Dependence of microwave dielectric properties on ceramic connectivity in ultralow-ε_r Al₂O₃ porous ceramics

INTRODUCTION

In recent years, the low-dielectric-constant (ε_r < 10) and ultralow-ε_r (ε_r < 5) microwave dielectric ceramics have attracted more and more attention mainly due to the rapid development of microwave communication technology toward higher working frequencies, especially millimeter-wave frequency band^[1-4]. On one hand, for the electromagnetic wave propagating in a dielectric substrate with the length of L, the phase delay is $$2 \pi f L /\left(\sqrt{\varepsilon_{r}} c_{0}\right)$$, where f and c₀ are the frequency and propagating speed of the electromagnetic wave in vacuum, respectively. Therefore, the phase delay increases proportionally with frequency, and it can be suppressed at higher working frequencies only by adopting the substrate materials with lower ε_r. On the other hand, microwave dielectric ceramics are also widely used in many key resonating units such as resonators and filters, whose sizes are mainly determined by the ε_r^[3,5]. For microwave communication mainly working at the sub-6 GHz band in the past decades, high-ε_r and middle-ε_r microwave dielectric ceramics are usually used in these resonating units for minimization^[5]. However, the minimization of devices and components is a natural consequence in millimeter-wave communication due to the rapid decrease in wavelength with frequency, and it is no longer an important aim for the resonating units in most cases. Instead, too small sizes of these resonating units may result in many serious problems, such as difficulty in precise size control and low carrying power. Therefore, low-ε_r and ultralow-ε_r microwave dielectric ceramics are also urgently demanded in these millimeter-wave resonating units to maintain reasonable and machinable sizes and high carrying power.

In the past decades, many low-ε_r microwave dielectric ceramics have been developed, including borates^[6-8], silicates^[9-12], aluminates^[13-15], phosphates^[16-18], molybdates^[19,20], fluorides^[21], and etc. However, the ε_r is higher than 5 for most of them, and very limited dense inorganic materials exhibit the ε_r lower than 5, such as SiO₂ polymorphs^[22,23], borosilicate glass^[24], and cordierite^[25,26]. Till now, amorphous SiO₂ with the ε_r of around 3.8 is still the most important and irreplaceable ultralow-ε_r inorganic material^[23]. The embarrassing situation for the research and application of ultralow-ε_r inorganic materials is attributed to the higher ionic polarization and higher density in comparison with organic materials, both of which tend to increase the ε_r^[4]. Therefore, developing dense inorganic materials with ultralow ε_r is a big challenge. An effective approach to reducing the ε_r of ceramics is to introduce pores, forming porous ceramics^[27-32], which can be regarded as “ceramic-air” composites. Since air has a very low ε_r of around 1 and is nearly lossless, porous ceramics offer a means to widely regulate the three key parameters of microwave dielectric properties: ε_r, Qf value, and temperature coefficient of resonant frequency (τ_f)^[2,3]. Moreover, introducing air into microwave dielectric ceramics can lighten microwave devices, lower costs, and improve portability^[33,34].

Although extensive research has been conducted on porous microwave dielectric ceramics, most studies have primarily focused on controlling porosity and its effects on microwave dielectric properties^[27,31,35], while other microstructural factors have often been overlooked. Among these factors, phase connectivity is an important microstructural factor, as it significantly influences the properties of composites and porous materials, such as mechanical properties^[36], electrical and thermal conductivities^[37], and piezoelectric properties^[38]. Our recent work revealed that the ceramic connectivity is also very important for the dielectric properties of BaTiO₃-epoxy composites^[39] and CaTiO₃ porous ceramics^[32] with high ε_r, since it dominates the electric field distributed in the constituting phases with the ε_r differing much from each other. For the low-ε_r and ultralow-ε_r porous ceramics, the difference between the ε_r of constituting phases (ceramic and air) becomes much less, so the effects of ceramic connectivity may be quite different. Furthermore, previous studies on porous ceramics have mainly addressed dielectric constant and dielectric loss, while little attention has been given to τ_f, a key parameter that determines the temperature stability of microwave dielectric ceramics. Therefore, it is not only of scientific significance but also important for practical application to systematically investigate how porosity and ceramic connectivity affect the ε_r, Qf value, and τ_f of low-ε_r and ultralow-ε_r porous ceramics.

