The Modeling of Electromagnetic Behavior in the High-Frequency Range of Al₂O₃ and TiO₂ Thermoplastic Composites in Support of Developing New Substrates for Flexible Electronics

Abstract

The paper describes the simulation of energy absorption in polymer micro-composites that include dielectric inserts (commercial Al₂O₃ and TiO₂ particles, with three particle sizes of 1, 5 and 25 µm, respectively). The investigated frequency spectrum, mainly from 0.001 to 100 GHz, is designed for various uses as substrates in electronic technologies. The electromagnetic simulation software chosen was CST Studio Suite, which evaluates the power loss at different frequencies, playing a crucial role in creating the ideal structure of these substrates. The effective limits of the electromagnetic simulation are specified. It is shown that a considerable increase in absorption occurs, by a factor of 12 to 120, depending on the dielectric material used for the inserts and the mass ratio applied in the insertion technique. Dielectrics with high permittivity provide higher absorption, but also create a nonuniform field distribution within the material, resulting in a high peak-to-average absorption ratio. In scenarios where this behavior is intolerable, the technology must be carefully tuned to improve the consistency of the insertions in the substrate material. The final outcomes of the simulations indicated that for creating new substrates for flexible electronics, polyethylene composites with TiO₂ insertions are suggested, particularly at lower concentrations of up to 7% and with a larger radius, such as 25 μm, which could offer significant economic advantages considering that the current concept advises the use of costly particles ranging from nanoscale particles to those 1 μm in size and a composition exceeding 10%.

1. Introduction

Substrates, commonly known as the base of electronic devices, are materials or frameworks on which electronic components are built. They offer structural reinforcement, enable electrical connections, and act as the surface for the complex circuitry. Flexible substrates represent a key area where the technology of flexible and stretchable electronic devices diverges from conventional electronic technology. Flexible substrates possess the same qualities as conventional rigid substrates, but along with typical attributes like insulation, high strength, and affordability, flexible substrates possess distinct qualities such as flexibility and stretchability, along with corrosion resistance [1,2,3,4]. The benefits pertain to the production of thin films, usually ranging from 0.2 to 1 mm, which lowers material costs and decreases the product’s weight, but also provides a fresh perspective on reducing the carbon footprint of electronic technology [5]. Flexible electronic systems can utilize substrates made from various thermoplastic materials, such as silicon–organic resin, polyethylene, polyimide, polydimethylsiloxane, or polyethylene terephthalate [6,7,8].

Recent research indicates that, in addition to exceptional flexibility and imposed dielectric parameters, stable chemical properties (including low moisture adsorption), mechanical strength, and good thermal stability (low thermal expansion coefficient) are other crucial requirements for mass production in the innovative electronics sector [2,4]. These characteristics are typically associated with polymer composite film technology and can be easily modified; however, they do not provide an accurate depiction of material behavior at high frequencies, which is more challenging to evaluate [9,10,11]. The estimation of transmission losses and waveform distortion in high-frequency (up to 10 GHz) environments may be initially more efficiently conducted through modeling with specialized software, providing essential insights regarding optimal composite structure and formulation.

The research focuses on modeling the electromagnetic properties in the high-frequency range of Al₂O₃ and TiO₂ thermoplastic composites, regarded as novel options for substrates in flexible electronics [12,13]. The advancement of thermoplastic composites featuring Al₂O₃ and TiO₂ inserts has regained scientists’ interest in recent years, whether separately for TiO₂ [14,15,16,17,18,19] and Al₂O₃ [20,21,22,23], or even in combination [24]. Due to their potential ability to operate effectively at high-speed signal frequencies, ranging from the upper-MHz to the GHz range, the understanding of their potential and specific requirements in high-frequency applications (e.g., base stations for flexible cell phones or communication and control systems—including for the automotive industry, wearable medical equipment, etc. [24,25,26,27,28,29]) is crucial for designing substrates that meet the demanding standards of advanced electronic technologies. The current paper represents a step forward in this area, addressing the modeling of electromagnetic performance concerning the composition of respective thermoplastic materials.

