Dielectric relaxation dynamics of high-temperature piezoelectric polyimide copolymers

Polyimide copolymers have been prepared based on different diamines as comonomers: a diamine without CN groups and a novel synthesized diamine with two CN groups prepared by polycondensation reaction followed by thermal cyclodehydration. Dielectric spectroscopy measurements were performed, and the dielectric complex function, ac conductivity and electric modulus of the copolymers were investigated as a function of CN group content in the frequency range from 0.1 to 107 Hz at temperatures from 25 to 260 °C. For all samples and temperatures above 150 °C, the dielectric constant increases with increasing temperature due to increasing conductivity. The α-relaxation is just detected for the sample without CN groups, being this relaxation overlapped by the electrical conductivity contributions in the remaining samples. For the copolymer samples and the polymer with CN groups, an important Maxwell–Wagner–Sillars contribution is detected. The mechanisms responsible for the dielectric relaxation, conduction process and electric modulus response have been discussed as a function of the CN group content present in the samples.


Introduction
Smart materials based on electroactive polymers (EAPs) have experienced increasing attention due to their large technological impact. In this way, new materials have been developed with excellent properties including low density, large electromechanical response and suitable mechanical and thermal properties, leading to materials with tailored responses for specific applications [1][2][3]. Many of those interesting qualities of EAPs are present in piezoelectric polymers, which show fast electromechanical response, relatively low power requirements and high forces [4]. Piezoelectricity is defined as the ability of some dielectric materials to change their polarization state when subject to mechanical stress [5][6][7] and, reciprocally, to undergo mechanical deformation by the application of an electric field [8]. In this way, a piezoelectric material can be used for both sensor and actuator applications.
Another polymer class with large interest for hightemperature applications are piezoelectric amorphous polymers, which show the glass transition at temperatures above 80°C [14,15].
The origin of the piezoelectricity in amorphous polymers is different than in semicrystalline polymers. In amorphous polymers, the polarization is not in an equilibrium state, but rather in a quasi-stable state due to the freezing of molecular dipoles. In these materials, the orientation of molecular dipoles is responsible for the piezoelectricity, and the glass transition temperature defines the poling process conditions [15,16]. Other relevant factors for the piezoelectric response on amorphous polymers are the presence of a sufficient concentration of dipoles, the ability to orient these dipoles and the stability of the locking of their orientation [14].
Due to the high glass transition temperature (T g = 176°C), thermal stability and excellent mechanical and dielectric properties, the aromatic polyimide [(b-CN)APB/ODPA] polymer is an interesting amorphous piezoelectric polymer suitable for the development of hightemperature piezoelectric and pyroelectric sensors and microelectromechanical system (MEMS) devices [22,23]. Further, aromatic polyimide shows nitrile groups with large dipole moment of 4.18 D that provide a strong interaction with the applied electric field [24].
The piezoelectric effect in (b-CN)-APB/ODPA is an order of magnitude lower than the one needed for the development of practical devices, but it exhibits the thermal stability necessary to withstand conventional MEMS processing, which is a drawback for fluoropolymers [25].
Poly2-6 and Poly2CN show low piezoelectric coefficient d 33 = 0.091 and 0.168 pC N -1 at room temperature, respectively. These values strongly depend on the number of dipole CN groups in the repetition unit, as well as on the imidization and the poling processes [28]. The dielectric and piezoelectric properties of these amorphous polyimides were addressed in [29], showing that the frozen-in polarization is mainly due to dipolar orientation of the dipolar CN groups, which in turn contribute to the piezoelectric response.
In this way and in order to improve the dielectric and piezoelectric response of these polyimides, new copolymers were produced based on different diamines as comonomers: a diamine without CN groups and a novel synthesized diamine with two CN groups. Thus, due to the large potential of this newly developed material, a deep characterization and understanding of the dielectric response is essential for the development of applications. This work reports on the dielectric relaxation spectrum, electrical conductivity and electric modulus of the different copolymers as a function of frequency and temperature.
Polyimides were synthesized using both single diamines (0CN diamine and diamine 2CN) and co-polyimides by using the diamines in different proportions. A two-step procedure was used. In the first step, a nucleophilic attack of amine groups toward carbonyl groups in the dianhydride produces the co-poly(amic acid), and in the second step, the cyclodehydration reaction caused by thermal treatment Diamine 0CN Diamine 2CN Fig. 1 Chemical structure of the used diamines gives rise to the co-polyimide. The general scheme of the synthesis of a co-polyimide is shown in Fig. 2.
After synthesis, the corresponding polyimides were obtained by thermal imidization of the poly(amic acid) and co-poly(amic acid).
With a silicon designed mold with a centered cavity of dimensions 45 9 45 mm, poly(amic acid)'s imidization and film fabrication were carried out simultaneously. Thus, 0.6 g of poly(amic acid) or co-poly(amic acid) was dissolved into 6 cm 3 N,N-dimethylacetamide (DMAc). Later, the poly(amic acid) was cast to form *150 lm films that were subsequently thermally imidizated, according to the thermal treatment represented in Fig. 3. The thermal treatment was chosen for solvent evaporations at a sufficiently low rate to avoid bubble formation during the curing step.