Al₂O₃ ceramic is an important low-ε_r microwave dielectric ceramic, which exhibits a low ε_r around 10, a high Qf value of 330,000-450,000 GHz, and a relatively large negative τ_f of about -60 ppm/°C^[27,40-42]. Al₂O₃ microwave dielectric ceramic is often adopted as the model material to reveal the effects of a series of factors such as porosity, grain size, impurity, and humility^[27,40,43]. Therefore, in the present work, two groups of Al₂O₃ porous ceramics with varying porosity and ceramic connectivity were prepared and used as the model materials. The microstructural evolution, microwave dielectric properties, and their dependence on ceramic connectivity were investigated and analyzed.

MATERIALS AND METHODS

Fine α-Al₂O₃ powder with an average particle size of 200 nm (> 99.9%, Sinopharm Chemical Reagent Co., Ltd.) was used for preparing Al₂O₃ porous ceramics. The raw powder was dried and mixed with 5 wt% polyvinyl alcohol solution as the binder. The powder was uniaxially pressed into compacts with 12 mm in diameter under 100 MPa for 1 min. The compacts were then sintered at different temperatures of 1,000-1,650 °C for 3 h in air with a heating rate of 5 °C/min and a cooling rate of 2 °C/min, and the as-prepared ceramics were denoted by Al₂O₃-A. For Al₂O₃-B porous ceramics, Al₂O₃ powder was mixed with 20-60 vol% NH₄HCO₃ powder (AR, Sinopharm Chemical Reagent Co., Ltd.) as the porogen because NH₄HCO₃ can be decomposed completely at low temperatures without any residual or reaction with Al₂O₃, and thus does not affect the sintering characteristics. The mixed powder was also pressed into compacts, which were sintered at the optimum temperature of 1,650 °C for 3 h after heating at 70 °C for 3 h to decompose NH₄HCO₃ and form large pores.

The bulk density and total porosity of the samples were measured using the geometric method. The open porosity was measured by the Archimedes drainage method, and the closed porosity was calculated from the total porosity and open porosity. The phase constitution of the Al₂O₃ raw powder and porous ceramics was identified by X-ray diffraction (XRD) using Cu Kα Radiation (Rigaku 2550/PC, Rigaku, Tokyo, Japan). The microstructure on the fractured surfaces of Al₂O₃-A and Al₂O₃-B ceramics was observed by field emission scanning electron microscopy (SEM, GeminiSEM 300, Carl Zeiss Microscopy GmbH, Germany). The Vickers hardness was measured by a Vickers indentation tester (MHVD-5IS, Shanghai Jujing Precision Instrument Manufacturing), and the measurement for Al₂O₃-B porous ceramics was conducted in the regions far from the large pores. The cylindrical samples with 9.5-12 mm in diameter and 3.8-6 mm in thickness were used to measure the microwave dielectric properties using a vector network analyzer (Agilent E8363B, Agilent Technologies Inc., Santa Clara, CA). The ε_r and Qf were measured at 10.2-15.5 GHz by a resonant cavity method^[44]. The τ_f was calculated and corrected from the resonant frequencies measured at 20 and 80 °C^[44].

