Liquid metal-based metamaterial with high-temperature sensitivity: Design and computational study

1 Introduction

A metamaterial is an artificial electromagnetic (EM) material with a periodic unit structure [1]. Due to the unique and fabulous ability to manipulate EM waves, the metamaterials have been utilized in several fields such as EM wave transmission modulation, EM shielding, and sensing [2,3,4,5]. By changing the unit cell structure or the dielectric properties of the substrate, one can easily realize the dynamic regulation of EM waves with the metamaterial, which sets a foundation of its application in sensing [6]. In recent years, metamaterial-based biology [7], pressure [8], and chemical [9] sensing has been widely investigated.

Among the sensing technologies, temperature sensing is a key research and has attracted considerable attention [10,11,12,13]. Generally, there are three main methods to realize metamaterial-based temperature sensing. One method is to use temperature-sensitive dielectric materials as the substrate. For example, materials such as liquid crystals [14,15], STO [16], and VO₂ [17] were used in thermal tunable metamaterials. It has the advantages of miniaturization and high flexibility. However, the variation of dielectric constant with temperature is usually small, and its nonlinearity is strong, resulting in low resolution/sensitivity of temperature sensing. Moreover, other defects such as small sensing dynamic range further restrict the development of practical applications of this method [18,19]. Another method is to include the nano-scale micro-electro-mechanical system (MEMS) into the metamaterial design. However, the complex preparation processes, low recognition degree, and high costs of the MEMS structures significantly hinder their practical applications [20,21]. Recently, the metamaterial-based temperature sensors have gradually turned to adopt materials with high thermal expansion coefficients. Due to the different thermal expansion coefficients of different materials, the heterogeneous composite structure produces deformation with the change of temperature, changing the equivalent EM parameters of the structure and hence realizing the detection of environmental temperature [12,22,23]. This approach has the advantages of ingenious structural design, continuous adjustability, and repeatability [24]. Meanwhile, the thermal bending will be induced in response to the temperature change by packing two or more materials with different thermal expansion coefficients into a resonant cell, as thermo-mechanical tuning is also an effective way for constructing reconfigurable metamaterials [25]. Thermo-mechanical tuning was first proposed at terahertz frequencies by Tao et al. [26]. Pitchappa et al. experimentally demonstrated thermally reconfigurable metamaterials with a tuning range of 37.5% [23]. However, the defects such as weak deformation and low-temperature sensing resolution for such configurations also hinder their application.

Mercury, as a liquid metal with strong volume expansion capacity with temperature changes, has been utilized in realizing temperature sensors for a long time. For example, the mercury-filled thermometer has been widely used for centuries. Recently, mercury has been used to construct metamaterials. Ma et al. proposed a thermally tunable metamaterial based on the toroidal resonators that could achieve high Q-factor sensing by verifying the temperatures [22]. Wang et al. investigated a temperature-controlled metamaterial absorber based on mercury [27]. By changing the background temperature, the absorption bandwidth can be tuned effectively. However, the method to increase the sensitivity of mercury-based metamaterial is still a focus of the community. In this article, a temperature sensor based on metamaterial design with high sensitivity is proposed. In the merit of high Q-factor resonance of the selected EM metamaterial and the excellent thermal expansion coefficient of mercury material, the temperature sensitivity of the proposed sensor can reach up to 27.64 MHz/°C within the range of 0–21.8°C, with a maximum figure-of-merit (FOM) of 12.5°C. The compact design strategy paves a new way for developing metamaterial-based temperature sensors with high sensitivity.

