Enthalpy and Dielectric Relaxation of Poly(vinyl methyl ether

Jul 24, 2018 - Both the enthalpy and dielectric relaxation are described via a stretched exponential function, exp(−(t/τ)βK), and the relaxation t...
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Enthalpy and Dielectric Relaxation of Poly(vinyl methyl ether) Kaito Sasaki,† Masanobu Takatsuka,‡ Rio Kita,†,‡ Naoki Shinyashiki,*,‡ and Shin Yagihara‡ Micro/Nano Technology Center and ‡Department of Physics, School of Science, Tokai University, 4-1-1 Kitakaname, Hiratsuka-shi, Kanagawa, Japan

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ABSTRACT: This study investigates the cooperative molecular dynamics of poly(vinyl methyl ether) (PVME) at temperatures above and below the glass transition temperature, Tg, using differential scanning calorimetry (DSC) and broadband dielectric spectroscopy (BDS). The DSC measurements of PVME aged at temperatures below Tg revealed the aging-time-dependent enthalpy relaxation. The BDS measurements revealed the structural α-relaxation process that originated from segmental chain motion at temperatures above Tg. Both the enthalpy and dielectric relaxation are described via a stretched exponential function, exp(−(t/τ)βK), and the relaxation time, τ, and the stretching index, βK, are obtained. τ and βK of the enthalpy relaxation obtained via DSC measurements are compatible with those of the structural α-relaxation obtained via BDS measurements. This similarity indicates that the molecular origin of the enthalpy relaxation is the same as that of the dielectric α-relaxation process of the polymer, i.e., its segmental chain motion.

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INTRODUCTION Investigation of the glass transition of polymers is essential for manufacturing polymer products and for elucidating the dynamics of macromolecules. When a polymer transforms between its glassy and liquid states, the structural α-process, i.e., the density fluctuation caused by molecular motion in the material, plays a key role because at temperatures near the glass transition temperature, Tg, the relaxation time of the α-process depends strongly on the temperature. The relaxation time of the α-process exhibits non-Arrhenius, Vogel−Fulcher−Tammann (VFT) temperature dependence.1−3 This VFT temperature dependence of the α-process implies the cooperativity of the molecular motion in which the cooperative rearranging region (CRR) increases with decreasing temperature.4 In general, the glass transition of a polymer is characterized by Tg. The thermal glass transition temperature, Tg,thermal, can be observed as a jump of the specific heat capacity, Cp, by using thermal techniques, such as differential scanning calorimetry (DSC), at a finite scanning rate. This jump in Cp is related to the freezing of molecular motion. In contrast, the dynamical glass transition temperature, Tg,dynamic, is the temperature wherein the relaxation time of the α-process reaches the range 102−103 s and is determined using dynamical techniques, such as broadband dielectric spectroscopy (BDS), nuclear magnetic resonance, ac calorimetry, and so on. Generally, Tg,dynamic roughly agrees with Tg,thermal. However, the time scale of Tg,thermal depends on the scanning rate and the behavior of the α-process.5 Therefore, in the strict sense, the time scale of Tg,thermal should not be blindly assumed to be 102−103 s. Figure 1 shows the temperature dependence of the specific volume, V, and enthalpy, H, of a material. Generally, Tg has been defined as a temperature that is intersection of the slope of liquid and glssy states. After continuous cooling from a liquid state to a glass state, the maintenance of the temperature at temperature below Tg induces a slight decrease of V and H toward the equilibrium statea phenomenon known as physical aging.6 © XXXX American Chemical Society

Figure 1. Schematic of the temperature dependence of the volume and enthalpy of the materials. The extension line of the equilibrium state is indicated by the dashed line.

