Modeling dielectric permittivity of polymer composites filled with transition metal dichalcogenide nanoparticles

Abstract

A model is developed for the dielectric permittivity of polymer nanocomposites reinforced with transition metal dichalcogenide fillers at microwave frequencies. The model takes into account aggregation of nanoparticles into clusters (that involve both filler and matrix components) and the aspect ratio of aggregates. The governing equations involve four material parameters that are found by matching observations on the real and imaginary parts of the dielectric permittivity of polymers reinforced with MoS₂ and WS₂ micro- and nanospheres, MoS₂ nanosheets and nanoflowers, and composite heterostructures formed by MoS₂ and MoS₂-CoS₂ nanoparticles with graphene and reduced graphene oxide. Good agreement is demonstrated between results of simulation and the experimental data at frequencies in the S, X, and Ku bands of the electromagnetic spectrum. It is shown that composite heterostructures have superior dielectric properties compared with those of neat transition metal dichalcogenide nanoparticles.

Keywords

Polymer composite transition metal dichalcogenide dielectric permittivity microwave frequency

Introduction

Development of novel communication technologies and wide use of electronic devices in commercial, civil, and military fields lead to an increasing production of electromagnetic (EM) radiation.¹ Electromagnetic interference (EMI) and pollution have become a serious public concern as they cause damage to telecommunication, induce malfunction in sensitive electronic devices, disturb information security, and adversely affect the human health.² EMI shielding (blocking EM radiation into sensitive areas) provides an effective way to diminish these harmful effects with the help of microwave absorbers that convert the EM energy into heat.

An ideal absorber should exhibit a wide absorption frequency range and high effectiveness of the EMI shielding.³ Development of cost-efficient, flexible, lightweight, and thermally and chemically stable polymer composites for enhanced EM attenuation have recently attracted substantial attention.⁴ At the initial stage, the research has mainly concentrated on the EM properties of composites with carbonaceous^5,6 and ferromagnetic⁷ particles. In the past few years, the focus has been shifted to polymers reinforced with “2D materials beyond graphene,”⁸ a wide family of monolayer nanomaterials that involve single-element 2D nanosheets (silicene, germanene, stanene, etc.),⁹ transition metal mono and dichalcogenides,^10,11 transition metal carbides (MXenes),¹² layered metal oxides and hydroxides,¹³ and their hierarchical structures.¹⁴ In particular, the Wang group has recently demonstrated exceptionally high EM absorption of polymers reinforced with (i) superparamagnetic cobalt ferrite nanospheres decorated with reduced graphene oxide (rGO),¹⁵ (ii) capsule-like structures of CoFe₂O₄ nanoparticles enriched with rGO and covered with poly(vinylpyrrolidone),¹⁶ (iii) binary composites of termochromic samarium-iron garnet Sm₃Fe₅O₁₂ with cobalt ferrite furnished with rGO,¹⁷ (iv) CoFe₂O₄ nanospheres embedded into MoS₂ nest-like nanostructures,¹⁸ and (v) graphene oxide aerogels loaded with cobalt ferrite nanoparticles.¹⁹

This study deals with dielectric properties and EMI shielding effectiveness of polymer composites filled with semiconductor transition metal dichalcogenide (TMDC) nanoparticles and their hybrid nanostructures. Mono- and few-layered TMDC nanosheets are prepared by micromechanical cleavage of bulk materials, ultrasonication-assisted liquid-phase exfoliation, chemical and electrochemical intercalation, colloidal synthesis, and chemical vapor deposition.²⁰ Experimental studies demonstrate good dielectric properties of polymer composites reinforced with TMDC nanosheets^21–23 and their ability to serve as effective EMI absorbers. However, to achieve satisfactory shielding performance, high loading of filler is required, which causes complications in processing of nanocomposites, leads to deterioration of their mechanical properties, and limits commercial applications.²⁴

To enhance the EMI shielding effectiveness of composites with TMDC nanofillers, (i) alternation of morphology^25–29 and (ii) functionalized hybridization of nanofiller^30–34 are suggested as the most efficient strategies. Although the above studies reveal a pronounced improvement of the EM attenuation, comparison of the proposed approaches is rather difficult as the experimental data are reported in different bands of the EM spectrum on absorbers with different thicknesses and different volume fractions on filler.

Mathematical modeling of the dielectric properties of polymer nanocomposites provides a helpful tool for their characterization and comparison of the EMI shielding effectiveness.⁵ A number of models have been developed in the past half-a-century for the dielectric permittivity of composite materials, refer Sheen et al.³⁵ and Araujo et al.³⁶ for their discussion and comparison. Simple phenomenological models for the dielectric properties of polymers reinforced with spherical particles were proposed in Almond et al.³⁷ and Drozdov and Christiansen.³⁸ A micromechanically-based effective-medium theory for the dielectric properties of composites with flake-like nanofillers was developed by Xia et al.³⁹ To the best of our knowledge, the ability of constitutive models to describe two sets of observations simultaneously (for the real and imaginary parts of the complex dielectric permittivity) on polymers reinforced with TMDC nanoparticles with sophisticated shapes (3D nano-flowers, hollow and hierarchical nanospheres, core-shells spheres coated with nanosheets, hierarchical heterostructures, etc.) has not yet been investigated.