2 Metamaterial-based sensor design and demonstration

Figure 1 shows the unit cell structure of the proposed mercury-based EM metamaterial. The resonant structure filled with mercury is embedded in the wrapping medium. The resonant structure is divided into upper and lower parts. In the upper part, both ends of the mercury-filled C-shaped pipe are sealed since the electrical length of the mercury bar cannot vary with the change of background temperature. However, since the liquid level can be raised with the increase of temperature, the mercury does not occupy the whole pipe in the initial state in the lower part. Furthermore, a cylindrical tank to store the mercury was placed at the bottom of the pipe. The dielectric constant and the loss tangent of the 4.5 mm-thick wrapping medium are 2.1 and 0.002, respectively. Moreover, the metallic groundsheet was modeled as a continuous copper film with the conductivity σ = 5.88 × 10⁷ S/m.

Figure 1

The unit cell of the metamaterial-based temperature sensor: (a) top view and (b) side view.

The thermal expansion of mercury can be described as follows:

where ΔV is the change of volume varied with the change of temperature, V ₀ is the initial volume of the material, ΔT is the variation of temperature, and γ is the expansion coefficient.

It can be obtained from equation (1) that V ₀ is proportional to ΔV; hence, the larger the initial volume of mercury, the larger the thermal expansion volume per unit temperature change. In this case, a relatively bigger mercury storage tank will lead to a greater change of the electrical length of mercury, which makes a bigger change of resonant frequency and thus a higher sensitivity.

Since the mercury filled in the upper part of the resonant structure is fixed, only the expansion of mercury in the lower part needs to be considered. From Figure 1, the initial volume of mercury filled in the lower part of the structure can be expressed as follows:

Hence, the relationship between the increasing temperature and the thermally expanded volume can be obtained by combining equations (1) and (2). To better show the influence of the temperature on the structural parameter of resonant structure, the change in the height of the liquid level (with the initial value of b ₂) versus the change of the temperature can be derived as follows:

For the proposed metamaterial-based sensor, the variable electrical length, e.g., the height of the mercury bar b ₂, influences the resonant frequency most. Hence, according to equation (3), Δl should be increased as much as possible to improve the temperature sensitivity. There are two methods to increase Δl for the proposed structure: by increasing the volume of the storage tank V ₀ or reducing the section width d of the mercury bar. Therefore, the optimized structural parameters of the proposed unit cell structure are as follows: a = 20, b = 16, b ₁ = 8, b ₂ = 6, c = 2, d = 0.2, h = 3, and h ₀ = 4.5 mm.

The absorption rate of the metamaterial can be calculated by the reflection parameter S ₁₁ and the transmission parameter S ₂₁ with A(w) = 1 − |S ₁₁(w)|² − |S ₂₁(w)|². Due to the existence of the ground plate, the transmission is zero and the absorption reads A(w) = 1 − |S ₁₁(w)|². The simulations were conducted using the finite element method (FEM), where periodic boundary conditions were applied for both x and y directions, and open boundary condition was utilized in the z-direction. In the simulation, the tetrahedral mesh was selected to model the proposed structure, while iterative accuracy of 1 × 10⁻⁶ was chosen to ensure the high precision of the simulation. Moreover, the EM response of the proposed structure was assumed to be excited by the plane wave with an electric field parallel to the x-axis. Figure 2 shows the reflection and absorption spectra of the proposed sensor in the initial state with a background temperature of 0°C. Figure 2 shows that there are two resonant peaks f ₁ and f ₂ located at 7.55 and 9.06 GHz, with absorption rates of 92.3 and 99.9%, respectively. These sharp resonant peaks with high Q-factors set the foundation for temperature sensing.

Figure 2

Reflection and absorption spectra of the metamaterial-based sensor under the initial state with a background temperature of 0°C.