From the viewpoint of thermal analysis, this process is called the enthalpy relaxation. The equilibrium state is represented by the extension of the low temperature side of the liquid state. In Figure 1, the aforementioned extension is indicated by the dashed line. The decreases of V and H are based on the αrelaxation process.7−10 During heating after aging, V and H overshoot the extension line. After V and H deviate downward from the extension line at temperatures below Tg, V and H gradually approach and eventually reach on the equilibrium line. In a DSC measurement, the enthalpy relaxation can be detected as a peak following the glass transition during heating, and the area of the peak depends on the aging time, ta, and on the aging temperature, Ta.7,11−14 Enthalpy relaxation has been widely investigated.15−29 As the common view, the enthalpy relaxation has been assumed to Received: April 11, 2018 Revised: July 12, 2018

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DOI: 10.1021/acs.macromol.8b00780 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules occur on a single time scale.15−25 However, some studies have suggested that enthalpy relaxation occurs on multiple time scales.26−28 A study clearly demonstrated that the enthalpy relaxation of poly(styrene) and poly(carbonate) proceeds via a two-step process.28 According to that study, the relaxation time of a slower step exhibits the same VFT temperature dependence of the α-process, and that of a faster step exhibits the Arrhenius temperature dependence. These results indicate that the enthalpy relaxation, particularly the slower step of the twostep process, is related to the α-process. However, the investigations of the compatibility of an absolute value of the time scale of the enthalpy relaxation and α-process are lacking, which precludes a detailed discussion on the molecular origin of the enthalpy relaxation. In this study, the relations between the α-process and the enthalpy relaxation of a polymer are investigated via BDS and DSC measurements, and it is revealed that the nature of enthalpy relaxation is the α-process. BDS is a powerful technique to investigate the glass transition of a polymer because of its broad time window, which covers picosecond to megasecond molecular dynamics from liquid to glassy states. For these experiments, poly(vinyl methyl ether) (PVME) is chosen as the investigated material because its α-process has already been investigated,30,31 and its Tg is approximately 243 K, which is convenient for our DSC and BDS measurements. Our results demonstrate that the enthalpy relaxation detected via DSC measurements exhibited the same molecular origin as the α -process, which is the origin of the glass transition detected via BDS measurements. In addition, our results strongly suggest that the assumption of the time scale of the glass transition obtained using thermal techniques needs more attention to compare with the relaxation time of the α-process.

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BDS measurements were performed in the frequency range from 10 mHz to 10 MHz using an Alpha-A analyzer (Novocontrol). Parallel gold-plated electrodes with a diameter of 20 mm were used, and the sample thickness was 0.1 mm. The temperature was controlled with a Quatro cryosystem (Novocontrol). BDS measurements were performed at 123−293 at 10 K intervals. The temperature was maintained for 30 min before the BDS measurements. This procedure was repeated for each measurement.

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RESULTS AND DISCUSSION Calorimetric Measurements. Figure 3 shows the DSC thermograms for PVME with various ta at Ta = 236 K. The step

Figure 3. DSC upscans at a heating rate of 10 K/min for PVME with various ta and at Ta = 236 K. The arrow indicates the evolution of the enthalpy relaxation with increasing ta. The inset shows the ta dependence of Tg.

in the thermograms indicates a change of the heat capacity, which is owing to the glass transition; the subsequent peak represents the enthalpy relaxation. The inset of Figure 3 shows the ta dependence of Tg. Tg was determined as the temperature wherein the heat flow crosses the middle of baselines of the highand low-temperature sides of the step. Figure 4 defines Tg. The

EXPERIMENTAL SECTION

The aqueous solution (50 wt %) of PVME (Sigma-Aldrich) used herein was purified with an ion-transfer resin to reduce its dc conductivity and electrode polarization (EP). A dehydrated sample was obtained by freeze-drying the aqueous solution of PVME under vacuum. Measurements were performed at 208−273 K using a DSC (PerkinElmer, DSC 7). The 21 mg sample was placed in an aluminum pan into a sample holder under a nitrogen atmosphere. The PVME sample was cooled from 298 K to the Ta and was then maintained at Ta for ta for aging. After the aging process, the sample was cooled from Ta to 208 K and then maintained for 5 min at the lowest temperature. The measurement was performed in a heating scan from 208 to 273 K. The cooling and heating scan rate, k, was 10 K/min. A typical temperature program is shown in Figure 2. Ta values were 234, 236, and 238 K, and ta ranged from 102 to 105 s. DSC 7 was calibrated for temperature and heat flow immediately before the measurement using dichloroethane and benzene.

