**4. Numerical analysis (NNA) of the acoustoelectric interaction in the sensing layer in the steady state conditions**

The acoustic-electric effect depends on the electrical charge profile distribution in the sensing layer on the absorbed gas particles' distance from the surface acoustic waveguide.

To determine the response of the detectors common impedance was designated. Impedance includes information about the profile of the concentration of gas molecules in the layer and has been implemented into the Ingebrigtsen formula [23]. This enables describe relative change velocity surface acoustic waves in steady state and transient mode. Analytical expressions clearly define the model of the SAW detectors. Based on this model numerical analysis detector's response was made. The results of numerical analyzes are shown in the next section.

The analytical model of a SAW gas detector having the described form was used in order to analyze it numerically. In order to analyze such a detector layer in the SAW gas detector we assumed that the film is a uniform stack of infinitesimally thin sheets with a variable concentration of gas molecules (**Figure 15**) and that it influences the electrical conductance [10].

$$\begin{split} \frac{\Delta \boldsymbol{\nu}}{\nu\_{0}} &= -\operatorname{Re} \left\{ \frac{\Delta k}{k\_{0}} \right\} \\ &= -\frac{K^{2}}{2} \left[ \frac{\sigma\_{\rm{T}\_{z}} \left( 1 + a \mathbf{C}\_{\rm{A},y=0} \right) + \sum\_{i=1}^{u-1} \sigma\_{\rm{T}\_{z}} \langle \boldsymbol{y}\_{i} \rangle \boldsymbol{f} \left( \boldsymbol{y}\_{i}, \sigma\_{\rm{T}\_{z}} \langle \boldsymbol{y}\_{i} \rangle \right) \right]^{2}}{\left[ \sigma\_{\rm{T}\_{z}} \left( 1 + a \mathbf{C}\_{\rm{A},y=0} \right) + \sum\_{i=1}^{u-1} \sigma\_{\rm{T}\_{z}} \langle \boldsymbol{y}\_{i} \rangle \boldsymbol{f} \left( \boldsymbol{y}\_{i}, \sigma\_{\rm{T}\_{z}} \langle \boldsymbol{y}\_{i} \rangle \right) \right]^{2} + \left[ 1 + \sum\_{i=1}^{u-1} \mathbf{g} \left( \boldsymbol{y}\_{i}, \sigma\_{\rm{T}\_{z}} \langle \boldsymbol{y}\_{i} \rangle \right) \right]^{2} (\nu\_{0} \text{Fs})^{2}}{\left( 1 \right)} \end{split} \tag{1}$$

where: n—number of sublayers and *FS* <sup>¼</sup> *<sup>ε</sup>*<sup>0</sup> <sup>þ</sup> *<sup>ε</sup><sup>T</sup> <sup>p</sup>* , i = 1, 2, 3, … n (the sublayers index), Δv/v0 and Δk/k0 relative changes of velocity and wave vector of SAW,

**Figure 15.** *Schematic diagram of the SAW detector model [2].*

respectively, σ0—electrical conductivity of sensing layer in air, K<sup>2</sup> —coefficient of electromechanical coupling,

$$
\sigma(y\_i) = \sigma\_0 \left[ \mathbf{1} + a \cdot \mathbf{C}\_{\mathbf{A}}(y\_i) \right] \tag{2}
$$

where: *a—*sensitivity coefficient of the sensing layer [1/ppm].

In the expression (1) the functions f(yi, σ(yi)) and g(yi, σ(yi)) are the results of the transformation of the individual sublayer on the surface of a detector waveguide (**Figure 15**) and it has the form:

$$f(\boldsymbol{y}\_i, \sigma(\boldsymbol{y}\_i)) = \frac{\mathbf{1} - \left[\text{tgh}\left(k\mathbf{y}\_i\right)\right]^2}{\left[\mathbf{1} + \text{tgh}\left(k\mathbf{y}\_i\right)\right]^2 + \left[\text{tgh}\left(k\mathbf{y}\_i\right) \cdot \frac{\sigma(\boldsymbol{y}\_i)}{e\_{\boldsymbol{\theta}\boldsymbol{\theta}\boldsymbol{\theta}}}\right]^2} \tag{3}$$

and

$$f(\boldsymbol{y}\_i, \sigma(\boldsymbol{y}\_i)) = \frac{\mathbf{1} - \left[\text{tgh}\left(k\mathbf{y}\_i\right)\right]^2}{\left[\mathbf{1} + \text{tgh}\left(k\mathbf{y}\_i\right)\right]^2 + \left[\text{tgh}\left(k\mathbf{y}\_i\right) \cdot \frac{\sigma(\boldsymbol{y}\_i)}{e\_0 \nu\_0}\right]^2} \tag{4}$$

