Effect of Reaction Time on Microwave Absorption Properties of Fe3O4 Hollow Spheres Synthesized via Ostwald Ripening

Hollow magnetic structures have great potential to be used in the microwave absorbing field. Herein, Fe3O4 hollow spheres with different levels of hollowness were synthesized by the hydrothermal method under Ostwald ripening effect. In addition to their microstructures, the microwave absorption properties of such spheres were investigated. The results show that the grain size and hollowness of Fe3O4 hollow spheres both increase as the reaction time increases. With increasing hollowness, the attenuation ability of electromagnetic wave of Fe3O4 spheres increases first and then decreases, finally increases sharply after the spheres break down. Samples with strong attenuation ability can achieve good impedance matching, which it does preferentially as the absorber thickness increases. Fe3O4 hollow spheres show the best microwave absorption performance when the reaction time is 24 h. The minimum reflection loss (RL (min)) can reach −40 dB, while the thickness is only 3.2 mm.


Introduction
It is well known that the structure of materials plays an important role in their thermo-mechanical behavior and transport properties. Hollow structures have shown great application prospects in lithium batteries, super capacitors, drug transport, biomedicine, gas sensors and other fields due to their advantages of large specific surface area and low density [1][2][3][4][5][6]. Template methods and template-free methods are usually used to construct hollow structures. Template-free methods usually adopt some special mechanisms, such as the Ostwald ripening effect [7], the Kendall effect [8], the current displacement effect [9], etc. Compared with template methods, template-free methods are more flexible and convenient. In recent years, with the wide application of communication technology and electromagnetic equipment in GHz, electromagnetic radiation has resulted in unprecedented pollution and seriously endangered human health [10][11][12][13]. To solve this problem, a lot of efforts have been directed towards the research and development of microwave absorbing materials (MAMs) [14,15]. As a traditional microwave absorbing material, Fe 3 O 4 has high saturation magnetization and Curie temperature at high temperature (585 • C). This stable and excellent magnetic property has attracted many scholars' attention. However, the high density of Fe 3 O 4 limits its further use as a MAM [16].
Fortunately, many studies have shown that hollowing of Fe 3 O 4 is an effective way to reduce material weight, produce special morphological effects and improve electromagnetic properties [17][18][19]. For example, Li et al. [20] prepared hollow Fe 3 O 4 spheres with a diameter of 450 nm, a wall thickness of 80 nm and a density of 3.28 g/cm 3 by solvothermal method, which was nearly 27% lower than the XRD results of Fe 3 O 4 with different reaction times are shown in Figure 1. The diffraction peaks of the samples correspond well to face-centered magnetite Fe 3 O 4 (JCPDS Card No. 99-0073), which indicates that the synthesized products have high purity. The diffraction peak intensities increase obviously as the reaction time increases. This means that the samples' crystallinity is improved as the reaction time increases. According to the calculation results of the Scherrer formula (D = Kγ/Bcosθ), the grain size of the sample at 8 h, 12 h, 16 h, 20 h and 24 h is 25.89 nm, 30.54 nm, 38.28 nm, 41.55 nm and 48.29 nm, respectively. The results show that the grain size increases as the reaction time increases. To further identify the purity of Fe 3 O 4 , RedOx tiration (potassium dichromate) was used. Table 1 calculates n(Fe 2+ ):n(Fe 3+ ), Fe σ O 4 (σ-nonstoichiometric) and Oxidation rate, respectively. Although Fe 3 O 4 may be oxidized during the experiment, the results in Table 1 show that their oxidation rates are below 10%, which indicates that the purity of Fe 3   To observe the morphology and internal structure of the samples, SEM and TEM analyses were carried out, and the results are illustrated in Figure 2. The prepared Fe3O4 spheres exhibit good dispersion and no agglomeration. As shown in the SEM images from Figure 2a-e, it can be seen that the surface of the Fe3O4 sphere is composed of many Fe3O4 grains, which shows the self-assembly effect of Ostwald ripening mechanism. Similar results can be seen from the TEM images, Figure 2f-j. In addition, the hollowness of Fe3O4 sphere increases gradually as the reaction time increases. The hollow size of Fe3O4 increases to breaking point as the reaction time reaches 24 h (S5). It is worth mentioning that although the formation of Fe3O4 hollow spheres reflects the typical Ostwald ripening phenomenon, the size of Fe3O4 spheres does not increase significantly during the whole reaction process. As can be seen from Figure 2k, the sizes of the Fe3O4 hollow spheres are mainly around 500 nm before they are broken. This may be due to the higher PVP concentration, which restricts the growth of Fe3O4 spheres during Ostwald ripening process [23].
