Dielectric relaxation of thin films of polyamide random copolymers

We investigate the relaxation behavior of thin films of a polyamide random copolymer, PA66/6I, with various film thicknesses using dielectric relaxation spectroscopy. Two dielectric signals are observed at high temperatures, the $\alpha$ process and the relaxation process due to electrode polarization (the EP process). The relaxation time of the EP process has a Vogel-Fulcher-Tammann type of temperature dependence, and the glass transition temperature, $T_{\mathrm{g}}$, evaluated from the EP process agrees very well with the $T_{\mathrm{g}}$ determined from the thermal measurements. The fragility index derived from the EP process increases with decreasing film thickness. The relaxation time and the dielectric relaxation strength of the EP process are described by a linear function of the film thickness $d$ for large values of $d$, which can be regarded as experimental evidence for the validity of attributing the observed signal to the EP process. Furthermore, there is distinct deviation from this linear law for thicknesses smaller than a critical value. This deviation observed in thinner films is associated with an increase in the mobility and/or diffusion constant of the charge carriers responsible for the EP process. The $\alpha$ process is located in a higher-frequency region than the EP process at high temperatures but merges with the EP process at lower temperatures near the glass transition region. The thickness dependence of the relaxation time of the $\alpha$ process is different from that of the EP process. This suggests that there is decoupling between the segmental motion of the polymers and the translational motion of the charge carriers in confinement.

Figure 1

The chemical formula of the amorphous polyamide copolymer PA66/6I.

Figure 2

Distribution of the charge between two electrodes under an applied electric field, with switching of the macroscopic dipole moment by reversal of the electric field. This figure has been modified from the original by Coelho.

Figure 3

X-ray scattering intensity with $q$ ranging from 0.1 to 38 ${\text{nm}}^{-1}$ at room temperature for the amorphous polyamide copolymer PA66/6I. The top figure is for the SAXS region, and the bottom one is for the WAXS region.

Figure 4

The dependence of the complex dielectric permittivity on the temperature at various frequencies, for films of the amorphous polyamide copolymer PA66/6I with a thickness of 680 nm. (a) The dielectric loss ${\varepsilon }^{\prime \prime }$; (b) the real part ${\varepsilon }^{\prime }$; and (inset) a magnified image of ${\varepsilon }^{\prime }$.

Figure 5

The dependence of the real and imaginary parts of the complex dielectric permittivity on the frequency at 400 K for various applied voltages from 0.1 to 0.5 V in thin films of the amorphous polyamide copolymer PA66/6I with a 40-nm thickness.

Figure 6

The dependence of the complex dielectric permittivity on the frequency for thin films of the amorphous polyamide copolymer PA66/6I with a thickness of 556 nm. (a) The real part of ${\varepsilon }^{*}$ and (b) the imaginary par of ${\varepsilon }^{*}$. The temperature ranges from 422.8 K (right) to 369.5 K (left).

Figure 7

The dependence of the logarithm of the real and imaginary parts of complex conductivity observed at 422 K for thin films of PA66/6I with a 556-nm thickness. The arrow shows the onset of the EP process.

Figure 8

The dependence of the real and imaginary parts of the complex dielectric permittivity on the frequency at various temperatures—(a) 422.8 K, (b) 413.2 K, (c) 403.5 K, and (d) 393.8 K—for thin films of the amorphous polyamide copolymer PA66/6I with a 556-nm thickness showing (red squares) the imaginary part ${\varepsilon }^{\prime \prime }$ and (green circles) the real part ${\varepsilon }^{\prime }$. The three different contributions to the imaginary part are also shown: The orange dashed-dotted line corresponds to the low-frequency component ${\varepsilon }_{\sigma }^{*}$, the blue dashed line corresponds to the EP process, and the pink dotted line corresponds to the $\alpha$ process. The curves for the three components and the two observed values of ${\varepsilon }^{\prime }$ and ${\varepsilon }^{\prime \prime }$ were evaluated by fitting the observed values to Eq. (18).

