Theoretical and Simulation Analysis of Static and Dynamic Properties of MXene-Based Humidity Sensors

Abstract

In this paper, the static and dynamic characteristics of the MXene-based IDE capacitive humidity sensor are investigated through theoretical modeling and simulation. It is found that the capacitance increases according to the thickness of the sensing film within a certain range while stopping increases along with the growth of the thickness when the thickness is over a threshold. When the thickness is at a tiny level, a larger thickness does not lead to a significant increase in the response time due to the diffusion mechanism of water molecules. When the thickness increases to certain extent, there is an evident relationship between the response time and the change of thickness. For the humidity-sensitive film, under the same relative humidity conditions, the capacitance has a positive correlation with temperature, and the response time shows the opposite trend. Subsequent studies on the sensitive mechanisms of MXene materials explain these phenomena and demonstrate the accuracy of the model. This provides a more accurate method for sensor design. The properties of the MXene capacitive humidity sensor can be optimized by changing its structure and adjusting material parameters.

1. Introduction

In meteorological monitoring, weather bureaus and marine monitoring applications rely on accurate humidity sensing. The accuracy of meteorological detection is directly related to the timeliness of disaster prevention. For this reason, humidity sensors that can be used in a wide humidity range and under conditions of sudden changes in temperature and humidity have received extensive attention [1,2,3,4,5]. Humidity sensors can be divided into resistive and capacitive [6]. Resistive humidity sensors tend to have lower manufacturing difficulty and are usually cheaper than capacitive humidity sensors. However, their shortcomings including poor sensing range, long response time, and low measurement accuracy widely limit their applications [7]. In contrast, the interdigitated electrode (IDE) capacitive humidity sensor has the characteristics of lower consumption, large output signal, and fast response time [8,9]. Consequently, most current humidity sensors prefer to use the IDE structure. It can be seen from the principle of capacitive sensors that the output value, sensitivity, response time, and other properties of the sensor are related to the structural and material parameters [10,11]. These parameters mainly affect the size and dielectric coefficient of the sensor. Generally speaking, the larger the dielectric coefficient, the better the capacitive response. The thinner the sensitive layer, the worse the sensitivity, but the shorter the response time.

Recently, the effects of sensor structures on humidity-sensing characteristics have been reported from various perspectives [9,12], but few works consider the effects of the film thickness. Since both capacitance and response time are related to thickness, thickness will inevitably affect the properties of the sensor [13,14]. During the past decade, a novel type of 2D transition metal carbide and carbon–nitrogen-labeled MXene families have been widely researched for their potential applications in humidity sensing, food analysis, clinical detection, and many other fields [15,16,17,18,19,20]. Similar to the common characteristic of 2D materials [21], many MXene-based humidity sensors have been demonstrated to have good conductivity [22], chemical stability [23], and fast-response properties [24]. From the mechanism analysis, the hydrophilic functional group on MXene adsorbs water molecules in the air, thus changing its physical characteristics, which means that it can be used as a humidity sensor [25]. Experimental results have also indicated that MXene is a good choice for humidity-sensitive materials [26,27]. For example, it has been reported that the response/recovery rate of a Ti₃C₂/Ag-based humidity sensor can reach 80/120 ms [28]. Additionally, many studies have shown that MXene-based humidity sensors also have good sensitivity, even reaching 317% at 90% relative humidity (RH) [29,30,31]. However, most researchers only analyzed the humidity-sensing properties of MXene based on the experimental results. Few attempts have been made on the MXene humidity-sensing mechanism through theoretical analysis [32,33,34]. Note that the theoretical models for other humidity-sensitive materials have been established, which facilitates the optimization of the sensor parameters [14,35]. In view of this, this paper aims to fill this gap through theoretical modeling and simulation.

This paper investigates the static and dynamic properties of MXene-based IDE capacitive humidity sensors through theoretical modeling and simulation. Coupled with the thickness structure of the sensors and the humidity-sensing properties of MXene, the static and dynamic properties of the sensor are analyzed by Fick’s law and other theories, and the effects of parameters on the humidity sensing properties are analyzed by simulation. The accuracy of the model is further demonstrated by the explanation of the humidity sensing mechanism. The main contribution of this paper is that it establishes a theoretical model for IDE capacitive humidity sensors based on MXene. Compared with the traditional model, this model can more accurately reflect the influence of structure and material parameters on the sensor properties, so that the calculated results can better reflect the real results. This provides a method for sensor design. The properties of an MXene capacitive humidity sensor can be adjusted by changing its structure, and can also be optimized by adjusting the material parameters.