Some analytical techniques and theoretical methods have remained steady as the dielectric inserts have been discovered to achieve satisfactory performance at microwave frequencies [30,31,32,33]. Nevertheless, analytical methods have proven effective only for straightforward structures and tend to fall short in the frequency range where the wavelength approaches the distances between particles and/or their size. Electromagnetic simulation relies on the methodology initially suggested by Nicolson, Ross, and Weir (the NRW procedure), which has been refined as discussed in references [34,35,36] to overcome the NRW procedure’s instability when dealing with low-loss materials, where measurement inaccuracies significantly impact the precision of the results.

The selected software for electromagnetic simulation was CST Studio Suite [37]. The simulation demonstrated below for thermoplastic composites with Al₂O₃ and TiO₂ inserts is groundbreaking and seeks to identify the ideal choices for particle size and power loss at different frequencies, serving as a foundation for advancing new substrates for flexible electronics within the high-frequency spectrum.

2. Electromagnetic Simulation

Usually, commercial Al₂O₃ and TiO₂ particles which can potentially be used for composite substrates present a mainly quasi-spherical shape and dimensions between 1 and 40 μm. A preliminary investigation into the influence of particle shape on electromagnetic field absorption is presented in Figure 1 (for spherical particles of TiO₂), where one can notice that the energy losses are volumetric rather than superficial.

Accordingly, in order to simplify the model, it is preferable to replace spherical particles with cubic (parallelepipedal) particles. A spherical particle can be replaced by a cube that provides a similar volume, as in Equation (1), or a cube that has the same external surface, as in Equation (2).

$$R_{cube\left(V\right)}^3={\displaystyle \frac{4\pi \cdot R_{sphere}^3}{3}}\to R_{cube\left(V\right)}=R_{sphere}\cdot \sqrt[3]{{\displaystyle \frac{4\pi }{3}}}$$

(1)

$$6\cdot R_{cube\left(S\right)}^2=4\pi \cdot R_{sphere}^2\to R_{cube\left(S\right)}=R_{sphere}\cdot \sqrt{{\displaystyle \frac{2\pi }{3}}}$$

(2)

where R_cube denotes the side of the cube that provides a similar volume/area to a sphere of radius R.

The main electromagnetic evaluation relies on the computed electromagnetic field values in the composite structure, as outlined in references [38,39,40].

The simulation of energy losses [×10⁻³ W/kg] vs. frequency [GHz] depicted in Figure 2 is further proof of this concept (where Cube(V) is a cube that provides a similar volume to the sphere; Cube(S) is a cube that provides a similar exterior area to the sphere), but better precision is achieved when a spherical particle is replaced by a cube that provides a similar volume, especially at higher frequencies, over 1 GHz. So, it can be admitted that the use of cubic/parallelepipedal particles models the absorption offered by spherical particles sufficiently precisely across the entire frequency range.

The simulation was further performed in such a way that particles occupy an arbitrary position so as not to exceed the limits of the structure. A parallelepiped (quasi-cubic) shape was chosen for the particles; all particle dimensions (L_p, W_p, and H_p), as in Equation (3), being between half and double the side of the cube corresponding to the sphere, with the reference sizes chosen for the study—1 μm, 5 μm, and 25 μm—being in line with the commercial dimensions of such particles. In Equation (3), x is a randomly generated number between 0 and 1. Equations (4) and (5) represent the theoretical limit for the volume of the individual parallelepipedal particles.

$$L_p,W_p,H_p=R_{cube}\cdot 2^{2x-1}$$

(3)

$$V_{min}={\displaystyle \frac{1}{8}}·R_{cube}^3;V_{ref}=R_{cube}^3;V_{max}=8·R_{cube}^3$$

(4)

$$V\in \left({\displaystyle \frac{1}{8}}·V_{ref};8·V_{ref}\right)$$

(5)

The number of parallelepipedal particles is limited due to practical reasons to about 125 (a test with 8000 particles revealed an excessive generation time and excessive memory requirements for the simulation). As a result, the cubical substrate domain must be reduced in order to offer the required mass ratio (MR) (and volume ratio as detailed in [39]) for a typical number of 125 cubical particles, see the values in

Table 1. Dimensions of structures, ~125 particles, TiO₂ in polymer matrices HDPE/LDPE.

