Research on the Geometry Control and Microwave Absorption Performance of Auxetic Materials

Abstract

There is great potential for the development of microwave-absorbing materials (MAMs) for structural regulation. Auxetic structures have excellent mechanical properties, which can be applied to multifunctional MAMs in various fields. Here, the microwave absorption performances of the auxetic structures were simulated using the High-Frequency Structure Simulator (HFSS), by regulating the structure, dielectric constant, layer number, and pore size. The simulation results show that increasing the dielectric constant, layer number, or decreasing pore size will lead to a decrease in the frequency of minimum reflection loss (RL_min). The main purpose of this study is to elucidate the influence of structure, dielectric constant, layer number, and pore size on the absorption performance of auxetic structures and obtain practical auxetic MAMs with a performance of RL_min < −30 dB and effective absorption bandwidth (EAB) > 3 GHz. Finally, practical auxetic MAMs between 8 and 18 GHz and MAMs optimized in dielectric constant were obtained, which were proven to have the advantages of lightweight characteristics, high absorption, and wide bandwidth. The four structures exhibit great RL_min values of −51.09, −55.52, −47.09, and −54.98 dB with wide EAB values of 3.25, 3, 4.75, and 4.5 GHz, demonstrating the strong electromagnetic wave absorption performance of auxetic structures. This work provides theoretical guidance for the study of auxetic structures in the field of microwave absorption and provides an effective approach for multi-disciplinary research on MAMs.

1. Introduction

With the explosive development of information technology, the application of electromagnetic equipment is becoming widespread on a global scale, leading to an increase in electromagnetic pollution [1,2,3,4]. Electromagnetic pollution has brought increasing interference to electronic devices and harm to human health, but traditional electromagnetic shielding materials cannot solve the secondary pollution problem caused by electromagnetic wave reflection. Therefore, it is imperative to develop effective microwave-absorbing materials (MAMs).

Structural absorbing materials are one kind of MAM that has been widely studied [5,6,7,8]. Such MAMs are mostly macroscopic porous materials, which can not only directly dissipate electromagnetic waves through the medium-like coating materials, but also achieve multiple reflections and transmissions of electromagnetic waves through the pores, thereby extending the propagation path of electromagnetic waves and enhancing the absorption ability. Gong et al. prepared a carbon fiber/polyamide/carbonyl iron composite with a flexible honeycomb structure via selective laser sintering (SLS) 3D printing and accomplished an RL_min of −47 dB and an effective absorption bandwidth (EAB) of 13.2 GHz under a high bending angle of 150° [9]. Xing et al. synthesized a gallium indium alloy (EGaIn)-doped SiBOC ceramic metamaterial based on the Schwarz P minimal surface, which has an EAB of 11.36 GHz and good mechanical properties [10]. Structural absorbing materials are often prepared by the template method, which involves immersing or covering MAMs on the corresponding porous structure template to induce molding [11,12,13,14,15]. However, the method is facing a lot of challenges. On the one hand, the template itself has a certain thickness and does not contribute to the absorbing performance. On the other hand, the materials prepared by this method gain poor mechanical properties, which limit their applications. Therefore, there is an urgent need for a porous material that can be integrally formed and has certain mechanical properties to reduce thickness and enhance the absorbing performance.

The auxetic structure is a new type of structure that has been widely studied due to its great impact resistance, strong energy absorption, and so on. It includes various structures such as concave structures [16,17,18,19], chiral structures [20,21,22,23], origami structures [24,25,26,27], and rotating structures [28,29,30,31], all of which share the common feature of a negative Poisson’s ratio. Since 1987, the excellent mechanical properties of auxetic structures have received wide attention [32,33,34]. The use of auxetic structures can improve the mechanical properties of MAMs, thereby making MAMs more compatible with the needs of practical applications. But traditional preparation processes such as cutting and grinding face difficulties in achieving precise control of structure parameters, which have a negative influence on the performance and application prospects of auxetic materials. To achieve high-precision material preparation, numerous manufacturing approaches have been considered. High-precision techniques such as 3D printing [35,36], subtractive methods, and injection molding, enable accurate preparation of macroscopic auxetic structures. For example, Han et al. developed a new wearable electronic, which used a re-entrant honeycomb structure and was printed directly on the substrate with direct ink writing technology. It has shown much better performance than other wearable electronics with uniform structures [37]. Following the identification of suitable preparation methods, the current priority is determining the structural parameters of auxetic MAMs.