2. Static Properties Analysis

2.1. Effect of Structure on Static Properties

In the considered case of the IDE capacitive humidity sensor, the electric potential and electric field can be well interpreted by the superposition principle if the charge distribution is known [36]. When there is an applied electric field, the electric potential distribution V(r) can be described by the following equations,

$$V(\overrightarrow{r})={\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{\sigma (\overrightarrow{r})}{4\pi \epsilon \left|\overrightarrow{r}-\overrightarrow{r^{\prime }}\right|}}}}{d}^3\overrightarrow{r^{\prime }}$$

(1)

where σ(r) denotes the distribution of the surface charges, r denotes 3D model sphere radius, and ε denotes the dielectric coefficient.

For the IDE with a long periodic structure, the edge effect at the end of the IDE is negligible, and its potential expression can be written as

V(x,y,z) = V(x,y,0)

(2)

Note that the entire potential distribution can be obtained by expressing the potential expression on the x−y plane. Therefore, for IDE with longer lengths, it is sufficient to discuss the potential V(x,y) in the x−y plane.

The two models used to describe IDE structure are shown in Figure 1.

Figure 1 depicts two IDE models at different positions for the sensitive layer. Where g denotes the distance between the two electrode fingers, and w denotes the width of the fingers. Take the interdigital electrode made of indium tin oxide(ITO) as an example. The configuration in Figure 1b has the simplest distribution of electric potential and electric field, which can be regarded as a parallel plate. The electric potential V_b(x,y) and the electric field E_b(x,y) of Figure 1b are shown in Equations (3) and (4),

$$V_b(x,y)=\frac{V_0}{g}X$$

(3)

$$E_b(x,y)=\frac{\partial V_b(x,y)}{\partial X}=\frac{V_0}{g}$$

(4)

In comparison, and when the sensitive layer is thicker than the electrode, the difference in capacitance is mainly caused by the structural change of (a). Using the Fourier series and applying the boundary conditions, the electric potential V_a(x,y) of (a) can be expressed as

$$V_a(x,y)=\frac{4V_0}{\pi }{\displaystyle \sum _{n=1}^{\infty }\frac{1}{2n-1}{J}_0[\frac{(2n-1)\pi g}{2(w+g)}]}\sin [\frac{(2n-1)\pi x}{(w+g)}]\exp [-\frac{(2n-1)\pi \left|y\right|}{(w+g)}]$$

(5)

where V₀ denotes the applied voltage, J₀ denotes the zeroth bessel function of the first kind, x denotes the distance along the horizontal axis in Figure 1, and y denotes the thickness of the sensitive layer.

The electric field E_a(x,y) and the surface charge distribution σ(x) are shown in Equations (6) and (7),

$$E_a(x,y)=-\nabla V_a(x,y)$$

(6)

$$\sigma (x)=-2\epsilon (\frac{{\partial }^2{V}_a(x,y)}{\partial y^2}),y=0$$

(7)

where ε denotes the dielectric coefficient of the sensitive layer.

Considering an applied voltage V₀ = 1 V, the g, and w are 10 μm and 50 μm, respectively. The electric potential, electric field, and the surface charge distributions in the x-direction under different thicknesses of the sensitive layer are analyzed, as shown in Figure 2.

The results of Figure 2 exhibit two interesting things. Firstly, the potential has a maximum value at the edge of the electrode, and the total potential drop is about 2 V, which is larger than the applied voltage. This is not unique; a similar phenomenon is also observed in the electric field distribution. This is attributed to the fact that the surface-charge distribution on the electrode surface is not uniform and that most charges accumulate at the edges. in addition, the electric field distribution is not uniform across different film thicknesses, and the electric field decreases rapidly as the film thicknesses increase.

Furthermore, the local current density (J) and current (i) are shown in Equations (8) and (9),

$$J={\sigma }_0•E$$

(8)

$$i={\displaystyle \int Jds=}{\displaystyle \int {\sigma }_{0}Eds}$$

(9)

where σ₀ denotes the local conductivity, and E denotes the electric field. If the cross-section is selected at x = 0, then V_a(0,y) = 0, the electric field E_a(x,y) in the x-direction can be expressed as Equation (10),

$$\overrightarrow{E_a}(x,y)\left|{}_{x=0}=-\frac{\partial V_a(x,y)}{\partial x}\right.\left|{}_{x=0}\right.•\overrightarrow{I}-\frac{\partial V_a(x,y)}{\partial y}\left|{}_{x=0}\right.•\overrightarrow{J}=-\frac{\partial V_a(x,y)}{\partial x}\left|{}_{x=0}\right.•\overrightarrow{I}$$

(10)

The current through the sensitive layer is shown in Equation (11),

$$i(h)=V_0\frac{1}{\pi }{\displaystyle {\int }_{y=0}^{y=h}{\sigma }_0(x,y)E_a(x,y)\left|{}_{x=0}dy\right.}$$

(11)

The current fraction can effectively predict the capacitance of the sensing layer. Maximum capacitance is considered to be achieved when all currents pass through the sensitive layer, the current fraction is shown in Equation (12),

$$i(h)/I=i(h)/i{(h)}_{\max }$$

(12)

where I denotes the total current flowing through the sensitive layer.

