Method for Calculation of the Lamellar Thickness Distribution of Not

Article pubs.acs.org/Macromolecules

Method for Calculation of the Lamellar Thickness Distribution of NotReorganized Linear Polyethylene Using Fast Scanning Calorimetry in Heating Yoshitomo Furushima,*,† Masaru Nakada,† Masataka Murakami,† Tsuneyuki Yamane,† Akihiko Toda,‡ and Christoph Schick§ †

Toray Research Center, Inc., 3-7, Sonoyama 3-chome, Otsu, Shiga 520-8567, Japan Graduate School of Integrated Arts and Sciences, Hiroshima University, Higashihiroshima 739-8521, Japan § Institute of Physics, University of Rostock, Albert-Einstein-Str. 23-24, 18059 Rostock, Germany ‡

ABSTRACT: The lamellar thickness distribution (LTD) of not-reorganized linear polyethylene was calculated on the basis of the melting point distribution using fast scanning calorimetry (FSC) by applying the following precautions. First, by taking sufficiently small sample mass and thickness, the influence of thermal lag during fast-heating is thought to be negligible. Second, by fastheating, reorganization and cold crystallization are strongly hindered or even suppressed. Third, the influence of superheating has been accounted for and corrected by deconvolution of the FSC curve using the calculated melting kinetics of single-sized lamellae consisting of folded-chain crystal. Under such precautions, the resulting FSC heating curve reflects the melting point distribution of metastable-but-not-reorganized folded-chain crystals. Finally, the Gibbs−Thomson equation was applied to calculate the LTD from the melting point distribution. The average lamellar thickness calculated was in good agreement with that determined by small-angle X-ray scattering and lowfrequency Raman spectroscopy. melting behavior of a semicrystalline polymer.5−8 The triple endotherms on the conventional DSC traces of the semicrystalline polymers could not be explained by cold crystallization and reorganization alone. They concluded that the endotherm at intermediate temperature is due to the melting of primary crystals without causing reorganization. On the other hand, Schick et al. reported that there is no evidence for the formation of multimodal distributions of crystals for isothermal melt crystallized PET. The single broad melting peaks at fast-heating rates show that broad monomodal distributions of lamellae thickness exist.9−11 According to their study, fast scanning calorimetry (FSC) can be used to estimate ZEP temperatures of semicrystalline polymers without the need for any chemical and physical treatments before the measurement. Furthermore, if the influence of superheating during melting is considered by accounting for the heating rate dependence of the melting temperatures, ZEP melting temperatures can be estimated by extrapolating the observed melting temperature to a heating rate of zero.12−17 An FSC method based on a chip calorimeter was developed by Schick et al.18,19 at the University of Rostock while recently a comparable instrument was developed by Mettler-Toledo: the

1. INTRODUCTION Analyses of the higher order structures of polymeric materials are important in industrial processes and material design as well as in academic pursuits. Conventional different scanning calorimetry (DSC) is a powerful method for estimating the crystallinity and melting temperature of semicrystalline polymers.1−3 However, the slow heating rate employed in conventional DSC changes the structures of semicrystalline polymers due to cold crystallization and reorganization. Therefore, it is difficult to estimate the zero-entropy-production (ZEP) melting temperature using conventional DSC. Here, ZEP melting is the melting of a metastable, folded-chain lamellar crystals without any increase in entropy (i.e., cold crystallization, reorganization, and superheating) during melting. Thus, if the lamellar thickness of the sample is distributed in a wide size range, the ZEP melting temperature is distributed in a wide temperature range, too. Todoki et al. reported a method for obtaining the ZEP melting temperatures of drawn polyamide 6 fibers.4 In this method, samples are irradiated with γ-rays in acetylene gas to introduce cross-links into the amorphous region and to suppress cold crystallization and reorganization. Unfortunately, this method is only applicable for polyamides and involves both chemical and physical treatments. Fast-heating is another way to determine the ZEP melting temperature. Marand et al. reported the heating rate dependent © XXXX American Chemical Society