Based on the results of SEM and TEM images, the formation process of Fe3O4 in this study can be explained as shown in Figure 2l. Firstly, iron ions and oxygen ions react continuously to form grains dispersed in solution under high temperature and pressure. These grains are extremely unstable due to the high surface free energy. To reduce the surface free energy, they will be aggregated into larger loose microspheres, which are the embryonic form of Fe3O4 spheres, as illustrated in Figure 2l-Ⅰ. From Figure 2l-Ⅱ to Ⅲ, with the prolongation of reaction time, the primary grains in outer layer of the microspheres contact with the solution sufficiently, and the absorption rate of iron and oxygen ions from the solution is faster, resulting in the larger grain size than the internal grain size. Then, the internal grains dissolve into the surrounding medium gradually, and re-precipitate on the surface of the external grains due to the higher surface free  To observe the morphology and internal structure of the samples, SEM and TEM analyses were carried out, and the results are illustrated in Figure 2. The prepared Fe 3 O 4 spheres exhibit good dispersion and no agglomeration. As shown in the SEM images from Figure 2a-e, it can be seen that the surface of the Fe 3 O 4 sphere is composed of many Fe 3 O 4 grains, which shows the self-assembly effect of Ostwald ripening mechanism. Similar results can be seen from the TEM images, Figure 2f-j. In addition, the hollowness of Fe 3 O 4 sphere increases gradually as the reaction time increases. The hollow size of Fe 3 O 4 increases to breaking point as the reaction time reaches 24 h (S5). It is worth mentioning that although the formation of Fe 3 O 4 hollow spheres reflects the typical Ostwald ripening phenomenon, the size of Fe 3 O 4 spheres does not increase significantly during the whole reaction process. As can be seen from Figure 2k, the sizes of the Fe 3 O 4 hollow spheres are mainly around 500 nm before they are broken. This may be due to the higher PVP concentration, which restricts the growth of Fe 3 O 4 spheres during Ostwald ripening process [23]. energy of internal grains. Therefore, the external grains grow further, leading to the formation of Fe3O4 spheres with hollow structure. In addition, there is residual stress in the external grains due to the continuous growth of Fe3O4 grains and the restraint of high PVP concentration. Excessive residual stress leads to the cracks at grain boundaries, resulting in crystal rupture, and the rupture of Fe3O4 spheres ultimately, as shown in Figure 2l-Ⅳ.   Table 2. The morphology of Fe3O4 has a significant effect on its magnetic properties. As the reaction time is prolonged, the defects of the sample decrease, the crystallinity is improved, and the super-exchange effect of the Fe-O-Fe bond is enhanced, resulting in a higher saturation magnetization [24]. However, the saturation magnetization of Fe3O4 is woken up as the reaction time is in the range of 12-24 h, which may be due to the hollow structure reducing the magnetic coupling between the Fe3O4 grains. The hollow structure may have a certain strengthening effect on the coercive force of Fe3O4, which is manifested in the increase in the coercive force of the Fe3O4 sphere at higher hollowness, and which suddenly decreases when the hollow sphere is broken. The report [25] shows that the critical size of superparamagnetism of Fe3O4 nanoparticles is 25 nm. Fe3O4 grains prepared in this paper are generally larger than this value, thus showing so it shows ferromagnetism. Based on the results of SEM and TEM images, the formation process of Fe 3 O 4 in this study can be explained as shown in Figure 2l. Firstly, iron ions and oxygen ions react continuously to form grains dispersed in solution under high temperature and pressure. These grains are extremely unstable due to the high surface free energy. To reduce the surface free energy, they will be aggregated into larger loose microspheres, which are the embryonic form of Fe 3 O 4 spheres, as illustrated in Figure 2l-I. From Figure 2l-II to III, with the prolongation of reaction time, the primary grains in outer layer of the microspheres contact with the solution sufficiently, and the absorption rate of iron and oxygen ions from the solution is faster, resulting in the larger grain size than the internal grain size. Then, the internal grains dissolve into the surrounding medium gradually, and re-precipitate on the surface of the external grains due to the higher surface free energy of internal grains. Therefore, the