Dielectric Permittivity, AC Electrical Conductivity and Conduction Mechanism of High Crosslinked-Vinyl Polymers and Their Pd(OAc)2 Composites

Semiconductor materials based on metal high crosslinked-vinyl polymer composites were prepared through loading of Pd(OAc)2 on both Poly(ethylene-1,2-diyl dimethacrylate) (poly(EDMA)) and poly(ethylene-1,2-diyl dimethacrylate-co-methyl methacrylate) (Poly(EDMA-co-MMA)). The thermochemical properties for both poly(EDMA) and poly(EDMA-co-MMA) were investigated by thermal gravimetric analysis TGA technique. The dielectric permittivity, AC electrical conductivity and conduction mechanism for all the prepared polymers and their Pd(OAc)2 composites were studied. The results showed that the loading of polymers with Pd(OAc)2 led to an increase in the magnitudes of both the dielectric permittivity and AC electrical conductivity (σac). The value of σac increased from 1.38 × 10−5 to 5.84 × 10−5 S m−1 and from 6.40 × 10−6 to 2.48 × 10−5 S m−1 for poly(EDMA) and poly(EDMA-co-MMA), respectively, at 1 MHz and 340 K after loading with Pd(OAc)2. Additionally, all the prepared polymers and composites were considered as semiconductors at all the test frequencies and in the temperature range of 300–340 K. Furthermore, it seems that a conduction mechanism for all the samples could be Quantum Mechanical Tunneling (QMT).


Introduction
Polymers have attracted a lot of interest in the development of modern technologies due to their easy synthesis, cheaper cost, high stability, noncorrosive nature, and low density, which makes them suitable materials for replacing metals and ceramics [1,2]. However, the electrical insulating nature of most polymers, their electrical conductivity σ ac in the range of 10 −12 -10 −16 S m −1 , has limited their technological and engineering applications. The insulating nature of the polymers can be improved by introducing highly electrically conductive fillers to the polymer matrices, having σ ac values in the range of 10 2 -10 7 S m −1 , which can facilitate the movement of charge carriers through the electron hopping or tunneling process [3]. These functional filler materials include carbon-based nanomaterials, ceramics, metals, or metal oxides [4]. As a result, semiconductors based on polymer composites with electrical properties close to metals and mechanical properties like plastics were produced [5][6][7]. These semiconductor composites can be applied in many Pd(OAc) 2 was loaded on poly(EDMA) and poly(EDMA-co-MMA) by soaking polymeric materials in a solution of Pd(OAc) 2 in methanol according to the procedure described in our previous work [25] (Scheme 1B). Each polymer, poly(EDMA) and poly(EDMA-co-MMA), (150 mg) was added to a solution of Pd(OAc) 2 (10 mg, 0.044 mmol) in methanol (50 mL) and stirred for 12 h to allow Pd(OAc) 2 molecules to go deep within the polymer matrices. After soaking for 12 h, Pd(OAc) 2 /poly(EDMA) and Pd(OAc) 2 /poly(EDMA-co-MMA) were separated from the unloaded Pd(OAc) 2 solution by centrifuge and washed with methanol several times to ensure that the unloaded Pd(OAc) 2 was removed from the polymers. Finally, Pd(OAc) 2 /poly(EDMA) and Pd(OAc) 2 /poly(EDMA-co-MMA) were dried under vacuum for 24 h, and the weight of the two composites became constant, which indicated the complete removal of all solvent molecules (the yield was 157.7 mg and 152.4 mg as black solid crystals for Pd(OAc) 2 /poly(EDMA) and Pd(OAc) 2 /poly(EDMAco-MMA), respectively). The loaded amount of Pd(OAc) 2 which was immobilized on poly(EDMA) and poly(EDMA-co-MMA) was determined by gravimetry after washing and drying the composites and was found to be 5.1% and 1.6% of the weight of poly(EDMA) and poly(EDMA-co-MMA), respectively. The structure and chemical properties of poly(EDMA), poly(EDMA-co-MMA) and their Pd composites were discussed in detail in our previous paper [25].