Figure 4. Definition of the glass transition temperature (Tg), the peak area, and the fictive temperature. The peak area is indicated by the yellow region.

peak indicates the enthalpy relaxation, and it dramatically increases in intensity with increasing ta. The area of the peak, Harea, was obtained using the following procedure. First, we estimated the baseline on the high-temperature side of the peak as a straight line via least-squares fitting. Second, we extrapolated the baseline to the lower-temperature side of the peak. Third, we obtained Harea as the upper area of the baseline. Figure 4 shows the definition of Harea. In this figure, the dashed line and the

Figure 2. Typical temperature program used for calorimetric measurements. B

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Macromolecules yellow-colored area indicate the baseline and Harea, respectively. Usually, H of the enthalpy relaxation, HER, is obtained as the difference in enthalpy between the aged and unaged samples, HER = Haged − Hunaged, as determined from their DSC thermograms. However, because of the limited temperature scanning rate, k, we could not acquire the DSC thermogram of a completely unaged sample. Therefore, we calculated the peak area using the aforementioned procedure. In addition, to characterize the enthalpy relaxation, we determined the fictive temperature, Tf, defined in Figure 4.32,33 Specifically, Tf is the temperature wherein the blue area and the sum of the yellow and red areas are equal. Figure 5a shows the ta dependence of Harea at various Ta (234, 236, and 238 K). The Harea clearly becomes saturated with

stretching index that accounts for the deviations from exponential behavior. The fitting results are shown in Figure 5a as curves. The data were well described by eq 1. The obtained a, τ, and βK,area values are listed in Table 1. According to the fitting results, τ and a decreased with increasing Ta. βK,area is almost independent of Ta or slightly decreases with decreasing T a. Table 1. Values of a, τarea, and βK,area Correspond to the Area of the Enthalpy Relaxation of PVME

a

Ta (K)

aa (AU)

log[τarea (s)]a

βK,areaa

234 236 238

7.6 5.2 3.3

3.9 3.2 2.6

0.43 0.49 0.48

Errors are less than ±8%.

Figure 5b shows the ta dependence of the normalized Harea at various Ta (234, 236, and 238 K). The data are normalized by a, which was obtained by a fitting procedure. These results clearly show the relaxation shift to long ta with decreasing Ta. Figure 5c shows the ta dependence of Tf at various Ta (234, 236, and 238 K). To characterize the evolution of Tf, we performed curve-fitting procedures using the following equation, which is similar to eq 1: l o o ij yz βK,Tf | o o o jj t zz o Tf = Tf, ∞ − ΔTf expo m −jj zz o } o o j z τ o o Tf { o o k (2) n ~ where ΔTf = Tf, ∞ − Tf,0

(3)

Here Tf,∞ and Tf,0 indicate the Tf for ta = ∞ and ta = 0, respectively; τTf is the relaxation time of Tf. For the fitting procedure, the Tf,0 value was fixed at 242.95 K which was obtained for the ta = 0 sample. The fitting results are shown in Figure 5c as curves. The data were well described by eq 3. The obtained values for τTf and βK,Tf are listed in Table 2. According to the fitting results, τTf increased with decreasing temperature, whereas βK,Tf was almost independent of Ta within the measurement error.

Figure 5. ta dependence of (a, b) the area of the enthalpy relaxation and (c, d) Tf for PVME at various Ta (234, 236, and 238 K). Panels b and d show the normalized value of the area and Tf, respectively. The dashed horizontal line indicates the Tf of PVME with ta = 0 s. The errors are smaller than the plot size.

Table 2. Values of τTf and βK,Tf for the Tf of PVME

increasing ta. This behavior implies that during aging at a temperature below Tg the state of the sample relaxes from a nonequilibrium to an equilibrium state via the enthalpy relaxation. In addition, the step height increases with decreasing Ta, which indicates how much the state of the sample relaxes toward the equilibrium state at each Ta. To characterize the evolution of Harea, curve-fitting procedures were performed using the following equation that involved a stretched exponential, i.e., the Kohlrausch correlation function:34 ÄÅ É βK,area |Ñ l ÅÅÅ ÑÑÑ o o i y o o Å ÑÑ jj t zz o − ÑÑ Harea = aÅÅÅ1 − expo m } j z jj τ zz o o ÅÅ ÑÑ o o o o area { k ÅÅÇ (1) n ~ÑÑÖ

a

Ta (K)

log[τTf (s)]a

βK,Tfa

234 236 238

3.6 3.1 2.6

0.44 0.54 0.51

Errors are less than ±11%.