RESULTS AND DISCUSSION

Figure 1 shows the relative density and porosities of Al₂O₃-A ceramics as functions of sintering temperature (T_s). With increasing T_s from 1,000 to 1,300 °C, the relative density of Al₂O₃-A ceramics increases slowly from 41.25% to 46.15%, and the trend speeds up for higher sintering temperatures. The closed porosity is within 5% for all the Al₂O₃-A ceramics, so the total porosity is dominated by the open one. The highest relative density of 91.73% is obtained for the sintering temperature of 1,650 °C. Therefore, the Al₂O₃-B ceramics are sintered at the optimum temperature of 1,650 °C, and Figure 1B shows the relative density and porosities of Al₂O₃-B ceramics as functions of NH₄HCO₃ volume fraction in the green compacts (V_NH4HCO3). The relative density of Al₂O₃-B ceramics decreases with V_NH4HCO3, and it can be tuned significantly from 78.76% to 48.31% by increasing V_NH4HCO3 from 20% to 60%. The decomposition of NH₄HCO₃ particles produces large pores, which tend to be separated from one another and, therefore, increase the closed porosity for the low V_NH4HCO3. When V_NH4HCO3 is high, however, the NH₄HCO₃ particles may contact the neighbors, and the decomposition of contacting particles results in the interconnected network of the formed large pores. Therefore, the closed porosity first increases and then decreases with V_NH4HCO3, although the total porosity always increases for Al₂O₃-B ceramics, as shown in Figure 1B. The XRD patterns of Al₂O₃ raw powder and some typical Al₂O₃-A and Al₂O₃-B ceramics are shown in Supplementary Figure 1. These XRD patterns are almost the same, and all the peaks can be indexed by α-Al₂O₃ (PDF card: 97-003-3639), indicating that the sintering temperature and porosity have no effect on the phase constitution and crystal structure.

Figure 1. Relative density and porosities of (A) Al₂O₃-A and (B) Al₂O₃-B porous ceramics.

The SEM images of the fractured surfaces of Al₂O₃-A [Figure 2] and Al₂O₃-B [Figure 3] ceramics clearly reveal the significant difference between the microstructures of the two groups of Al₂O₃ ceramics. When the sintering temperature is as low as 1,000 °C, the small and rounded particles with a size of around 200 nm and their aggregations are observed in Al₂O₃-A, which inherits the microstructure of Al₂O₃ raw powder. The microstructure does not change much at a sintering temperature of 1,200 °C, except that the aggregations become more clear, which is consistent with the slowly increasing relative density shown in Figure 1A and indicates the initial stage of sintering. A significant microstructural evolution can be observed at a sintering temperature of 1,400 °C, at which the original particles with round shapes disappear. Instead, the particles become a little larger and their edges are somewhat sharp, indicating the appearance of initial grains. Furthermore, the initial grains are bonded to some extent and the sintering necks between grains appear, although a lot of small pores seem to hinder the bonding of grains. With further increase in the sintering temperature up to 1,600 °C, the grains with sharp edges and the bonding between neighbored grains become more distinct, together with significant grain growth, clear grain boundaries, and exclusion of pores. Here, connectivity is employed to represent the phase continuity or degree of interconnection within a phase. For the present Al₂O₃ porous ceramics, the ceramic connectivity refers to the extent of bonding between Al₂O₃ particles or grains. Therefore, the above discussion reveals that the connectivity of ceramic phase is enhanced with relative density when the sintering temperature increases from 1,000 to 1,600 °C. The microstructure changes little at a higher sintering temperature of 1,650 °C, except that the pores continue to decrease, indicating the final stage of sintering. Additionally, the average grain size increases with sintering temperature from 0.44 to 4.23 um, as shown in Figure 4, and the grain size distribution is illustrated in Supplementary Figure 2. In comparison, the microstructure changes very differently with relative density for Al₂O₃-B ceramics, as shown in Figure 3. The large pores with the size of several to tens of micrometers are observed in Al₂O₃-B ceramics, which are formed by decomposing the NH₄HCO₃ large particles as the porogen. As discussed above, the relative density of Al₂O₃-B ceramics decreases significantly from 78.76% to 48.31% by adjusting the NH₄HCO₃ volume fraction from 20% to 60%. Interestingly, the microstructure far from the large pores in Al₂O₃-B ceramics depends little on relative density and is similar to that of Al₂O₃-A ceramic sintered at 1,650 °C. Since the porogen of NH₄HCO₃ is decomposed completely at only 70 °C without any residual or reaction with Al₂O₃, it is expected to have a limited effect on the sintering characteristics and microstructure of Al₂O₃-B ceramics except the produced large pores. Therefore, the ceramic connectivity is dominated by the sintering temperature rather than the porogen volume fraction, and the ceramic connectivity is similar for all the Al₂O₃-B ceramics. Moreover, the ceramic connectivity in Al₂O₃-B porous ceramic is significantly improved compared to that in Al₂O₃-A counterpart with a similar density. For example, the ceramic connectivity in Figure 3C is much better than that in Figure 2C, although the corresponding relative density of Al₂O₃-B (48.31%) is a little lower than that of Al₂O₃-A (53.00%). As shown in Figure 4 and Supplementary Figure 3, the average grain size of Al₂O₃-B porous ceramics is also higher and more insensitive to relative density compared with Al₂O₃-A, which is attributed to the higher sintering temperature of 1,650 °C for the former. Figure 5 shows the Vickers hardness as a function of relative density. It is worth noting that the Al₂O₃-B ceramics are inhomogenous in microstructure because of the large pores, so the measurement was conducted in the regions far from the large pores, and the Vickers hardness of Al₂O₃-B ceramics exhibits a much larger deviation compared with Al₂O₃-A. In spite of the large deviation, the Vickers hardness increases with relative density for both the Al₂O₃-A and Al₂O₃-B ceramics, and the latter exhibits a significantly higher Vickers hardness than the former with a similar relative density. The large difference in Vickers hardness is attributed to the different ceramic connectivity between the two groups of ceramics, which is also expected to affect the microwave dielectric properties significantly.