	RSph [μm]	1	5	25
	RCube [μm]	1.612	8.06	40.3
Substrate	Insertion MR [%]	L = W = H [μm]
HDPE	10%	27.857	139.28	696.42
	7%	31.636	158.18	790.89
	3%	42.415	212.07	1060.4
LDPE	10%	28.081	140.41	702.03
	7%	31.892	159.46	797.31
	3%	42.762	213.81	1069.1

and

Table 2. Dimensions of structures, ~125 particles, Al₂O₃ in polymer matrices HDPE/LDPE.

	RSph [μm]	1	5	25
	RCube [μm]	1.612	8.06	40.3
Substrate	Insertion MR [%]	L = W = H [μm]
HDPE	10%	26.904	134.52	672.59
	7%	30.544	152.72	763.61
	3%	40.936	204.68	1023.4
LDPE	10%	27.12	135.6	677.99
	7%	30.792	153.96	769.79
	3%	41.271	206.36	1031.8

. Taking into account that the particles’ volume and dimensions may not be equal, the relationships in Equations (3)–(5) above cannot be applied to calculate the required number of particles, so particle generation continues incrementally until their total volume reaches the required value. As a result, an example of a typical simulated structure is the one shown in Figure 3 (for TiO₂, with a reference sphere radius of 5 μm, the mass and volume ratio specified, and the random generation results in less than 125 particles for the required particle volume).

The boundary conditions and the generation of the plane wave characteristics of the electromagnetic field were determined by choosing the conditions of an electric wall (x_min and x_max) and, respectively, a magnetic wall (y_min and y_max). The conditions for the input and output ports are labeled “open”. Details about the procedure and related conditions can be found in [40]. Port definition was realized by positioning the ports at a fixed distance from the substrate (dependent on the wavelength, but not on the structure content). As a result, the ports were defined; namely, a free section of material (void) was introduced to move the port to a distance equal to half the side of the substrate/insert cube analyzed. Additionally, for the ports, the reference plane was moved to the boundary of the respective cube, as shown in Figure 4.

Even if at first glance some uncertainties in the simulation are evident, we point out that the use of simulation in defining the composite substrate technology will obviously also be accompanied by limited possibilities in terms of controlling the position of the particles inside the substrate.

The electric and magnetic fields at the ports can be visualized (Figure 5), thus verifying the judicious choice of boundary conditions generated as an interaction with a plane wave. Analysis of the vector distribution of power transfer (Poynting vectors) shows that dielectric inserts reduce the electromagnetic power transfer through the material, leading to additional energy losses.

The polymer matrices that host the inserts are low- and, respectively, high-density polyethylene (LDPE and HDPE), for which CST software does not have models in its own library, but a new model was made based on a first-order Debye equation, as described in [40], and presented in

Table 3. LDPE and HDPE matrix properties.

Matrix	LDPE	HDPE
Model type	Normal	Normal
Thermal conductivity	0.48 [W/K/m]	0.51 [W/K/m]
Density	945 kg/m3	964 kg/m3
Dielectric permittivity	2.2	2.4
Magnetic permeability	1	1
Specific heat	1.9 kJ/K/kg	1.9 kJ/K/kg

. The CST software has models in its own library for Al₂O₃ and, for TiO₂, high-permittivity parameters from [41] were used, as in

Table 4. Inserts properties.

Material	Al2O3	TiO2
Model	Normal	Normal
Dielectric permittivity	9.4	63.7
Magnetic permeability	1	1
Thermal conductivity	0.88 kJ/K/kg	0.69 kJ/K/kg
Density	3800 kg/m3	4230 kg/m3
Specific heat	0.88 kJ/K/kg	0.69 kJ/K/kg

.