The objective of this study is three-fold: (i) to develop a model for the complex dielectric permittivity of polymer composites that takes into account aggregation of TMDC particles into clusters (due to the van der Waals interaction between particles) and non-spherical shapes of the aggregates, (ii) to determine adjustable parameters in the governing equations by fitting observations on composites with various types of TMDC nanofillers in the S, X, and Ku bands of the EM spectrum, and (iii) to compare dielectric properties of neat TMDC nanoparticles and composite TMDC–graphene heterostructures.

To derive a constitutive model, we treat a composite as a two-phase medium consisting of clusters of aggregated nanoparticles homogeneously distributed in a polymer matrix. Following Xue,⁴⁰ we presume each cluster to involve both (filler and polymer) components. The dielectric permittivity of a cluster is connected with the intrinsic dielectric permittivity of filler by the Maxwell-Garnett formula. At small volume fractions of non-spherical clusters in a polymer matrix, the dielectric permittivity of a composite is determined by means of the Fricke model.⁴¹ Its dielectric permittivity at an arbitrary concentration of filler is found with the help of the integration embedding scheme.⁴² The effect of frequency of the EM field on the intrinsic dielectric permittivity of TMDC nanoparticles is described within the Debye model.⁴³

Model

The complex dielectric permittivity $\overset{\land}{ε}$ of an isotropic medium reads $\overset{\land}{ε} = ε_{0} ε$ , where $ε_{0} = 8.85 \cdot 10^{- 12}$ F/m is the dielectric permittivity of vacuum. The relative dielectric permittivity ε is given by

ε = ε' - ι ε,

(1)

where ε′ and ε″ are the real and imaginary parts of ε, and

ι = \sqrt{- 1}

. The quantities ε′, ε″ are functions of the angular frequency

ω = 2 π

, where f stands for the frequency of EM radiation.

A composite is treated as a two-phase medium, where filler particles with relative dielectric permittivity ε_f are distributed in a polymer matrix with relative dielectric permittivity ε_m. The volume fraction of filler is denoted by φ. With reference to Xue,⁴⁰ we suppose that filler particles aggregate into clusters that involve both filler and polymer components. Aggregation of TMDC particles in polymer matrices is confirmed by transmission electron microscopy (TEM) images, as well as by observations in tensile tests, which show a pronounced deterioration of the mechanical properties of nanocomposites with volume fraction of filler exceeding 1 vol%.^44,45 The effective volume fraction of clusters is defined as

φ_{a} = K φ,

(2)

where

K \geq 1

is a coefficient characterizing the presence of polymer molecules in clusters, and the subscript index “a” stands for aggregates. Keeping in mind that the volume fraction of filler in an individual cluster equals

\frac{1}{K}

, we find from equation (2) that the volume fraction of polymer in a cluster ψ reads

ψ = 1 - \frac{1}{K}

(3)

A cluster is treated as a composite material where polymer inclusions (with relative dielectric permittivity $ε_{m}$ and volume fraction ψ) are homogeneously distributed in a filler matrix with relative dielectric permittivity $ε_{f}$ . The relative dielectric permittivity $ε_{a}$ of a cluster is determined by means of the Maxwell-Garnett formula

ε_{a} = ε_{f} \frac{ε_{m} + 2 ε_{f} + 2 ψ (ε_{m} - ε_{f})}{ε_{m} + 2 ε_{f} - ψ (ε_{m} - ε_{f})}

Insertion of equation (3) into this equation results in

ε_{a} = Λ ε_{f}, Λ = \frac{3 K ε_{m} + 2 (ε_{f} - ε_{m})}{3 K ε_{f} - (ε_{f} - ε_{m})}

(4)

When K = 1 (no clustering of filler), equation (4) implies that $ε_{a} = ε_{f}$ . When K > 1, the dielectric permittivity of filler can be found by solving equation (4) with respect to ε_f. Setting $Λ = Λ_{1} - ι Λ_{2}$ , we arrive at the set of nonlinear equations

Λ_{1} ε_{f}' - Λ_{2} ε_{f}^{″} = ε_{a}', Λ_{2} ε_{f}' + Λ_{1} ε_{f}^{″} = ε_{a}^{″}

(5)

where

\begin{matrix} Λ_{1} = \frac{X_{1} Y_{1} + X_{2} Y_{2}}{Y_{1}^{2} + Y_{2}^{2}}, Λ_{2} = \frac{X_{2} Y_{1} - X_{1} Y_{2}}{Y_{1}^{2} + Y_{2}^{2}}, \\ X_{1} = 3 K ε_{m}' + 2 (ε_{f}' - ε_{m}'), X_{2} = 3 K ε_{m}^{″} + 2 (ε_{f}^{″} - ε_{m}^{″}), \\ Y_{1} = 3 K ε_{f}' - (ε_{f}' - ε_{m}'), Y_{2} = 3 K ε_{f}^{″} - (ε_{f}^{″} - ε_{m}^{″}) \end{matrix}

(6)

We now suppose that clusters of filler with relative dielectric permittivity $ε_{a}$ are homogeneously and randomly distributed in a matrix with relative dielectric permittivity $ε_{m}$ . For definiteness, each cluster is treated as a spheroid with semi-axes a₁ and $a_{2} = a_{3}$ . Under the conventional assumptions that (i) the characteristic size of clusters is small compared with the wavelength of the EM field and (ii) multipole interactions between the clusters are disregarded, the relative dielectric permittivity of the composite ε_c is determined by the Fricke model⁴¹

ε_{c} = ε_{m} [1 + \frac{B φ_{a} (ε_{a} - ε_{m})}{ε_{a} + (B - 1) ε_{m} - φ_{a} (ε_{a} - ε_{m})}]

(7)