Measurements
Thermal gravimetric analysis (TGA) was carried out by using Rigaku Thermo plus TG8120 and DSC8230 apparatuses under the flow of nitrogen gas (20 mL/min) and a heating rate of 10 K/min on an aluminum crucible from room temperature to 750 K. The AC electrical conductivity was measured by Hioki 3532-50 LCR hitester in a temperature range of 300-400 K. The samples were comprised of a pellet with a thickness of 1 mm and a surface area of 1.3 cm 2 . During the measurements, the samples were fixed between two copper electrodes.

Thermal Gravimetric Analysis
The thermal gravimetric analysis (TGA) and derivative thermal gravimetric (DTG) of poly(EDMA) and poly(EDMA-co-MMA) are presented in Figure 1. The TGA curves for both polymers showed a slight decrease in weight loss percentage starting from 340 K to about 500 K; then, the weight loss increased by higher rates from about 500 K to 730 K. The DTG curves for both polymers show a main degradation peak at temperatures of 670 K and 640 K with weight loss percentages of 91.27% and 97.34% for poly(EDMA) and poly(EDMA-co-MMA), respectively. The low degradation weight loss percentage starting at 340 K in both polymers could be attributed to a loss of moisture or residual organic solvents from the matrices, while the high-rate degradation weight loss percentage starting at about 500 K could be attributed to the degradation of the backbone of both polymers. In addition, the two polymers are chemically stable up to 460 K and 480 K for poly poly(EDMA) and poly(EDMA-co-MMA), respectively. poly(EDMA-co-MMA), respectively. The low degradation weight loss percentage starting at 340 K in both polymers could be attributed to a loss of moisture or residual organic solvents from the matrices, while the high-rate degradation weight loss percentage starting at about 500 K could be attributed to the degradation of the backbone of both polymers. In addition, the two polymers are chemically stable up to 460 K and 480 K for poly poly(EDMA) and poly(EDMA-co-MMA), respectively. The activation energy and thermodynamic parameters of the main thermal stage degradation for poly(EDMA) and poly(EDMA-co-MMA) are determined by using the Coats-Redfern method [26,27]. According to Coats-Redfern, the mathematical formula of the first-order reaction is given by Equation (1): where α is the fraction of sample decomposed at temperature T, A ' is the Arrhenius constant, R is the general gas constant, θ is the heating rate and E * is the activation energy. The value of α is calculated according to Equation (2): The activation energy and thermodynamic parameters of the main thermal stage degradation for poly(EDMA) and poly(EDMA-co-MMA) are determined by using the Coats-Redfern method [26,27]. According to Coats-Redfern, the mathematical formula of the first-order reaction is given by Equation (1): where α is the fraction of sample decomposed at temperature T, A is the Arrhenius constant, R is the general gas constant, θ is the heating rate and E * is the activation energy. The value of α is calculated according to Equation (2): where W o , W t and W f are the initial weight of the sample, weight of the sample at any given temperature and the final weight of the sample after completion of the reaction, respectively. By applying Equation (1) on the experimental data of TGA and plotting the relation between log − log(1−α) T 2 and 1/T, a straight line was produced for both poly(EDMA) and poly(EDMA-co-MMA), and the values of E * and A were calculated from the slope and intercept with the Y-axis ( Figure S1 in the supporting information). The entropy (∆S * ), enthalpy (∆H * ), and change in free energy (∆G * ) of the activation are determined for both poly(EDMA) and poly(EDMA-co-MMA) by Equations (3)-(5) [28]: where h, K B are the Planck and Boltzmann constants, respectively. The thermal activation energy, Arrhenius constant and thermodynamic parameters for both poly(EDMA) and poly(EDMA-co-MMA) are summarized in Table 1. From these results, the degradation of both polymers is a nonspontaneous and endothermic process, which is confirmed by the positive values of ∆G * and ∆H * , respectively [28]. Table 1. Thermal activation energy and thermodynamic parameters of poly(EDMA) and poly(EDMAco-MMA).