Characterization
Dielectric spectroscopy measurements were carried out using an impedance analyzer Alpha-S trough the measurement of the capacitance (C) and the loss factor (tan d). The temperature control was performed by a Quatro Cryosystem from Novocontrol GmbH. Circular conducting gold electrodes (10 mm diameter) were deposited onto both sides of each sample, to form a parallel plate capacitor. The electrodes were deposited by magnetron sputtering with a Polaron Coater SC502 under an argon atmosphere. The sample cell with active head dielectric converter was mounted on a cryostat (BDS 1100) and exposed to a heated gas stream evaporated from a liquid nitrogen Dewar. The isothermal experiments were performed from 25 to 260°C (thermal stability: 0.5°C) in 5°C steps. The complex dielectric permittivity e Ã ¼ e 0 À ie 00 was determined as a function of frequency (10 -1 -10 7 Hz) through the following equations: where C is the capacitance, e 0 is real part of the dielectric constant, e 00 is imaginary part of the dielectric constant, e 0 is the permittivity of the free space (8.85 9 10 -12 F m -1 ) and d and A are the sample thickness and electrode area, respectively.
3 Results and discussion

Overall dielectric response
The dielectric response will be discussed, highlighting first the temperature-dependent features and then the frequency dependence ones. Figure 4 shows the evolution of real (e 0 ) and imaginary parts (e 00 ) of the dielectric function as a function of temperature at different frequencies for the different samples. For all samples, the real part of the dielectric constant increases as frequency decreases for all temperature range due to the increasing inability of the dipoles to orient with the rapidly varying applied electric field as frequency increases [30]. The same trend is found for the dc conductivity contribution to the imaginary part of the dielectric permittivity, the peak due to dipolar relaxation appearing in e 00 shifting to higher temperatures with increasing frequency.