Under the assumption that the dielectric permittivity of clusters exceeds strongly that of the matrix, the coefficient B is given by

B = \frac{4 - 3 L}{3 L (1 - L)}

(8)

where L is expressed in terms of the ratio

a_{1} / a_{2}

of semi-axes of spheroidal clusters

\begin{matrix} L = \frac{1}{{sin}^{2} ϕ} - \frac{{cos}^{2} ϕ}{2 {sin}^{3} ϕ} \ln (\frac{1 + sin ϕ}{1 - sin ϕ}), \\ cos ϕ = \frac{a_{2}}{a_{1}} (a_{1} > a_{2}), \\ L = \frac{2 ϕ - sin 2 ϕ}{2 {sin}^{3} ϕ} cos ϕ, \\ cos ϕ = \frac{a_{1}}{a_{2}} (a_{1} < a_{2}) \end{matrix}

(9)

In particular, for spherical particles with $a_{1} = a_{2}$ , equations (8) and (9) imply that $L = \frac{2}{3}$ and B = 3.

Equation (7) is fulfilled when the volume fraction of clusters is small. To extend this relation to higher values of φ, we apply the integration embedding method.⁴² Starting with a matrix with relative dielectric permittivity $ε_{m}$ , we replace a small volume of its material $Δ v_{1}$ (per unit volume of the composite) with filler, and calculate the dielectric permittivity of the composite $ε_{c, 1}$ by means of equations (2) and (7). Afterwards, a new matrix is considered with dielectric permittivity $ε_{c, 1}$ , in which another infinitesimal volume $Δ v_{2}$ is replaced with filler, and its dielectric permittivity $ε_{c, 2}$ is calculated by using the same approach. In this process, the dielectric permittivity of the composite grows from its initial value $ε_{m}$ to the final value ε_c corresponding to a given volume fraction of filler φ. Omitting simple algebra (see Appendix 1 for details), we arrive at the set of nonlinear differential equations for the real and imaginary parts of the dielectric permittivity ε_c

\begin{matrix} \frac{{d ε}_{c}'}{d φ} = \frac{BK P_{1}}{(1 - φ) P}, \frac{{d ε}_{c}^{″}}{d φ} = \frac{BK P_{2}}{(1 - φ) P}, \\ ε_{c}' (0) = ε_{m}', ε_{c}^{″} (0) = ε_{m}^{″} \end{matrix}

(10)

where

\begin{matrix} P_{1} = [ε_{c}' (ε_{a}' - ε_{c}') - ε_{c}^{″} (ε_{a}^{″} - ε_{c}^{″})] (ε_{a}' + (B - 1) ε_{c}') \\ + [ε_{c}' (ε_{a}^{″} - ε_{c}^{″}) + ε_{c}^{″} (ε_{a}' - ε_{c}')] (ε_{a}^{″} + (B - 1) ε_{c}^{″}), \\ P_{2} = [ε_{c}' (ε_{a}^{″} - ε_{c}^{″}) + ε_{c}^{″} (ε_{a}' - ε_{c}')] (ε_{a}' + (B - 1) ε_{c}') \\ - [ε_{c}' (ε_{a}' - ε_{c}') - ε_{c}^{″} (ε_{a}^{″} - ε_{c}^{″})] (ε_{a}^{″} + (B - 1) ε_{c}^{″}), \\ P = (ε_{a}' + (B - 1) ε_{c}')^{2} + (ε_{a}^{″} + (B - 1) ε_{c}^{″})^{2} \end{matrix}

(11)

Equations (10) and (11) provide the governing equations for the complex dielectric permittivity of a nanocomposite. They involve four adjustable parameters with the following physical meaning: B is a measure of non-sphericity of clusters, K denotes volume fraction of polymer in clusters of filler, and $ε_{a}', ε_{a}^{″}$ are the real and imaginary parts of the relative dielectric permittivity of clusters. The initial conditions $ε_{m}^{'}$ and $ε_{m}^{″}$ for equation (10) are determined in a test on a polymer matrix.

Given $ε_{a}'$ and $ε_{a} ″$ , the real and imaginary parts $ε_{f}'$ , $ε_{f} ″$ of the intrinsic dielectric permittivity of filler are found from nonlinear equations (5) and (6). Following common practice in the analysis of dielectric response at microwave frequencies,⁴³ we disregard the effect of angular frequency ω on properties of the polymer matrix and presume the complex dielectric permittivity of filler to obey the Debye model

ε_{f} = ε_{f \infty} + \frac{ε_{f 0} - ε_{f \infty}}{1 + ι ω τ}

(12)

where

ε_{f 0}

and ε_f∞ stand for real dielectric permittivities at small (

ω \to 0

) and high (

ω \to \infty

) frequencies, and τ is the characteristic time for dielectric relaxation. Equation (12) means that the intrinsic electrical conductivity of TMDCs is rather poor, and tunneling (hopping of electrons from one nanoparticle to another) provides the main mechanism for AC conductivity in clusters.⁴⁶ It follows from equation (12) that the parameters

ε_{f}'

and

ε_{f}^{″}

obey the equations

\begin{matrix} ε_{f}' = ε_{f \infty} + \frac{ε_{f 0} - ε_{f \infty}}{1 + {(ω τ)}^{2}}, \\ ε_{f}^{″} = \frac{(ε_{f 0} - ε_{f \infty}) ω τ}{1 + {(ω τ)}^{2}} \end{matrix}

Combination of these relations yields

\begin{matrix} ε_{f}' = C_{0} + C_{1} \frac{ε_{f}^{″}}{f}, \\ C_{0} = ε_{f \infty}, \\ C_{1} = \frac{1}{2 π τ} \end{matrix}

(13)

Fitting of observations

In the analysis of experimental data, we focus on observations at frequencies f = 4, 6, 8, 10, 12, 14, and 16 GHz, which correspond to the S, X, and Ku bands of the EM spectrum.