Figure 2. SEM images of fractured surfaces of Al₂O₃-A ceramics sintered at (A) 1,000 °C, (B) 1,200 °C, (C) 1,400 °C, (D) 1,450 °C, (E) 1,500 °C, (F) 1,550 °C, (G) 1,600 °C, and (H) 1,650 °C.

Figure 3. SEM images of fractured surfaces of Al₂O₃-B ceramics with relative densities of (A) 78.76%, (B) 65.29%, and (C) 48.31%. The insets show the fractured surfaces with higher magnification.

Figure 4. Average grain size of Al₂O₃-A and Al₂O₃-B ceramics with different relative densities.

Figure 5. Vickers hardness of Al₂O₃-A and Al₂O₃-B ceramics with different relative densities.

Figure 6A shows the ε_r of Al₂O₃-A ceramics sintered at different temperatures. With increasing T_s from 1,000 to 1,300 °C, the ε_r increases slightly from 3.00 to 3.49, and it increases more rapidly to 8.99 with further increasing T_s up to 1,650 °C, which is consistent with the relative density shown in Figure 1. The relationship between ε_r and relative density for Al₂O₃-A ceramics can be observed more clearly in Figure 6B, where the ε_r increases monotonously and smoothly with relative density. Figure 6B also shows that the ε_r of Al₂O₃-B ceramics exhibits similar dependence on relative density to Al₂O₃-A counterparts, while the ε_r of the former is always slightly higher than that of the latter with a similar relative density, and the difference is larger for the lower relative density. As discussed above, the Al₂O₃-B ceramic exhibits a larger grain size and improved ceramic connectivity than the Al₂O₃-A counterpart with a similar density because of the higher sintering temperature. Since the low-loss microwave dielectric ceramics are generally non-polar materials, their ε_r is determined by the electric and ionic polarizations and, hence, insensitive to grain size. Therefore, the larger ε_r of Al₂O₃-B ceramics is attributed to better ceramic connectivity. Such results are consistent with those in CaTiO₃ porous ceramic^[32], although the difference in ε_r between the two groups of Al₂O₃ ceramics is not so dramatic in the present work. Since CaTiO₃ has a high ε_r of 183^[32] and the difference between the ε_r of the ceramic phase and air is much larger for the CaTiO₃ porous ceramics, the electric field distribution in the constituting phases is more sensitive to the continuity of ceramic phase, leading to the much stronger dependence of ε_r on the ceramic connectivity. Moreover, the ultralow ε_r (< 5) can be obtained in both Al₂O₃-A and Al₂O₃-B ceramics when the relative density is lower than 60%, proving the feasibility of achieving ultralow ε_r by introducing pores into low-ε_r ceramics.

Figure 6. (A) ε_r of Al₂O₃-A ceramics sintered at different T_s. (B) ε_r of Al₂O₃-A and Al₂O₃-B ceramics with different relative densities and the predicted ε_r by several dielectric mixing rules.