Figure 5d shows the ta dependence of the normalized Tf at various Ta (234, 236, and 238 K). The data were normalized by Tf,∞ and Tf,0, which were obtained by a fitting procedure. The relaxation shifts to a longer ta with decreasing Ta. Figure 6 shows the Ta dependence of a; a linear correlation was observed. The line was obtained by least-squares fitting. Herein, Tc is defined as the intersection of the line and a = 0; thus, Tc is 241 K. At temperatures higher than Tc, the aging has no effect on the thermal properties, specifically, V and H, of the material; i.e., above Tc, the equilibration of the system is

where a indicates the height of the step, τarea is the relaxation time of the enthalpy relaxation, and βK (0 < βK ≤ 1) is the C

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ω is the angular frequency, σ is the dc conductivity, and βK (0 < βK ≤ 1) is the stretching index indicating the broadness of the symmetric relaxation curves. This equation includes a Cole− Cole relaxation function 35 for the EP contribution, a Kohlrausch−Williams−Watts function34,36 for the α-process, and a dc conductivity term. The contributions of dc conductivity and EP are small, and these appear at frequencies 3 decades lower than that of the loss peak of the α-process. Then the contributions can be neglected on the curve fitting for characterization of the α-process. The obtained τα and βK values are shown in Figure 8 and discussed in detail. Figure 6. Ta dependence of a. The line was obtained by a least-squares fit.

achieved within the measurement time scale. Therefore, the enthalpy relaxation can be found in a limited temperature range, below Tc. This lack of aging effect means that Tc corresponds to the border between the liquid and glass states. As mentioned in the Introduction, a multistep process of enthalpy relaxation has been reported.26−28 However, our data do not show a multistep process for PVME, indicating that PVME does not undergo the multistep process or that the Ta of our measurement is not sufficiently low for the multistep process to be observed. Dielectric Measurements. Figure 7a shows the frequency dependences of the imaginary parts of the complex permittivity

Figure 8. (a) DSC thermogram for PVME with ta = 0 s. Temperature dependence of (b) the relaxation time and (c) the βK of PVME. The curves were determined by the VFT fits. The errors of the plots without error bar are smaller than the plot size.

Molecular Origin of the α-Process and the Enthalpy Relaxation. Figure 8a shows the DSC thermogram of PVME with ta = 0 s. The orange vertical-dashed line indicates the Tg,thermal, where Tg,thermal was determined in the same manner as previously described. Figure 8b shows the reciprocal temperature dependences of τα obtained via BDS measurements and the τarea and τTf obtained from DSC measurements. To describe the temperature dependence of τα, τarea, and τTf, curve fitting using the VFT equation1−3 was performed. The VFT equation is expressed as follows:

Figure 7. Frequency dependence of the imaginary parts of permittivity for PVME at various temperatures between 253 and 313 K in steps of 10 K. The plot color indicates the measured temperatures. Arrows indicate peak frequencies for the α-process. The errors are smaller than the plot size.

for PVME at 253−313 K. At 313 K, a broad and symmetric relaxation process is observed at 1 MHz. This relaxation process is the α-process, which is caused by the segmental motion of PVME. To characterize the α-process, we performed curve-fitting procedures using eq 4 as follows: ÄÅ ÉÑ ∞Å ΔεEP ÅÅ dΦα ÑÑÑ ε* = ε∞ + ε + Δ − Å α ÅÅ dt ÑÑÑ 0 Å 1 + (iωτEP)βEP ÑÖ Ç σ × exp( −jωt ) dt + iωε0 (4)

ij B yzz τ = τ∞ ,VF expjjj z j T − T0 zz (5) k { where T is the temperature and τ∞,VF, B, and T0 are empirical VFT parameters. In Figure 8b, the dashed and solid curves were obtained by fitting for τα only and for both BDS and DSC measurements (τα, τarea, and τTf), respectively. The obtained VFT parameters are listed in Table 3. The VFT parameters for only τα are comparable with those determined in previous study involving BDS.37 As shown in Figure 8b, the solid curve representing the VFT fitting result for both BDS and DSC measurements well describes τα, τarea, and τTf in the entire temperature range. In general, the temperature range of the step of the glass transition in the DSC thermograms depends on the scanning

