Research Article Received: 2 September 2012 Revised: 27 November 2012 Accepted: 30 November 2012 Published online in Wiley Online Library: (wileyonlinelibrary.com) DOI 10.1002/pi.4479 New method to analyze dielectric relaxation processes: a study on polymethacrylate series Silvia Soreto Teixeira,a Carlos J. Dias,b Madalena Dionisioc and Luı́s C. Costaa∗ Abstract The relaxation properties of polymethacrylates of the n-alkyl series with n = l, 2 and 4 (poly(methyl methacrylate) (PMMA), poly(ethyl methacrylate) (PEMA) and poly(n-butyl methacrylate) (PnBMA)) have been measured and analyzed in order to relate their properties to the size of the lateral side chains. The n-alkyl series has been regarded as a model system and was used in this work to test a graphical data analysis method. Essentially, four relaxation processes were detected in the three polymers: the γ , β, α and αβ relaxations, in increasing order of temperature. It was found that the γ relaxation has a low activation energy, of around 36.3–38.5 kJ mol−1 , independent of the side chain, exhibiting low entropy of activation values when referring to the Eyring description of the activation parameters. The β relaxation was found to be similar in PMMA and PEMA, showing an activation energy of 88.8 kJ mol−1 , increasing to 112.8 kJ mol−1 in PnBMA. The activation entropy was also found to be low for this relaxation, although greater than that for the γ relaxation. In contrast, the α relaxation is quite different in these polymers. We observed a gradual shift in the glass transition temperature towards lower temperatures as the side chain increases in length. The manner in which the α transition makes its way into the dielectric spectra is more abrupt in PMMA than in PnBMA, denoting a higher fragility in the former polymer. Finally, there is a significant difference in the coalescence scenarios of the α and β relaxations for temperatures higher than the glass transition temperature, where they give rise to the so-called αβ relaxation. c 2013 Society of Chemical Industry
Keywords: impedance spectroscopy; dielectric relaxation; glass transition; polymethacrylates INTRODUCTION Polymers are very complex materials compared with low molecular weight compounds. The large number of macromolecular chains is responsible for a great number of conformations with consequences at the level of chain flexibility.1 Temperature has an important influence in that flexibility, and consequently this behavior is reflected in the dielectric measurements by broadband dielectric spectroscopy. In fact, this technique allows us to understand the polarization mechanisms present in polymers, i.e. the charge migration and the mechanism due to the orientation of permanent dipoles.2 – 6 Usually, polymers present several relaxation processes, known as α, β and γ . The α process is observed at lower frequencies or higher temperatures and is usually related to the dynamic glass transition, i.e. the main intermolecular internal friction mechanism due to translational movement of molecular chains.7 γ relaxation is observed at higher frequencies or lower temperatures and is typical of local motion. Secondary relaxation, known as β relaxation, has its origin in the molecular fluctuations of the chain segments. It is accepted that β relaxation observed in polymethacrylates arises from the rotational motion of the side chain about the C–C bond which links to the main chain.8 This relaxation motion of the side chain is governed essentially by intra-chain interaction, in contrast to the micro-Brownian motion of the main chain for the α relaxation.9 The characteristic relaxation time of the β process shows an Arrhenius-like temperature dependence, whereas the α process is better described by a Vogel–Fulcher–Tammann law. Usually, both Polym Int (2013) processes are well separated. Due to the different temperature dependences of their relaxation times, however, they tend to merge when the timescales attain the same order of magnitude. At highest temperatures, only one process is observable, known as the αβ process.10 Other techniques such as DSC,11 thermally stimulated depolarization currents (TSDC)12,13 and mechanical spectroscopy14 have also been used to confirm and support the evidence collected by dielectric spectroscopy. A graphical technique for the analysis of dielectric spectroscopy data is presented in this work whereby the dielectric spectra obtained for the imaginary part of the permittivity are mathematically transformed to give a relaxation distribution as a function of the Gibbs activation energy. This representation allows a better visualization and characterization of the relaxation processes in the material as a function of temperature together with a clear indication of the glass transition temperature and its associated segmental relaxation. ∗ Correspondence to: L. C. Costa, Physics Department and I3N, University of Aveiro, 3810–193 Aveiro, Portugal. E-mail: kady@ua.pt a Physics Department and I3N, University of Aveiro, 3810-193, Aveiro, Portugal b Materials Science Department and I3N, New University of Lisbon, 2829-516, Caparica, Portugal c Chemistry Department, FCT, New University of Lisbon, 2829-516, Caparica, Portugal www.soci.org c 2013 Society of Chemical Industry