It is well known that the ε_r of dielectric composites and porous materials can be predicted by many dielectric mixing rules, including parallel model, serial model, logarithmic model^[45], effective medium approximation (EMA)^[46], Heidinger’s approximation^[47], etc. Additionally, Bosman and Havinga’s correction^[48] described by

is often used for predicting the dielectric constant of porous materials, where p is the porosity, ε_r,theo is the theoretical dielectric constant of the fully densified material, and ε_r is the measured dielectric constant of the material with the porosity p. Bosman and Havinga’s correction is usually reliable when the porosity is not high, and the theoretical dielectric constant of Al₂O₃ is calculated to be 10.12 from the measured ε_r (8.99) and porosity (8.27%) of Al₂O₃-A ceramic sintered at 1,650 °C, which is used for the following prediction. Figure 6B shows the predicted ε_r of Al₂O₃ porous ceramics with different relative densities from these dielectric mixing rules. The predicted results by the parallel model, serial model, logarithmic model, and Bosman and Havinga’s correction differ much from the experimental ones, especially for the low relative density. Although Heidinger’s approximation and effective medium approximation provide much better prediction, they still fail to explain the difference between Al₂O₃-A and Al₂O₃-B porous ceramics. Such a failure is not surprising, since all the above mixing rules are empirical or based on some assumptions in microstructure, and only the relative density acts as the independent variable. However, the above discussion on microstructure reveals the significant difference in microstructure and ceramic connectivity between Al₂O₃-A and Al₂O₃-B ceramics, which should be considered for a more reliable prediction.

Our previous work shows that the parameter α in the exponential model

can be used to describe the connectivity of constituting phases in dielectric composites and porous materials^[32], where V is the volume fraction, and the numeric subscripts mean phases 1 and 2. The volume fraction of ceramic phase (phase 1) is just the relative density for porous ceramics, and phase 2 is air with ε_r,2 = 1 and V₂ = 1 - V₁, so Eq. (2) can be rewritten as

The fitted parameter α is calculated for the present Al₂O₃-A and Al₂O₃-B ceramics from the measured ε_r and relative density, and the calculated α as a function of relative density is shown in Figure 7. With increasing T_s from 1,000 to 1,650 °C and relative density from 41.25% to 91.73% for Al₂O₃-A ceramics, the calculated α increases greatly from 0.162 to 0.488 by about three times. In comparison, Al₂O₃-B ceramics always exhibit high α values of 0.461-0.478 in spite of the varying relative density from 78.76% to 48.31%, which are very close to the highest α of Al₂O₃-A ceramic (0.488) sintered at 1,650 °C. Combining with the microstructural analysis shown in Figures 2 and 3, it is clear that the ceramic connectivity and α are dominated by the sintering temperature and insensitive to relative density. As revealed in CaTiO₃ porous ceramics, the improved ceramic connectivity enhances the electric field in ceramic phase and its contribution to the dielectric response of porous ceramics, and therefore increases the final dielectric constant^[32]. Such an effect is similar in Al₂O₃ porous ceramics, although the higher ε_r of Al₂O₃-B than Al₂O₃-A is not so dramatic compared to the CaTiO₃ porous ceramics due to the much lower ε_r of Al₂O₃ (10.12) than CaTiO₃ (183)^[32].

Figure 7. α of Al₂O₃-A and Al₂O₃-B ceramics as a function of relative density.

Since Al₂O₃ has a negative τ_f, all the Al₂O₃-A and Al₂O₃-B ceramics in the present work exhibit negative τ_f, as shown in Figure 8A and B. With increasing T_s from 1,000 to 1,650 °C and relative density from 41.25% to 91.73% for Al₂O₃-A, τ_f decreases significantly from -15.8 to -58.5 ppm/°C. In comparison, τ_f decreases more slightly with relative density for Al₂O₃-B, and the difference between the τ_f values for the relative densities of 48.31% and 91.73% is only 12.7 ppm/°C. Similarly, the parallel model, serial model, and logarithmic model fail in predicting the τ_f of Al₂O₃ porous ceramics. Since ε_r is independent of grain size for low-loss microwave dielectric ceramics, as discussed above, its temperature coefficient (τ_ε) and, hence, the τ_f are also insensitive to grain size. Therefore, it can be inferred that τ_f is also strongly dependent on ceramic connectivity. The expression for the τ_ε of a dielectric composite can be obtained by derivating Equation 2 with respect to temperature as^[32]

Figure 8. (A) τ_f of Al₂O₃-A ceramics sintered at different T_s. (B) τ_f of Al₂O₃-A and Al₂O₃-B ceramics with different relative densities, and the predicted τ_f by Equation 6 and other dielectric mixing rules.

