Dielectric properties of polymethacrylate-grafted carbon nanotube composites

Abstract

The dielectric properties of a series of multi-walled carbon nanotubes on which poly(cyclohexyl methacrylate)s were densely grafted (PCHMA-CNTs) were systematically characterized over the broad frequency range of 10⁻²–10⁸ Hz. PCHMA-CNT was used as an ideal polymer/CNT nanocomposite system in which the CNT is homogeneously dispersed and the polymer/CNT interface is strong. When the volume fraction of the CNT Φ_CNT < 0.1 PCHMA-CNT exhibited a relatively high dielectric constant ε′_r and a quite low dielectric loss factor ε′′_r, while when Φ_CNT > 0.1 a very large ε′_r was observed at low frequencies. The Cole–Cole relaxation model was applied to the impedance spectra of PCHMA-CNT in order to investigate the correlation between the tunneling conduction and the very large ε′_r. Also, from thus-obtained relaxation frequencies, the inter-nanotube distance in the PCHMA-CNT system was estimated. As a result, a new mechanism was proposed for the very large ε′_r: the locally enhanced electric field by the tunneling conduction results in the very large ε′_r at the low frequencies along with the increase in ε′′_r. Furthermore, the CNT was demonstrated to be five times more effective than barium titanate (BT) for the enhancement in ε′_r of polymer materials because of the higher electric field enhancement effect of the CNT than that of the BT.

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Introduction

Dielectric materials are mainly used to store electrical energy as capacitors nowadays.¹ Among the dielectric materials, polymer materials have the advantages of high flexibility, lightness, durability, and processability. Highly flexible dielectrics are also promising for other applications such as dielectric actuators and sensors.^2–6 However, the low dielectric constant ε′_r of the polymer materials strongly limits the extent of the applications. Because of the high ε′_r of ceramic materials, therefore, polymer/ceramic composite materials have been much developed over the last few decades.^7–11 Barium titanate (BT), one of the ferroelectric metal oxides, is most widely used for enhancing ε′_r. For practical use, however, over 50 vol% of the BT is often required to be added despite the high ε′_r of the BT. Such high BT loading spoils the superior properties of polymeric materials described above.

Recently, polymer/carbon nanotube (CNT) nanocomposites have received much attention as dielectric materials.^12–23 A very large ε′_r was often observed at low frequencies with only a small percentage volume of CNT loading.^12–21 According to the previous studies, it is considered that the very large ε′_r is attributed to space charge polarization and/or interface polarization between the CNT and the polymer matrix.^{13,14,17,18,23} However, almost no evidence of such polarizations has been reported. Because the dielectric behavior of the polymer/CNT nanocomposites strongly depends on the dispersivity of the CNT and/or the interface interaction between the CNT and the polymer matrix, fundamental understanding of the mechanism of the increase in ε′_r remains unclear. This is a serious issue that needs to be overcome for material design of the polymer/CNT nanocomposites. We suggest that the ideal polymer/CNT nanocomposite in which the CNT is homogeneously dispersed and the polymer/CNT interface is strong should give a much better understanding of the very large ε′_r.

The second issue in the polymer/CNT nanocomposites is a great increase in dielectric loss factor ε′′_r along with the increase in ε′_r.^12–21 The increase in ε′′_r mainly results from a leak current passing through the nanocomposites. The polymer/CNT nanocomposites become conductive owing to the formation of a three-dimensional conductive network of the CNT within the polymer matrix when the CNT content exceeds a critical value, known as a percolation threshold.^24–26 At over the percolation threshold, the CNTs approach each other within several nanometers, allowing efficient electron hopping between the nanotubes (tunneling conduction).^15,27–29 The tunneling conduction causing the leak current cannot be avoided even if the CNT is covered with an organic layer having a thickness of a few nanometers.^13,19,29 To impede the tunneling current, Nan et al. covered CNTs with a thick polymer shell. The physically adsorbed polymer chain on the CNT brings a low ε′′_r as well as a relatively high ε′_r and breakdown strength.^22,23