external grains grow further, leading to the formation of Fe 3 O 4 spheres with hollow structure. In addition, there is residual stress in the external grains due to the continuous growth of Fe 3 O 4 grains and the restraint of high PVP concentration. Excessive residual stress leads to the cracks at grain boundaries, resulting in crystal rupture, and the rupture of   Table 2. The morphology of Fe 3 O 4 has a significant effect on its magnetic properties. As the reaction time is prolonged, the defects of the sample decrease, the crystallinity is improved, and the super-exchange effect of the Fe-O-Fe bond is enhanced, resulting in a higher saturation magnetization [24]. However, the saturation magnetization of Fe 3 O 4 is woken up as the reaction time is in the range of 12-24 h, which may be due to the hollow structure reducing the magnetic coupling between the Fe 3 O 4 grains. The hollow structure may have a certain strengthening effect on the coercive force of Fe 3 O 4 , which is manifested in the increase in the coercive force of the Fe 3 O 4 sphere at higher hollowness, and which suddenly decreases when the hollow sphere is broken. The report [25] shows that the critical size of superparamagnetism of Fe 3 O 4 nanoparticles is 25 nm. Fe 3 O 4 grains prepared in this paper are generally larger than this value, thus showing so it shows ferromagnetism.  The frequency dependence of complex permittivity (εr = ε′ − jε″), dielectric loss tangent and Cole-Cole semicircle of samples in the range of 2-18 GHz are illustrated in Figure 4. Generally speaking, the real part of permittivity ε′ is produced by various displacement polarization within the material, it represents the energy storage term of the material. The imaginary part of permittivity, ε″, is produced by various relaxation polarizations caused by steering polarization in the material, it cannot keep up with the change of external high frequency electric field, and represents the loss term of material. As shown in Figure 4a,b, the complex permittivity of Fe3O4 hollow spheres with different morphologies varies significantly with the increase of reaction time. The variation ranges of real and imaginary parts of complex permittivity are 6.1-11.8 and 0.2-1.5, respectively. The dielectric properties of different samples vary greatly and show no obvious regularity, which is related to the complex variation of polarization and conductivity affected by the grain size of Fe3O4 and the state of dispersion and aggregation.
The dielectric loss tangent (tan δE = ε″/ε′) reflects the dielectric loss ability of material. Tan δE of Fe3O4 samples at different time are shown in Figure 4c. ε′ changes gently with frequency; the trend of tan δE is similar to ε″. Tan δE decreases with the prolongation of reaction time (S1-S3) before the rupture of Fe3O4 spheres, and increases slightly as the reaction time reaches 20 h (S4). There are two obvious loss peaks in tan δE curve with frequency after the rupture of Fe3O4 spheres (S5), at 10.86 GHz and 16.82 GHz, respectively. The appearance of the loss peak increases the dielectric loss capacity of S5 at this frequency greatly. According to Debye relaxation theory, the real and imaginary parts of complex permittivity can be transformed into the following form [26]: In Formula (1), f is the frequency, ε ∞ and εs are relative dielectric permittivity at the  The frequency dependence of complex permittivity (ε r = ε − jε"), dielectric loss tangent and Cole-Cole semicircle of samples in the range of 2-18 GHz are illustrated in Figure 4. Generally speaking, the real part of permittivity ε is produced by various displacement polarization within the material, it represents the energy storage term of the material. The imaginary part of permittivity, ε", is produced by various relaxation polarizations caused by steering polarization in the material, it cannot keep up with the change of external high frequency electric field, and represents the loss term of material. As shown in Figure 4a maintained at about 10 initially, then decreases to 8.9 from 10.3 to 11.4 GHz, resulting in a value that corresponds to a loss peak of ε″ (10.86 GHz). Simply estimated, the maximum value of the loss peak is much larger than that of (εs − ε ∞ )/2, indicating that the dielectric behavior is not a typical Debye relaxation. This behavior may correspond to a resonance process similar to dielectric relaxation. Overall, we consider that the dielectric loss of Fe3O4 is not Debye relaxation, but resonant dielectric response.   