The structure, pore size, thickness, and layer numbers of auxetic structures will all affect their microwave absorption performance. Meanwhile, the porosity also needs to be taken into consideration, as it is related to economic benefits. Therefore, achieving excellent microwave absorption performance through structure design is the key to research. In practical applications, RL_min < −30 dB is considered to be a great absorption performance. However, it is laborious to select conforming structures and matching materials only by experiments. Accordingly, instead of repetitive experiments, it is practicable to conduct a series of simulations to analyze performance simply and cost-efficiently. The High-Frequency Structure Simulator (HFSS) is a simulator with a strong post-processor, which can effectively solve the simulations of auxetic MAMs [38,39,40]. By utilizing it, we can easily find out the mechanism of auxetic MAMs and let the mechanism guide actual experiments. In recent studies of MAMs, the real part of permeability is near 1 and the imaginary part is near 0, which shows a low effect on microwave absorption [41,42,43,44,45]. In this case, the material’s dielectric constant can be used to represent the material’s property. Therefore, we can analyze the material simply by regulating its dielectric constant.

Herein, the dielectric constant regions were simulated by MATLAB. After ascertaining the range of dielectric constant, the parameters of auxetic MAMs such as structure, dielectric constant, pore size, and layer numbers were simulated and optimized by the HFSS, and then the influence of four parameters on microwave absorption was clarified, which leads to a process of structure and parameter selection with good absorption performance. The simulation results indicate the absorption performance of RL_min < −30 dB and EAB > 3 GHz can be achieved in the frequency range of 8–18 GHz. The real part of the dielectric constant (ε′) and the imaginary part of the dielectric constant (ε″) of auxetic MAMs are quietly decreased from 40 and 12.5 to 30 and 7.5 by changing each layer of the structures. At last, the auxetic structures with desired microwave absorption performance and low cost were obtained. In summary, this study demonstrates a predictive framework for structures, dielectric constants, and geometric parameters (layer number and pore size) with microwave absorption frequency/performance, ultimately yielding auxetic metamaterials exhibiting core advantages of lightweight characteristics, high absorption, and wide bandwidth. The methodology enables reverse engineering of structures, dielectric constants, and geometric parameters based on application-specific frequency requirements, thereby providing theoretical guidance for microwave absorption research. Furthermore, leveraging the intrinsic mechanical superiority (negative Poisson’s ratio) inherent to auxetic structures, this work proposes a novel cross-disciplinary development strategy for MAMs while delivering innovative engineering-oriented technical solutions.

2. Simulations

2.1. Parameters

Structure: Rhombus (RH), concave quadrilateral (CQ), concave hexagonal (CH), and swastika shape (SW) structures as shown in Figure 1 were used as minimum periodic modeling elements. The swastika shape considered herein has been used solely for its physical properties.

Dielectric constant: For dielectric materials, the absorption performance does not change at a fixed permeability, so the influence of the materials on the absorption performance can be considered as the influence of the dielectric constants. In order to match the dielectric constant of existing MAMs, the real part of the dielectric constant (ε′) at 2 GHz is set to 5–50 as a value is taken every 5; the imaginary part (ε″) is 2.5–25 and the value is taken every 2.5. The dielectric constants decrease with increasing frequency.

Layers: In order to ensure the uniformity of parameters, the unit thickness is consistently set to 0.6 mm, referring to the 23G needle nozzle, and the total thickness is the unit thickness multiplied by the layer numbers. Different layers are staggered and arranged periodically, with each adjacent layer having a difference of 1 divided by the layer numbers in the y-axis direction. Therefore, the influence of thickness on the absorption performance can be expressed as the influence of the layer numbers. In order to match the thickness of prevailing MAMs, the layer numbers are set to three, four, and five.

Pore size: The pore size is defined by the maximum length of the pores. In this simulation, the pore size of each unit is set to 1–5 mm and chosen every 0.5 mm to evaluate the RL. In addition, the pore wall thickness of the structure is uniformly set to 0.6 mm.