Another simple method to perfectly prevent the tunneling conduction of the polymer/CNT nanocomposites is polymer grafting on the surface of the CNT.^29–33 Unlike a physically adsorbed polymer chain, the covalently attached polymer chain would be more effective in the insulation of the CNT. We have recently developed a surface-initiated atom-transfer radical polymerization (ATRP) method using an originally designed ATRP initiator, p-(bromomethyl)benzyl 2-bromoisobutylate.^34,35 The ATRP initiator moiety was covalently attached on the surface of the CNT (CNT-Br), and high molecular weight poly(cyclohexyl methacrylate)s were densely grafted on the CNT-Br (PCHMA-CNTs).²⁹ It was characteristic that all the polymer chains were tethered on the CNT in the PCHMA-CNT system, and the grafted polymer chains served as a matrix, implying that the CNT should be homogeneously dispersed in the polymer matrix. The electrical resistance of PCHMA-CNT under direct current (DC) was compared to that of two types of conventional nanocomposites prepared by blending PCHMA with the CNT or CNT-Br (PCHMA/CNT and PCHMA/CNT-Br). At a comparable volume fraction of the CNT, DC volume resistivity of PCHMA-CNT was 14 orders of magnitude higher than that of PCHMA/CNT. This is because the grafted polymer with a combination of high molecular weight and high grafting density isolates individual CNTs at a long distance in the PCHMA-CNT system. In addition, it was indicated that PCHMA-CNT had a strong polymer/CNT interface according to dynamic mechanical analysis.

We believe our PCHMA-CNT system is the best candidate of the ideal polymer/CNT nanocomposite as mentioned above, and the most suitable for the characterization of the dielectric behavior. In this study, therefore, the dielectric properties of PCHMA-CNT are systematically characterized over the broad frequency range of 10⁻²–10⁸, and compared to that of PCHMA/CNT-Br. The Cole–Cole relaxation model is applied to the impedance spectra of the composites in order to obtain information of the tunneling conduction. From the information, the inter-nanotube distance is also estimated. As a result, a new mechanism is proposed for the very large ε′_r. Furthermore, the performance of PCHMA-CNT as a dielectric material is evaluated by comparing with that of a BT-filled PCHMA composite (PCHMA/BT).

Results and discussion

A series of PCHMA-CNTs with various molecular weights from 60 000 to 190 000 are listed in the ESI.† Conventional nanocomposites (PCHMA/CNT and PCHMA/CNT-Br) were designed to have the same volume fraction of CNT, Φ_CNT, as PCHMA-CNT. In the blending procedure, a preliminary blending method using freeze-drying was used in order to obtain as good dispersion of the CNTs as possible.³⁵ These nanocomposite samples were molded by compression of 10 MPa at 130 °C prior to electrical measurements. The dispersivity of the CNT in the polymer matrix for the three types of nanocomposites thus produced was observed using a scanning electron microscope (SEM). In this observation, the surfaces of the nanocomposites had been slightly etched with oxygen plasma in order that the PCHMA polymer on the surface was removed. Fig. 1 shows SEM images of the nanocomposites with Φ_CNT = 0.055. For PCHMA/CNT and PCHMA/CNT-Br, the CNTs are inhomogeneously dispersed at a submicron scale. This inhomogeneity is typical of polymer/CNT composites prepared by blending, owing to the poor polymer/CNT compatibility. In contrast, the dispersivity of the CNT is surprisingly good for PCHMA-CNT. In the PCHMA-CNT system, the PCHMA matrix cannot be separated from the CNT because all the PCHMA chains are tethered on the CNT. The excellent CNT dispersivity for PCHMA-CNT is also confirmed by an SEM image with low magnification (ESI†). We suggest polymer-grafting should be the best method of producing the perfectly homogeneous dispersion of CNT in a polymer matrix.

Fig. 1 SEM images of three types of nanocomposites with Φ_CNT = 0.055: (a) PCHMA/CNT, (b) PCHMA/CNT-Br, (c) PCHMA-CNT.

