1 Introduction

The percolating theory was brought up initially in disordered systems to analyze sudden change of physical properties, broadening its application in heterogeneous materials. The percolation transition was generally presented in heterogeneous multicomponent composites. Actually, filler in composites was gradually contacted with each other leading to a continuous cluster throughout the composites when increasing filler content approaches fc [1]. Accompanied with microstructure changing, percolative composites also underwent sudden change in some physical properties. For instance, positive/negative permittivity was tuned by controlling the functional phase content below/above fc. Therefore, metacomposites with negative permittivity can be designed and fabricated following the percolative composite route.

Metamaterials with negative permittivity \({\epsilon }^{\text{'}}\) have been extended into multiple novel applications which hardly can be realized by conventional materials [2]. Specifically, applications in perfect lens, invisible cloaking, wireless power transfer (WPT), magnetic resonance imaging and colossal permittivity materials [3] have been developed due to their exotically physical properties (e.g., negative refraction index, reversed Doppler effect and Vavilov–Cherenkov effect). In fact, the fascinating performance of metamaterials generally originates from their artificial and periodical metal structures (split-ring resonators, fishnets, wires or cut-wire pairs) [4,5,6]. Great achievements of metamaterials have been reported in recent years, some characteristics of metamaterials can be summarized. For example, in order to tune the negative electromagnetic parameters, shape, size or geometric arrangement in metamaterials should be redesigned, leading to complex constructing process [3, 6,7,8,9]. Besides, anisotropic electromagnetic response in metamaterials will usually trigger an adverse impact when applied in electronics [10]. Metacomposites with random functional units were promising candidates to broaden the scope of metamaterials [11, 12], which can be fabricated by typical and traditional preparation technology of materials and could be designed on the basis of percolation theory.

When constructing metacomposites, metallic fillers (e.g., Fe, Ni, Cu or Ag) were usually served as conductive functional phases hosted in insulating matrixes. Negative parameters corresponding to the chemical composition and microstructure of metacomposites could be easily tuned [13, 14]. Negative permittivity could also be observed in metacomposites consisting of metallic alloys or amorphous alloys (e.g., Fe50Ni50, FeNiMo or Fe78Si9B13) [15,16,17]. However, development of metal-based metacomposites were limited by the excessive power loss in composites and electromagnetic interference (EMI) to surrounding metal electronics [15, 18]. Under this circumstance, we presented a new route to metacomposites by consisting of nonmetallic conductive functional phases, “MAX” phases. Mn+1AXn phase was a group of layered ternary materials, where n is 1, 2 or 3, M is an early transition metal, A is an A-group element, and X is either C or N. MAX phases have triggered tremendous attentions due to their promising applications in structural reinforced ceramic matrix composites (CMCs), battery electrodes materials and supercapacitors [19,20,21]. MAX phases presented combination properties in metal and ceramic, including high-temperature oxidation resistance, damage tolerance, machinability, great electrical conductivity, and excellent irradiation/corrosion resistance [22,23,24,25,26,27,28,29,30]. Among all the MAX phases, Ti3SiC2 exhibits great electrical conductivity and structural performance which makes it a promising candidate for preparing metacomposites [19]. Poly(vinylidene fluoride) (PVDF) was a semicrystalline thermoplastic polymer with high piezo- and pyroelectric coeefficients, great thermal and chemical resistance. PVDF has been employed into percolative composites with colossal permittivity by containing functional phases (e.g., BaTiO3, PZT) [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51]. Therefore, PVDF and Ti3SiC2 MAX phase was selected as matrix and functional phase respectively to fabricate percolative metacomposites towards negative permittivity.

In this work, Ti3SiC2/PVDF metacomposites with different Ti3SiC2 content were prepared. The electrical and dielectric properties were investigated at radio frequency range (20 MHz–1 GHz). The different variation trends of ac conductivity spectra verified the percolation threshold. Negative permittivity behavior was observed in composites above fc. Equivalent circuit models were applied to analyze the impedance response of Ti3SiC2/PVDF composites.