3 Results and discussion

To verify the sensing performance of the proposed structure, EM responses under different background temperatures were simulated. The change in the height of liquid level Δl with temperature was first calculated using equation (3). Then, the lower part structure with a different mercury bar length of b ₂ + Δl was simulated to show the resonant frequency variation of the metamaterial at different temperatures. Figure 3(a) shows the relationship between the background temperature and the length of mercury bar b ₂ after the thermal expansion. It can be seen from the figure that b ₂ increases from 6 to 7 mm, corresponding to the increase in the background temperature from 0 to 21.8°C. Figure 3(b) illustrates the simulated reflection peak frequencies of mode f ₂ under different temperatures. The variation of resonant peaks of mode f ₁ is also included as a subfigure in Figure 3(b). For mode f ₂, the resonant frequency redshifts from 9.06 to 8.47 GHz with the increase in the temperature from 0 to 21.8°C, with a maximum frequency tunability of 6.51%. However, the frequency shifts of mode f ₁ in the temperature region of 0–21.8°C are much smaller compared with mode f ₂. For example, the calculated frequency tunability of mode f ₁ is only 0.5%. A bigger frequency tunability leads to a better sensor resolution as mode f ₂ is more suitable for temperature sensing. Moreover, Figure 3(b) shows that the resonant intensity of mode f ₁ decreases rapidly with the increase of temperature, which significantly degrades its application in sensing. Hence, mode f ₁ will not be further discussed in the following sections. Figure 3(c) shows the absorption spectra of the proposed sensor at different temperatures. It can be seen from the figure that the overall absorption rate maintains above 90% in each case. Hence, the structure can also be used as a temperature-controlled tunable EM wave absorber.

Figure 3

Simulation results for the tunability of mercury-based metamaterial at different temperatures: (a) mercury bar length b ₂ and (b) reflection and (c) absorption spectra.

To investigate the physical mechanism of the proposed structure, the electric field distribution and the surface current distribution were simulated on the mercury pattern and the ground plane, respectively. Figure 4(a) and (b) show the electric field and also show the surface current distributions of the metamaterial sensor at the frequency of 9.06 GHz. This figure also shows that there are two electric dipoles located at the bottom of the resonant structure, which excites strong electric resonances. Figure 4(b) further indicates that the currents are mainly concentrated in the lower part of the resonant structure as the resonant frequency is more sensitive to the mercury bar length b ₂. Hence, the temperature sensitivity of the proposed structure can be enhanced. The current flow on the ground plate shown in Figure 4(c) is opposite to that on the mercury pattern, which leads to the existence of an equivalent current loop and the excitation of magnetic resonance. Therefore, both the electric and the magnetic resonances are responsible for the strong absorption of the temperature sensor.

Figure 4

Numerical simulation results. (a) Distributions of the electric field and the surface current on (b) mercury pattern layer and (c) metal ground plate. The frequency is 9.06 GHz.

It is well known that the resonance frequency of third-order resonances is approximately three times higher than that of the fundamental mode. However, in the TPA structure, the location of the third-order resonances shows an obvious red shift. To further investigate this phenomenon, we simulated the magnetic field distribution on the surface of some adjacent stacking patches at 27.0 GHz, and the results are shown in Figure 4. As shown in Figure 4b–d, the magnetic field represented as three stripes is originated from the third-order resonances. Furthermore, the distribution of magnetic field is mainly along the diagonal line of the square patch under TE-polarized incidence, which is very different from the circumstances in traditional PA structures. The typical magnetic field distribution of third-order resonance in the nontwisted PA structure is shown in Figure 4e. Compared with the nontwisted PA structure, the M-field spot on TPA’s patch is much larger. Hence, the resonance frequency of the third-order mode in TPA is much lower than that in PA, which enables its coupling with fundamental modes within a broadband region (22.0–35.0 GHz) and finally leads to an extra broadband absorption.