A preliminary example of power transfer (and implicit energy accumulation within the particles) is presented in Figure 6 as a plot of the directional electromagnetic energy flux (Poynting vector) for a composite of HDPE with 3% TiO₂ at 10 GHz.

3. Results and Discussion

Analysis of the power loss density (see example in Figure 7 for 3% mass ratio 5 μm TiO₂ particles in HDPE) shows significant absorption inside the particles. Moreover, the specific distribution of the particles generates local maxima at specific points where the fields are concentrated, making the results difficult to interpret.

We chose to use the specific absorption rate (SAR) computed by the software as a macroscopic indicator of loses inside the material. Due to the small dimensions of the simulated substrate (Table 1 and Table 2), a significantly smaller averaging mass was used.

The computed SAR still shows local maxima (see example in Figure 8 for 10% mass ratio 25 μm Al₂O₃ particles in HDPE); however, averaging, as in the computation of Total SAR, will show the overall effect of the insertions (including of course the effect of the displacement of some of the substrate material with those corresponding losses when using insertions).

Also, as SAR computations are inherently connected to the material mass, different materials for substrate and insertions, as well as different mass ratios for the insertions, will affect the absorption results. For every combination of mass ratio/substrate material/insertion material we made a reference analysis of the losses in a substrate without insertions with exactly the same dimensions, which later will accommodate those specific insertions. The effect of a specific set of insertions is computed as a relative increase in the total specific absorption rate (Total SAR), see the relationship in Equation (6), allowing for comparisons of different substrate materials (HDPE/LDPE), different insertion materials (Al₂O₃/TiO₂), and different mass ratios (3/7/10%).

$$\Delta S/S={\displaystyle \frac{S-S_{ref}}{S_{ref}}}\cdot 100\left[\%\right]$$

(6)

We expect an increase in mass ratio for the insertion to generate increased absorption as in Figure 9 and Figure 10. Also, an increase in the dimensions of the insertions was expected to have the same effect, higher localized absorption.

The dielectric properties of the insertions are also important. In Figure 11 and Figure 12, we plot SAR for Al₂O₃. The dielectric model offered by CST results in decreased absorption at low frequencies (100 MHz in Figure 11) while, at high frequencies (1 GHz in Figure 12), absorption is increased significantly and also shows the same pattern of localized increase as for TiO₂. In order to detect the influence of the frequency, the structure was the same (mass ratio and radius unchanged); also, the cutting plane for the figures was kept the same.

Figure 13 plots the Total SAR increase for various substrate materials and mass ratios in the mixture. The effect, especially for the high permittivity material (TiO₂), is a significant increase (loses are increased by a factor between 35 and 120 depending on the mass ratio, for both substrate materials). The results are confirmed by the related literature [42,43,44,45], where the effect of TiO₂ addition into other type of composites is presented. Even the low-permittivity material (Al₂O₃) shows an absorption increased by a factor between 12 and 40, an aspect in line with previous findings for other composite types [46,47,48].

Figure 14 shows the clear increase with the mass ratio of the absorption (the same conclusion as in the qualitative analysis in Figure 9 and Figure 10).

Figure 15 shows that, for low-loss substrates (HDPE and LDPE), the absorptive properties are governed by the insertion and not substrate. The difference between the two substrate materials is unsignificant and can be attributed to the random positioning of the insertions at every analysis.

Figure 16 shows the ratio between the localized Peak SAR and the average Total SAR for the entire structure. While there is a clear influence of the random positioning of the particles, we can find ratios between 200 and 700 for TiO₂ and between 30 and 100 for Al₂O₃. This result is consistent with the previous observations over the electromagnetic field’s structure; the concentration of energy dissipation can reach significant values, depending on the random distribution of the particles. In applications where uniform absorption is required, the distribution of the insertions must be controlled by the insertion technology.