Porosity: The porosity is the ratio of the volume of the structure to the total volume of space. The porosity of the structure in normal multiple layers is equivalent to that in a single layer, while the porosity of different structures in multiple layers is calculated by adding up the volume of the multiple layers before division.

2.2. Simulation Process

MATLAB is used for inputting the absorption formula code and outputting the RL curves. The absorption formulas are as follows:

$$Z=\left|{\displaystyle \frac{Z_{in}}{Z_0}}\right|=\sqrt{\left|{\displaystyle \frac{{\mu }_r}{{\epsilon }_r}}\right|}\mathit{tanh}\left[j\left({\displaystyle \frac{2\pi fd}{c}}\right)\sqrt{{\mu }_r{\epsilon }_r}\right],$$

(1)

$$RL=20\mathit{lg}\left|{\displaystyle \frac{Z_{in}-Z_0}{Z_{in}+Z_0}}\right|,$$

(2)

where Z is the matching impedance, Z_in is the input characteristic impedance, Z₀ is the free space impedance, ε_r is the complex dielectric constant, μ_r is the complex magnetic permeability, f is the frequency, d is the thickness, and c is the speed of light. After obtaining the calculated theoretical RL curves, we can deduce the theoretical dielectric constant adaptation region within the range of RL < −10 dB through the RL curves in order to further determine the dielectric constant region used in the simulation. After that, the HFSS is used to simulate auxetic MAMs. Firstly, the materials in three, four, and five layers are modeled with each layer having the same structure shown in Figure 1, and then an appropriately sized air box is added to the model to simulate the application environment. Afterward, master and slave boundaries are set around the air box to specify periodic units to simulate the auxetic MAMs. The Floquet port is set at the top surface to specify the incident wave, and then the structures are simulated to obtain the RL curves. On this basis, any one layer of the multilayers is replaced and the simulation continues according to the above steps. Finally, the auxetic MAMs are obtained. After each parameter change, the parameters and the obtained RL and EAB are incorporated into Origin software, which eventually contains all the data. The COMSOL is used to simulate the electromagnetic power loss density of three-layered auxetic MAMs. The models and dielectric constants are imported from the HFSS. The electric and magnetic boundaries are set along the x and y directions. An excitation port is set on the upper surface of the air box. The electromagnetic power loss density images of four structures at frequencies of 2 GHz, 6 GHz, 10 GHz, 14 GHz, and 18 GHz are generated by the simulation. The schematic diagram of the research process is shown in Figure S1.

3. Results and Discussion

3.1. The Influence of Dielectric Constant on Absorption Performances

To select the fitting dielectric constant of MAMs used in the simulation that can ensure effective microwave absorption, MATLAB is put into operation. By inputting the codes to MATLAB based on Equations (1) and (2), the dielectric constants of RL< −10 dB at a thickness of 1–4 mm were calculated. The corresponding dielectric constant regions between 2 and 18 GHz are shown in Figure 2a–d. By analyzing the regions, the ranges of dielectric constant can be preliminarily determined. It can be seen that the dielectric constant decreases as frequency increases. This is due to the low change rate of the polarization intensity inside the material to keep up with the change rate of the external electric field, resulting in the hysteresis phenomenon. The relationship between dielectric constant and frequency is as follows:

$${\epsilon }_r={\displaystyle \frac{\epsilon }{{\epsilon }_0}}=\left(1+i{\displaystyle \frac{\sigma }{{\epsilon }_0\omega }}\right),$$

(3)

where ε is the absolute dielectric constant, ε₀ is the vacuum dielectric constant, σ is the conductivity, ω = 2πf, and f is the frequency. The commonly used dielectric constant refers to ε_r. Equation 3 also indicates the relationship between dielectric constant and frequency. When the frequency of the electric field changes, the conductivity cannot keep up with the frequency change caused by the variable electric field, resulting in a gradual decrease in ε_r. The above theory is consistent with the simulation results, proving the accuracy of the simulation. In Figure 2a–d, the dielectric constant decreases rapidly with increasing thickness. The ε′ region shows extremely large values and the ε″ region is almost from 0 to 30. However, the dielectric constants are too large to achieve. This is due to the performances of the structure, thickness, and pore size, which have great effects on microwave absorption as well. In a recent study, it was found that materials with ε′ under 50 and ε″ under 25 are common and easy to prepare. As a result, we used such dielectric constants to represent the materials.