Received: October 15, 2015 Revised: November 23, 2015

A

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Macromolecules Flash DSC 1.20 The accuracy of the electrical and thermal characterization using the chip calorimeter for Flash DSC 1 has been discussed in a previous study.20−22 It has become clear that the intrinsic melting behaviors of semicrystalline polymers do not cause cold crystallization and reorganization during heating using FSC. Schick et al. reported the crystallization and melting behaviors of polymeric materials (e.g., poly(ethylene terephthalate) (PET), isotactic polypropylene (iPP), poly(butylene terephthalate) (PBT), polyamide, and polycaprolactone (PCL)) using fast scanning calorimetry.23−27 The author of this study also succeeded in determining the ZEP melting temperature of polyacrylonitrile (PAN).15 However, when considering the ZEP melting temperature distributions of polymeric materials, thermal lag generates serious problems during FSC measurements. Toda et al. reported superheating and thermal lag effects during melting under fast-heating for polymers and metals.12−14,28 According to their study about the thickness-dependent melting behavior using FSC, the thermal lag during fast-heating is negligible if the sample thickness is sufficiently small (e.g., below 1 μm). Under these conditions, the observed broad distribution of melting temperature arises from the broad size distribution of the metastable, folded-chain lamellar crystals. Moreover, if the influence of not only thermal lag but also superheating is excluded from the FSC curve of a semicrystalline polymer, ZEP melting temperature and its temperature distribution will be obtained. Using the ZEP melting temperature distribution data, it will be possible to determine the lamellar thickness distributions (LTDs) of several polymers such as polyethylene (PE), PP, and PET. The LTDs have been determined for wellcrystallized PE and PP using conventional DSC measurements.29−34 However, these results seem to overestimate the LTD, likely due to insufficient consideration of cold crystallization, reorganization, superheating, melting kinetics, and thermal lag. In this study, a method for obtaining the LTD of linear PE was developed using FSC. The melting kinetics were considered, and a deconvolution method was adopted to estimate the distribution of ZEP melting temperatures. Finally, the distribution of ZEP melting temperatures was converted to LTD using the Gibbs−Thomson equation. Small-angle X-ray scattering (SAXS) and low-frequency Raman spectroscopy were used to confirm the average lamellar thickness of the sample.

The sample was microtomed from a pellet at room temperature to a thickness of 1 μm. The sample was placed directly on the sensor of the FSC instrument and premelted at 160 °C to establish a good thermal contact between the sample and the sensor. A sample mass of 9 ng was estimated from the ratio of the heats of fusion obtained from conventional DSC and FSC measurements. Here, the sample mass determined from the area of the microscope image is consistent with that estimated from the heats of fusion. If the sample thickness is sufficiently small, the total thermal gradient across the sample will be less than 1 °C, even at the fastest applied heating rate of 10 000 °C s−1.12−14 Consequently, it is not unrealistic to examine the melting behavior at fast-heating rates in this study. The measured sample was cooled from the molten state (i.e., held at 160 °C for 5 min) to room temperature at a cooling rate of 10 °C min−1 using a Flash DSC 1 before the FSC measurements. SAXS measurements were carried out at room temperature on line BL03XU in the synchrotron radiation X-ray facility SPring-8 at the Advanced Softmaterial BL Consortium. An R-AXIS VII imaging plate (Rigaku Co., Tokyo, Japan) was used as the detector system. The acquisition time and wavelength λ were 10 s and 0.1 nm, respectively. The q range covered in the SAXS measurements was 0.04−3.5 nm−1, where q is the length of a scattering vector defined by q = 4π sin(θ)/λ (here 2θ is the scattering angle). The SAXS data were corrected for the sample absorption and background scattering from air and the windows of vacuum passes. The low-frequency Raman spectrum was obtained using a Jobin Yvon T64000 triple Raman spectrophotometer with 1800 grooves mm−1. The 514.5 nm line of an argon ion laser was used as the excitation source. The spectra were recorded under the condition of pseudoback scattering in the lower frequency region of 50−200 cm−1. All observations were performed at room temperature.