Figure 9

The dependence of the complex dielectric permittivity on the frequency for thin films of the amorphous polyamide copolymer PA66/6I with a thickness of 56 nm. (a) The real part of ${\varepsilon }^{*}$ and (b) the imaginary part of ${\varepsilon }^{*}$. The temperature ranges from 417.6 K (right) to 368.7 K (left).

Figure 10

The dependence of the real and imaginary parts of the complex dielectric permittivity on the frequency at various temperatures—(a) 417.6 K, (b) 407.8 K, (c) 397.9 K, and (d) 388.2 K—for thin films of the amorphous polyamide copolymer PA66/6I with a 56-nm thickness. The meanings of the symbols and curves are identical to those in Fig. 8.

Figure 11

Dispersion map of the relaxation rate $f_{\mathrm{ep}}$ of the EP process and the relaxation rate $f_{\alpha }$ of the $\alpha$ process, for thin films of the amorphous polyamide copolymer PA66/6I with various thicknesses ranging from 20 to 556 nm. The values of $f_{\mathrm{ep}}$ and $f_{\alpha }$ are evaluated as $f_{\mathrm{ep}}={\left(2\pi {\tau }_{\mathrm{ep}}\right)}^{-1}$ and $f_{\alpha }={\left(2\pi {\tau }_{\alpha }\right)}^{-1}$, where ${\tau }_{\mathrm{ep}}$ and ${\tau }_{\alpha }$ are the fitting parameters obtained by fitting the observed ${\varepsilon }^{*}$ to Eq. (18). The curves are governed by the VFT law. The curves of the $\alpha$ process were obtained under the condition that the Vogel temperature $T_0$ was fixed at the value obtained by fitting the temperature dependence of ${\tau }_{\mathrm{ep}}^{-1}$ to Eq. (21).

Figure 12

The dependence of the Vogel temperature $T_0$ () and the glass transition temperature $T_{\mathrm{g}}$ () on the film thickness, as determined by the temperature dependence of the relaxation time ${\tau }_{\mathrm{ep}}$ of the EP process for thin films of the amorphous polyamide copolymer PA66/6I. Here, $T_{\mathrm{g}}$ is defined such that ${\tau }_{\mathrm{ep}}\left(T_{\mathrm{g}}\right)={10}^3$ s. The value of $T_{\mathrm{g}}$ as evaluated by DSC measurements is also shown.

Figure 13

The dependence of the apparent fragility index $m$ on the film thickness, based on the temperature dependence of the relaxation time of the electrode polarization process. Here $T_{\mathrm{g}}$ is defined as the temperature at which the relaxation time ${\tau }_{\mathrm{ep}}$ is equal to ${10}^3$ s.

Figure 14

The dependence of the dielectric relaxation strength $\mathrm{\Delta }\varepsilon$ at 403 K on the film thickness for the EP process () and the $\alpha$ process () in thin films of the amorphous polyamide copolymer PA66/6I with various thicknesses. The slope of the straight line for the EP process is equal to unity and that for the $\alpha$ process is almost equal to zero.

Figure 15

The dependence of (a) the relaxation rate $f_{\mathrm{ep}}$ of the EP process and (b) the relaxation rate $f_{\alpha }$ of the $\alpha$ process on the film thickness—at 413 K, 393 K, and 373 K—for thin films of PA66/6I with various thicknesses ranging from 556 to 20 nm. The straight lines in (a) were obtained by fitting for three values of the thin films of PA66/6I with a thickness larger than 99 nm. The slopes are fixed to be $-1$.

Figure 16

The dependence of the charge carrier mobility $\mu$ on the inverse of the absolute temperature $1/T$ for thin films of the amorphous polyamide copolymer PA66/6I with various thicknesses ranging from 556 to 40 nm. The curves were obtained by fitting the observed values to the VFT equation for the mobility, Eq. (15). In the inset, the dependence of the equilibrium concentration of charge carriers $n_0$ on $1/T$ is also given for the same thin films of the amorphous polyamide copolymer PA66/6I.

Figure 17

The dependence of the diffusion constant of the charge carriers $D$ on the inverse of the absolute temperature $1/T$ for thin films of the amorphous polyamide copolymer PA66/6I with various thicknesses ranging from 556 to 40 nm.