On this basis, the HFSS simulation results were obtained by directly inputting the dielectric constant and permeability to define the material’s properties. The effects on the microwave absorption performance of ε′ and ε″ were obtained only by changing the dielectric constants of four structures. In Figure 2e–h, the RL curves of four five-layered structures with dielectric constants of (40, 15), (45, 15), (50, 15), (50, 17.5), and (50, 20) with a pore size of 3 mm are listed as an example, where the first digit represents ε′ and the second digit represents ε″. It can be seen that all structures exhibit the same rule: when ε′ increases, the frequency of RL_min decreases; while ε″ increases, the frequency remains almost unchanged. In a word, ε′ simultaneously affects the RL_min and the range of frequency, while ε″ affects only the RL_min. Additionally, the RL curves of four structures with a pore size of 4 mm and their four-layered structures with a pore size of 3 mm are also listed in Figure S2 in the Supplementary Materials. It shows that whether changing the layer number or the pore size, the conclusion is still valid. In applications, RL_min < −30 dB is considered to be a great absorption performance. As shown in Figure 2e–h, RL_min < −30 dB is difficult to reach without changing other parameters. To find out the rule of the change in RL_min, a new simulation was set that the ε″ is fixed and the ε′ is increased by 2.5 each time, which was then alternated to test the other one. Finally, the data are presented in Figure 3.

As shown in Figure 2e–h, changing only the dielectric constant cannot ensure RL_min < −30 dB. To reach the goal of RL_min < −30 dB, the pore size inevitably changes. It can be seen from Figure 3 that as ε′ increases, the corresponding pore size of RL_min < −30 dB decreases. As ε″ increases, the corresponding pore size of RL_min increases. Other data show the same rule above. In addition, the RL_min of CQ at the condition of (40,15), 2.5 mm, is −55.52 dB, and the RL_min of SW at the condition of (30,15), 5 mm, is −54.98 dB in Figure 3, which shows great microwave absorption performance. However, as mentioned in Figure 2e–h earlier, ε″ only causes a change in the reflection loss and does not alter its frequency. Therefore, the change in the frequency of RL_min is only due to the change in the corresponding pore size. In summary, ε′ affects the RL_min and the frequency of RL_min, while ε″ affects only the RL_min.

3.2. The Influence of Layer Number on Absorption Performance

The layer number represents the thickness, whose change always accompanies the change in frequency and RL_min. It is obvious in Figure 4 that only when the dielectric constant reduces or the pore size increases as the layer numbers decrease can the RL_min still be less than −30 dB. To prove this, the additional data are shown in Figure S3 in the Supplementary Materials. In Figure S3a, as the layer numbers change, the parameters remain almost unchanged. In Figure S3c, RL_min < −30 dB cannot be met by only changing the layer number. And in Figure S3b,d, we enlarged the pore size and decreased the dielectric constant as the layer number decreased. As a result, the microwave absorption performance met the need of RL_min < −30 dB, which is consistent with the conclusion in Figure 4. In the meantime, the changes above will force the EAB to become higher as well. So, the frequency of RL_min is significantly increased when the layer number decreases. The coherent subtraction phenomenon occurs with the generation of a standing wave when the structure thickness reaches one-quarter of the wavelength, resulting in enhanced microwave absorption. The formula for calculating the matching thickness is as follows:

$$d=n\lambda /4=nc/(4f{\left(\left|{\mu }_r\right|\left|{\epsilon }_r\right|\right)}^{{\displaystyle \frac{1}{2}}})\left(n=\mathrm{1,3},5\dots \right),$$

(4)

where λ is the matching wavelength. It can be observed that when other conditions in the formula remain unchanged, the longer the wavelength, the greater the required matching thickness, which can confirm the above viewpoint. In addition, when the layer numbers reduce, the cost of MAMs reduces as well. Therefore, it is necessary to obtain the required absorption frequency band by selecting the appropriate layer numbers.