The relationship between the sensitivity and the thickness of the sensitive layer is theoretically calculated and simulated. The simulation model is shown in Figure 3.

The comparisons between the theoretical analysis and the simulation results of thickness versus normalized current fraction and capacitance are shown in Figure 4. The current fractional trend is calculated by theoretical calculation and is represented by the black line in Figure 4, the capacitance trend is calculated by the model simulation and is represented by the red line. First, it is found that as the thickness of the sensitive layer increases, the ascending trends of the current fraction and capacitance become slow. Increasing sensitive film thickness does not always bring about an increase capacitance and sensitivity, and, as the thickness increases, it will also bring problems such as slower response time.

Moreover, for the given sensitive layer with the IDE width and gap of 50 μm, 80% of the current flows in the 52 μm thick layer, and when the IDE width and gap are reduced to 10 μm, 80% of the current flows in the 11 μm thick layer. The smaller the width and gap of IDE, the smaller the corresponding optimal sensitive layer thickness. The simulation results also support this conclusion. This can be explained by the current being distributed more densely when the width or gap between IDEs is smaller.

2.2. Effect of Material on Static Properties

The properties of sensitive materials are also closely related to the static properties of the sensor. In this section, the theoretical analysis of the MXene-based capacitive humidity sensor is carried out, and the effects of the change of the dielectric coefficient on the static properties of the sensor are analyzed. The Clausius–Mossotti equation [37,38] can describe the relationship between the dielectric constant of the sensitive layer and the humidity, which is formulated as,

$$\frac{N\alpha }{3{\epsilon }_0}=\frac{{\epsilon }_r(RH)-1}{{\epsilon }_r(RH)+2}-\frac{{\epsilon }_r(0)-1}{{\epsilon }_r(0)+2}$$

(13)

where N denotes the concentration of water vapor adsorbed in the sensitive layer, α denotes molecular polarizability, ε_r(RH) denotes the dielectric coefficient at a certain RH, ε_r(0) denotes the dielectric coefficient at room temperature without RH, and ε₀ denotes the vacuum permittivity. In this model, α is 3 × 10⁻¹², ε_r(0) of MXene without humidity is 8.69, and ε₀ is 8.85 × 10⁻¹² F/m.

The concentration N is obtained from the ideal gas law, as shown in Equation (14),

$$N=\frac{RH\cdot P}{R\cdot T}$$

(14)

where P denotes saturated vapor pressure, R denotes the molar gas constant, and T denotes the temperature. In this model, R is 8.314. The saturated vapor pressure is related to temperature, and the empirical equation is summarized in the following equation,

$$\ln P=b_0+\frac{b_1}{T}+b_2\ln T$$

(15)

All the coefficients are normalized and set as b₀ = 49.34, b₁ = −6651.95, and b₂ = −4.54. Based on Equations (14) and (15), the relationship between the model dielectric constant versus humidity and temperature can be obtained by simulation, as shown in Figure 5a.

The dielectric coefficient is input into the model, and the relationship between normalized capacitance versus temperature and humidity can be obtained by simulation, as shown in Figure 5b.

As shown in Figure 5b, the capacitance of the MXene-based capacitive humidity sensor shows a non-linear ascending trend with humidity at a certain temperature. In the low humidity state, the capacitance increases slowly, and the growth rate increases with the humidity. The growth rate of capacitance with humidity also increases with temperature. This behavior can be explained by the change of N. When the temperature increases, the concentration of water molecules increases under the same humidity, which leads to the surge of water molecules absorbed by the sensitive layer and finally results in the high level of the capacitance.

