(𝑠)=𝑠+2𝛿𝜔𝑠+𝜔 𝜔 (11) 𝐺(𝑠)=𝐺∙𝐺=𝜔 𝜔𝑠+2𝛿𝜔𝑠+𝜔 𝑠+2𝛿𝜔𝑠+𝜔 (12) The frequency values obtained for the resonance (maximum value) and anti-resonance (minimum value) of the respective curves in Figure 8a, coincide with those obtained in the fittings (𝑓 𝐹𝑖𝑡 and 𝑓 𝐹𝑖𝑡 in Figure 8a). In the combined curve, however, the maximum and minimum are shifted with respect to these values, resulting in the values of the resonance and anti-resonance frequencies (𝑓 𝐷𝑎𝑚𝑝𝑒𝑑 and 𝑓 𝐷𝑎𝑚𝑝𝑒𝑑 in Figure 8a), which match the experimental data. The mass increase on the resonator produces both a decrease of the maximum of the resonance frequency (𝑓) and a decrease of the quality factor. Figure 9 illustrates the correlation between both parameters. However, the observed reduction of 𝑓 combines the effect of mass loading (Equation (2)) and the effect described in Figure 8a. The analysis of the resonance curves through the fitting procedure allows the determination of 𝑓, which is, in principle, the one that is considered in Equation (2). Figure 8. ( a ) Frequency response of the system for resonance and anti-resonance behavior, and the total response affected by the damping. Corresponding resonance and anti-resonance frequencies are indicated; ( b ) Evolution of the difference between the experimental resonance frequency fmax and the resonance frequency obtained through the fitting to Equation (9) as the damping parameter δr increases, for the case of concentration 50 mM. The shift in the resonance frequency becomes greater as the damping increases. The curves in Figure 8a were obtained with the magnitude of the frequency response corresponding to the transfer functions representing systems of: a pure resonance ( G1 ), a pure anti-resonance ( G2 ), and the combination of both ( G3 ), which represents the observed resonance-antiresonance behaviour. G1(s)=ω2 r s2+2δrωrs+ω2 r (10)