For each set of data, we begin with matching the experimental dependencies $ε_{c}' (φ)$ and $ε_{c}^{″} (φ)$ at f = 10 GHz. The quantities B, K, $ε_{a}'$ , and $ε_{a}^{″}$ are found by the nonlinear regression method to minimize the expression

\sum [(ε_{cexp}' - ε_{csim}')^{2} + (ε_{cexp}^{″} - ε_{csim}^{″})^{2}]

(14)

where summation is performed over all φ under consideration,

ε_{cexp}', ε_{cexp}^{″}

are real and imaginary parts of the relative dielectric permittivity measured in a test, and

ε_{csim}', ε_{csim}^{″}

are determined by numerical integration of equations (10) and (11) with the help of the Runge–Kutta method.

Afterwards, the coefficients B and K are fixed. For each frequency f under investigation and the corresponding set of experimental dependencies $ε_{c}' (φ)$ and $ε_{c}^{″} (φ)$ , the parameters $ε_{a}'$ and $ε_{a}^{″}$ are determined from the condition of minimum of expression (14). The intrinsic dielectric permittivities of nanoparticles $ε_{f}'$ and $ε_{f}^{″}$ are found by solving equations (5) and (6) by the Newton–Raphson algorithm. After determining $ε_{f}'$ and $ε_{f}^{″}$ at each frequency f separately, we plot $ε_{f}'$ versus $ε_{f}^{″} / f$ , approximate the data by equation (13), where C₀ and C₁ are found by the least-squares technique, and calculate $ε_{f \infty}$ and τ.

Paraffin wax reinforced with MoS₂ microspheres and nanosheets

To examine the difference between the dielectric properties of composites reinforced with spherical micro-particles and nanosheets, we approximate experimental data reported in Ning et al.²¹

First, observations in Figure 1(a) are matched on paraffin wax reinforced with MoS₂ powder (an average diameter of 1–2 μm). The powder and paraffin wax were blended with ether and ultrasonicated. After cooling and evaporation of ether, the mixture was compressed to produce composite specimens. Given mass fraction of filler m, its volume fraction φ is calculated as

φ = \frac{ρ m}{1 - m (1 - ρ)}, ρ = \frac{ρ_{m}}{ρ_{f}}

(15)

where

ρ_{m} = 0.9

g/cm³ is density of the matrix and

ρ_{f} = 5.06

g/cm³ stands for density of the filler.

Figure 1.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c}^{″}$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data²¹ on paraffin wax reinforced with MoS₂ micro-particles at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13).

In Figure 1(a), the real

ε_{c}'

and imaginary

ε_{c}^{″}

parts of the dielectric permittivity of composites at frequency f = 10 GHz are plotted versus volume fraction of filler φ together with results of simulation with the material parameters listed in Table 1. To reduce the number of parameters to be found, we presume clusters of micro-particles to be spherical and set B = 3. Figure 1(a) reveals acceptable agreement between the experimental data and their description by the model. The same accuracy of matching observations is reached at other frequencies f as well. For reasons of brevity, the corresponding dependencies are not reported. After calculation of parameters

ε_{f}'

and

ε_{f} ″

from equations (5) and (6) at each frequency f separately, we plot the data in Figure 1(b) and approximate them by equation (13) with the coefficients

ε_{f \infty}

and τ collected in Table 2.

Table 1.

Material parameters for composites at frequency f = 10 GHz.

Fig.	Filler	Matrix	B	K	$ɛ_{a}'$	$ɛ_{a}^{″}$
1	MoS₂ microspheres	Wax	3.0	2.6	16.0	6.0
2	MoS₂ nanosheets	Wax	15.0	2.6	19.0	7.0
3	MoS₂ nanospheres	PVDF	3.0	4.5	20.0	32.0
4	MoS₂ nanoflowers	PANI	60.0	1.0	99.0	188.0
5	MoS₂-rGO nanosheets	Wax	140.0	2.1	124.0	85.0
6	MoS₂-graphene nanosheets	Wax	35.0	1.8	176.0	118.0
7	MoS₂-rGO nanospheres	PVDF	3.0	2.0	152.0	132.0
8	MoS₂-CoS₂-rGO nanospheres	Wax	3.0	7.1	49.0	73.0
9	WS₂ nanosheets	Wax	35.0	2.7	35.0	19.0

rGO: reduced graphene oxide; PVDF: poly(vinylidene fluoride); PANI: polyaniline.

Table 2.

Parameters in the Debye formula for the dielectric permittivity of filler.

Fig.	Filler	$ɛ_{f \infty}$	$τ \cdot 10^{11}$ s
1	MoS₂ microspheres	46.5	13.6
2	MoS₂ nanosheets	52.7	6.6
3	MoS₂ nanospheres	100.9	22.4
4	MoS₂ nanoflowers	65.1	8.7
5	MoS₂-rGO nanosheets	268.5	6.2
6	MoS₂-graphene nanosheets	356.5	13.1
7	MoS₂-rGO nanospheres	296.4	6.8
8	MoS₂-CoS₂-rGO nanospheres	877.8	3.0
9	WS₂ nanosheets	104.0	4.3

rGO: reduced graphene oxide.