By changing the layer numbers and maintaining the values of ε_r and pore size, the rule of layers is studied. In Figure 5, the ε_r is (50, 15) and the pore size is 3 mm, which exhibits a generalized rule of layers. It can be seen that the frequency of RL_min decreases with the increase in layers and reaches its minimum at five layers, indicating that the change in the layer numbers can significantly affect the frequency. Moreover, the layer number at the best RL_min depends on the structure of the MAMs. In addition, if changing ε_r and pore size, the best layer numbers may change as well. Therefore, only proper layer numbers can enhance the microwave absorption performance.

3.3. The Influence of Pore Size on Absorption Performance

Pore size plays a crucial role in the propagation of electromagnetic waves. In the observation area far away from the scatterer, the spatial distribution of scattered waves can be represented by the function of $f(\theta ,\phi )e^{i\omega \tau (r_s,r)}/\left|r_s-r\right|$, where θ is the polar angle, φ is the azimuth, and f(θ, φ) describes the spatial distribution of scattered waves and relates to the geometric shape of MAMs [46]. When the size of the structure is much larger than the wavelength, the structure can be seen as a plane. Since the dielectric constant of the MAM is much higher than that of air, the reflection coefficient of electromagnetic waves is high, which makes electromagnetic waves propagate mainly by reflection. When the size approaches the wavelength, electromagnetic waves not only turn into diffracted waves and propagate around the edges of the absorption materials, but also get absorbed by the material through multiple reflections and transmission on the pore wall or inside the material. When the size is much smaller than the wavelength, most electromagnetic waves can only directly incident on the structure and be absorbed after multiple reflections and transmissions inside the structure. As for the pore size, when the pore size is small, the porosity of the structure turns low and the wavelength is much smaller than the pore size, resulting in direct absorption of electromagnetic waves as they pass through. Low-frequency electromagnetic waves have lower energy and are more prone to attenuation. When the pore is large, the porosity turns high, and the proportion of electromagnetic waves diffracted along the structure edge increases as well. When electromagnetic waves receive higher frequencies, the electric field changes faster and greater, which causes more energy to be consumed in the medium. In this circumstance, electromagnetic waves can better reflect multiple waves at the pore wall, enhancing the absorption ability. Therefore, when the pore size is large, the frequency at the RL_min is higher.

Studies on MAMs concentrate on 2–18 GHz, with corresponding wavelengths of 16.7–150 mm. In previous research, pore size should be less than or equal to λ/5, so we chose 1–5 mm as the pore size range. In order to investigate the influence of pore size, we conducted a series of simulations to test the RL_min and EAB of every pore size with the dielectric constant and the fixed layer number. The RL curves of five-layered structures under different pore sizes when ε_r is (50, 15) are shown in Figure 6. It is clear that the frequency of RL_min becomes higher as the pore size increases. The lack of data below 2.5 mm for SW is because, according to the definition of pore size, the pore size of the SW structure should be greater than 2.4 mm. Therefore, the smallest pore size is set to 2.5 mm. From the simulation results, it can be seen that changes in pore size can significantly alter the frequency and reflection loss. The simulated pore sizes are much smaller than the wavelength at 1-5 mm, which indicates that the incident electromagnetic wave mainly entered the material from the surface, and the remaining part will diffract along the edge of the pore, and then reach the pore wall for the process of reflection and transmission, and finally be absorbed.

To elucidate performance variations among auxetic structures, we conducted comparative simulations of four distinct structures under identical conditions. Figure S4 employs four parametric combinations to contrast the absorption characteristics across these auxetic structures. The results reveal a consistent descending frequency hierarchy: RH > CQ > CH > SW under equivalent parameters. Furthermore, the optimal absorption performance for each structure necessitates unique parametric combinations, thereby substantiating significant performance differentiation among the four auxetic structures.

To clearly analyze the microwave absorption performance of each layer and clarify the argumentation above, the electromagnetic power loss density was introduced and simulated to explain the results. As the materials of the structures are all the same, every layer should have the ability to both attract microwaves into the materials and absorb them. Therefore, the three-layered structures were chosen to show the rule. The pore sizes are fixed to not influence the conclusion. In Figure 7, the electromagnetic power loss density of different three-layered structures with a dielectric constant of (50, 15) and a pore size of 3 mm is presented. In this simulation, we chose a frequency every 4 GHz to represent the variation of 2–18 GHz. It can be seen that the color turns redder when the electromagnetic power loss density becomes higher. In this case, the structures show higher loss density in all three layers near the frequency of RL_min, while only one or two layers have high density loss at the rest of the frequencies, which can support the above argumentation.