In the present work, phase 2 is air with V₂ = 1 - V₁, ε_r,2 = 1, and τ_ε,2= 0 for Al₂O₃ porous ceramics. Combining with

for both the porous and fully dense Al₂O₃ ceramics, the τ_f of the Al₂O₃ porous ceramics can be obtained as

By using τ_f,1 = -60.2 ppm/°C and α_L,1 = 8.1 ppm/°C for fully dense Al₂O₃ ceramic and assuming that the α_L of porous ceramics is proportional to relative density, the τ_f of Al₂O₃-A and Al₂O₃-B porous ceramics can be predicted from the relative density and α, and the results are also shown in Figure 8B. The predicted τ_f decreases with relative density for both Al₂O₃-A and Al₂O₃-B ceramics, and Al₂O₃-A exhibits a higher predicted τ_f than Al₂O₃-B with similar density, which is consistent with the experimental results and attributed to the poorer ceramic connectivity and lower α for the former. The predicted τ_f values fit the experiments well for all the Al₂O₃-B porous ceramics and the Al₂O₃-A counterparts with high relative density. However, when the relative density of Al₂O₃-A ceramics is lower than 60%, the difference between the predicted and experimental results becomes large, indicating that the exponential model is more suitable for predicting the τ_f of Al₂O₃ porous ceramics with relatively good ceramic connectivity. Similarly to ε_r, the effect of ceramic connectivity on the τ_f of Al₂O₃ porous ceramics is not so dramatic as that for CaTiO₃^[32] because of the much lower ε_r of Al₂O₃ and less sensitive electric field distribution to the continuity of ceramic phase. Additionally, it is well known that the low-ε_r microwave dielectric ceramics usually exhibit large negative τ_f values that deteriorate the temperature stability for practical application. The above discussion shows that the pores introduced in microwave dielectric ceramics can not only decrease the dielectric constant, but also improve the temperature stability.

As discussed above, the ε_r and τ_f of Al₂O₃ porous ceramics are directly determined by the relative density and ceramic connectivity. In comparison, the Qf value of porous ceramics is more complex since many microstructural factors may result in extrinsic dielectric loss and thus decrease the Qf value. Without considering the effects of microstructural factors, the Qf value of a composite can be expressed as

from the exponential rule^[32]. For the porous ceramics, phase 2 is air with V₂ = 1 - V₁, ε_r,2 = 1, and tanδ₂ = 0, so the theoretical Qf value of porous ceramics is

By using ε_r,1 = 10.12 and Qf₁ = 402,000 GHz, one can find in Figure 9A that the predicted Qf value of Al₂O₃ porous ceramics increases as the relative density decreases, since the decreased relative density means the decreased content of lossy Al₂O₃ and increased content of lossless air. However, the measured Qf values of Al₂O₃ porous ceramics are always much lower than that of the dense Al₂O₃ ceramic and the predicted Qf values. Such a contradiction is not surprising since the Qf value of Al₂O₃ itself is very high, and the extrinsic dielectric loss caused by microstructural factors may greatly exceed the ultra-low dielectric loss of Al₂O₃ and hence dominate the dielectric loss and Qf value of Al₂O₃ porous ceramics. Moreover, the Al₂O₃-B ceramics exhibit similar Qf values of around 40,000 GHz, which are insensitive to the varying relative density from 48.31% to 78.76%, and 1.5-4 times higher than those of the Al₂O₃-A ceramics with the similar relative densities. Differently from Al₂O₃-B, the Qf value is strongly dependent on the relative density for Al₂O₃-A porous ceramics. Such results further verify the dominant role of the microstructural factors. Supplementary Figure 4 also shows the calculated dielectric loss of Al₂O₃-A porous ceramics at 13 GHz as a function of relative density. Since the Qf value is defined as the product of frequency and reciprocal dielectric loss, the dielectric loss shows a reverse trend to the Qf value. As analyzed from the SEM images, the grain size and ceramic connectivity are the main microstructural factors for the Al₂O₃-A and Al₂O₃-B porous ceramics. For Al₂O₃-A, both the grain size and ceramic connectivity increase with higher sintering temperatures and relative densities. In contrast, for Al₂O₃-B, these factors remain largely unaffected by changes in relative density. The increase in grain size generally increases the dielectric loss and decreases the Qf value for the dense Al₂O₃ ceramics^[27], which may be responsible for the decreasing Qf value of Al₂O₃-A ceramics with increasing sintering temperature from 1,000 to 1,300 °C and relative density from 41.25% to 46.15%, as shown in Figure 9B. However, the negative effect of grain size on Qf value is inconsistent with the experimental results for the Al₂O₃-A ceramics with higher relative densities and Al₂O₃-B ceramics, which are much more important in the present work. It can be inferred that the Qf value of these ceramics is predominantly determined by the ceramic connectivity, as the ceramic-pore interfaces introduce extrinsic dielectric loss through microwave scattering, and these interfaces can be greatly decreased by enhancing the ceramic connectivity.