3. RESULTS AND DISCUSSION 3.1. Conventional DSC. Figure 1a shows the conventional DSC cooling trace of a linear PE sample molded for 5 min at

2. EXPERIMENTS Linear PE (NBS SRM 1475) with a density of 0.978 g cm−3 at 23 °C35 was used in this study. The average molecular weights were Mw = 52 000 and Mw/Mn = 2.9.35 The measured sample was cooled from the molten state (i.e., held at 160 °C for 5 min) to room temperature at a cooling rate of 10 °C min−1. The samples for conventional DSC, SAXS, and low-frequency Raman spectroscopy measurements were prepared under the temperature control of conventional DSC. In contrast, the thin sample for the FSC measurement was prepared separately before the measurement. Conventional DSC measurements were carried out using a Q1000 differential scanning calorimeter from TA Instruments. Dry nitrogen gas was purged through the cell at a flow rate of 50 mL min−1. The sample mass was 10 mg, and the scanning rate (β) was 10 °C min−1. No changes in sample mass were observed after the measurements. FSC measurements were carried out using a Flash DSC 1 from Mettler-Toledo; the instrument was equipped with a chip sensor, namely Multistar UFS1 XENZ14 30439. The sampling rate was 10 kHz. The value of β was varied from 1000 to 10 000 °C s−1 with a constant flow (10 mL min−1) of dry nitrogen gas over the chip sensor.

Figure 1. Conventional DSC traces for a linear PE sample (a) cooled from the molten state (160 °C) at a cooling rate of 10 °C min−1 and (b) reheating run at a heating rate of 10 °C min−1. Green dotted line is used to determine the heat of fusion of 212 J g−1.

160 °C. An exothermic peak due to crystallization was observed. The onset and peak exotherm temperatures were 121 and 114 °C, respectively. Figure 1b shows the reheating trace. The endotherm covered a wide temperature range, and the endotherm peak (Tm) was observed at 134 °C. The total heat of fusion of the sample was 212 J g−1. The total crystallinity, Φ, was calculated as follows: Φ= B

ΔHm ΔHm0

× 100 (1) DOI: 10.1021/acs.macromol.5b02278 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules where ΔHm (212 J g−1) is the observed total heat of fusion and ΔH0m (293 J g−1) is the heat of fusion for perfect crystalline PE2. To be exact, crystallization determination as described above is known to be inadequate because the temperature dependence of heat of fusion and intersection between the baseline and the trace are not considered in eq 1. However, according to a previous study,36 these influences for calculating the crystallinity are negligible; the limiting factor as regards accuracy is the instability of the calorimeter. For this reason, we adopt eq 1 for calculating the crystallinity. Finally, a Φ of 72% was determined from conventional DSC. 3.2. FSC. Figure 2 shows the heating rate dependence of FSC traces after crystallizing the sample from 160 °C to room

calculate the size distribution of lamellar crystals, first, the influence of superheating was calibrated from the FSC curves as mentioned in the next section. 3.3. Superheating and Melting Kinetics. The influence of superheating is considered in this section. The heating rate dependence of the melting temperature of the superheated crystals was originally suggested by Schawe et al.17 and extended to general cases by Toda et al.16 as follows: Tm = TZEP + Acβ z

with 0 < z ≤ 0.5

(2)

where Ac is a constant, z is an exponent, β is a heating rate, and TZEP is the ZEP melting temperature. Figure 3 shows the

Figure 3. Melting peak temperature of a linear PE sample (filled squares) plotted against heating rate, β, with a power, z, of 0.37. The best linear fit is Tm = 125 + 0.144β0.37.