Dielectric Permittivity
The dielectric permittivity of a material (ε) is considered a complex quantity with a real part (ε r ) and imaginary part (ε i ) and is given by Equation (6) [29,30]: The values of the real and imaginary parts of dielectric permittivity ε r and ε i are calculated from the value of capacitance measured in parallel mode (C p ) and the loss tangent (tan δ). The values of C p and tan δ are measured for poly(EDMA), poly(EDMAco-MMA) and their Pd(OAc) 2 composites in the temperature range of 300-400 K and frequency range of 0.5-1000 KHz. The values of ε r and ε i are calculated according to the following Equations (7) and (8) [29,30]: where ε o is the permittivity of free space, and d and A are the sample's thickness and cross-section area, respectively. Polymers and other dielectric compounds exhibit several relaxation and loss (dissipation) modes that appear as maxima in the dielectric spectrum depending on the type of material, temperature and the applied electric field. In polymers, relaxation as well as dielectric loss may be due to the motion of relatively long chain movements in the amorphous region. When the frequency of the applied external field is comparable to the rate of the internal motions, the loss function (tan δ) and consequently the imaginary dielectric constant (ε i ) tend to increase, and a maximum may be observed [31].
The dependences of ε r and ε i on temperature and frequency for poly(EDMA), poly(EDMAco-MMA) and their Pd(OAc) 2 composites are presented in Figures 2 and 3, respectively. Both ε r and ε i decrease when increasing the test frequency from 0.5 to 1000 KHz. On the other hand, the dependences of ε r and ε i on temperature exhibit an irregular behavior in the temperature range under consideration for the prepared polymers and their Pd(OAc) 2 composites (a peak value around 340 K). Taking into account the absence of any peaks in DTG with a slight decrease in TGA curves in the temperature range of 300-500 K, the peak value at this temperature could be attributed to a loss of moisture or residual organic solvents [28,32,33]. Furthermore, the loading of both polymers by Pd(OAc) 2 leads to an increase in the value of ε r ( Figure 2B,D) and ε i ( Figure 3B,D). This behavior of the dielectric permittivity with the frequency and temperature for both polymers and their Pd(OAc) 2 composites has been reported for other polymers and organic composites [24,[33][34][35][36][37][38][39]. of any peaks in DTG with a slight decrease in TGA curves in the temperature range of 300-500 K, the peak value at this temperature could be attributed to a loss of moisture or residual organic solvents [28,32,33]. Furthermore, the loading of both polymers by Pd(OAc)2 leads to an increase in the value of εr ( Figure 2B,D) and εi ( Figure 3B,D). This behavior of the dielectric permittivity with the frequency and temperature for both polymers and their Pd(OAc)2 composites has been reported for other polymers and organic composites [24,33-

AC Electrical Conductivity
The AC electrical conductivity (σac) is expressed as a function of both the value of the imaginary dielectric permittivity (εi) and angular frequency (ω = 2ᴫF), and is calculated according to Equation (9) [29,30] Figure 4 shows the dependence of σac on both the temperature in the temperature range of 300-400 K and the test frequency in the range of 0.5 KHz-1 MHz for the prepared polymers and their Pd(OAc)2 composites. For all samples under testing, σac increases when