We proceed with matching observations on paraffin wax reinforced with MoS₂ nanosheets. The nanosheets were prepared by intercalation of MoS₂ powder with lithium ions (in a solution of N-butyl lithium) followed by exfoliation under sonication in an aqueous solution. The nanofiller was mixed with paraffin wax in a solution of ether and ultrasonicated. After cooling and evaporation of ether, the mixture was pressed at room temperature to prepare specimens.

The real $ε_{c}'$ and imaginary $ε_{f}^{″}$ parts of the dielectric permittivity of nanocomposites at frequency f = 10 GHz are reported in Figure 2(a) together with results of simulation with the adjustable parameters listed in Table 1. In the fitting procedure, we set K = 2.6 (the same value as in Figure 1(a)), and approximate observations at each frequency f with the help of three parameters: B, $ε_{a}'$ , and $ε_{a}^{″}$ . Afterwards, we calculate $ε_{f}'$ and $ε_{f}^{″}$ from equations (5) and (6), plot $ε_{f}'$ versus $ε_{f}^{″} / f$ in Figure 2(b), and match the data by equation (13) with the coefficients listed in Table 2.

Figure 2.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c} ″$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data²¹ on paraffin wax reinforced with MoS₂ nanosheets at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13).

Poly(vinylidene fluoride) reinforced with MoS₂ nanospheres

To assess the difference between the dielectric response of composites filled with MoS₂ microspheres and hierarchical nanospheres, we fit experimental data reported in Zhang et al.²⁵

MoS₂ nanospheres (an average diameter of 200 to 300 nm) were synthesized by reaction of sodium molybdate (Na₂MoO $4 \cdot$ 2H₂O) with L-cysteine in an aqueous solution (20 h at 200℃). After washing and drying of the precipitate, it was mixed with a solution of poly(vinylidene fluoride) (PVDF) in N, N-dimethylformamide (DMF), sonicated, and dried in an oven. Composite specimens were manufactured by pressing the mixture at room temperature. Volume fraction of filler φ is calculated from equation (15) with $ρ_{m} = 1.78$ g/cm³ (the density of most commercial PVDF polymers).

We begin with fitting observations in Figure 3(a), where the real and imaginary parts of the dielectric permittivity of composites are reported at frequency f = 10 GHz. The best-fit adjustable parameters are collected in Table 1. Then, the experimental dependencies $ε_{c}' (φ)$ and $ε_{c}^{″} (φ)$ measured at the other frequencies are approximated. Each set of observations is matched separately with the help of two parameters, $ε_{a}'$ and $ε_{a}^{″}$ . By using these data, we calculate $ε_{f}'$ and $ε_{f}^{″}$ for each frequency f from equations (5) and (6) and plot $ε_{f}'$ versus $ε_{f}^{″} / f$ in Figure 3(b). The data are approximated by equation (13) with the coefficients collected in Table 2.

Figure 3.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c}^{″}$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data²⁵ on PVDF reinforced with MoS₂ nanospheres at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13). PVDF: poly(vinylidene fluoride).

Polyaniline reinforced with MoS₂ nanoflowers

To evaluate the effect of electrical conductivity of the polymer matrix on the dielectric properties of nanocomposites, we approximate observations reported in An et al.²⁷

Polyaniline–MoS₂ composites were prepared in a two-step process. At the first step, MoS₂ nanoflowers were synthesized by chemical reaction of ammonium molybdate tetrahydrate ((NH₄)₆Mo₇O₂₄) with thiourea (CN₂H₄S) in an aqueous solution of sodium dodecyl benzene sulfonate (18 h at 220℃). The precipitate was collected by centrifugation, washed, and dried. At the other step, the nanofiller was mixed with aniline monomers in an aqueous solution of dodecyl benzenesulfonic acid and ammonium persulfate. After in-situ chemical oxidative polymerization of polyaniline (PANI), the product was dried under vacuum. Samples for dielectric tests were fabricated by blending nanoparticles with paraffin wax (in proportion 1:4 by weight) and compressing the mixture at room temperature. Volume fraction of filler φ is calculated from equation (15) with $ρ_{m} = 0.94$ g/cm³ (an average value for paraffin wax and PANI).

The real and imaginary parts of the dielectric permittivity of composites at frequency f = 10 GHz are presented in Figure 4(a) together with results of simulation with the material parameters collected in Table 1. According to this table, K = 1, which means that MoS₂ and PANI are chemically incompatible, and clusters of filler are formed by nanosheets only. Afterwards, the experimental dependencies $ε_{c}' (φ)$ and $ε_{c}^{″} (φ)$ at the other frequencies f are matched. Each set of observations is approximated separately with the help of two parameters, $ε_{a}'$ and $ε_{a}^{″}$ (which coincide with $ε_{f}'$ and $ε_{f}^{″}$ at K = 1). The plot of $ε_{f}'$ versus $ε_{f}^{″} / f$ is depicted Figure 4(b), where the data are fitted by equation (13) with the coefficients listed in Table 2.

Figure 4.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c} ″$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data²⁷ on paraffin wax-polyaniline mixture reinforced with MoS₂ nanoflowers at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13).

Polymers reinforced with MoS₂–graphene hybrids

To study how the presence of graphene and rGO in MoS₂–carbon nanohybrids affects the dielectric properties of nanocomposites, four sets of experimental data are matched. In calculations of volume fraction of filler φ, we use density of MoS₂, as densities of MoS₂–rGO hybrids are not provided.