Temperature dependence
The dielectric response is practically constant up to 150°C for all samples, independently of the frequency (Fig. 4). No significant differences are observed between the dielectric response of the 0CN/2CN (60/40) and 0CN/ 2CN (50/50) copolymer so just the data for the 50/50 copolymer will be shown.
The dielectric constant for 0CN/2CN (50/50) copolymer shows a low-frequency peak (up to *100 Hz) around 175°C as shown in Fig. 4c. The reason of this peak in both copolymers is due to a decrease in the ionic mobility related to imidization of the dianhydride groups [28]. This decrease cannot be attributed to a relaxation process, as it is characterized by a peak in the real part of the dielectric constant.
The room temperature values of the dielectric response are summarized in Table 1 for a frequency of 1 kHz, together with the values at 200°C. The room temperature values are in agreement with the values found in the literature for the polyimides [31]. The differences observed of the dielectric constant for the 0CN and 2CN polymers at room temperature are fully attributed to the presence of the nitrile groups in the 2CN polymer.
For temperatures above 150°C and for all samples, the dielectric constant increases with increasing temperature due to enhancement of the dipolar mobility and increased conductivity, i.e., dielectric losses ( Fig. 4b, d, f), as reflected in the comparison of the values of the dielectric response at room temperature and 200°C presented in Table 1.
The insert of Fig. 4b for the 0CN polymer shows the arelaxation above 150°C related to the glass transition and attributed to the cooperative segmental motions of the amorphous sample [32].
The dynamics of the a-relaxation observed in the 0CN polymer ( Fig. 4b) was analyzed in the scope of the Vogel-Fulcher-Tammann (VTFH) formalism [33]: where s is the relaxation time, E VFTH is the VFTH energy, k B is the Boltzmann constant and T 0 is the critical temperature at which molecular motions become infinitely slow [34]. From the VTFH fitting parameters, the fragility parameter [35] can be calculated: where m is an indication of the steepness of the variation of the material properties (viscosity, relaxation time, etc.) as T g is reached. A high m value defines a fragile material, whereas a strong material is characterized by small m values [36].
The m(T g ) value was determined at the glass transition temperature (T g ) where the relaxation time is equal to 100 s.
The a-relaxation observed for the 0CN polymer was thus obtained from the fitting with Eq. 3 shown in Fig. 5. The obtained fitting parameters are given in Table 2.
The results shown in Table 2 indicate that the cooperative motions of the amorphous chains (s 0 value) are very slow and with high VFTH energy. The glass transition temperature determined through this formulation is very similar to the value determined by DSC technique (T g = 176°C) [27]. The fragility parameter for the 0CN polymer is equal to the one observed for the semicrystalline polymer a-PVDF [37]. T 0 is typically defined as the real glass transition temperature, which is found to be 30-70°C below the measured T g . For 0CN polymer, T 0 is below T g and the difference of both temperatures is 56°C [32]. For   2CN polymer and the respective copolymers, the a-relaxation is not detected due to the overlapping of the electric conductivity contributions, particularly high at high temperatures and low frequencies.

Frequency dependence
In the frequency domain representation (Fig. 6), e 00 versus frequency at isothermal conditions is used for the identification of the dipolar relaxations. Amorphous polymers show different motional processes visible the e 00 versus m representation as peaks (Fig. 6). Figure 6 shows the imaginary part of the dielectric function for the different samples. For the 0CN polymer (Fig. 6a), the a-relaxation is clearly identified. A detailed representation of this relaxation is represented in Fig. 6b (above 195°C), where it is observed that the a-relaxation is shifted to higher frequencies as temperature increases. This relaxation is related to a long timescale and corresponds to an overall structure rearrangement of the system [38].
For the 0CN polymer (Fig. 6a) and for low frequencies and temperatures below 195°C, the conductivity contribution in the frequency domain is prominent, being this contribution the main one for the rest of the samples (Fig. 6c, d). The a-relaxation is not resolved in the copolymers and the 2CN polymer due to the overlapping of dielectric relaxation and conductivity contributions. The conductivity contribution decreases with decreasing temperature, being larger for low frequencies. Further, it can be stated an increase in the conductivity associated with the addition of the nitrile (-CN) dipolar group.
As it will also be shown later, the conductivity effect contributing to the increase in the dielectric loss in all samples (Fig. 6) is also influenced by the Maxwell-Wagner-Sillars process, i.e., the charges blocked at internal phase boundaries, related to the different constitutes of the copolymers, submicropores or at the interface between the sample and the electrodes (electrode or space charge polarization) [39].