Electrical impedance Z depends on the frequency f of an applied electric field. The magnitude of impedance |Z| for the three types of nanocomposites is plotted as a function of f in the range of 10²–10⁶ Hz in Fig. 2. PCHMA/CNT with Φ_CNT = 0.055 shows very low |Z|, independent of f over the frequency range. This is typical behavior of conductive polymer/CNT nanocomposites. Due to the coating of the CNT with the organic thin layer (≈2 nm), PCHMA/CNT-Br has much higher |Z| than PCHMA/CNT at Φ_CNT = 0.055. However, |Z| of PCHMA/CNT-Br is rapidly decreased with the increase in Φ_CNT, indicating that such a thin layer on the CNT in PCHMA/CNT-Br easily allows the tunneling conduction.²⁹ At Φ_CNT = 0.094, no frequency dependence of |Z|, conductor behavior, is observed in the low frequency range of 10²–10³ Hz. In comparison at the same Φ_CNT, |Z| of PCHMA-CNT is higher than that of PCHMA/CNT-Br, because the CNT is covered with the thick polymer shells in the PCHMA-CNT system. At Φ_CNT = 0.055 and 0.094, |Z| of PCHMA-CNT linearly increased with the decrease in f in all of the frequency range, similar to that of PCHMA. On the other hand, weak frequency dependence of |Z| is observed for PCHMA-CNT with Φ_CNT = 0.14 at below 10⁴ Hz. Further characterization of these electrical behaviors was carried out by the introduction of the real and imaginary parts of Z. Because a material can be simply modeled as the electric circuit composed of a resistor and a capacitor connected in parallel, we can divide Z into the resistance R_p and the capacitive reactance X_cp as:

1/Z = 1/R_p + 1/(jX_cp); −X_cp = 1/(2πfC_p), (1)

where j is the imaginary unit and C_p is the capacitance. The two components, R_p and −X_cp for PCHMA/CNT-Br and PCHMA-CNT are shown in Fig. 3. Notice that X_cp has a minus sign. The corresponding spectra of PCHMA/CNT cannot be exhibited because the reactance component for PCHMA/CNT was too small to be measured precisely. For PCHMA-CNT with Φ_CNT = 0.094, −X_cp is much smaller than R_p in all the frequency range, which means almost all electric current is passed through the capacitor in the parallel resistor-capacitor (RC) circuit. That is, the PCHMA-CNT composite is a dielectric. In addition, −X_cp of the composite has a relationship described by −X_cp ∝ f⁻¹ because C_p is almost constant. Similar dielectric behavior is also observed for PCHMA and PCHMA-CNT with Φ_CNT = 0.055. Unlike these PCHMA-CNT composites, PCHMA-CNT with Φ_CNT = 0.14 has a transition from a dielectric (R_p > −X_cp) to a conductor (R_p < −X_cp) at ∼10³ Hz. This is because R_p at Φ_CNT = 0.14 is small especially at the low frequencies due to the tunneling conduction. In our previous study, the dominant tunneling conduction was clarified for PCHMA-CNT with Φ_CNT over 0.1.²⁹

Fig. 2 Frequency f dependence of magnitude of impedance |Z| of PCHMA (diamond) and three types of nanocomposites with Φ_CNT = 0.055 (circles), Φ_CNT = 0.094 (triangles), and Φ_CNT = 0.14 (squares). White symbols, PCHMA/CNT; gray symbols, PCHMA/CNT-Br; black symbols, PCHMA-CNT.

Fig. 3 f dependence of (a) resistance R_p and (b) capacitive reactance X_cp of PCHMA (diamond) and two types of nanocomposites with Φ_CNT = 0.055 (circles), Φ_CNT = 0.094 (triangles), and Φ_CNT = 0.14 (squares). Gray symbols, PCHMA/CNT-Br; black symbols, PCHMA-CNT.

The transition from a dielectric to a conductor results from a Z relaxation related to the tunneling conduction that occurs at a microcapacitor formed between two CNT electrodes.¹⁵ To characterize the Z relaxation in detail, we again assume the parallel RC circuit with a resistance R and a capacitance C. Z is expressed using the Cole–Cole model as:³⁶