The dielectric loss tangent (tan δ E = ε"/ε ) reflects the dielectric loss ability of material. Tan δ E of Fe 3 O 4 samples at different time are shown in Figure 4c. ε changes gently with frequency; the trend of tan δ E is similar to ε". Tan δ E decreases with the prolongation of reaction time (S1-S3) before the rupture of Fe 3 O 4 spheres, and increases slightly as the reaction time reaches 20 h (S4). There are two obvious loss peaks in tan δ E curve with frequency after the rupture of Fe 3 O 4 spheres (S5), at 10.86 GHz and 16.82 GHz, respectively. The appearance of the loss peak increases the dielectric loss capacity of S5 at this frequency greatly. According to Debye relaxation theory, the real and imaginary parts of complex permittivity can be transformed into the following form [26]: In Formula (1), f is the frequency, ε ∞ and ε s are relative dielectric permittivity at the high-frequency limit and static dielectric permittivity, respectively, and τ is the relaxation time. Therefore, ε and ε" can be further transformed:

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According to Formulas (2) and (3), the relationship between ε and ε" can be further deduced: According to Formula (4). If the polarization of the permittivity is caused by Debye relaxation process, it will be shown as a Cole-Cole semicircle. Based on the above theory, 3D illustration of Cole-Cole curves is made as shown in Figure 4d. First, for S1-S4, no Debye semicircles exist, and the Cole-Cole curves are just twisted spirals. Even for S5, it is not an ideal Debye semicircle. According to Formulas (2) and (3), if the dielectric spectrum satisfies the Debye relaxation equation, the maximum value of ε" appears and ε" = (ε s − ε ∞ )/2 when 2πf = τ −1 . However, the ε of S5 is maintained at about 10 initially, then decreases to 8.9 from 10.3 to 11.4 GHz, resulting in a value that corresponds to a loss peak of ε" (10.86 GHz). Simply estimated, the maximum value of the loss peak is much larger than that of (ε s − ε ∞ )/2, indicating that the dielectric behavior is not a typical Debye relaxation. This behavior may correspond to a resonance process similar to dielectric relaxation. Overall, we consider that the dielectric loss of Fe 3 O 4 is not Debye relaxation, but resonant dielectric response. Figure 5 displays the frequency dependence of complex permeability (µ r = µ − jµ"), magnetic loss tangent and C 0 of samples in the range of 2-18 GHz. From Figure 5a,b, it can be seen that real part of permeability (µ ) and imaginary part of permeability (µ") curves exhibit a similar trend, decreasing first and then remaining stable. This indicates that the magnetic loss ability of Fe 3 O 4 is mainly reflected in the S band (2-4 GHz) and C band (4-8 GHz), and Ostwald ripening has less effect on the magnetic properties than on the dielectric properties of Fe 3 O 4 . Similarly to dielectric loss, the magnetic loss tangent (tan δ M = µ"/µ ) reflects the magnetic loss ability of materials, and tan δ M can also be observed in Figure 5c. The results show that there is a loss peak at the tested frequency of tan δ M , at which is much larger than that of tan δ E . Tan δ M decreases slowly and finally approaches to tan δ E as the frequency increases. Therefore, it can be judged that the loss mechanism of Fe 3 O 4 is mainly magnetic loss. For ferrite materials, magnetic loss is usually caused by hysteresis loss, eddy current loss, domain wall displacement and natural resonance. However, hysteresis loss can be neglected in the case of weak applied electric field, and domain wall displacement only occurs in the MHz range, so the magnetic loss mechanism in Fe 3 O 4 is generally related to eddy current loss and natural resonance. Material thickness (d) and conductivity (σ) are two main factors affecting eddy current loss, which can be expressed as follows [27]: If the magnetic loss is caused entirely by the eddy current loss, the right side of Equation (5) is constant, so no the variation of C 0 with frequency should be revealed in a constant. In other words, the eddy current loss should be dominant in the frequency range of relatively stable and small fluctuations. As shown in Figure 5d, C 0 curves of Fe 3 O 4 fluctuate greatly at low frequencies and tend to be stable at the middle and high frequencies, indicating that the magnetic loss should be caused by natural resonance at low frequency, and as the frequency increases, it gradually changes into eddy current loss. Combined with Figure 5c, the corresponding frequencies of loss peaks caused by natural resonance at low