Furthermore, the volumetric distribution of power loss density within the material containing inserts continues to exhibit a significantly nonuniform loss distribution. One could attempt to manage the spatial arrangement of the inserts during manufacturing to direct the waves towards an area with superior absorbent qualities; however, this could lead to greater technological complexity and higher material expenses, with a limited improvement of electromagnetic parameters.

At the macroscopic level, for the entire structure we can investigate the electromagnetic field absorption features using the S parameters [49]. The relationships can also be used to extract the absorption characteristics of the homogeneous dielectric equivalent to the substrate with inserts.

$$R={\left|S_{11}\right|}^2$$

(7)

$$T={\left|S_{21}\right|}^2$$

(8)

$$A=1-R-T=1-{\left|S_{11}\right|}^2-{\left|S_{21}\right|}^2$$

(9)

From Figure 17, it is obvious that a content of 3% TiO₂ with a dimension of 1 μm presents a major influence mainly in composites with HDPE, in the case of the S₁₁ parameter, especially in the frequency domain up to 2 GHz.

From Figure 18, it was noticed that even at higher contents of, e.g., 7%, the addition of Al₂O₃ with higher dimensions (here 25 μm) presents a limited influence upon the S parameters in the entire frequency range, an aspect which can be correlated with the results in Figure 13.

As long as the effect of Al₂O₃ insertions is less relevant compared to TiO₂ insertions (as also suggested by Figure 16), Figure 19 and Figure 20 analyze the effect of particles dimension and content upon the S parameters. At higher dimensions, the effect is relevant mainly at higher frequencies, exceeding 1 GHz. On the other hand, an increase in TiO₂ particle (with the dimension of 25 μm) content to over 3% brings no significant benefit in terms of the evolution of S parameters, especially at frequencies higher than 0.5 GHz.

The consolidated results of the simulations led to the conclusion that, for the development of new substrates for flexible electronics, composites of polyethylene with TiO₂ insertions are recommended, especially at lower contents of up to 7% and with a higher radius, e.g., of 25 μm, which may bring important economic benefits when taking into account that current concepts recommend expensive particles with nanoscale to 1 μm dimensions and a content of over 10%. This study may represent a step forward in designing new cost-effective substrates for flexible electronics, and, based on such simulations, in establishing a more convenient design for experiments for innovative materials, being a more effective way than using more expensive matrices to test associations with TiO₂, and more costly experimental procedures to assess the electromagnetic features, as presented in [50,51,52,53,54].

The results in Figure 17, Figure 18, Figure 19 and Figure 20 together with the relationships in Equations (7)–(9) can be used to compute and measure absorption if the insertion technology ensures a quasi-uniform distribution, for similar particles for electronic technology use.

4. Conclusions

The article outlines the simulation of absorbed energy in polymeric micro-composites containing dielectric inserts (commercial Al₂O₃ and TiO₂ particles, featuring three particle sizes, 1, 5, and 25 µm, respectively). The examined frequency range, mainly of 0.001–100 GHz, is tailored for numerous applications of substrates in electronic technologies. The electromagnetic simulation software selected was CST Studio Suite, analyzing power loss at different frequencies, and serving as a necessary step in designing the optimal structure of such substrates. The real constraints in electromagnetic simulation are detailed.

It was shown that we achieved a significant increase in absorption, with a factor between 12 and 120, depending on the dielectric material used for the insertions and the mass ratio used in the insertion technology. High-permittivity dielectrics offer higher absorption, but also generate a nonuniform distribution of the field inside the material, leading to increased peak-to-average absorption ratios. In applications where such behavior is not acceptable, the technology must be controlled in detail to increase the uniformity of the insertions inside the substrate material.

For the development of new substrates for flexible electronics, composites of polyethylene with TiO₂ insertions are recommended, especially at lower contents of up to 7% and a higher radius, e.g., of 25 μm.

If nonuniformity can be controlled, the composite material can be treated as a homogenous dielectric, with the computation or measurement of the scattering parameters allowing for the definition of an equivalent dielectric material which can be used in more complex structure simulations, for similar particles for electronic technology use.