By incorporating the variation of resonant frequency with different temperatures shown in Figure 3(a), the temperature sensitivity of the proposed structure can be calculated. The sensitivity can be defined as S = ∆f/∆T, where ∆f is the change of resonant frequency, while ∆T is the change of the background temperature. Figure 5(a) presents the linear fitting result of the temperature sensitivity. It can be seen from the figure that the calculated temperature sensitivity of the proposed structure is 27.64 MHz/°C within the region of 0–21.8°C. Hence, the sensor can achieve the detection of the background temperature with high sensitivity. Meanwhile, the precision for sensing application was determined using the Q-factor of the sensor, defined as Q = f ₀/∆f, where f ₀ is the resonant frequency and ∆f represents the 3 dB resonance frequency bandwidth. As shown in Figure 3(a), with the variation of temperature, both the resonant frequency f ₀ and the 3 dB bandwidth ∆f change correspondingly. Hence, the Q-factor of the sensor varies in the range of 110–983. The high Q-factor within the overall operation range enables the precision sensing of the proposed temperature sensor.

Figure 5

(a) Relationship between the resonant frequency and background temperature changing ΔT with (a) mercury bar section width d of 0.2 mm and (b) different mercury bar section widths. (c) Calculated temperature sensitivity with different values of section width d.

The influence of structural parameters (e.g., the section width of mercury bar d) on sensing performance is further analyzed. Figure 5(b) shows the change of resonant frequencies with different temperatures under different values of section width d of mercury bar. It is observed that the resonant frequencies redshift with the increase of temperature. However, there is a wider frequency modulation bandwidth for structures with smaller mercury bar section width, which leads to a higher sensitivity. Figure 5(c) shows that the temperature sensitivity of the structure with d of 0.2 mm is 27.64 MHz/°C. When the mercury bar section width increases to 0.6 mm, the sensitivity decreases rapidly to 3.41 MHz/°C. Further decrease of d will lead to a higher sensitivity. However, the sample preparation would be more difficult for such a configuration. It should be noted that the width of the mercury bar d has a strong influence on the sensitivity of the proposed structure. The volume change of the mercury will be enlarged with the increase of the width d, which also leads to an increase of the measurable temperature limit. However, the sensitivity of the absorber will decrease as the resonant frequency shift per temperature is smaller. On the contrary, reducing the width d will enhance the sensitivity at the cost of a lower temperature limit.

The proposed metamaterial-based temperature sensor can be fabricated with the aid of 3D printing technology. In this case, the dielectric material of the substrate should be printable. For instance, the polytetrafluoroethylene (PTFE) can be used as the wrapping medium, which acts as a good mercury container with similar dielectric properties to the design (ε _PTFE = 2.1 + 0.001i). To fabricate such a sample, the multi-material hybrid micro-droplets jetting modeling (MHMJM) technology, which enables two-material hybrid printing via two non-contact nozzles, can be utilized to form the PTFE and the mercury layer simultaneously [28,29].

Table 1 compares the performance of the proposed temperature sensor with some reported sensors. It should be noted that the FOM, defined as sensitivity/∆f, was included in the table to show the precision of temperature sensing. It can be observed from Table 1 that the proposed sensor outperforms the recently reported temperature sensors in terms of sensitivity and FOM. Hence, more precision sensing can be achieved with the configuration.

4 Conclusion

A metamaterial-based temperature sensor using the thermally tunable liquid metal of mercury is proposed in this article. The thermal expansion of mercury leads to a deformation of the metamaterial unit cell structure, hence adjusting the resonant frequency and setting a foundation for temperature sensing. In the merit of the high Q-factor resonance of the selected metamaterial configuration and the excellent thermal expansion coefficient of mercury material, the temperature sensitivity of the proposed sensor can reach up to 27.64 MHz/°C within the range of 0–21.8°C, with the maximum FOM of 12.56. The simulation results demonstrate that the proposed structure can maintain an absorption rate of above 90%. Moreover, the proposed structure can also be treated as a temperature-controlled tunable EM wave absorber with a frequency tunability of 6.51%. Meanwhile, by reducing the mercury bar width, the temperature sensitivity of the proposed structure can be further enhanced. The simulated electric field and surface current distributions show that the high sensitivity of the structure is originated from the excited dual-dipole mode. The proposed metamaterial-based sensor provides an alternative for future applications in high-performance temperature sensing.