3.4. The Influence of Porosity on Absorption Performance

As analyzed earlier in Section 3.1, Section 3.2 and Section 3.3, it can be concluded that each parameter can shift the frequency of RL_min independently. When other parameters remain constant, the frequencies of RL_min vary differently with their structures in the order of RH > CQ > CH > SW. When the pore size is fixed at 3 mm in Figure 6, the order can be clearly proved as the frequencies of RL_min are 9.5 GHz, 8.1 GHz, 7.15 GHz, and 6.5 GHz. But in Figure 6a,b, it can be seen that when the dielectric constant and layer number are the same, RH’s frequency of RL_min is smaller than CQ’s at 1 mm, which is different from other phenomena. Therefore, there must be a relationship between pore size and structure that forces the frequency of RL_min to be changed.

To integrate pore size and structure, porosity is used to analyze the relationship between them. The porosities of each structure at different pore sizes are depicted in Figure 8a. It can be seen that the porosities are RH > CQ > CH > SW. In addition, the porosity relationship of the four structures basically does not change with the increase in pore size. Only when the pore size is about 1–1.5 mm is the porosity of CQ higher than that of RH. Combining the data from Figure 6 and Figure 8a, it can be found that the absorption frequency of RH at 1 mm is similar to that of CH at 2 mm, and they have similar porosity as well. The same conclusion can be clearly seen in Figure S5 in the Supplementary Materials. As all four structures show different pore sizes, their porosities calculated by the data of Figure 8a are similar. Therefore, the changes in structure and pore size on the frequency of RL_min are equivalent to the changes in porosity. It also indicates that increasing porosity will also increase the frequency of RL_min. In this sense, another analysis was employed to substantiate the effect of porosity. In Figure 8b, the porosity and EAB at various pore sizes and structures were obtained by selecting data with RL < −10 dB at the same dielectric constant. As the porosity increases, the frequency of the EAB rises. The repeated experiments demonstrate the universality of this law in Figure S6 in the Supplementary Materials. Thus, it can be fully demonstrated that the phenomenon that frequency varied with the pore size is essentially caused by the porosity. Importantly, porosity can also affect the weight and material costs. Structures with high porosity are lighter, which can reduce costs and processing time. To obtain an MAM that balances both absorption performance and practical applications, it is significant to have higher porosity while meeting requirements of lightweight characteristics, strong absorption, wide EAB, and thin thickness.

3.5. Design of Practical Auxetic MAMs

Based on the above results, absorption performance is affected by the dielectric constant, layer number, structure, and pore size (porosity). By adjusting the parameters, we can gain practical auxetic MAMs with good microwave absorption performance. As studied above, RL_min < −30 dB shows good microwave absorption performance. Meanwhile, EAB > 3 GHz exhibits great potential in practical applications. In accordance with the above two requirements,

Table 1. Practical auxetic MAMs at RL_min < −30 dB and EAB > 3 GHz.