3. Dynamic Properties Analysis

3.1. Effect of Structure on Dynamic Properties

For the dynamic properties of the sensor, the transfer process of water molecules in the sensitive layer can be described by Fick’s second law [39], as shown in the following equation,

$$\frac{\partial c}{\partial t}=\frac{D{\partial }^{2}c}{\partial x^2}$$

(16)

where c denotes concentration of water vapor, t denotes time, D denotes the diffusion coefficient, and x denotes the diffusion distance. In the three-dimensional model, it can be rewritten as the following equation,

$$\frac{\partial c(x,y,z,t)}{\partial t}=\frac{D_x{\partial }^{2}c(x,y,z,t)}{\partial x^2}+\frac{D_y{\partial }^{2}c(x,y,z,t)}{\partial y^2}+\frac{D_z{\partial }^{2}c(x,y,z,t)}{\partial z^2}$$

(17)

Equation (17) implies that in the process of gas diffusion, the response time of the sensitive layer is affected by the distance and the diffusion coefficient. The final distance of the gas diffusion process is the thickness of the sensitive layer, so the thickness is closely related to the response time.

As shown in Figure 6, when the temperature is 298 K and D = 11.01 × 10⁻¹⁰ m²/s, the relationship between the response time and the thickness can be obtained through simulation analysis.

It can be seen from Figure 6 that with the increase of sensitive layer thickness, the response time increases, and the growth rate gradually increases. The thickness of the sensitive layer has a significant effect on the dynamic properties. Therefore, controlling the thickness of the sensitive layer can effectively improve the dynamic properties of the sensor, and, by adjusting the amount of dispensing, the sensitive layer thickness can be controlled.

3.2. Effect of Material on Dynamic Properties

For material properties in Equation (17), the relationship between diffusion coefficient D and the temperature is shown in the following equation,

$$D=D_{\mathrm{ref}}{\left(\frac{T}{T_{\mathrm{ref}}}\right)}^{1.75}$$

(18)

where D_ref is the reference diffusion coefficient between the water molecule and MXene sensitive layer, and T_ref is reference temperature. At T_ref = 298 K, D_ref is 11.01 × 10⁻¹⁰ m²/s.

The diffusion coefficient is input into the model, and the relationship between normalized response time versus temperature and humidity can be obtained by simulation analysis, as shown in Figure 7.

The diffusion coefficient increases and the response time decreases with increasing temperature. This demonstrates when the temperature increases by 110 K, the response time decreases by about 8%, which can be attributed to the increasing temperature speeding up the movement of water molecules and accelerating the water diffusion between the gas and the sensitive layer.

4. Results and Discussion

The properties and related parameters of the MXene-based IDE capacitive humidity sensor are shown in Figure 8.

The electrical output of the sensor depends on the humidity of the environment in which the sensor is located. In different humidity environments, the dielectric coefficient and capacitance of the sensitive material are different because the sensitive material is limited by different concentrations of water molecules. In addition, the capacitance of the sensor is affected by the actual current. As the current increases, so does the sensor’s capacitance, but the electric field is not necessarily fully contained in the sensitive layer. Only the current in the sensitive layer affects the capacitance of the sensor.

For the effect of structure on static properties, the thickness affects the current fraction passing through the sensitive layer. The greater the thickness, the greater the current passing through the sensitive layer and the greater the capacitance. However, when the thickness increases to a certain extent, the current has all passed through the sensitive layer. At this phase, increasing the thickness does not increase the capacitance. This shows that the traditional method of improving the sensitivity by increasing the film thickness has limitations. When the sensitivity cannot be improved by increasing the film thickness, other methods, such as improving the dielectric coefficient of the material through material modification, should be used to further improve the sensitivity.

Under the same relative humidity, the concentration of water molecules increases with temperature, due to the variance of the static properties of the material. The increase in water molecules leads to an increase in the dielectric coefficient of humidity-sensitive materials, which increases the capacitance of the humidity-sensor output. The results show that temperature can be used to adjust the characteristics of sensor materials, and the dielectric coefficient of humidity-sensitive materials can be adjusted by adding heating elements and other methods. From this, it can also be seen that the dielectric coefficient of humidity-sensitive materials is related to the materials’ properties, and similar effects can be achieved by adjusting the dielectric coefficient through other methods.

For the effect of structure on dynamic properties, when the thickness is small, increasing the thickness will not lead to a significant increase in the response time. The reason for that is the thickness of the sensitive layer is very small, and the water molecules are still mainly absorbed on the surface of the sensitive layer. In such conditions, it is the characteristics of the material itself that affect the electrical properties, but when the thickness increase continuously, the response time increases with the thickness because water molecules pass through the surface and combine with the hydrophilic groups inside the film. The time required for complete binding is positively related to the thickness. At this phase, the main effect on the electrical properties is the concentration of water molecules combined with the sensitive layer.

For the effect of material on dynamic properties, when the temperature increases, the water molecules move more violently, leading to an increase in the diffusion coefficient. The time for the water molecules to completely combine with the hydrophilic groups in the sensitive layer decreases, so the response time decreases.