frequencies are 2.86 GHz (S1), 4.12 GHz (S2), 4.16 GHz (S3), 3.77 GHz (S4) and 4.55 GHz (S5), respectively. The difference of natural resonance frequencies between different samples is caused by the effective anisotropy field of magneto-crystals with different morphologies of Fe 3 O 4 [28]. In addition, C 0 approaches zero at the middle and high frequencies, indicating that the eddy current loss is very weak. This is consistent with the weak magnetic loss at the corresponding frequency in Figure 5c. It is noteworthy that there is also a loss peak at 12 GHz in S1, which is caused by exchange resonance. A relevant report shows that there is exchange resonance as the grain size of Fe 3 O 4 is about 10 nm [29]. In addition, S1 is the early formation stage of the Fe 3 O 4 hollow spheres; there are some grains about 10 nm in size, so the speculation of exchange resonance in S1 is valid. samples is caused by the effective anisotropy field of magneto-crystals with different morphologies of Fe3O4 [28]. In addition, C0 approaches zero at the middle and high frequencies, indicating that the eddy current loss is very weak. This is consistent with the weak magnetic loss at the corresponding frequency in Figure 5c. It is noteworthy that there is also a loss peak at 12 GHz in S1, which is caused by exchange resonance. A relevant report shows that there is exchange resonance as the grain size of Fe3O4 is about 10 nm [29]. In addition, S1 is the early formation stage of the Fe3O4 hollow spheres; there are some grains about 10 nm in size, so the speculation of exchange resonance in S1 is valid. The energy of the electromagnetic wave is transferred to heat energy by dielectric loss and magnetic loss as the electromagnetic wave enters into the absorber. The attenuation constant α determines the capability attenuation characteristics of materials to attenuate the electromagnetic wave, which can be expressed by the following formula [30][31][32] In Formula (6), c and f represent the velocity of light and frequency, respectively. The attenuation coefficient α of different Fe3O4 samples in the frequency range of 2-18 GHz can be visually expressed by the contour plot map, as shown in Figure 6. The result shows that the attenuation ability of Fe3O4 to electromagnetic wave is mainly reflected in the low and high frequency bands, while the attenuation ability of Fe3O4 to the medium frequency band is weak. With increasing reaction time, the attenuation coefficient of the intermediate frequency band decreases gradually, reaches the minimum at 20 h (S4), and increases after the sphere ruptures at 24 h (S5). To study the attenuation ability of different samples in each work frequency bands, the integral values of α in each band are calculated. The specific results are shown in Table 3. When the reaction time is 8-16 h (S1-S3), the total integral value of α in the range of 2-18 GHz increases, then decreases sharply (S4). When the sphere ruptures, the total integral value rises immediately. Based on the above calculation results, the electromagnetic wave absorption capacity of the samples can be arranged in the following order: S5 > S3 > S2 > S1 > S4. The energy of the electromagnetic wave is transferred to heat energy by dielectric loss and magnetic loss as the electromagnetic wave enters into the absorber. The attenuation constant α determines the capability attenuation characteristics of materials to attenuate the electromagnetic wave, which can be expressed by the following formula [30][31][32]; In Formula (6), c and f represent the velocity of light and frequency, respectively. The attenuation coefficient α of different Fe 3 O 4 samples in the frequency range of 2-18 GHz can be visually expressed by the contour plot map, as shown in Figure 6. The result shows that the attenuation ability of Fe 3 O 4 to electromagnetic wave is mainly reflected in the low and high frequency bands, while the attenuation ability of Fe 3 O 4 to the medium frequency band is weak. With increasing reaction time, the attenuation coefficient of the intermediate frequency band decreases gradually, reaches the minimum at 20 h (S4), and increases after the sphere ruptures at 24 h (S5). To study the attenuation ability of different samples in each work frequency bands, the integral values of α in each band are calculated. The specific results are shown in Table 3. When the reaction time is 8-16 h (S1-S3), the total integral value of α in the range of 2-18 GHz increases, then decreases sharply (S4). When the sphere ruptures, the total integral value rises