RLmin (GHz)	Layer Number	Structure	εr	Pore Size (mm)	Porosity (%)	EAB (GHz)
8.2	5	CH	(35, 12.5)	2.5	45.21	3
8.25	5	CQ	(40, 15)	2	51.69	3
8.4	5	RH	(35, 12.5)	1.5	51.02	3.1
8.4	5	CQ	(50, 20)	3.5	59.77	3.2
8.7	5	SW	(50, 20)	5	59.90	3.25
8.75	5	CQ	(50, 22.5)	4	61.97	3
8.9	5	CH	(45, 17.5)	4.5	58.62	3.2
9	5	CH	(40, 15)	4	55.98	3.3
9	5	RH	(40, 15)	2	59.17	3.25
9.1	5	CQ	(50, 22.5)	4.5	63.95	3.2
9.25	5	CH	(45,17.5)	5	60.95	3.3
9.25	5	CQ	(50, 22.5)	5	65.75	3.2
9.3	4	CH	(50, 15)	3	49.40	3
9.75	4	CH	(50, 17.5)	3.5	52.94	3.1
9.9	4	CQ	(50, 15)	2.5	54.65	3.05
10	4	SW	(50, 15)	4.5	56.31	3.1
10.5	4	RH	(40, 12.5)	1.5	51.02	3.1
10.5	4	CH	(50, 17.5)	4	55.98	3.05
10.55	4	CQ	(50, 15)	3	57.34	3.1
10.85	4	RH	(45, 15)	2	59.17	3.05
10.95	4	CH	(50, 17.5)	5	60.95	3.35
11.2	4	RH	(50, 17.5)	2.5	65.03	3.75
11.75	3	CQ	(50, 10)	1	45.02	3
12.4	3	CH	(50, 12.5)	2.5	45.21	3
12.5	3	RH	(50, 12.5)	1.5	51.02	3.1
12.85	3	SW	(50, 12.5)	4	52.04	3.8
12.85	3	SW	(45, 10)	3.5	46.88	3.7
13.4	3	CH	(50, 12.5)	3.5	52.94	3.6
13.9	3	CH	(50, 15)	4	55.98	4.2
14	3	RH	(50, 15)	2	59.17	4.3
14.2	3	SW	(50, 17.5)	5	59.90	4.5
14.3	3	CH	(50, 15)	4.5	58.62	4.5
14.5	3	CQ	(45, 12.5)	3	57.34	4.6
14.7	3	CQ	(50, 15)	4	61.97	4.6
15	3	CH	(50, 15)	5	60.95	4.8
15.2	3	CQ	(50, 15)	4.5	63.95	4.8
15.9	3	CQ	(45, 17.5)	4.5	63.95	4.25
16	3	RH	(45, 15)	2.5	65.04	4.1
16.3	3	CQ	(45, 17.5)	5	65.76	4
16.4	3	RH	(50, 17.5)	3	69.44	4
17.2	3	CQ	(40, 15)	5	65.76	3.5
17.45	3	RH	(50, 17.5)	3.5	72.87	3
17.8	3	CH	(35, 12.5)	5	60.95	3

lists the practical auxetic MAMs at different frequencies. It can be seen that the layer number decreases with the frequency of RL_min increasing. With the same layer numbers, it can be seen that materials with higher porosity have the RL_min at higher frequencies, but the porosity of the SW structure increases more slowly, so the SW structures are concentrated in relatively low frequencies. The other three structures have suitable parameters that meet the need above in 8–18 GHz. The porosities of different structures are analogous at similar frequencies, and the anomalous porosity occurs only when the dielectric constants show great variety. These conclusions are consistent with the above results on the mechanism of the effects of layer number, dielectric constant, and porosity on MAMs.

In recent studies, the obtained MAMs often have a high RL_min or wide EAB, but the absorption frequency of MAMs is usually uncontrollable. Table 1 shows the great potential of the auxetic MAMs in microwave absorption. As long as the absorption frequency is definite, the structure and pore size of the material can be regulated by high-precision preparation technology, and printing materials with a suitable dielectric constant can be selected to finally obtain the MAMs. However, the MAMs in Table 1 have higher dielectric constants than many MAMs, which makes the material selection difficult. In order to reduce the required dielectric constant, each layer of the three-layered structures is varied individually and their microwave absorption performances are simulated. The optimized three-layered auxetic MAMs are summarized in Table S1. It can be seen that in Table 1 the ε′ are mostly 40–50 and the ε″ are mostly 12.5–22.5, but those are apparently lower in Table S1 in the Supplementary Materials. The ε′ is down to 30 and the ε″ is down to 7.5, which are basically similar to modern MAMs. As the dielectric constants become lower, more kinds of materials can be applied to assemble auxetic structures and it makes auxetic structures more practical to absorb microwaves. Anyway, changing each layer of the structure is a useful way to reduce the dielectric constant. The elevated dielectric constant ranges presented in Table 1 and Table S1 are specifically attributed to layer number selections within the structural design paradigm. Table 1 and Table S1 prioritize thinner multilayer configurations under lightweight design principles at higher frequencies, while thicker architectures with equivalent absorption performance (RL_min < −30 dB, EAB > 3 GHz) at reduced ε′/ε″ values are intentionally omitted. Figure S7 demonstrates that five-layered auxetic MAMs with reduced dielectric constants (ε′ = 10, ε″ = 5) consistently achieve the target absorption performances (RLmin < −30 dB, EAB > 3 GHz). Numerous additional configurations satisfy these performance criteria across diverse material parameters, such as RH (15, 7.5) at 2 mm, CH (5, 2) at 1 mm, SW (5, 2) at 2.5 mm, etc. This study demonstrates that material compatibility can fully satisfy current high-precision preparation conditions while maintaining target electromagnetic performance metrics. To strengthen the conclusion, we compared the auxetic MAMs in this study against recently reported MAMs in