AC Electrical Conductivity
The AC electrical conductivity (σ ac ) is expressed as a function of both the value of the imaginary dielectric permittivity (ε i ) and angular frequency (ω = 2πF), and is calculated according to Equation (9) [29,30]: Polymers 2021, 13, 3005 8 of 15 Figure 4 shows the dependence of σ ac on both the temperature in the temperature range of 300-400 K and the test frequency in the range of 0.5 KHz-1 MHz for the prepared polymers and their Pd(OAc) 2 composites. For all samples under testing, σ ac increases when increasing the test frequency; however, it demonstrates two different behaviors depending on the temperature. In the first stage of heating up to 340 K, a semiconductor behavior is observed, while at a higher temperature, σ ac decreases as the temperature increases more (metallic behavior), and this behavior is in agreement with other reported polymers and organic materials [24,32,34,39,40]. On the other hand, the existence of Pd(OAc) 2 slightly enhances the conductivity of the two polymers. For example, the peak value at a test frequency of 1 MHz for poly(EDMA) was 1.38 × 10 −5 S m −1 and increased to 5.84 × 10 −5 S m −1 for Pd(OAc) 2 /poly(EDMA)-5.1%. Additionally, it increased from 6.40 × 10 −6 to 2.48 × 10 −5 S m −1 for poly(EDMA-co-MMA) and Pd(OAc) 2 /poly(EDMAco-MMA)-1.6%, respectively. From these results, we can conclude that poly(EDMA), Pd(OAc) 2 /poly(EDMA)-5.1%, poly(EDMA-co-MMA), and Pd(OAc) 2 /poly(EDMA-co-MMA)-1.6% exhibit a semiconductor behavior in the temperature range of 300-340 K at all the test frequencies, with an enhancement of the electrical conductivity value by adding Pd(OAc) 2 to the polymer matrices.    Figure 5B,D). It appears that the loading of Pd(OAc)2 has a stronger effect on σac at a higher temperature and frequency. This result can be explained due to the increase in charge carriers, since the loading of metal ions leads to an increase of free charge carriers, which, accordingly, leads to an increase of electrical conductivity. This result is in good agreement with the results of other polymers doped with different metal ions [18,19,24,33,38,39].  Figure 5 illustrates the effect of loading poly(EDMA) and poly(EDMA-co-MMA) with Pd(OAc) 2 on the electrical conductivities across the entire test frequency range at room temperature ( Figure 5A,C) and at 340 K ( Figure 5B,D). It appears that the loading of Pd(OAc) 2 has a stronger effect on σ ac at a higher temperature and frequency. This result can be explained due to the increase in charge carriers, since the loading of metal ions leads to an increase of free charge carriers, which, accordingly, leads to an increase of electrical conductivity. This result is in good agreement with the results of other polymers doped with different metal ions [18,19,24,33,38,39].
with Pd(OAc)2 on the electrical conductivities across the entire test frequency range at room temperature ( Figure 5A,C) and at 340 K ( Figure 5B,D). It appears that the loading of Pd(OAc)2 has a stronger effect on σac at a higher temperature and frequency. This result can be explained due to the increase in charge carriers, since the loading of metal ions leads to an increase of free charge carriers, which, accordingly, leads to an increase of electrical conductivity. This result is in good agreement with the results of other polymers doped with different metal ions [18,19,24,33,38,39].  The thermal activation energy (∆E ac ) for the AC electrical conductivity is determined by applying the following Arrhenius equation [28,34]: where σ o is the pre-exponential constant and k B is the Boltzmann constant. The Arrhenius equation is applied for the experimental data in the temperature range of 300-340 K (semiconductor behavior) and at selected test frequencies for all specimens under consideration. Figure 6 shows the relation between ln(σ ac ) and 1/T for all the prepared polymers and their Pd(OAc) 2 composites. As seen from Figure 6, straight lines with negative slopes are produced, in which ∆E ac can be calculated according to Equation (10). Figure 7 presents the dependence of ∆E ac on the frequency for poly(EDMA), Pd(OAc) 2 /poly(EDMA)-5.1%, poly(EDMA-co-MMA) and Pd(OAc) 2 /poly(EDMA-co-MMA)-1.6%. It is clear from this figure that ∆E ac has values in the range of 0.1-0.4 eV, depending on both the test frequency and the type of polymeric material or composite.
under consideration. Figure 6 shows the relation between ln(σac) and 1/T for all the prepared polymers and their Pd(OAc)2 composites. As seen from Figure 6, straight lines with negative slopes are produced, in which ∆Eac can be calculated according to Equation (10). Figure 7 presents the dependence of ∆Eac on the frequency for poly(EDMA), Pd(OAc)2/poly(EDMA)-5.1%, poly(EDMA-co-MMA) and Pd(OAc)2/poly(EDMA-co-MMA)-1.6%. It is clear from this figure that ∆Eac has values in the range of 0.1-0.4 eV, depending on both the test frequency and the type of polymeric material or composite.

Conduction Mechanism
The conduction mechanism of semiconductors is investigated through theoretical models based on the dependence of the AC electrical conductivity on the test frequency according to the following Equation (11): where A is a constant depending on the temperature, and S is the exponent whose value and behavior with the temperature determine the suitable conduction mechanism. The most frequently reported conduction mechanisms are the small polaron tunneling (SPT) Figure 7. Dependence of ∆Eac on the frequency.