Figure 2. Heating rate dependence of FSC traces for a linear PE sample. The curves were normalized by the sample mass of 9 ng. Black, red, green, and blue traces are the results for heating rates of 1000, 2000, 3000, and 5000 °C s−1, respectively.

relationship between peak melting temperature and βz for the linear PE sample. A linear relation was obtained at an Ac value of 0.144 and a z value of 0.37. The z value obtained in this study is in good agreement with the reported values.12 In that study, the sample also crystallized in the same temperature region as found in this study. Besides, a TZEP of 125 °C was obtained by extrapolating the plots to a heating rate of zero (yaxis intercept in Figure 3). The calculated heat flow rate of melting (F0(Δt)) was derived using superheating parameters (i.e., Ac of 0.144 and z of 0.37) derived from the peak endotherm temperatures. First, the change in crystallinity, ϕ, of crystalline polymers, having singlesized lamellae, can be expressed16 as follows:

−1

temperature at a cooling rate of 10 °C min . The total heat of fusion was independent of the heating rate; however, due to the influence of superheating, the melting peak temperature shifted toward higher temperature with increasing heating rate. Note that superheating is the melting of a crystal at a temperature above that expected under equilibrium conditions. Furthermore, the peak melting temperatures in Figure 2 are lower than the Tm obtained from conventional DSC (compare Figure 2 with Figure 1b). The width of the melting peak also becomes narrower than that from conventional DSC. These results suggest that cold crystallization and reorganization are suppressed during fast-heating. Furthermore, according to Toda et al.,12 reorganization during the heating process was fully suppressed at the crystallization temperature above 100.5 °C, when the heating rate increases above 200 °C s−1. As shown in the cooling curve of Figure 1b, the crystallization peak temperature of this sample is 114 °C. Note that the linear PE used in this study is the same as that from Toda et al.12 These results also suggest that reorganization during heating is fully inhibited for the FSC measurements. In addition, it is known that since the sample mass and thickness are sufficiently small, the broad, heating rate dependent melting behavior of the polymer crystal is not due to thermal lag.12−14,28 The influence of thermal lag is negligible for the sample mass of 9 ng used in this study. Here, the sample mass was calculated from the heat of fusion from the FSC measurement by comparison to that from the conventional DSC of 212 J g−1. For the aforementioned reasons, FSC traces in Figure 2 strongly suggest the presence of a broad size distribution for the metastable lamellar crystals of the linear PE sample. To

dϕ = −a(ΔT ) y ϕ dt

with y ≥ 1

(3)

ΔT = β Δt

(4)

Then, the melting trace of DSC can be calculated as follows: F0(Δt ) = ΔHm

dϕ = −ΔHma(β Δt ) y ϕ d(Δt )

(5)

where a and y are constants, ΔT is the degree of superheating, ΔHm (212 J g−1) is the observed total heat of fusion from conventional DSC, β is a heating rate, and Δt is the superheated time. Equation 3 is the kinetic equation of crystallinity, which can be rewritten as follows: ⎡ ⎛ ⎞ y + 1⎤ Δt ϕ = ϕ0 exp⎢ −⎜ ⎟ ⎥ ⎢⎣ ⎝ τc ⎠ ⎥⎦ C

(6) DOI: 10.1021/acs.macromol.5b02278 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules τc =

⎛ y + 1 −y⎞1/(y + 1) ⎜ ϕ ⎟ ⎝ a ⎠

3.4. Deconvolution Method. A deconvolution method was adopted to estimate the distribution of ZEP melting temperatures. The calculated DSC traces shown in Figure 4 are considered to be the thermal responses of the single-sized lamellae with melting temperature Tm. Then, the observed DSC trace is described below. First, the total crystallinity, Φ(t), is described as

(7)