The conductivity of the detector layer depends on the temperature:

$$
\sigma\_{T\_2} = \sigma\_{T\_1} \exp\left(\frac{E\_\mathrm{g}}{2k\_B} \cdot \frac{T\_2 - T\_1}{T\_1 T\_2}\right) \tag{5}
$$

where T1 = 300 K, *σ<sup>T</sup>*<sup>1</sup> = *σ*<sup>0</sup> are the conductance of the layer, respectively, at temperature the T1 and T2, kB—the Boltzmann constant, vo—SAW velocity, k—acoustic wave number (*k=2π/λ*), *E*g—the width of the energy gap of detector layer material. The expressions (1, 3, 4) make it possible to determine the use of the iteration method response of the surface acoustic wave in the steady-state (t ! ∞).

For the selected parameters (thickness, concentration, temperature, sensitivity *a*, and conductivity) of bilayer sensor structure (Nafion+Polyanyline), the numerical analysis were made and the influence of carbon monoxide gas concentration on detection response was determined.

**Figure 16** presents the numerical analysis detector response depending on the thickness detector layer. From the analysis, it follows that the optimum thickness is approx. 90 nm. The software was written in Python. In the range of thicker layers

#### **Figure 16.**

*Changes of velocity (relative) vs. thickness from 2 nm to 180 nm, Nafion+Polyanyline. Assumed in the analysis: Sensitivity coefficient a =* �*4.25 ppm*�*<sup>1</sup> , <sup>σ</sup><sup>s</sup> = 2* � *<sup>10</sup>* �*<sup>12</sup> S, M = 28 g/Mol (CO), concentration: 5, 10, 15 ppm. Numerical results.*

*Numerical Analysis of the Steady State in SAW Sensor Structures with Selected Polymers… DOI: http://dx.doi.org/10.5772/intechopen.109367*

above 180 nm interaction decreases, which confirms obtained empirical results. From the characteristic, it follows that for the small concentration (below 5 ppm) measurements of carbon monoxide using an SAW detector for the layer above 180 nm and thicker will be useless (measurement temperature of 35°C, see **Figures 5, 6a**). This fact very explicitly confirms the experiment mentioned above for thickness layer of 180 nm and at the temperature of 35°C.

The numerical analysis of the influence of the CO concentration on the response of the detector in a layer thickness of 180 nm for a temperature of 35° C and 42° C (**Figure 6b**) was made (**Figure 17**).

A separate place is devoted to researching the temperature properties of the layer. It is important here to note that the selection of the operating temperature of the detector depends not only on the type of detector layer but also on the porosity and roughness (**Figure 18a** and **b**). Temperature change also allows to determine the operating point of the detector and also has an impact on the speed of response and recovery of the detector layer. Changing the operating point of the detector by increasing the temperature from 35°C to 42° C, should increase the detection properties of the detector in accordance with the experiment (**Figure 6a** and **b**). The effect of temperature on the response of the gas detector is also confirmed by the analysis of the properties of the temperature detector (**Figure 8**). The numerical analysis of the influence of the temperature on the response of the detector in a layer thickness of 180 nm for a temperature of 35°C (308 K) and 42°C (315 K) was made (**Figure 19**). Numerical analysis shows that based on the numerical model of a gas detector with a layer of Nafion + Polyaniline gas impact in the case of measuring the temperature increase of 35–42°C will result in a marked increase in the detector response.

The results of numerical analysis (**Figure 19**) coincide approximately with the results of the experiment (see **Figures 5**–**8**).