www.soci.org S. Soreto Teixeiraet al. Figure 1. PMMA, PEMA and PnBMA structures. EXPERIMENTAL RESULTS The glass transition temperatures obtained using the DSC technique were 114.6 ◦ C for PMMA, 71.5 ◦ C for PEMA and 24.5 ◦ C for PnBMA. These values were calculated according to the method of tangents. wileyonlinelibrary.com/journal/pi Figure 2. Real part of the complex permittivity as a function of frequency, for the three polymers. 1 T = 160 °C ε" The polymers studied, poly(methyl methacrylate) (PMMA), poly(ethyl methacrylate) (PEMA) and poly(n-butyl methacrylate) (PnBMA), belong to the family of poly(n-alkyl methacrylate)s and have the general formula (CH2 –C(CH3 (COOR))) (Fig. 1). The difference between these polymers is the length of the lateral side chain R, R1 and R2, respectively. PMMA was obtained from Aldrich (St. Louis, USA) and has a rather low molecular weight, Mw < 100 000 g mol−1 . PEMA and PnBMA were from Aldrich, with Mw = 340 000 g mol−1 and Mw = 320 000 g mol−1 , respectively. The samples were in powder form and films were obtained using a hydraulic press and were then metallized through a sputtering DC technique performed by Fisons Polaron SG502. DSC was only used for the determination of the glass transition temperature (T g ) in a Setaram DSC 131 calorimeter fitted with a liquid nitrogen cooling accessory. Dry high purity N2 gas was purged through the sample during measurements. Samples were cooled to −40 ◦ C and the thermograms were collected in a subsequent heating run at 10 ◦ C min−1 . The experimental setup used to perform the TSDC measurements was designed in the laboratory.15 This measurement system consisted of a bath cryostat covering a temperature range between −193 and 87 ◦ C, an IT54 Oxford Research temperature controller, a PS325-SRS high voltage supply to polarize samples between 0 and 2500 V, and a Keithley-617 electrometer detecting currents in the range 10 –14 –10 –4 A with software to register the current as a function of temperature. The system was heated to the polarization temperature (T p ) defined for each polymer (T p = T g + 25), at a constant rate of 4 ◦ C min−1 . For PMMA, T p was only 12 ◦ C above T g due to the resistance of the polymer at higher temperature. The polarization electric field was about 100 kV m−1 , which ensures the linearity of the response. The dielectric measurements were carried out using an AlphaN broadband impedance analyzer (Novocontrol GmbH), covering the frequency range from 10−1 to 106 Hz in increasing temperature steps from −85 ◦ C up to 160 ◦ C (selection of temperatures differing by 10 ◦ C between two measurements). The sample was introduced between two gold-plated electrodes (diameter 20 mm) of a parallel plate capacitor. The sample cell (BDS 1200) was mounted in a cryostat (BDS 1100) and exposed to a heated gas stream evaporated from a liquid nitrogen Dewar. The temperature control was performed within ±0.5 ◦ C with the Quatro Cryosystem from Novocontrol. We used the control and acquisition of data software WinData from Novocontrol. 0.1 T = -50 °C 10-1 100 101 102 103 104 105 106 f (Hz) Figure 3. Imaginary part of the complex permittivity as a function of frequency at several temperatures, for PMMA. Figure 2 shows the real part of the complex permittivity as a function of frequency, for the three polymers under study, at temperatures above the glass transition and below the melting temperature, where Maxwell–Wagner–Sillar interfacial polarization can be clearly observed.16 Figures 3, 4 and 5 show the imaginary parts of the complex permittivity as a function of frequency at several temperatures, for PMMA, PEMA and PnBMA, respectively. The real (ε’) and imaginary (ε’’) parts of the complex dielectric constant can be fitted by the sum of the well-known Havriliak–Negami (HN) function:17 ε = ε∞ + ε σ/ε0
+   αHNi βHNi jω 1 + (jωτ HNi ) i (1) where i is the number of the relaxation process, ε is the dielectric strength, α HN and β HN are the shape parameters (0 < α HN < 1; β HN < 1), τ HN is the characteristic relaxation time, related to the frequency of the maximum of the loss peak, and ε∞ is the high frequency dielectric constant. At the highest temperatures, data are influenced by a low frequency conductivity contribution, and c 2013 Society of Chemical Industry