Figure 9. (A) Experimental and predicted Qf values of Al₂O₃-A and Al₂O₃-B ceramics with different relative densities. (B)Qf value of Al₂O₃-A ceramics sintered at different T_s.

By integrating the above discussion on the ε_r, τ_f, and Qf value, the optimum microwave dielectric properties with ε_r = 4.16, Qf = 38,400 GHz, and τ_f = -44.3 ppm/°C are obtained in Al₂O₃-B ceramic with a low relative density of 48.31%, indicating that the microwave dielectric properties of ultralow-ε_r porous ceramics can be optimized by regulating both the porosity and ceramic connectivity. Furthermore, the microstructural effects revealed in the present work provide valuable guidance for the future development of ultralow-ε_r porous ceramics. Considering the substantial impact of ceramic connectivity on the Qf value of porous ceramics, good ceramic connectivity is essential for achieving high Qf values. However, the enhanced ceramic connectivity also leads to the increase in both ε_r and |τ_f|, so it is crucial to select the ceramics with both low ε_r and small τ_f in the dense state for obtaining the ultra-low ε_r and improving the temperature stability.

CONCLUSIONS

In conclusion, two groups of Al₂O₃ porous ceramics are prepared by incomplete sintering at low temperatures and sintering at the optimal temperature with the introduction of NH₄HCO₃ porogen, respectively. The introduction of pores leads to a significant decrease in ε_r, Qf, and |τ_f|, and Al₂O₃-B ceramic exhibits a slightly higher ε_r, a larger |τ_f|, and a much higher Qf value than Al₂O₃-A with similar density. The microstructural analysis clearly reveals that the ceramic connectivity and relative density of Al₂O₃-A porous ceramics are improved significantly with increasing sintering temperature, while the excellent ceramic connectivity of Al₂O₃-B changes little with the porogen content and relative density. The ceramic connectivity is dominated by sintering temperature rather than relative density, and can be described by the parameter α in the exponential rule, which increases significantly with relative density for Al₂O₃-A but changes little for Al₂O₃-B. The significantly improved ceramic connectivity in Al₂O₃-B porous ceramics enhances the contribution of ceramic phase to the ε_r and τ_f of porous ceramics, although such effects are not so dramatic as those in CaTiO₃ porous ceramics because of the much lower ε_r of Al₂O₃. Moreover, the improved ceramic connectivity increases the Qf value by decreasing the ceramic-pore interfaces that cause extrinsic dielectric loss. The good combination of microwave dielectric properties with ε_r = 4.16, Qf = 38,400 GHz, and τ_f = -44.3 ppm/°C is obtained in Al₂O₃-B ceramic with a low relative density of 48.31% and good ceramic connectivity. The important effects of ceramic connectivity revealed in this work contribute to the deep understanding of the tight relationship between microwave dielectric properties and microstructures in porous ceramics, and also inspire the development of ultralow-ε_r dielectric materials by regulating both the porosity and ceramic connectivity.