We begin with observations on paraffin wax reinforced with MoS₂-rGO nanosheets.³⁰ GO sheets were synthesized by a modified Hummers method, freeze-dried to obtain a powder, dispersed in water, and centrifuged to remove agglomerates. Sodium molybdate dihydrate (Na₂MoO·H₂O) was mixed with GO powder in an aqueous solution and freeze-dried to prepare Na₂MoO₄–GO hybrid. The powder was transferred to a tube furnace, where it reacted with CS₂ vapor (2 h at 650℃) to produce MoS₂ and to reduce GO thermally. After washing and drying, MoS₂-rGO powder was mixed with molten paraffin wax, cooled, and pressed at room temperature to manufacture composite specimens.

The real and imaginary parts of the dielectric permittivity of nanocomposites at frequency f = 10 GHz are plotted versus volume fraction of filler φ in Figure 5(a) together with results of simulation with the material parameters reported in Table 1. After fitting the experimental diagrams at all other frequencies f under consideration, we determine $ε_{f}' and ε_{f}^{″}$ from equations (5) and (6), and plot $ε_{f}'$ versus $ε_{f}^{″} / f$ in Figure 5(b). The data are approximated by equation (13) with the coefficients collected in Table 2.

Figure 5.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c} ″$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data³⁰ on paraffin wax reinforced with MoS2–rGO hybrid nanosheets at frequency f = 10 GHz. Solid lines: results of simulation. (a) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13). rGO: reduced graphene oxide.

We proceed with matching observations on paraffin wax reinforced with MoS₂–graphene nanosheets.³² MoS₂ nanosheets were prepared by dispersion of molybdenum disulfide powder in 1-methyl-2-pyrrolidone (MP) solution followed by sonication, centrifugation, and drying under vacuum. Graphene nanosheets were manufactured by a similar technique: graphite powder was dispersed in MP solution with addition of hexadecyl trimethyl ammonium bromide (CTAB) as a surfactant, sonicated, centrifuged, and dried. MoS₂–graphene hybrid nanofiller was produced by hydrothermal reaction. MoS₂ and graphene nanosheets (in proportion 4:1 by weight) were dispersed in ethanol solution and held for 12 h at 200℃. After cooling, the precipitate was centrifuged, washed, and dried. Composite specimens were manufactured by pressing mixtures of paraffin wax with MoS₂–graphene nanosheets.

Observations on the nanocomposites at frequency f = 10 GHz are presented in Figure 6(a) together with results of simulation with the material parameters reported in Table 1. After fitting the experimental diagrams at all frequencies f under investigation, we calculate $ε_{f}'$ and $ε_{f}^{″}$ from equations (5) and (6), plot $ε_{f}'$ versus $ε_{f}^{″} / f$ in Figure 6(b), and approximate the data by equation (13) with the coefficients listed in Table 2.

Figure 6.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c}^{″}$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data³² on paraffin wax reinforced with MoS₂–graphene hybrid nanosheets at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13).

We now approximate experimental data on PVDF loaded with rGO-coated MoS₂ nanospheres.³¹ The nanospheres (an average diameter of 150–200 nm) were prepared by means of a hydrothermal method: chemical reaction in a suspension of sodium molybdate dihydrate with L-cysteine (20 h at 200℃) followed by centrifugation, washing, and drying. GO nanosheets were synthesized by a modified Hummers method. MoS₂ nanospheres and GO nanosheets were mixed (in proportion 1:2 by weight) in an aqueous solution. To reduce GO chemically, the dispersion was heated to 90℃ and hydrazine hydrate was added. Composite specimens were manufactured by mixing the nanofiller with a solution of PVDF in DMF, followed by sonication and drying at an elevated temperature.

The real and imaginary parts of the dielectric permittivity of nanocomposites at frequency f = 10 GHz are reported in Figure 7(a) together with results of simulation with the adjustable parameters reported in Table 1. The plot of $ε_{f}'$ versus $ε_{f}^{″} / f$ is presented in Figure 7(b). The data are fitted by equation (13) with the coefficients collected in Table 2.

Figure 7.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c} ″$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data³¹ on PVDF reinforced with rGO-coated MoS₂ nanospheres at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13). rGO: reduced graphene oxide; PVDF: poly(vinylidene fluoride).

Finally, we match observations on paraffin wax reinforced with rGo-coated CoS₂@MoS₂ nanospheres.³³ To synthesize CoS₂ nanospheres, cobalt chloride hexahydrate (CoCl $2 \cdot$ 6H₂O) powder and sulfur were dispersed in ethylene glycol. After addition of polyvinyl pyrrolidone, the suspension was transferred into an autoclave and heated for 12 h at 200℃. The precipitate was centrifuged, washed, and dried. Flower-like CoS₂@MoS₂ nanospheres were produced by mixing the CoS₂ powder with sodium molybdate dihydrate and L-cysteine in an aqueous solution of CTAB, followed by chemical reaction (12 h at 200℃) in an autoclave. GO nanosheets were prepared by a modified Hummers method. To synthesize ternary rGO-coated CoS₂@MoS₂ nanospheres, the CoS₂@MoS₂ powder was mixed with GO nanosheets in water at pH = 2.5, and the suspension was heated (24 h at 200℃). The product of the hydrothermal reaction was centrifuged, washed, and dried. Composite samples were manufactured by mixing the ternary hybrid nanoparticles with paraffin wax and pressing.