Polym Int (2013) New method to analyze dielectric relaxation processes www.soci.org Figure 4. Imaginary part of the complex permittivity as a function of frequency at several temperatures, for PEMA. Figure 6. Imaginary part of the complex permittivity for PnBMA, at T = 30 ◦ C, and best fit, showing two relaxation processes. DISCUSSION 1 A new technique for the analysis of dielectric spectroscopy data is presented, where the measured dielectric spectra are mathematically transformed to give the relaxation distribution as a function of the Gibbs activation energy. The frequency translation from any given temperature to another so-called reference temperature (T ref ) is achieved for processes without a change in G according to the equation21 T = 100 °C ε" 0.1 ′ fp = T = -50 °C kTref 2π h  2π fp h kT T/Tref (5) 0.01 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 f (Hz) Figure 5. Imaginary part of the complex permittivity as a function of frequency at several temperatures, for PnBMA. an additional term σ /jωε0 was added, where ε0 is the vacuum permittivity and σ is the DC conductivity of the sample.18 For cooperative processes the temperature dependence of the relaxation time obeys the Vogel–Fulcher–Tamman (VFT) law:19   B (2) τ = τo exp T − To where B is a constant, τ 0 is the relaxation time for infinite frequency and T 0 is defined as the Vogel temperature, 30 to 70 ◦ C below T g . By replacing the VFT law in the activation energy equation we find RB ∂ ln τ = (3) Ea (T) = R ∂ (1/T) (1 − T0 /T)2 where R is the ideal gas constant, 8.324 J mol−1 . Finally the fragility index is calculated according to Angell’s equation20 and Eqn (3):   Ea Tg ∂ log τ    = m=  (4) ∂ Tg /T T=Tg ln 10 RTg where T g is estimated by replacing τ in Eqn (2) by 100 s. Polym Int (2013) It should be pointed out that local processes have in general low activation entropy and therefore their values for the Gibbs activation energy remain fairly constant with temperature. The most notable deviation is that of the α process where high activation entropy processes are present. Figure 6 shows the imaginary part of the complex permittivity for PnBMA at T = 30 ◦ C and the quality of the fit, where two relaxation processes can be observed. Figure 7 presents the dielectric loss spectra for PMMA using this new method for temperatures between −50 ◦ C and 160 ◦ C, in temperature steps of 10 ◦ C, where the various relaxations present in the spectrum, i.e. the γ , β, α and αβ relaxations,22,23 are readily apparent. Figures 8 and 9 represent the dielectric spectra for PEMA and PnBMA using the same procedure but for different temperature ranges. These ranges have been chosen to cover the glass transition temperature suitably for each of the polymers. The results from fitting the dielectric spectra can be examined in Tables 1 and 2 where the relaxation parameters for the three polymers are listed. In Table 1 one can see that the activation energies for the β process in PMMA and PEMA are similar, while that of PnBMA is slightly higher. However, two regimes can be distinguished for the β process in PMMA and PEMA (see Figs 10 and 11) such that, for the regime closer to the glass transition temperature, the activation energies of these polymers approach that of PnBMA (Fig. 12). The value obtained for PnBMA is also close to its glass transition. The relaxation strength for the β relaxation is about the same for PMMA and PEMA and lower for PnBMA probably due to a lower dipole density originating from its long side chain. c 2013 Society of Chemical Industry
wileyonlinelibrary.com/journal/pi www.soci.org S. Soreto Teixeiraet al. Table 1. Values of E a , f 0 and ε for β and γ processes in PMMA, PEMA and PnBMA β process PEMA PMMA 81.5 ± 0.4 a117.0 (8.6 ± 2.0) x 1015 [1.6 : 2.9] E a ± E a (kJ mol−1 ) f 0 (Hz) ε a PnBMA 83.2 ± 1.2 a112.9 (1.5 ± 0.7) x 1016 [1.8 : 3.1] γ process PEMA PMMA 112.9 ± 4.2 38.5 ± 1.2 21 34.7 ± 1.2 12 (4.7 ± 8.6) x 10 [0.9 : 1.6] PnBMA (7.3 ± 4.0) x 10 [0.2 : 0.3] 35.1 ± 3.3 13 (1.4 ± 0.8) x 10 [0.2 : 0.3] (3.4 ± 5.5) x 1013 [0.009 : 0.1] Two regimes are considered in the β process for PMMA and PEMA. 1 1 αβ β αβ T = 100 °C β ε" ε" 0.1 γ 0.1 γ T = -50 °C T = 160 °C 0.01 T = -85 °C -1 10 0 10 1 10 2 10 3 10 4 5 10 10 6 7 10 10 8 10 10-1 100 101 102 f (Hz) 104 105 106 107 108 f (Hz) Figure 7. Dielectric loss spectra for PMMA using the new method. Figure 9. Dielectric loss spectra for PnBMA using the new method. αβ 1 Table 2. Values of T g , T 0 , E a , f 0 , fragility index (m) and T g obtained by DSC measurements for the α process in PMMA, PEMA and PnBMA T = 140 °C β PMMA ◦ ε" 103 0.1 γ T = -80 °C 0.01 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107 108 f (Hz) Figure 8. Dielectric loss spectra for PEMA using the new method. The same reasoning can be applied to the relaxation strength of the γ relaxation in these polymers. The activation energies for the γ relaxation are fairly similar in PEMA and PnBMA and slightly higher for PMMA, implying that this relaxation is not greatly affected by the length of the side chain. In fact, this relaxation has been attributed to local movements of the 0-R moiety. The pre-exponential factor for this relaxation is consistent with a low activation entropy, S. The α relaxation