immediately. Based on the above calculation results, the electromagnetic wave absorption capacity of the samples can be arranged in the following order: S5 > S3 > S2 > S1 > S4. study the attenuation ability of different samples in each work frequency bands, the integral values of α in each band are calculated. The specific results are shown in Table 3. When the reaction time is 8-16 h (S1-S3), the total integral value of α in the range of 2-18 GHz increases, then decreases sharply (S4). When the sphere ruptures, the total integral value rises immediately. Based on the above calculation results, the electromagnetic wave absorption capacity of the samples can be arranged in the following order: S5 > S3 > S2 > S1 > S4.   On the basis of the complex permittivity and complex permeability data, the reflection loss (R L ) of Fe 3 O 4 can be deduced from the transmission line theory [33]: Among them, Z in stands for the input impedance of absorbing material, Z 0 is the impedance of free space, c is the speed of light in vacuum, d is the thickness of absorber, and f represents microwave frequency. Based on the above formulas, R L can be simulated as the thickness is in the range of 0-10 mm and the frequency is 2-18 GHz, as depicted in Figure 7a-e. The absorption peak of the sample moves to low frequency gradually as the thickness increases, which can be explained by the equation [34]: f = c/2πdµ", where f represents the optimal matching frequency, d is the optimal matching thickness. Therefore, the absorption band can be adjusted by changing the thickness of Fe 3 O 4 to meet the actual needs. Moreover, when the thickness of the sample exceeds a certain value (S1-4.8 mm, S2-4.3 mm, S3-3.8 mm, S4-4.8 mm, S5-4.2 mm), there are absorption peaks at both low and high frequencies, which indicates that Fe 3 O 4 is promising as a low frequency and high frequency compatible microwave absorbing material. In addition, R L is very sensitive to the change of thickness and can produce very strong reflection loss at a specific thickness. That can be explained by a quarter-wavelength model [35]: Where |εr| and |μr| are the modulus of εr and μr at fm, respectively. If the thickness of the absorber is equal to the calculated tm, the interference effect will occur, and the electromagnetic wave will be attenuated greatly. When n = 1, 3 and 5, the frequency dependence of tm is calculated and plotted on the contour maps of S1-S5 in Figure 7. It can be observed that almost all the points of RL (min) on the curves of tm. Thus, we can judge that the absorption peaks of Fe3O4 samples are aroused by thickness resonance to the specific frequency microwave. As the thickness of the sample increases, the resonance thicknesses are satisfied, with the quarter-wavelength model appearing in turn, which are 3.2 mm-S5, 3.6 mm-S3, 3.7 mm-S2, 5.2 mm-S1 and 5.4 mm-S4, respectively. The order is consistent with the order of attenuation ability mentioned above and the RL at resonance thickness are shown in Figure 8a. The absorption band of samples at the resonance thickness is mainly concentrated at low frequency, it might be related to the strong natural resonance loss at low frequency. Additionally, RL (min) of each sample can approach −40 dB, which is equivalent to 99.99% of the electromagnetic wave energy absorbed, indicating that Fe3O4 possesses great potential as an excellent low-frequency microwave absorption material. The normalized characteristic impedance Z is a key parameter in reducing the reflection of the electromagnetic wave, which can be expressed by the following equation [36]: Input impedance Zin should be equal to the free space impedance Z0, so that the electromagnetic wave can enter into the absorber completely and can be totally attenuated completely; thus, Z should be as close as possible to 1. As shown in Figure 8b, at the optimum thickness and frequency of the sample, the Z values corresponding to the loss peaks are closed to 1, indicating that they are  Where |ε r | and |µ r | are the modulus of ε r and µ r at f m , respectively. If the thickness of the absorber is equal to the calculated t m , the interference effect will occur, and the electromagnetic wave will be attenuated greatly. When n = 1, 3 and 5, the frequency dependence of t m is calculated and plotted on the contour maps of S1-S5 in Figure 7. It can be observed that almost all the points of R L (min) on the curves of t m . Thus, we can judge that the absorption peaks of Fe 3 O 4 samples are aroused by thickness resonance to the specific frequency microwave.