Z = R_∞ + R/{1+(jω/ω₀)^β}; ω = 2πf, (2)

where R_∞ is the resistance at infinite frequency, and ω₀ is the relaxation frequency of the RC circuit. The exponent parameter β (0 < β ≤ 1) indicates stretching of the relaxation. When β = 1, eqn (2) denotes a single relaxation, the Debye relaxation. Table 1 summarizes parameters obtained by fitting eqn (2) to the experimental results. Examples of the Cole–Cole plots are shown in the ESI.† The fitting was not applicable to PCHMA-CNT with Φ_CNT below 0.1, because ω₀ was too low to be observed. At Φ_CNT = 0.14, PCHMA-CNT has low ω₀ than PCHMA/CNT-Br, resulting from the difference of magnitude of R. Assuming that the relaxation of Z is the Debye relaxation, the relationship ω₀ = 1/(RC) is concluded. Therefore, we can estimate the average value of the internanotube distance d of the microcapacitor with a tunneling resistance R_t and a capacitance C_μ using the following equation:^15,37where e is the charge of the electron, ħ is the Planck constant, ε₀ is the dielectric constant of vacuum, ε′_r is the dielectric constant of a matrix polymer, and m_e is the electron mass. k₀ describes the potential barrier with a potential height of V against the electron. The average internanotube distance d̄ calculated using the potential height of 0.2 eV³⁸ and ε′_r of 2.7 is plotted in Fig. 4. d̄ decreases with the increase in Φ_CNT (Fig. 4a), which is reasonable and consistent with a previous result that has been reported by Klüppel et al.¹⁵ It is more important that R is exponentially increased with the increase in d̄ at such nanometer scale (Fig. 4b), which is well-known as a quantum effect. This fact means that if many microcapacitors with different d are connected in a series like a real composite system, the applied voltage is concentrated at some of the microcapacitors with the longest d; in a series circuit, the applied voltage is distributed to the microcapacitors in proportion to their tunneling resistance R_t. This locally amplified electric field would give an apparent increase in the total capacitance of the composite. In the case of the corresponding macroscale capacitors, the electric field is not amplified because the resistance is proportional to the gap width between the electrodes.

Table 1 Parameters of the Cole–Cole relaxation equation for two series of PCHMA/CNT-Br and PCHMA-CNT

Sample	Φ CNT	ω 0 [rad s^−1]	R [Ω m]	β
PCHMA/CNT-Br	0.020	8.2 × 10^−1	1.9 × 10^9	0.69
	0.055	6.6 × 10^1	1.2 × 10^7	0.72
	0.094	1.3 × 10^4	3.3 × 10^4	0.70
	0.143	2.0 × 10^6	1.8 × 10^2	0.72
PCHMA-CNT	0.112	8.2 × 10^−2	7.2 × 10^9	0.72
	0.124	6.3 × 10^0	1.1 × 10^8	0.71
	0.143	1.2 × 10^3	6.4 × 10^5	0.68

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Fig. 4 Relationships of the average internanotube distance d̄ for two types of nanocomposites to (a) Φ_CNT and (b) R. Gray symbols, PCHMA/CNT-Br; black symbols, PCHMA-CNT.

The apparent enhanced capacitance implies an increase in ε′_r. Fig. 5 shows frequency dependence of ε′_r and ε′′_r of PCHMA/CNT-Br and PCHMA-CNT in the wide range of 10⁻²–10⁸ Hz. For PCHMA/CNT-Br, ε′_r and ε′′_r are significantly increased with the decrease in f. These behaviors of ε′_r and ε′′_r at the lower frequencies than ω₀ were consistent with the theoretical curves calculated from eqn (2).

ε′_r = sin(πβ/2)/Rε₀ω₀^βω^1−β + ε′_r∞, (4)

ε′′_r = 1/Rε₀ω + cos(πβ/2)/Rε₀ω₀^βω^1−β, (5)

where ε′_r∞ is the dielectric constant at infinite frequency. At the higher frequencies than ω₀, however, eqn (4) and (5) were not consistent with the experimental spectra (ESI†). This could be attributed to negligible tunneling conduction, because −X_cp of the polymer bulk between the CNTs is lower than R_t at the higher frequencies than ω₀. In other words, the electric current is mainly passed through the polymer matrix, not through the tunneling conduction.

Fig. 5 f dependence of (a) dielectric constant ε′_r and (b) dielectric loss factor ε′′_r of PCHMA (diamonds) and two types of nanocomposites with Φ_CNT = 0.055 (circles), Φ_CNT = 0.094 (triangles), and Φ_CNT = 0.14 (squares). Gray symbols, PCHMA/CNT-Br; black symbols, PCHMA-CNT.