shows a nonlinear dependence of relaxation time as a function of the inverse of temperature, thus obeying the VFT equation. The activation energy of this process at T g is wileyonlinelibrary.com/journal/pi T g (DSC) ( C) T g (◦ C) T 0 (◦ C) E a (kJ mol−1 ) f 0 (Hz) m 114.6 99.5 44.8 618 5.0 x 1010 86 PEMA 71.4 57.5 1.8 485 8.7 x 1010 77 PnBMA 24.5 23.6 −23.1 430 7.5 x 109 76 greater than 418 kJ mol−1 , being higher for PMMA (618 kJ mol−1 ), then PEMA (485 kJ mol−1 ) and finally PnBMA (430 kJ mol−1 ). The fragility (m) of a polymer is a measure of the abruptness of a given rubber material to vitrify in a small temperature range. One sees that PMMA shows the highest values of fragility followed by PEMA and PnBMA. Therefore PMMA has a behavior that departs strongly from the Arrhenian which indicates that a stronger cooperativity exists between the main chains in PMMA. Again this can be explained in terms of its shorter side chain. The region in which the α and β relaxations couple to give rise to the αβ relaxation is not the same in these polymers, as can be observed in Fig. 5. In this case, PMMA and PEMA exhibit a coupling scenario where it is clear that the β relaxation acts as a precursor of the cooperative α process, for higher temperatures. The scenario obtained for PnBMA is slightly different, presenting a lower cooperativity in the coupling of these relaxations, because c 2013 Society of Chemical Industry
Polym Int (2013) New method to analyze dielectric relaxation processes Figure 10. Log f max as a function of the inverse of temperature for PMMA. www.soci.org Figure 12. Log f max as a function of the inverse of temperature for PnBMA. Figure 13. Thermally stimulated currents for the three polymers. Figure 11. Log f max as a function of the inverse of temperature for PEMA. when the α process becomes visible subvitreous processes are not affected. In Fig. 13 are shown the thermally stimulated currents of these polymers. Two peaks are distinguished, called β and α, in order of increasing temperature. The α peak is located at the calorimetric glass transition temperature, corresponding to the main relaxation or α relaxation.24 The β peak, in polar polymers, mainly arises from localized rotational fluctuations of the dipoles and therefore it is also referred to as the dipolar relaxation process or β relaxation.2,25 The shift of the α relaxation to lower temperature with increase of the side chain is confirmed. CONCLUSIONS In this work, two different methods of analyzing the dielectric spectra of the polymethacrylate series have been used. These methods do not work as alternatives but provide different views of the same experimental data. The new method eases the identification of the relaxation processes present and their localization, while the other method allows for a quantitative expression of dielectric spectra analysis, yielding values of the relaxation frequencies and relaxation strength as a function of temperature. Polym Int (2013) Four relaxation processes were detected in the three polymers: the γ , β, α and αβ. The γ relaxation has a low activation energy, independent of the side chain, exhibiting low entropy of activation when referring to the Eyring description of the activation parameters. The β relaxation presents similar activation energies in PMMA and PEMA, but an increased activation energy in PnBMA. The activation entropy was also low for this relaxation. The α relaxation is quite different. A gradual shift in the glass transition temperature towards lower temperature as the length of the side chain increases is observed. Finally, there is a significant difference in the coalescence scenarios of the α and β relaxations for temperatures higher than the glass transition temperature, where they give rise to the αβ relaxation. REFERENCES 1 Schonhals A, Molecular dynamics in polymer model systems, in Broadband Dielectric Spectroscopy, ed. by Kremer F and Schonhals A. Springer, Berlin (2003). 2 Merino EG, Atlas S, Raihane M, Belfkira A, Lahcini M, Hult A, et al, Europ Polym J 47: 1429–1446 (2011). 3 Pei X and Xingyuan Z, Europ Polym J 47: 1031–1038 (2011). 4 Hassan MK, Abukmail A and Mauritz KA, Europ Polym J 48: 789–802 (2012). 5 Hedvig P, Dielectric Spectroscopy of Polymers. Adam Hilger, Bristol (1977). c 2013 Society of Chemical Industry
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Polym Int (2013) Contents Published online in Wiley Online Library: (wileyonlinelibrary.com) DOI 10.1002/pi.4479 Research Article New method to analyze dielectric relaxation processes: a study on polymethacrylate series 000 A new method eases the process of identifying the relaxation processes, while the usual one allows for quantitative expressions of dielectric spectra analysis, giving off values of the relaxation parameters. Polym Int (2013) Silvia Soreto Teixeira, Carlos Dias, Madalena Dionisio and Luı́s C. Costa∗ www.soci.org J. c 2013 Society of Chemical Industry