Overall conductivity
In order to further explain relaxation dynamics of charge carriers and the characteristics of the ionic conduction with certain influence of a MWS process, it is more appropriate the representation of frequency and temperature dependence of complex conductivity, r Ã , which can be calculated from the dielectric function through the following equation: Thus, the real part of the conductivity is given by: and the imaginary part of the conductivity is given by: where e 0 (8.85 9 10 -12 F m -1 ) is the permittivity of free space and x ¼ 2pf is the angular frequency. The real and imaginary parts on the complex electrical conductivity, r Ã ðxÞ, are plotted in Fig. 7 as a function of frequency for several temperatures.
For all samples, it is observed two well-identified regimes that depend on the temperature and frequency (Fig. 7). For low temperatures up to T = 135°C, the conductivity increases rapidly with increasing frequency. For temperatures above T = 185°C, it is observed a plateau, in which the conductivity (Fig. 7a, c, e, g) is nearly independent of frequency in the low-frequency region, being this regime thus dominated by the dc conductivity.   At higher frequencies and temperatures, a power law dependence of the ac electrical conductivity is observed, which is related to hopping transport of localized charge carriers [40]. The electrical conductivity is larger for the samples with larger CN content. There are also found deviations from a pure plateau region in the low-frequency conductivity spectrum (Fig. 7) that can be attributed to electrode polarization effects and to the charge blocking at the interphase regions as well as the imperfections (i.e., submicropores due to the solvent evaporation process) that occur due to the polymer processing method.
This behavior can be thus ascribed to contribution of MWS and, at still higher frequencies, to dipolar relaxation modes. It is evident that 2CN polymer (Fig. 7e) and the copolymers (Fig. 7c) show higher conductivity due to the presence of the nitrile groups in the polymer chain in comparison with the 0CN polymer (Fig. 7a).
The presence of the nitrile group affects also the MWS relaxation that is related to the existence and blocking of free charge carriers inside the system. The MWS relaxation becomes increasingly important at high temperatures, reflecting the enhancement of the mobility of charge carriers.
The ac conductivity at constant temperature as a function of frequency can be expressed as [41]: where r dc is the DC conductivity, A is the pre-exponential factor and n is the fractional exponent (0 \ n \ 1) that depends on the temperature. The value of n is used to a better understanding of the conduction or relaxation mechanism in insulating materials [42]. Figure 8, left, shows that r dc increases with increasing temperature in a similar way for all samples. This behavior reflects the mechanism of charge transport of carriers. There are not significant differences in values or behavior for the two copolymers, the values of the conductivity being intermediate between the ones obtained for the pure polymers. With respect to the two homopolymers, 2CN polymer shows higher r dc than 0CN polymer in all temperature range, attributed to the presence of the nitrile group.
In relation to the n parameter, it decreases with increasing temperature until 227°C for all polyimide films. Independently of the sample type, the n parameter ranges between 0.6 and 0.9, as corresponds to a hopping conductivity process (Fig. 8, right) [43].
In this way, the transport mechanism of the samples is thermally activated hopping across an energy barrier. For Dielectric relaxation dynamics of high-temperature piezoelectric polyimide copolymers 737 the evaluation of the dependence of the ac conductivity with temperature at different frequencies, the Dyre model [44] (random free energy barrier model or symmetric model) was applied [45]: where B is a pre-exponential factor identified as the attempt frequency, E a is the activation energy of the process, T is temperature and k B is the Boltzmann constant.
The application of the Dyre model in order to obtain the dependence of the electrical conductivity on the reciprocal temperature at different frequencies is shown in Fig. 9, and the activation energy, E a , calculated is plotted in Fig. 10. Figure 9 shows the ac conductivity for the 0CN polymer (Fig. 9a), with noteworthy differences from the remaining polymers in both temperature dependence behavior (Fig. 9b, c) and activation energy (Fig. 10).   It can be observed that for the copolymers samples and the 2CN polymer, the activation energy is larger (with the exception of the higher frequencies) and decreases with increasing frequencies. This fact is linked to the mobility of nitrile groups, determining the electrical response of these samples. The activation energy for 0CN shows a strong decrease for low frequencies with respect to the other samples, increasing for frequencies above 10 Hz. This variation can be ascribed the presence of free charges contributing to the MWS relaxation processes, as shown in the next section.