**Figure 17.**

*Changes of velocity (relative) vs. concentration from 2 to 25 ppm in synthetic air, Nafion+polyaniline at temperatures <sup>35</sup>°C and 42°C. assumed in the analysis: H = 180 nm (thickness), sensitivity coefficient a = 106 ppm<sup>1</sup> (T = 35°C), a = 107 ppm<sup>1</sup> (T = 42°C), <sup>σ</sup><sup>s</sup> = 2 <sup>10</sup> <sup>12</sup> S, M = 28 g/Mol (CO). Numerical results.*

**Figure 18.**

*The polymer layer (RR)-P3HT surface 3D-AFM view (a) graphic illustration of the average profile radius (b).*

#### **Figure 19.**

*Changes of velocity (relative) vs. temperature from 300 K to 344 K, Nafion+polyaniline, assumed in the analysis: A = 1 ppm<sup>1</sup> , <sup>σ</sup><sup>s</sup> = 2 <sup>10</sup> <sup>12</sup> S, M = 28 g/Mol (CO), concentration 5 ppm, thickness 80, 100, 180 nm. Numerical results.*

The sensitivity of sensing structures to external gas atmospheres depends on the temperature of the structure. The problem of the structure temperature is also important in the aspect of "detoxifying" its electrical properties after measurements. Our previous research [2] shows that detoxification is much more effective at elevated temperatures. On the other hand, however, the elevated temperature affects the mechanical degradation of the structure and causes irreversible changes in its physicochemical properties. This problem is particularly important when the sensing layers are made of organic semiconductors. Polyaniline and Nafion are organic semiconductors. In order not to destroy the examined structures, the measurements were made at relatively low temperatures: 35°C (308 K) and 42°C (315 K).

Analyzes for the sarin simulator—DMMP gas was also performed. The roughness of the polymer [25] detector layer for the average height of the layer profile from 60 nm to 960 nm (max. average height of the layer profile) theoretically and numerically was performed. The thickness using the atomic force microscope (AFM) profile analysis was estimated too. The acoustoelectric interaction of the surface wave with charge carriers distributed in the detector layer according to the profile resulting from the diffusion of gas molecules from the surrounding atmosphere was analyzed numerically in the Python programming environment, using the expression (1)–(5) [10]. The results of the numerical analysis can be the basis for the optimization of the layer parameters in terms of the maximum sensitivity of the detector. **Figures 20**, **21**, and **22**–**24** show exemplary results of the numerical analysis.

The best results in the range of 60–960 nm thicknesses have been achieved. In **Figure 20** analysis for a polymer (RR)-P3HT was performed. For the assumed DMMP gas ambient parameters (temperature, concentration 3 ppm) and layer parameters

#### **Figure 20.**

*Changes of velocity (relative) vs. thickness, from 80 nm to 960 nm, layer (RR)-P3HT, gas DMMP: σ<sup>s</sup> = 5 10<sup>4</sup> S, M = 124.08 g/Mol, K2/2 (quartz) =0.09%, C = 3 ppm.*

*Numerical Analysis of the Steady State in SAW Sensor Structures with Selected Polymers… DOI: http://dx.doi.org/10.5772/intechopen.109367*

#### **Figure 21.**

*Changes of velocity (relative) vs. roughness, temp. 305 K (32°C), (RR)-P3HT, a = 1 ppm<sup>1</sup> , <sup>σ</sup><sup>s</sup> = 5 <sup>10</sup><sup>4</sup> S, DK = 106 nm<sup>2</sup> s 1 , Eg = 2.7 eV, M = 124.08 g/Mol, DMMP concentration: 2 ppm, thickness 500 nm (H), K2/2 (quartz) =0.09%, measurement AFM Ra = 76 nm.*

#### **Figure 22.**

*Temperature range: 293–322 K (20–49°C)—Curve (theoretical numerical calculations): <sup>σ</sup><sup>s</sup> = 5 <sup>10</sup><sup>4</sup> S, M = 124.08 g/Mol, K2/2 (quartz) =0.09%, C = 2 ppm.*