Table 2. Comparison between auxetic MAMs and traditional MAMs.

Material	Thickness (mm)	EAB (GHz)	RLmin (dB)	Ref.
RGO–NW–CNT composites	4	6.3	−35	[47]
Co2P	1.1	2.4	−39.3	[48]
RF/SiO2 aerogel	1.95	2.8	−36.42	[49]
Fe3O4–graphite composites	4	3.3	−40.6	[50]
CQ (40, 15), 2.5 mm	3	3	−55.52	This work
SW (30, 15), 5 mm	3	4.5	−54.98	This work

. Our auxetic MAMs outperform others in both RL_min and EAB, demonstrating their practical value.

In summary, we obtained practical auxetic MAMs with achievable dielectric constant, lightweight characteristics, high absorption performance, and wide EAB, which make much practical sense in MAM research and industry.

To enhance the reliability of our conclusions, we conducted a robustness analysis on auxetic MAMs. Given the high precision of preparation technology with maximum fabrication tolerance below 0.1 mm, Figure 9 presents RL curves through parameter sweep analyses with 0.1 mm pore size gradients for four three-layered auxetic MAMs at (35, 10). As demonstrated, the RL_min of these four structures occur at 10.3, 15.45, 15.2, and 15.2 GHz, respectively, and their RL_min are −33.48, −52.85, −45.1, and −47.96 dB. At the critical RLmin proximity (0.1 mm tolerance band), localized RL_min variations of 4.11, 6.24, 3.81, and 4.93 dB/0.1 mm were observed. The results indicate deviations within approximately 5 dB, exhibiting favorable stability. This outcome correlates directly with porosity. Under these conditions, the corresponding pore sizes for the four structures measure 1.2, 2.2, 1.9, and 3.7 mm, respectively. Cross-referencing these pore sizes with Figure 8a demonstrates that increased porosity induces greater variations in absorption performance, which is consistent with the established conclusion.

4. Conclusions

Here, the simulation of auxetic MAMs is carried out using the HFSS and COMSOL to elucidate the absorption mechanism of such structures and summarize their absorption performances. RH at (25,10), 1 mm, exhibits an RL_min of −51.09 dB and an EAB of 3.25 GHz; CQ at (40,15), 2.5 mm, exhibits an RL_min of −55.52 dB and an EAB of 3 GHz; CH at (10, 5), 5 mm, exhibits an RL_min of −47.09 dB and an EAB of 4.75 GHz; and SW at (25,10), 1 mm, exhibits an RL_min of −54.98 dB and an EAB of 4.5 GHz. The structures above are all five-layered structures. Practical auxetic MAMs in 8–18 GHz were also obtained. It was found that reducing the ε′, the layer number, or increasing the pore size (actual porosity) leads to an increase in the frequency of RL_min and EAB. Under the same parameters, the effective absorption frequency decreases in the sequence of RH, CQ, CH, and SW. In addition, auxetic MAMs optimized in a dielectric constant by changing each layer of the structure were also obtained. By adjusting the structure and parameters, it is possible to obtain a result close to the desired absorption frequency, which shows good application flexibility of the design method and provides guidance for subsequent research in the MAM field. The inherent outstanding mechanical properties of the auxetic MAMs show great potential for applications in multiple fields. Notably, this work is the first to conduct a feasibility analysis of auxetic MAM absorption, providing theoretical guidance for subsequent studies in this field.