Electric modulus formalism
In order to further explore the conduction behavior (ionic conductivity and interfacial polarization, the so-called Maxwell-Wagner-Sillars (MWS) effect), the electric modulus formalism, M Ã [46], is the most appropriate representation and can be used for differentiate dielectric relaxation processes from long-range conductivity [47].
Typically, for samples with higher conductivity, one high peak related to the conductivity is detected in the frequency spectra of the imaginary electric modulus formalism, M 00 in the same temperature interval where the conductivity process is shown in the imaginary part of the dielectric function (e 00 ). The electrical modulus, M Ã , is defined as the reciprocal of the complex relative permittivity, e Ã x ð Þ: where M 0 and M 00 are the real and imaginary components of the complex electric modulus, respectively. For a dipolar relaxation process, Eq. (10) yields a relaxation peak in M 00 and a drop of M 0 with decreasing frequency. On the other hand, for a pure dc conduction process in which e 0 is independent of frequency and e 00 is inversely proportional to frequency, Eq. (11) yields the same frequency dependence than in a Debye single relaxation time process. The imaginary component of dielectric modulus can be used to define a relaxation time for conductivity MWS processes also yield narrow relaxation processes in electric modulus formalism while presenting a clear increase in e 0 with decreasing frequency.
Deviations with respect to the Debye single relaxation time model in the electric field relaxation due to motions of charge carriers are generally described by the empirical Kohlrausch-Williams-Watts (KWW) function [48,49]: where s M and b are the conductivity relaxation time and the Kohlrausch exponent, respectively. The smaller the value of b, the larger the deviation of the relaxation with respect to a Debye-type relaxation (b = 1). Values of the Kohlrausch exponential parameter b in the KWW function were approximately estimated by means of [50]: where w is the peak's full width at half maximum. The temperature dependence of the real M 0 and imaginary M 00 electric modulus for various frequencies is shown in Fig. 11 for all samples.
The M Ã spectra of the 0CN sample clearly show the space charge mechanism at low frequencies appearing as a step in M 0 and a narrow peak in M 00 (the value of the Kohlrausch parameter calculated for this peak using Eq. 14 is b = 0.9) followed by the dipolar relaxation process appearing as a second step in M 0 and a broad peak in M 00 (Fig. 11a, b). The relaxation time for conductivity s Cond is related to the frequency at which the peak in M 00 appears, f max , (Eq. 12) through s Cond = 1/2pf max . The temperature dependence of f max is represented in Fig. 5.
In the 2CN samples, space charge and dipolar relaxation are closer to each other and partially overlap. In addition, the low-frequency M 00 peak due to free charge motions is broader than in the 0CN samples (Kohlrausch parameter calculated for this peak using Eq. 14 is b = 0.60 at 155°C). The peak of this process appears at lower temperatures than in the 0CN sample and shows a smaller temperature variation indicating that lower energy barriers hinder the motions of space charge carriers in the 2CN sample. The presence of the high-frequency relaxation can be noticed above 175°C by the loss of the symmetry of the conductivity peak by the lifting of its high-frequency side. At higher temperatures, the shape of the M 00 plot clearly shows the overlapping of two processes. Some indications of the presence of this highfrequency peak could be observed in the permittivity plot in Fig. 6e. Interestingly, the drop of the real part of permittivity that takes place in the 2CN sample in the temperature interval between 225 and 250°C that was ascribed to the imidization of dianhydride groups [28] is not reflected in the dielectric modulus formalism. The behavior of the copolymer is in between that of the two components. The space charge M 00 peak is even broader than in 2CN sample and Kohlrausch parameter is only b = 0.5. The presence of the high-frequency dipolar relaxation is now clear above 200°C, and the sharp decrease in e 0 due to the imidization of dianhydride groups has not a counterpart in M 0 . Instead, for the copolymers and 2CN polymer (Fig. 11b-d), two peaks appear above 205°C, which shift to higher frequency with increasing temperature, both samples showing similar relaxation process behavior. The peak observed for all temperatures at low frequencies represents the movement of charge carriers over a long distance, i.e., charge carriers can perform successful hopping from one site to a neighboring site [51]. For higher frequencies, the observed peak indicates the presence of charges confined to local motions [47].

Conclusions
Thin films of polyimide polymers have been produced by polycondensation reaction followed by thermal cyclodehydration, based on different diamines as comonomers: a diamine without CN groups and a novel synthesized diamine with two CN groups. The dielectric properties of the samples are determined by the presence of the CN groups. The dielectric constant, e 0 , increases with increasing temperature for all samples. The a-relaxation is only detected for 0CN polymer being overlapped by the large conductivity contributions in the remaining samples. The conductivity behavior reflects the mechanism of charge transport behavior. For copolymers and 2CN polymer at high temperature, the spectrum is dominated by ionic conductivity and interfacial polarization. The differences observed for the activation energy between the samples in the conductivity process are attributed to the presence of the MWS relaxation process. The presence of the CN group in the copolymers and 2CN polymer increases the MWS contribution as detected in the electric modulus formalism plots.