2 Materials and Methods

Ti3SiC2 (purity > 98%, average size ~ 44 \({\upmu }\)m) were purchased from Haoxinano Technology Co., Ltd. Poly(vinylidene fluoride) were acquired from Shenzhen Boyi Plastic Raw Material Co., Ltd. The composites with Ti3SiC2 content of 10 wt%, 30 wt%, 50 wt%, 60 wt%, 70 wt%, 80 wt%, and 85 wt% were prepared by blending and compression molding procedure. Field emission scanning electron microscopy (FESEM) and X-ray diffractometer (XRD) with Cu Kα radiation were applied to characterize microstructure and phase composition of composites. The electrical and dielectric properties of composites was tested by Agilent E4991A precision impedance analyzer. The detailed testing and calculating process was presented in Supplementary Information.

3 Results and Discussion

3.1 Microstructure and Composition Characterization

Figure 1 shows SEM images of Ti3SiC2/PVDF metacomposites at different Ti3SiC2 filling content. The isolated Ti3SiC2 particles are randomly distributed in PVDF matrix. As increasing Ti3SiC2 content, the Ti3SiC2 particles gradually interconnect with each other leading to the formation of three-dimensional (3D) conductive networks in Ti3SiC2/PVDF composites (85 wt%). Figure 2 shows XRD patterns of Ti3SiC2 and Ti3SiC2/PVDF composites with different Ti3SiC2 content. The XRD pattern of PVDF materials was indexed by α, β, and γ crystal phases of PVDF. The peak at 18.4° was corresponding to the α-phase. The peaks of 20.8° and 26.6° indicated the β-phase superposition and γ-phase diffraction respectively. As increasing Ti3SiC2 content, the diffraction peaks of PVDF were wakening while the diffraction peaks of Ti3SiC2 were enhancing. FT-IR experiments of raw PVDF materials were performed as shown in Fig. 3a. The FT-IR spectra showed the typical absorption peaks of α-phase, β-phase, and γ-phase of PVDF at about 613 cm−1, 488 cm−1 and 841 cm−1, respectively. The bending of C–C–C is observed at 1071 cm−1, and the peak of CH2 appears at 1403 cm−1. Figure 3b shows the simultaneous DSC/TGA curves of PVDF raw materials measured at heating rate 10 K/min in air. As temperature rising, the weight of PVDF materials kept unchanged below 350 °C and then starting losing weight. DSC curve shows an endothermic peak at about 380 °C and an exothermic peak at about 485 °C which could be attributed to the decomposition of PVDF. PVDF becomes exothermic over 450 centigrade, suggesting that the heat quantity originating from PVDF decomposed reaction was larger than that from PVDF melting.

Fig. 1
figure 1

SEM images of Ti3SiC2/PVDF metacomposites with Ti3SiC2 content of 10 wt% (a), 30 wt% (b), 50 wt% (c), 70 wt% (d), 80 wt% (e), and 85 wt% (f)

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Fig. 2
figure 2

XRD patterns of Ti3SiC2 and Ti3SiC2/PVDF composites with different Ti3SiC2 content

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Fig. 3
figure 3

FT-IR curves of PVDF raw materials (a) and simultaneous DSC/TGA curves for PVDF raw materials (b)

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3.2 Conductivity Behavior

Figure 4 shows the frequency dependences of ac conductivity (σac) in Ti3SiC2/PVDF metacomposites with different Ti3SiC2 content. σac increased on frequency rising when Ti3SiC2 contents were lower than that of 80 wt% in composites, while σac decreased with frequency for composites at higher filling content. It is noteworthy that σac sharply increases (\({\epsilon }^{\text{'}}\) sharply decreases) when Ti3SiC2 content range from 80 to 85 wt% as shown in Fig. 5d. Further to say, the different variation trends of σac versus frequency indicated different conductive mechanisms. Percolation behavior occurred in Ti3SiC2/PVDF composites on raising Ti3SiC2 content. The percolation threshold fc was between 80 and 85 wt%, which was verified by the different conductive model. For composites below fc, the σac − f relationship can be expressed as:

Fig. 4
figure 4

Frequency dispersions of ac conductivity for the Ti3SiC2/PVDF composites

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Fig. 5
figure 5

Frequency dependences of real permittivity (\({\epsilon }^{\text{'}}\)) for Ti3SiC2/PVDF composites (a, b). Frequency dispersions of the imaginary permittivity (\({\epsilon }^{{\prime \prime }}\)) (c). Variation trends of ac conductivity and real permittivity at 20 MHz with different Ti3SiC2 filling content (d)