The real and imaginary parts of the dielectric permittivity at frequency f = 10 GHz are reported in Figure 8(a) together with results of simulation with the parameters collected in Table 1. As the magnetic permeability of the nanocomposites is close to that of air (see Figure 7(d) and (e) in Zhu et al.³³), it is disregarded in modeling. At each frequency f under investigation, we find $ε_{f}'$ and $ε_{f}^{″}$ from equations (5) and (6), plot $ε_{f}'$ versus $ε_{f}^{″} / f$ in Figure 8(b), and approximate the data by equation (13) with the coefficients listed in Table 2.

Figure 8.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c} ″$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data³³ on paraffin wax reinforced with rGO-coated CoS₂@MoS₂ nanospheres at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13). rGO: reduced graphene.

Polymers reinforced with WS₂ nanosheets

To compare dielectric properties of polymer composites filled with MoS₂ and WS₂ nanoparticles, experimental data are fitted on paraffin wax reinforced with WS₂ nanosheets.²²

To prepare few-layer WS₂ nanosheets, WS₂ powder was dispersed in DMF and sonicated. The precipitate was centrifuged, washed, and dried. Composite samples were produced by mixing the nanofiller with paraffin wax and pressing.

The real and imaginary parts of the dielectric permittivity at frequency f = 10 GHz are plotted versus volume fraction of filler φ in Figure 9(a). Volume fraction of nanosheets is calculated from equation (15) with ρ = 0.9 and $ρ_{f} = 7.5$ g/cm³. The data in Figure 9(a) are approximated by the model with the material parameters listed in Table 1. After matching the experimental curves at all frequencies f under consideration, the quantities $ε_{f}'$ and $ε_{f}^{″}$ are determined from equations (5) and (6), and $ε_{f}'$ is plotted versus $ε_{f}^{″} / f$ in Figure 9(b). The data are fitted by equation (13) with the coefficients collected in Table 2.

Figure 9.

(a) Real $ɛ_{c}'$ and imaginary $ɛ_{c} ″$ parts of dielectric permittivity versus volume fraction of filler φ. Symbols: experimental data²² on paraffin wax reinforced with WS₂ nanosheets at frequency f = 10 GHz. Solid lines: results of simulation. (b) Real part $ɛ_{f}'$ of the intrinsic dielectric permittivity of filler versus the ratio $ɛ_{f}^{″} / f$ . Circles: treatment of observations. Solid line: approximation of the data by equation (13).

Discussion

Figures 1(a) to 9(a) demonstrate good agreement between the experimental data on the real and imaginary parts of dielectric permittivity of nanocomposites and results of simulation, which confirms the ability of the model to describe observations at microwave frequencies.

Figures 1(b) to 9(b) show that the Debye model (equation (13)) predicts adequately the effect of frequency f on the real $ε_{f}'$ and imaginary $ε_{f}^{″}$ parts of the intrinsic dielectric permittivity of nanofiller in the S and X bands of the EM spectrum. Figures 1(b), 3(b), 6(b) to 9(b) reveal that this model can describe the dielectric properties of filler in the Ku band as well, whereas Figures 2(b), 4(b), and 5(b) demonstrate noticeable deviations of the experimental data in this band from theoretical predictions. The latter may be ascribed to a pronounced decay in $ε_{c}'$ at frequencies f above 12 GHz reported in literature^21,27,30 and not observed in other experimental studies. To ensure better fit of observations in Figures 2(b), 4(b) and 5(b), the Debye formula (equation (13)) is to be replaced with the Havriliak–Negami equation following the approach of Samir et al.⁴⁷

Deviations from the Debye model are conventionally attributed⁴⁸ to the influence of DC electrical conductivity of filler, whose effect is accounted for by the additional term

- ι σ_{DC} / (ε_{0} ω)

in equation (12). To evaluate the importance of this term, we calculate the quantities

ε_{f}', ε_{f}^{″}

and the AC conductivity of filler

σ_{f} = 2 π ε_{0} f ε_{f}^{″}

(16)

at frequency f = 10 GHz and collect the data in Table 3. This table shows that the intrinsic conductivity of TMDC particles

σ_{f}

varies between 1.1 (microspheres) and 40.5 (rGo-coated hierarchical nanospheres) S/m. These values exceed by 2 to 3 orders of magnitude the electrical conductivity

σ_{DC}

of MoS₂ nanosheets (

2.9 \cdot 10^{- 2}

S/m) and hierarchical nanospheres (

9.4 \cdot 10^{- 3}

1.4 \cdot 10^{- 2}

S/m),²⁶ which confirms that

σ_{DC}

weakly affects the dielectric response in the Ku band.

Table 3.

Intrinsic dielectric permittivity and AC conductivity of filler at frequency f = 10 GHz.

Fig.	Filler	$ɛ_{f}'$	$ɛ_{f} ″$	$σ_{f}$ S/m
1	MoS₂ microspheres	48.4	20.2	1.1
2	MoS₂ nanosheets	58.6	23.6	1.3
3	MoS₂ nanospheres	109.1	198.9	11.1
4	MoS₂ nanoflowers	99.0	188.0	10.4
5	MoS₂-rGO nanosheets	321.8	224.9	12.5
6	MoS₂-graphene nanosheets	384.0	258.6	14.4
7	MoS₂-rGO nanospheres	375.0	329.3	18.3
8	MoS₂-CoS₂-rGO nanospheres	471.9	729.4	40.5
9	WS₂ nanosheets	136.1	78.1	4.3

rGO: reduced graphene oxide.