To sum up, as the thickness of the sensitive layer increases, the sensitivity of the sensor can be increased until the current completely passes through the sensitive layer. However, this also causes the response time to increase. In addition, an increase in temperature can lead to an increase in sensitivity and a decrease in response time. The essence of this phenomenon is that it affects the dielectric and diffusion coefficients of the material. If the material parameters are changed through material modification, similar results can also be obtained.

The phenomena above can be explained by the humidity-sensing mechanism of MXene. MXene absorbs water molecules in the air, causing changes in its dielectric coefficient. In different humidity environments, different numbers of water molecules are attached to the membrane, so the humidity sensor outputs different capacitance values. The adsorption of water molecules on MXene is shown in Figure 9.

In a low-humidity environment, due to the abundant hydrophilic functional groups on the surface of the humidity-sensitive material, water molecules can easily adsorb to the humidity-sensitive material through physical adsorption, thereby changing its dielectric coefficient. However, in a low-humidity environment, more free water molecules cannot be provided, and the water molecules adsorbed on the surface of the humidity-sensitive material are only at sites such as hydrophilic functional groups and do not form a continuous water film. At this time, it is the intrinsic properties of the material that affect the conduction.

With the increase in ambient humidity, a large number of water molecules occupy the hydrophilic functional group on the surface of humidity-sensitive materials. At the same time, the adsorption between water molecules has begun due to the action of hydrogen bonds. Because of the presence of certain water molecules on the surface of humidity-sensitive materials, water molecules are more easily absorbed by the environment. Finally, the adsorption of humidity-sensitive materials by water molecules in the air reaches a saturation state in this high-humidity environment. The adsorption forms a water film covering the surface of a humidity-sensitive material. At this time, under the action of an electric field, the water-membrane layer will electrolyze to form conductive hydronium ions. At this time, the humidity sensor has two conduction mechanisms, one is intrinsic to the material and another is the electrolytic conduction by liquid aqueous electrolysis [28,34]. This is why the capacitive humidity sensor based on MXene increases nonlinearly with humidity at a certain temperature. The higher the temperature or the thinner the material, the more efficiently the water molecules binds to the film, indicating that the response time decreases when the thickness decreases or the temperature increases. However, decreasing the thickness leads to a decrease in the absorption of water molecules, and increasing the temperature leads to an increase in the absorption of water molecules. This is the reason why the sensitivity decreases with decreasing thickness and increases with increasing temperature.

5. Conclusions

In this paper, the static and dynamic characteristics of the MXene-based IDE capacitive humidity sensor are investigated through theoretical modeling and simulation. Structural and material parameters were found to affect sensitivity and response time. The trends of sensor properties with each parameter are more accurately analyzed.

• The greater the thickness, the more current flows through the sensitive layer and the greater the capacitance. However, unlike the traditional perception, when the thickness increases to a certain extent, the growing thickness does not increase the capacitance. This is because all current has passed through the sensitive layer at this point.
• When the thickness is at a tiny level, increasing the thickness does not lead to a significant increase in the response time due to the diffusion mechanism of water molecules. However, with the increase in thickness, the response time is obviously related to the change.
• In the same relative humidity conditions, the concentration of water molecules increases with temperature. The increase in water molecules results in a rise in the dielectric coefficient of the humidity-sensitive material so that the capacitance of the humidity sensor output has a positive correlation with temperature.
• When the temperature increases, the water molecules move more vigorously, resulting in a rise in the diffusion coefficient. The reaction response time is shortened by reducing the time for water molecules to fully bind to the hydrophilic groups in the sensitive layer.

Subsequent studies on the material-sensitive mechanism of MXene explain these phenomena and demonstrate the accuracy of the model. Through this model, we are able to design and test the structural and material parameters of the sensor. This can help to improve the properties of the sensor.

When it is necessary to improve the sensor properties, it is often desired to improve the sensitivity and reduce the response time. Through the numerical simulation of the structure and material through this theoretical model, the sensors can be designed in the theoretical stage and the structural adjustment and material modification can be carried out in the follow-up. Moreover, this model is different from the simple theoretical analysis. For example, from this model, it can be seen that the sensitivity does not increase with the increase of the thickness. When the thickness is small, increasing the thickness will not lead to a significant increase in the response time. This paper also accurately analyzes the influences of temperature on sensitivity and response time. These makes the simulation of the sensor properties more accurate. Through this model, the influence of the design on the properties of the sensor can be simulated, so that the properties can be controlled. In addition, the changes in the structure and material parameters predicted by this model can also be applied to MXene-based gas sensors and mechanical sensors, etc.