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$${\sigma _{ac}}={\sigma _{dc}}+A{\left( {2\pi f} \right)^n}$$
(1)

where σdc is direct current conductivity, f is the frequency, A is the pre-exponential factor and n is the fractional exponent (0 < n < 1). For composites (10 wt%, 30 wt%, 50 wt% and 60 wt%), ac conductance was primary over whole test frequency as shown in Fig. 4. As increasing Ti3SiC2 content to 70 wt% and 80 wt%, dc conductance dominated at low frequency region and ac conductance was primary at high frequency range. The fitting parameters were also confirmed the above analysis shown in Table S3. Experiencing an external electric field (especially high-frequency electric field), free electrons can “jump” across adjacent Ti3SiC2 particles which was denoted as hopping conduction behavior. When the Ti3SiC2 content exceeded fc, the Ti3SiC2 particles were interconnected to each other leading to the formation of 3D conductive networks throughout the composites. MAX phase presented metallic conduction (denoted as metal-like conduction behavior), i.e., σac was almost independent of frequency at low frequencies while σac decreased at high frequencies. Skin effect was applied to explain the metallic conduction. The skin depth was expressed as:

$$\delta ={\left( {\frac{2}{{\omega \mu {\sigma _{dc}}}}} \right)^{\frac{1}{2}}}$$
(2)

where δ is the skin depth, ω is the angular frequency, σdc is the dc conductivity, and \(\mu\)is the static permeability of the composites. The increasing frequency reduced the skin depth resulting in enhancement of the skin effects. The metal-like conductive behavior of composites with Ti3SiC2 content of 85 wt% was explained by Drude model:

$${\sigma _{ac}}=\frac{{{\sigma _{dc}}\omega _{\tau }^{2}}}{{{\omega ^2}+\omega _{\tau }^{2}}}$$
(3)
$${\sigma _{dc}}=\frac{{N{e^2}\tau }}{m}=\frac{{\omega _{p}^{2}\tau }}{{4\pi }}$$
(4)

where σdc is the dc limitation in conductivity, \({\omega _\tau }\)\(({\omega _\tau }={1 \mathord{\left/ {\vphantom {1 \tau }} \right. \kern-0pt} \tau })\) is the relaxation rate, and \({\omega _p}\) describes the oscillator strength.

3.3 Negative Permittivity Behavior

Frequency dependences of the real permittivity (\({\epsilon }^{\text{'}}\)) of Ti3SiC2/PVDF with different Ti3SiC2 content were showed in Fig. 5a, b. For composites below the percolation threshold fc, the values of \({\epsilon }^{\text{'}}\) were positive and enhanced with increasing Ti3SiC2 content at 20 MHz–1 GHz region shown in Fig. 5a, d, which could be ascribed to increasing interface connection of isolated Ti3SiC2 particles and PVDF matrix in the composites. Further, interfacial polarization in these micro-capacitors formed by Ti3SiC2 particles and PVDF matrix, denoted as Maxwell–Wagner–Sillars effect, was responsible for the improvement of permittivity.

As analyzed above, negative permittivity behavior observed over fc in composites, was ascribed to the formation of 3D interconnected Ti3SiC2 networks. Ti3SiC2 networks in composites with metallic conduction, generally presented low frequency plasmonic state leading to plasma-type negative permittivity behavior. The plasma-type negative permittivity behavior was theoretically described by Drude model as follows:

$${\varepsilon ^ * }=\varepsilon ^{\prime} - i\varepsilon ^{\prime\prime}=1 - \frac{{\omega _{p}^{2}}}{{{\omega ^2}+i\omega {\Gamma _D}}}$$
(5)
$$\varepsilon ^{\prime}=1 - \frac{{\omega _{p}^{2}}}{{{\omega ^2}+\Gamma _{D}^{2}}}$$
(6)
$${\omega _p}=\sqrt {\frac{{{n_{eff}}{e^2}}}{{{m_{eff}}{\varepsilon _0}}}}$$
(7)

where, ΓD is the damping constant, ωp = 2πfp is plasmons angular frequency, neff is effective concentration of electron, and meff is effective weight of electron. However, as shown in Fig. S1, Drude model was not in agreement with the negative permittivity spectra at low frequency regions, suggesting that there should be another generation mechanism. Considering the combinative metallic and ceramic properties of MAX phase, there may be impactions of induced electric dipole at low frequency range. Thus, we combined Lorentz model with Drude model to explain negative permittivity behavior. The Lorentz model was expressed as:

$${\varepsilon ^ * }=\varepsilon ^{\prime} - i\varepsilon ^{\prime\prime}=1+\frac{{\omega _{p}^{2}}}{{\omega _{0}^{2}+{\omega ^2}+i{\Gamma _L}\omega }}$$
(8)
$$\varepsilon ^{\prime}=1+\frac{{\omega _{p}^{2}\left( {\omega _{0}^{2} - {\omega ^2}} \right)}}{{{{\left( {\omega _{0}^{2} - {\omega ^2}} \right)}^2}+{\omega ^2}\Gamma _{L}^{2}}}$$
(9)

where ω\((\omega =2\pi f)\) is the angular frequency, \({\omega _0}\)\(({\omega _0}=2\pi {f_0})\) is the characteristic frequency, \({\omega _p}\)\(({\omega _p}=2\pi {f_p})\) is the angular plasma frequency, and \({\Gamma _L}\) represents the damping constant. The Lorentz type dielectric resonance was resulted from the induced electric dipole in the isolated Ti3SiC2 particles. As shown in Fig. 5b, negative permittivity spectra was fitted well by combination of Drude model and Lorentz model.

Dielectric loss in composites evaluated by imaginary permittivity (\({\epsilon }^{{\prime \prime }}\)) was an important performance when applied in electronic devices. In percolative composites, electric field frequency and concentration of conductive fillers were primary influencing factor to dielectric loss. Generally, dielectric loss mainly includes the conduction loss \({\varepsilon ^{\prime\prime}_C}\), polarization loss \({\varepsilon ^{\prime\prime}_P}\) and dipole loss \({\varepsilon ^{\prime\prime}_D}\), which was expressed as:

$$\varepsilon ^{\prime\prime}={\varepsilon ^{\prime\prime}_C}+{\varepsilon ^{\prime\prime}_D}+{\varepsilon ^{\prime\prime}_P}$$
(10)

At 20 MHz–1 GHz region, the conduction loss and dipolar loss were primary loss. \({\varepsilon ^{\prime\prime}_C}\) originating from leakage current among conductive fillers was expressed as:

$${\varepsilon ^{\prime\prime}_C}=\frac{{{\sigma _{dc}}}}{{2\pi f{\varepsilon _0}}}$$
(11)

where σdc is a constant. Thus, \({\varepsilon ^{\prime\prime}_C}\) was inversely related to f\(({\varepsilon ^{\prime\prime}_C} \propto {f^{ - 1}})\). Figure 5c presented frequency dependent \({\epsilon }^{{\prime \prime }}\)for the Ti3SiC2/PVDF composites with different filling content. \({\epsilon }^{{\prime \prime }}\)was evidently enhanced ascribing to the incorporation of conductive Ti3SiC2 particles. For the composites below fc, \({\epsilon }^{{\prime \prime }}\) spectra exhibited liner decrease trend in low frequency. With increasing frequency, relationship of \({\epsilon }^{{\prime \prime }}\) versus f presented nonlinear increasing trend. In other words, the dominant role in dielectric loss changed from the \({\varepsilon ^{\prime\prime}_C}\) to the \({\varepsilon ^{\prime\prime}_D}\) with frequency rising.

3.4 Impedance and Equivalent Circuit Analysis

For Ti3SiC2/PVDF composites with positive permittivity, the reactance showed negative values at 20 MHz–1 GHz region (Fig. 6a). The relationship for different circuit elements was expressed as:

Fig. 6
figure 6

Nyquist plots (a, b) for the Ti3SiC2/PVDF composites with different Ti3SiC2 content

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$$Z=\frac{{\mathop U\limits^{ \bullet } }}{{\mathop I\limits^{ \bullet } }}=R+i\left( {{X_L} - {X_C}} \right)=Z^{\prime}+iZ^{\prime\prime}$$
(12)
$$\varphi {\text{=arctan}}\frac{{\mathop {{U_X}}\limits^{ \bullet } }}{{\mathop {{U_R}}\limits^{ \bullet } }}=\arctan \frac{{\mathop {\mathop {{U_L}}\limits^{ \bullet } - \mathop {{U_C}}\limits^{ \bullet } }\limits^{{}} }}{{\mathop {{U_R}}\limits^{ \bullet } }}$$
(13)