Conduction Mechanism
The conduction mechanism of semiconductors is investigated through theoretical models based on the dependence of the AC electrical conductivity on the test frequency according to the following Equation (11): where A is a constant depending on the temperature, and S is the exponent whose value and behavior with the temperature determine the suitable conduction mechanism. The most frequently reported conduction mechanisms are the small polaron tunneling (SPT) model, correlated barrier hopping (CBH) model and quantum mechanical tunneling (QMT) model [29,30]. In the SPT model, the value of S and σ ac are calculated according to the following Equations (12) and (13): σ ac(SPT) = π 4 24 where W H is the barrier height for infinite site separation, τ o is the relaxation time, e is the charge of electron, a is the spatial extent of polaron, N(E F ) is the density of states at the Fermi level and R w is the tunneling distance. For the CBH model, the electrons in charged defect states hop over the Coulomb barrier with height W, which is calculated according to Equation (14): where W m and R are the maximum barrier height and the distance between hopping states, respectively. The values of S, σ ac and the hopping length (R ω ) are determined using Equations (15)- (17): According to the QMT model, S is independent of temperature and is given by Equation (18): Figure 8 shows the relation between log(σ ac ) and log(ω) at the temperature range of 300-400 K. This relationship produces a straight line in which its slope is equal to the value of S. The behavior of S with the temperature suggests the probable conduction mechanism for the materials. The dependence of S on the temperature for all the prepared polymers and their Pd(OAc) 2 composites is presented in Figure 9. It is clear that S is nearly independent of the temperature for all polymers and composites and has values in the range of 0.67-0.85 depending on the polymer type. This behavior and these values of S strongly point to Quantum Mechanical Tunneling (QMT) being the best conduction mechanism (Equation (18)) to describe the electrical conduction process in all the samples under consideration [29,30]. QMT has been reported to be the operating conduction mechanism in many polymers and organic compounds such as copolymer (N, N -bissulphinyl-mbenzenediamine-p-phenylenediamine) [28], quinoline Schiff base complexes [35] and 2-Hydroxy-1-naphthylideneaniline [40]. of 0.67-0.85 depending on the polymer type. This behavior and these values of S strongly point to Quantum Mechanical Tunneling (QMT) being the best conduction mechanism (Equation (18)) to describe the electrical conduction process in all the samples under consideration [29,30]. QMT has been reported to be the operating conduction mechanism in many polymers and organic compounds such as copolymer (N, N′-bissulphinyl-m-benzenediamine-p-phenylenediamine) [28], quinoline Schiff base complexes [35] and 2-Hydroxy-1-naphthylideneaniline [40].

Conclusions
In conclusion, poly(EDMA), poly(EDMA-co-MMA) and their Pd(OAc)2 were confirmed as semiconductor materials based on the dependence of their σac on both the temperature and test frequency. The TGA analysis showed that the polymers were chemically stable up to 460 K and 480 K for poly(EDMA) and poly(EDMA-co-MMA), respectively. Poly(EDMA), Pd(OAc)2/poly(EDMA)-5.1%, poly(EDMA-co-MMA), and Pd(OAc)2/poly(EDMA-co-MMA)-1.6% exhibited a semiconductor behavior in the temperature range of 300-340 K at all the tested frequencies, with an enhancement of the value

Conclusions
In conclusion, poly(EDMA), poly(EDMA-co-MMA) and their Pd(OAc) 2 were confirmed as semiconductor materials based on the dependence of their σ ac on both the temperature and test frequency. The TGA analysis showed that the polymers were chemically stable up to 460 K and 480 K for poly(EDMA) and poly(EDMA-co-MMA), respectively. Poly(EDMA), Pd(OAc) 2 /poly(EDMA)-5.1%, poly(EDMA-co-MMA), and Pd(OAc) 2 /poly(EDMA-co-MMA)-1.6% exhibited a semiconductor behavior in the temperature range of 300-340 K at all the tested frequencies, with an enhancement of the value of the electrical conductivity by adding Pd(OAc) 2 to the polymer matrices. The value of σ ac increased from 1.38 × 10 −5 to 5.84 × 10 −5 S m −1 and from 6.40 × 10 −6 to 2.48 × 10 −5 S m −1 for poly(EDMA) and poly(EDMA-co-MMA), respectively, at 1 MHz and 340 K after loading Pd(OAc) 2 . The activation energy of the electrical conductivity ∆E ac for all the polymers and their Pd(OAc) 2 composites showed values in the range of 0.1-0.4 eV, depending on both the test frequency and the type of polymeric material or composite. The dependence of the S values on the temperature strongly points to Quantum Mechanical Tunneling (QMT) being the best conduction mechanism to describe the electrical conduction process in all the samples under consideration.