From the obtained results mentioned above, the relaxation related to the tunneling conduction at the microcapacitor is understood as follows. At the much higher frequencies than ω₀, Z of the polymer bulk determines the overall Z of the composite because −X_cp of the polymer matrix between the CNTs is much lower than R_t. Therefore, the tunneling conduction can be ignored. As the −X_cp is increased with the decrease in f, the −X_cp becomes equal to R_t. The microcapacitors with the smaller d (with the smaller R_t) are electrically connected by the tunneling conduction. These short circuits give the system the apparent enhanced capacitance because the electric field becomes concentrated only at the residual microcapacitors with longer d. Finally, at the much lower frequencies than ω₀, R_t of the microcapacitor with the longest d dominates R_p, resulting in no frequency dependence of R_p (= R). In many previous studies on polymer/CNT nanocomposites, the very large ε′_r at low frequencies has been attributed to space charge polarization and/or interface polarization between the CNT and the polymer matrix.^{13,14,17,18,23} Therefore, it has also been suggested that the interaction at the polymer/CNT interface and the dispersivity of the CNT in the polymer matrix are important for the enhancement in ε′_r.^{13,16,18,19,23} However, the discrepant phenomena have been found for our PCHMA-CNT system. ε′_r of PCHMA-CNT with Φ_CNT = 0.055 is almost independent of f even at the low frequency although the dispersivity of the CNT is surprisingly good (Fig. 1c). Furthermore, the polymer/CNT interface should be strong because of polymer grafting. On the other hand, PCHMA-CNT with Φ_CNT = 0.14, where the tunneling conduction is observed, exhibits very large ε′_r at the low frequencies, as previously reported. As a result, we suggest that the increase in ε′_r should result from not the interface polarization and/or the space charge polarization but the apparently enhanced capacitance by locally amplified electric field mentioned above. This explanation is consistent with no increase in ε′_r at the low frequencies for PCHMA-CNT with Φ_CNT = 0.055. Also, this electric field enhancement effect can account for the increase in ε′_r∞ of the polymer/CNT composites relative to the neat polymer (ESI†). In this case, the electric field enhancement originates from the heterogeneity of the polymer/CNT composite regardless of the tunneling conduction. In other words, a large electric field fluctuation originally exists in the polymer/CNT composites. The field fluctuations in composite materials have been demonstrated using simulated methods.^39,40

For the PCHMA/CNT-Br system, R_p is usually much smaller than −X_cp, suggesting that the microcapacitor is a resistor rather than a capacitor, i.e., it is a conductor. In contrast, PCHMA-CNTs with Φ_CNT = 0.055 and 0.094 have very low ε′′_r in all the frequency range (Fig. 5b). This is because d is large enough to prevent the tunneling conduction. ε′_r and the dissipation factor (tan δ = ε′′_r/ε′_r) of PCHMA/CNT-Br and PCHMA-CNT at 1 MHz are plotted in Fig. 6. For PCHMA-CNT, tan δ is below 0.01 when Φ_CNT is less than 0.1. At Φ_CNT = 0.094, ε′_r of 22.5 and tan δ of 0.010 are obtained. This tan δ value is small even compared with low-loss polymer/CNT composites that have been reported.^19–23 The dielectric property of the low-loss PCHMA-CNT is comparable to that of PCHMA filled with barium titanate (PCHMA/BT). In Fig. 7a, the plots of log ε′_rversus Φ_CNT and Φ_BT show linear relationships for both the composites. The slope of the relationship for PCHMA-CNT is five times larger than those for PCHMA/BT, implying the CNT has higher electric field enhancement effect than the BT owing to the high aspect ratio of the CNT.³⁹ This finding is a major advantage of PCHMA-CNT over PCHMA/BT for polymer material design. According to Fig. 7a, if ε′_r of PCHMA-CNT is enhanced by about ten times, it can be achieved with only 10 vol% of the CNT. In addition, tan δ of PCHMA-CNT is as low as that of PCHMA/BT (Fig. 7b), and PCHMA-CNT has smaller frequency dependence of the dielectric properties than PCHMA/BT (Fig. 7c). For PCHMA/BT, due to the dielectric relaxation related to the BT particle, tan δ is much increased at the low frequencies.

Fig. 6 (a) ε′_r and (b) dissipation factor (tan δ) of PCHMA (diamonds) and two types of nanocomposites (circles) at 1 MHz as a function of Φ_CNT. Gray circles, PCHMA/CNT-Br; black circles, PCHMA-CNT.