∫

ÄÅ É ÅÅ i y βK ÑÑÑ ÅÅ jj t zz ÑÑ Φα (t ) = expÅÅÅ−jj zz ÑÑÑ ÅÅ jk τα z{ ÑÑ ÅÅÇ ÑÑÖ where ε0 is the permittivity in a vacuum, ε∞ is the limiting highfrequency permittivity, τα is the relaxation time of the α-process,

where

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same as that of the dielectric α-relaxation process of the polymer, i.e., its segmental chain motion.

Table 3. VFT Parameters and τ of the α-Process at Tg for PVME BDS, area, and Tf BDS ref 37, BDS

B (K)

T0 (K)

log[τ∞,VF (s)]

τα,DSCa (s)

2.1 × 103 1.3 × 103 1481.5

184 202 202

−14.3 −12.4 −12.9

11 8 8.2

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Ph +81 (0)463 58 1211 ext 3706.

τα,DSC are obtained by eq 6.

a

ORCID

Kaito Sasaki: 0000-0002-2365-650X Rio Kita: 0000-0002-9683-5840 Naoki Shinyashiki: 0000-0003-0486-2911 Shin Yagihara: 0000-0003-2927-6134

rate, k. In the liquid state, equilibrium of the α-process is always maintained. When the system approaches the glassy state with continuous cooling, τα increases drastically because it obeys the VFT temperature dependence, and the α-process requires a long time for equilibration. After the temperature crosses the Tg, the α-process freezes and the system can no longer maintain equilibrium. Consequently, a step appears in the DSC thermogram. Therefore, the time scale of the α-process at Tg,thermal, τα,DSC, can be estimated from the VFT parameters and k.5 τα,DSC can be estimated by the following equation:5 B kτα ,DSC ≃ 1 (Tg,thermal − T0)2 (6)

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was partly supported by JSPS KAKENHI Grants 16K05522, 15K13554, and 24350122 and by the MEXTSupported Program for the Strategic Research Foundation at Private Universities, 2014−2018.

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From eq 6 with VFT parameters, B and T0, we can obtain τα,DSC. The values of τα,DSC calculated via eq 6 are listed in Table 3. The τα,DSC is 11 s for the VFT parameters obtained via BDS and DSC measurements and 8 s for the parameters obtained via BDS measurement alone. In Figure 8b, a range of τα,DSC is indicated by the solid blue circle. The τα,DSC is well located on the VFT curves. Notably, τα,DSC differs from the conventionally defined time scale of the α-process at Tg in the range from 102 to 103 s, which is indicated by the gray band in Figure 8. The time scale of the glass transition obtained via DSC measurements has been assumed to range from 102 to 103 s. This assumption has been widely applied. However, herein, the time scale from 102 to 103 s for Tg is not compatible with any of the VFT fitting curves. This incompatibility is understandable because of the assumption based on k = 0.01 K/s (0.6 K/min). By contrast, the current DSC measurements were performed at k = 0.167 K/s (10 K/ min). Therefore, τα,DSC, i.e., the time scale of the glass transition, should be in the range from 1 to 10 s. Thus, our results strongly suggest that the time scale of Tg,thermal should not be blindly assumed to be 102−103 s. In addition, as shown in Figure 8c, the βK values that obtained via BDS measurements agree well with those obtained via DSC measurements. The results presented herein suggest that the origins of both relaxation processes observed via BDS and DSC measurementsspecifically, the dielectric α-process and the calorimetric enthalpy relaxationare phenomenologically the same. On the basis of the BDS measurements, the α-process has been associated with the segmental motion of PVME. Several properties of the enthalpy relaxation agree with those of the αprocess. Therefore, we conclude that the origin of the enthalpy relaxation is the segmental motion of PVME, which is the same as the origin of the α-process.

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CONCLUSION DSC and BDS measurements were performed to characterize PVME to enable a discussion of the dynamics of the structural relaxation process. Our results demonstrated that the temperature dependence of τ and βK of the enthalpy relaxation in PVME agrees well with those of the α-process. This agreement implies that the molecular origin of the enthalpy relaxation is the E

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