As the thickness of the sample increases, the resonance thicknesses are satisfied, with the quarter-wavelength model appearing in turn, which are 3.2 mm-S5, 3.6 mm-S3, 3.7 mm-S2, 5.2 mm-S1 and 5.4 mm-S4, respectively. The order is consistent with the order of attenuation ability mentioned above and the R L at resonance thickness are shown in Figure 8a. The absorption band of samples at the resonance thickness is mainly concentrated at low frequency, it might be related to the strong natural resonance loss at low frequency. Additionally, R L (min) of each sample can approach −40 dB, which is equivalent to 99.99% of the electromagnetic wave energy absorbed, indicating that Fe 3 O 4 possesses great potential as an excellent low-frequency microwave absorption material. The normalized characteristic impedance Z is a key parameter in reducing the reflection of the electromagnetic wave, which can be expressed by the following equation [36]:

Conclusions
Fe3O4 hollow spheres were fabricated with different reaction times by Ostwald ripening process. The microstructure and electromagnetic properties have been investigated systemically. Fe3O4 spheres all have hollow structure at different reaction times ranging from 8 h to 24 h, the size of Fe3O4 spheres remains at about 500 nm in diameter. The grain size and hollowness of Fe3O4 spheres increase as the reaction time. Fe3O4 hollow spheres break down as the reaction time reaches 24 h. With increasing in hollowness, the electromagnetic wave attenuation ability of Fe3O4 spheres increases first and then decreases, and increases sharply after the spheres finally break down. There is an obvious effect on the dielectric and magnetic properties of Fe3O4 spheres with different morphologies, resulting in a significant difference in the attenuation ability of different samples to electromagnetic waves. The electromagnetic attenuation ability of Fe3O4 spheres is S5 > S3 > S2 > S1 > S4. Sample with strong attenuation ability can achieve impedance matching preferentially with an increase in thickness. In addition, Fe3O4 hollow nanospheres exhibit good microwave absorption properties due to the strong natural resonance loss and interference cancellation of electromagnetic wave. RL (min) can reach −40 dB as the thickness is only 3.2 mm and the reaction time is 24 h.

Conflicts of Interest:
The authors declare no conflict of interest.  Input impedance Z in should be equal to the free space impedance Z 0 , so that the electromagnetic wave can enter into the absorber completely and can be totally attenuated completely; thus, Z should be as close as possible to 1. As shown in Figure 8b, at the optimum thickness and frequency of the sample, the Z values corresponding to the loss peaks are closed to 1, indicating that they are well matched with the free space impedance. In addition, by comparing the attenuation ability, the relationship between resonance thickness and Z, it is easy to draw the conclusion that the impedance matching of Fe 3 O 4 hollow spheres can be achieved in a thinner case when the attenuation ability is strong. There is an obvious effect on the dielectric and magnetic properties of Fe 3 O 4 spheres with different morphologies, resulting in a significant difference in the attenuation ability of different samples to electromagnetic waves. The electromagnetic attenuation ability of Fe 3 O 4 spheres is S5 > S3 > S2 > S1 > S4. Sample with strong attenuation ability can achieve impedance matching preferentially with an increase in thickness. In addition, Fe 3 O 4 hollow nanospheres exhibit good microwave absorption properties due to the strong natural resonance loss and interference cancellation of electromagnetic wave. R L (min) can reach −40 dB as the thickness is only 3.2 mm and the reaction time is 24 h.