#### **Figure 23.**

*Curve (theoretical numerical calculations) for parameters: <sup>σ</sup><sup>s</sup> = 5 <sup>10</sup><sup>4</sup> S, M = 124.08 g/Mol, K2 /2 (quartz) =0.09%, C = 2 ppm.*

(sensitivity, conductivity, diffusion parameters, substrate—quartz), the graph shows the value of about 3 <sup>10</sup><sup>6</sup> for a layer with a thickness of 500 nm. The changes resulting from the numerical model are at a relative level of changes of the order of 10<sup>6</sup> . The results shown in **Figure 20** show that there is an optimal thickness of the detector layer for which the acoustoelectric impact (change in the velocity of the acoustic wave) is the greatest. Qualitatively, the existence of an optimal layer thickness was confirmed empirically in Ref. [7, 8].

**Figure 24.**

*Changes (relative) of velocity vs. concentration, temp. 305 K (32°C), (RR)-P3HT, a = 1.75 ppm<sup>1</sup> , σ<sup>s</sup> = 5 10<sup>4</sup> S, DK = 106 nm<sup>2</sup> s 1 , Eg = 2 eV, M = 124.08 g/Mol, gas DMMP, thickness 500 nm, K2/2 (quartz) = 0.09%.*

The thickness of the layer affects the response of the detector layer, also in this case. In our case, we identified the height of the layer with the average radius of the detector layer profile. This is a particularly favorable circumstance because in the perturbation theory, the concept of surface conductivity is used. The following graph was obtained by equating the average radius of the Ra layer with the thickness of the layer *σ<sup>s</sup>* = *σ Ra*. This fact gave our considerations a further direction enabling the development of a mathematical model to take into consideration roughness. In this case, most of the volume of the layer is close to the surface of the piezoelectric layer of the quartz base. The diffusion of larger molecules is not difficult, however, in layers of greater thicknesses above about 650 nm, the interaction is not as strong as in smaller ones, eg. 100 nm. The electrical conductivity of the deeper parts of the layer changes, and their contribution to the detector response is significant. In the case of detectors based on polymer layers, the shape of the surface and its roughness are significant. The above characteristics of the detector response depending on the average radius of the layer's profile were shown (**Figure 18**). The average radius of the Ra layer profile by the height of the H-layer for the better presentation was normed (*R*a/H— **Figure 21**). In the case of the average profile radius *R*<sup>a</sup> of 0, the layer "disappears", boundary conditions correspond to the lack of interaction of the analyte with the detector layer. In the case when the average radius of the *R*a—layer profile reaches the height of the maximum H. This is a theoretical case when we have an ideal detector layer in the image of a "rectangular"—and the surface that is not rough—a theoretical case. Therefore, the roughness has decisive for the response of the DMMP gas detector based on the optic layer using (RR)-P3HT and this aspect will be developed in further studies. In the case of DMMP, the optimal layer thickness is approx. 500 nm. These analyzes were carried out for other analytes, such as: H2, CO2, NO2, NH3. Numerical NNA analysis also showed the existence of an optimal thickness at which the interaction is greatest. The optimal thickness for LiNbO3 (piezoelectric) layer and interaction with gases H2, CO2, NO2, NH3 was approx. 100 nm [9]. As a rule, the diffusion of larger molecules is difficult, additional in layers of higher thicknesses, the entire volume of the layer is not saturated. However, in the case of polymers, it is the other way round. The electrical conductivity of the deeper parts of the layer also changes, so their contribution to the detector response is significant. This state describes the solution of the general diffusion equation for polymers [26].

The results of the experiment to the theory were compared. **Figure 22** shows theoretical numerical analysis in the temperature range of about 293–322 K. Below a *Numerical Analysis of the Steady State in SAW Sensor Structures with Selected Polymers… DOI: http://dx.doi.org/10.5772/intechopen.109367*

comparison of numerical calculations based on the assumed layer model with empirical measurements was made. Good consistency of row magnitude was achieved. The results of the experiment coincide with the theory and are convergent. The calculations for a DMMP concentration of 2 ppm were made.

In **Figure 24** the theoretical dependence of the detector response depending on the concentration in the range from 1 to 3 ppm was shown. The results of empirical research concern the concentration of 1.5–3 ppm and at 32°C were made. All tests on a piezoelectric substrate made of quartz were done.