Table 2 reveals that for all TMDC nanofillers under consideration, the characteristic relaxation time τ is of order of $10^{- 11}$ s, which is typical of polarization relaxation on localized point defects.⁴⁹ Variations in the relaxation times for different nanostructures (from $3.0 \cdot 10^{- 11}$ to $22.4 \cdot 10^{- 11}$ s) are rather weak, and they may be explained by the influence of severe mechanical deformation under preparation of nanosheets on the rates of dielectric relaxation.⁵⁰

The following conclusions are drawn from Tables 2 and 3:

(I) Transformation of MoS₂ microspheres into few-layer nanosheets²¹ leads to a modest increase in the complex dielectric permittivity of filler (at f = 10 GHz, $ε_{f}'$ grows by 21%, and $ε_{f}'$ grows by 17%, while $ε_{f \infty}$ increases by 13%). Weak changes in dielectric properties may be explained by breakage of fragile MoS₂ nanosheets subjected to high stresses under ultrasonication.⁵¹

(II) The real and imaginary parts of the dielectric permittivity of WS₂ nanosheets²² exceed those of MoS₂ nanosheets²¹ by a factor of 2 to 3. This conclusion is in accord with the experimental data reported in Wang et al.,²³ where $ε_{c}'$ and $ε_{c}^{″}$ for nanocomposites with these fillers are directly compared.

(III) Hybrid fillers involving TMDC together with graphene and rGO nanosheets demonstrate superior dielectric properties compared with pure TMDC ( $ε_{f}'$ increases by a factor of 3 to 10, $ε_{f}^{″}$ grows by 2 to 30 times, and $σ_{f}$ increases by an order of magnitude). This may be explained by a strong effect of high electrical conductivity of carbon nanoparticles on the intrinsic dielectric permittivity of hierarchical heterostructures.

To evaluate the effect of aggregation of filler (characterized by the parameter K in the model) on the complex dielectric permittivity of composites, numerical analysis is conducted for a composite with the paraffin wax matrix reinforced with rGo-coated CoS₂@MoS₂ nanospheres³³ at frequency f = 10 GHz. Results of simulation with the material constants reported in Tables 1 and 3 are depicted in Figure 10, where the real (Figure 10(a)) and imaginary (Figure 10(b)) parts of the dielectric permittivity are plotted versus volume fraction of filler φ at K values ranging from K = 1 (no aggregation) to K = 10 (high degree of clustering). This figure shows that an increase in K induces a pronounced (by an order of magnitude) growth of $ε_{c}'$ and $ε_{c}^{″}$ for a fixed concentration of filler.

Figure 10.

Real $ɛ_{c}'$ and imaginary $ɛ_{c}^{″}$ parts of complex dielectric permittivity versus volume fraction of filler φ. Symbols: results of simulation with various K values for paraffin wax reinforced with rGO-coated CoS₂@MoS₂ nanospheres at frequency f = 10 GHz.³³ rGO: reduced graphene oxide.

Conclusions

A model is developed for the complex dielectric permittivity of polymer nanocomposites that takes into account non-sphericity of filler particles and their aggregation into clusters containing both filler and matrix components. The dielectric permittivity of composites with small volume fractions of aggregates (spheroids with arbitrary aspect ratios) are determined with the help of the Fricke model. Differential equations for the real and imaginary parts of the dielectric permittivity of a composite with an arbitrary volume fraction of filler φ are derived by means of the integration embedding method. These relations involve four material constants with transparent physical meaning: (i) B characterizes the aspect ratio of clusters, (ii) K describes volume fraction of polymer molecules in clusters and serves as a measure of wetting of filler particles by the matrix, and (iii) $ε_{f}'$ and $ε_{f}^{″}$ are the intrinsic dielectric parameters of filler. The coefficients B and K are independent of frequency of the EM field, while evolution of $ε_{f}'$ and $ε_{f}^{″}$ with f is governed by the Debye model.

Material parameters are found by matching observations on the real and imaginary parts of the dielectric permittivity (two sets of data are fitted simultaneously) of polymers reinforced with MoS₂ and WS₂ micro- and nanospheres, MoS₂ nanosheets and nanoflowers, and composite heterostructures formed by MoS₂ and MoS₂-CoS₂ nanoparticles with graphene and rGO nanosheets. Figures 1 to 9 demonstrate good agreement between results of simulation and experimental data at microwave frequencies.

It is shown that composite heterostructures formed by TMDC nanoparticles together with graphene and rGO nanosheets have superior dielectric properties compared with neat TMDC nanospheres and nanosheets.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Innovationsfonden (Innovation Fund Denmark, projects 5152-00002B and 9091-00010B).

ORCID iD

AD Drozdov

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Modeling dielectric permittivity of polymer composites filled with transition metal dichalcogenide nanoparticles

Abstract

Keywords

Introduction

Model

Fitting of observations

Paraffin wax reinforced with MoS2 microspheres and nanosheets

Poly(vinylidene fluoride) reinforced with MoS2 nanospheres

Polyaniline reinforced with MoS2 nanoflowers

Polymers reinforced with MoS2–graphene hybrids

Polymers reinforced with WS2 nanosheets

Discussion

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

References

Paraffin wax reinforced with MoS₂ microspheres and nanosheets

Poly(vinylidene fluoride) reinforced with MoS₂ nanospheres

Polyaniline reinforced with MoS₂ nanoflowers

Polymers reinforced with MoS₂–graphene hybrids

Polymers reinforced with WS₂ nanosheets