Fig. 7 Comparison of dielectric properties between PCHMA-CNT (black circle) and PCHMA/BT (white triangles). (a) ε′_r as a function of Φ_CNT and Φ_BT, and (b) tan δ/ε′_r plot at 1 MHz. The values for PCHMA are plotted with diamond symbols. (c) f dependence of ε′_r and tan δ of PCHMA-CNT with Φ_CNT = 0.094 and PCHMA/BT with Φ_BT = 0.50.

In summary, the dielectric properties of PCHMA-CNT were systematically characterized by focusing on the tunneling conduction. When Φ_CNT < 0.1, PCHMA-CNT exhibited relatively high ε′_r and quite low ε′′_r, while when Φ_CNT > 0.1, a very large ε′_r was observed at low frequencies. The Cole–Cole relaxation model was applied to the impedance spectra of PCHMA-CNT, and from the obtained relaxation frequencies, the average internanotube distance d̄ in the PCHMA-CNT system was estimated. The relationship between d̄ and the resistance R of PCHMA-CNT strongly supports the tunnneling conduction that occurs in the PCHMA-CNT system when Φ_CNT > 0.1. As a result, a new mechanism is proposed for the very large ε′_r; the locally enhanced electric field by the tunneling conduction results in the very large ε′_r at the low frequencies along with the increase in ε′′_r. Furthermore, we have demonstrated that the CNT is five times more effective than the BT for the enhancement in ε′_r of polymer materials because of the higher electric field enhancement effect of the CNT than that of the BT.

Experimental

Sample preparation

MWCNTs with an average diameter and length of 10 nm and 1.5 μm, respectively, were obtained from Nanocyl SA. ATRP-initiator-modified MWCNT (CNT-Br) was synthesized as reported previously.²⁹ For the synthesis of PCHMA-CNT, the surface-initiated polymerization was carried out. For example, N,N-dimethylacetamide (DMAc) (20 ml) was added to CNT-Br (0.188 g) and CuBr (7.2 mg, 50 μmol) in a vacuum. After sonication of the mixture, 2,2′-bipyridyl (bpy) (15.6 mg, 100 μmol) in cyclohexyl methacrylate (10 ml, CHMA) was added, and the solution was kept at 60 °C for 6 h. The resultant polymer mixture was precipitated in methanol and freeze-dried by benzene. Conventional nanocomposites (PCHMA/CNT, PCHMA/CNT-Br and PCHMA/BT) were prepared by blending PCHMA with CNT, CNT-Br, or BT, where BT particle with an average size of less than 100 nm (Aldrich) was used. First, the PCHMA and the CNTs or the BT were added in benzene and dispersed by sonication. The dispersion was quickly frozen by liquid nitrogen, and then freeze-dried under vacuum. Thus-obtained composite samples were crushed into powders. These nanocomposite samples were molded into 15 mm square specimens with a thickness of ∼0.9 mm by compression of 10 MPa at 130 °C. For impedance measurements in the frequency range of 10⁻²–10⁶ Hz, two gold electrodes 11 mm square were deposited on the top and bottom of the specimen at a distance of 2 mm from the edge of the specimen.

Measurements

The specimens were observed using a scanning electron microscope (JEOL, JEM-2100F) operated at an accelerating voltage of 2 kV. For impedance measurement, three types of impedance analyzers, Solartron 126096W, Agilent E4980A, and Agilent E4991A, were used depending on the frequency range, 10⁻²–10² Hz, 10²–10⁶ Hz, and 10⁶–10⁸ Hz, respectively.

Acknowledgements

SEM observations were carried out by Ms Yoriko Matsuoka at Toyota Central R&D Labs Inc., and impedance measurement was supported by Dr Kensuke Wada at Toyota Central R&D Labs Inc. The author appreciates their help.

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Footnote

† Electronic supplementary information (ESI) available: Characteristics of a series of PCHMA-CNTs, SEM and TEM images of PCHMA-CNT, Cole–Cole plots of the impedance spectra, comparison between the experimental and theoretical spectra, and mechanism of the increase ε′_r in the polymer/CNT system. See DOI: 10.1039/c2ra22164k

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