### **5. Summary**

In research layers (PANI) for different thicknesses: 100, 180 nm, Nafion approx. 300 nm were prepared. The main target was to research the interaction of carbon monoxide and a layer of PANI change were measured. Influence the temperature on the bilayer structure of Nafion+Polyaniline at T = 35° C and T = 42° C were examined. Visible effects were observed at low CO concentrations—5 ppm. During the experiment, representative samples of the PANI with a Nafion layer thicknesses of 100 and 180 nm were investigated. Very interesting properties were examined at layer 180 nm depending on the interaction temperature. Very clearly and a sharp rise detection of carbon monoxide properties as a result of the growth of the temperature of about 7°C (temperature change from 35°C to 42°C) was observed. Exactly ambient temperatures were measured. Empirical results with layer basis on PANI + Nafion (100, 180 nm) compared with Polypyrrole detector layer (thickness 80 nm). It is useful to measure a higher level of the concentration of CO above 25 ppm. Extensive numerical analyses on CO of the SAW response parameter like: thickness, concentration, and temperature at low concentrations (5 ppm) were conducted. Numerical researches were performed in a steady state. As shown above, PANI and PPy nano-layers have a lot of valuable physical properties useful for applications in various fields of science. These materials are used in medical and biological applications, eg. for the detection of both chemicals, e.g. the selected metal cations, lactic acid as well as biological samples tumor cells. Polyaniline nano-layers can develop a detector that can detect lactic acid in the range of physiological concentrations informing about the different types of diseases. Empirical studies can observe specific changes in the electrical conductivity of micro and PANI nanolayers in contact with tumor cells and healthy [13].

Detector studies of photoconductive (RR)-P3HT were performed as a potential material for detecting trace amounts of DMMP compound vapors (2 ppm) in air using the SAW method. Polymer (RR)-P3HT possesses significant sensitivity to trace amounts of DMMP, only when additional white light was used. Increasing the light flux also causes the sample temperature to rise and to obtain larger frequency variations (130–300 Hz) for the same concentration of 2 ppm DMMP in the air. Estimated response and regeneration times for this DMMP concentration are respectively 10– 20 sec. and 7 min [1]. Extensive NNA of the SAW detector response depending on parameters: concentration, roughness, temperature, and thickness were conducted. Numerical research was performed in a steady state. NNA was performed using proprietary software written in Python. The layer thickness decides the maximum range of the detected gas concentration [10].

Also, the choice of the temperature of the polymer detector is important—the optimal work temperature depends on the type of the detector layer and its roughness. The change in temperature allows to determine the point of detector work, and has

also an impact on the speed of its response and the recovery time of the sensing properties of the detector layer [9]. Empirical results of a SAW detector with (RR)- P3HT were achieved for three thicknesses (100,350,550 nm) of the layers. The main target was to verify empirically the analytical model of the SAW gas detector affected by: 2, 3 ppm DMMP in the air. Empirical results confirmed the usefulness of the elaborated analytical model for the investigation of the SAW detector in the design stage. In particular, the influence of the concentration and thickness (existing optimal thickness) was confirmed. The essential parameter of the polymer detector is the roughness of the sensing layer—under investigation.

The results of numerical acoustoelectric analysis of the SAW detector investigations confirmed theoretical were showed. Different sources of light (different of wavelengths light LED, see **Figure 13**) were selected and different wavelengths (histogram) were checked.

The study has shown that the parameters of the detector layer for the SAW detector should be individually adjusted, according to the type of the detected gas and the applied sensing layer. The exposure time of polymer layers was also empirically selected for about 10 min for each current value of 100, 200, 300 mA and the detector layer was illuminated with LED light of different wavelengths and different light energy. From the tests, the best results for yellow light: 76.6–87.4 [lm] and wavelength = 560 nm were obtained. The author's main goal was to numerically analyze phenomena based on empirical investigation and to create a predictive model that would show the tendency of changes before they appear. Extensive numerical analyses of the SAW detector response depending on parameters like: concentration, roughness, temperature, and thickness were conducted. Numerical investigation were performed in the steady state conditions in Python.