That is to say, for composites below fc, capacitive reactance was stronger than inductive reactance (Z″ = XL− XC < 0) indicating capacitive character. Equivalent circuit models was applied to analyze impedance response of Ti3SiC2/PVDF composites. Equivalent circuit model of composites below fc consists of a series resistor (Rs) and a parallel connection of a resistor (Rp) and a capacitor (Cp) (inset of Fig. 7a). Rp, originating from the leakage current of composites, decreased with increasing conductive Ti3SiC2 particles in PVDF matrix. Cp, mainly deriving from the micro-capacitors constructed by the Ti3SiC2 and PVDF particles in composites, increased on increasing Ti3SiC2 content. Noteworthy, Cp sharply increased near fc. While capacitive reactance for composites above fc was less than inductive reactance (Z″ = XL− XC > 0) leading to inductive character. Equivalent circuit model of composites above fc consists of resistors (Rp, R1 and R2), capacitor (Cp), and inductors (L1 and L2) as shown in Fig. 7a. As illustrated in Fig. 8a, b, isolated Ti3SiC2 particles distributed in PVDF matrix were equivalently forming to capacitors, while conductive paths of inductors in composites were formed by connective Ti3SiC2 particles.

Fig. 7
figure 7

Frequency dependent impedance (a) and phase angle φ (b) for Ti3SiC2/PVDF composites. Phasor diagrams of voltage versus current were presented in b. \(\mathop U\limits^{ \bullet }\), \(\mathop I\limits^{ \bullet }\), \(\mathop {{U_R}}\limits^{ \bullet }\), \(\mathop {{U_L}}\limits^{ \bullet }\), \(\mathop {{U_X}}\limits^{ \bullet }\) and \(\mathop {{U_C}}\limits^{ \bullet }\) are voltage or current phasor for different circuit elements, \(\varphi\) is the impedance angle, \({X_L}\), \({X_C}\) and X are reactance

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Fig. 8
figure 8

Schematic evolution of the microstructure corresponding with variation of capacitor (a) and inductor (b)

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Figure 7b presented frequency dependent φ for Ti3SiC2/PVDF composites with different filling content. When composites experiencing ac electric field, the samples was considered as a typical RLC circuit consisting of capacitors (C), resistors (R) and/or inductors (L) shown in Fig. S3 in Supplementary Materials. When sample is pure resistor, the voltage and current is synchronous and φ = 0°. Generally, when sample presents capacitive or inductive, the current and voltage is unsynchronized. Specifically, when the current and voltage flows through a capacitor, voltage lags current by 90° (φ = − 90°). While, the current lags voltage by 90° (φ = 90°) as the current and voltage flows through an inductor. In Ti3SiC2/PVDF composites, φ shifts from negative to positive on increasing Ti3SiC2 content, indicating transition of capacitive to inductive. For composite below fc, the φ values are range from − 90° to 0°. Under this circumstance, \({\mathop U\limits^{ \bullet } _L}<{\mathop U\limits^{ \bullet } _C}\), thus \(\mathop {{U_X}}\limits^{ \bullet } <0\), suggests that voltage phase falls behind the current phase, capacitors dominate in the circuit and composite manifests capacitive character. Correspondingly, composites above fc present inductive character when phase shift angle φ values range from 0° to 90°. Under this circumstance, \({\mathop U\limits^{ \bullet } _L}>{\mathop U\limits^{ \bullet } _C}\), thus \(\mathop {{U_X}}\limits^{ \bullet }>0\), indicates that current phase lags behind voltage phase, inductors dominates in the circuit and composite manifests inductive character.

4 Conclusion

In conclusion, Ti3SiC2/PVDF percolative metacomposites towards negative permittivity were prepared. Conductive mechanism changes when increasing Ti3SiC2 content over fc. Negative permittivity behavior was explained by Lorentz and Drude model, suggesting the combinative contribution of induced electric dipole resonance and low-frequency plasmonic oscillation at radio-frequency region. Equivalent circuit analysis to impedance response of metacomposites manifested correspondence between capacitive-inductive characteristic change and positive–negative permittivity change. This work facilitates clarifying the generation mechanism of negative permittivity which will greatly extend applications of MAX phase in metacomposites.