The characteristic time, τc, can be determined from the shift in temperature, ΔTm, due to superheating. Both Ac and z were obtained from the linear extrapolation of ΔTm vs βz (see Figure 3). According to Toda,28 the values y and a can be calculated from the experimental values of Ac and z as follows: ⎛ y + 1 ⎞1/(y + 1) ΔTm = βτc = ⎜ β⎟ = Ac β z ⎝ a ⎠

z=

∫ ϕ(t , Tm) dTm

(12)

where ϕ(t,Tm) is the crystallinity of single-sized lamellae with melting temperature Tm at time t. ϕ(t,Tm) is written as follows:

(8)

1 ≤ 0.5 y+1

⎛ y + 1 ⎞1/(y + 1) ⎟ Ac = ⎜ ⎝ a ⎠

Φ(t ) =

(9)

(10)

Accordingly, eq 5 can be rewritten using eqs 6 and 7 as follows: ⎤ ⎡ aβ y F0(Δt ) = −ΔHma(β Δt ) y ϕ0 exp⎢ − (Δt ) y + 1⎥ ⎦ ⎣ y+1

⎡ aβ y ⎤ ϕ(t , Tm) = ϕ0(tm) exp⎢ − (Δt ) y + 1⎥ ⎣ y+1 ⎦

(13)

⎧ ΔT for Δt > 0 ⎪(t0 + t ) − tm = β Δt = ⎨ ⎪ ⎩0 for Δt ≤ 0

(14)

where ϕ0(tm) corresponds to the distribution function of the single-sized lamellae with melting temperature Tm (= βtm). In the linear system, the relation between the distribution function ϕ0(tm) and calorimeter-recorded line profile F(t) is given by the equation

(11-1)

Here, y and a are calculated from eqs 9 and 10, respectively. F0(Δt) corresponds to the calculated heat flow rate of melting (i.e., the calculated DSC trace) of the single-sized lamellae as a function of time. As mentioned above, the kinetic parameters Ac and z were derived from the heating-rate dependence of the melting peak temperature. Equation 11-1 can also be written as a function of ΔT (= βΔt) as follows:

F(t ) = −ΔHm =

⎡ ⎤ a F0(ΔT ) = −ΔHma(ΔT ) y ϕ0 exp⎢ − (ΔT ) y + 1⎥ ⎣ (y + 1)β ⎦

dΦ = −ΔHm dt

∫ ddt ϕ(t , Tm) dTm

∫ βϕ0(tm)F0(Δt ) dtm

(15)

where t is time, and F0(Δt) is the impulse response for the melting of single-sized lamellae with melting temperature Tm. The melting kinetics calculated from eqs 11 were adopted for F0(Δt). ϕ0(tm) corresponds to the distribution function of the ZEP melting temperature, and F(t) is the observed FSC trace (see Figure 2). Equation 15 is also considered to be the fundamental equation for convolution. Here, the deconvolution technique allows the determination of ϕ0(tm) from F(t) and F0(Δt) using Fourier transformation.37 This technique has been applied for the impulse response of the calorimeter in quasiisothermal systems.38,39 The present study is the first example of applying this method to FSC melting traces. The fundamental equations for the deconvolution are as follows:

(11-2)

Figure 4 shows the heating rate dependence of the calculated melting traces. It is clear that if the LTD is monodispersed, the

∫0 ∫0

∫0 Figure 4. Heating rate dependence of the calculated DSC curves for a linear PE composed of single lamellae crystals (i.e., lamellar crystals having the same thickness as each other). Black, red, green, and blue traces are the results for heating rates of 1000, 2000, 3000, and 5000 °C s−1, respectively.

∞

∞

∞

F(t )e−iωt dt = F ̅ (ω)

(16)

βϕ0(tm)e−iωt dtm = ϕ0̅ (ω)

(17)

F0(Δt )e−iωt dΔt = F0̅ (ω)

(18)

F ̅ (ω) = F0̅ (ω) × ϕ0̅ (ω)

(19)

where i is √−1 and ω is angular frequency. Using eqs 15−19, the distribution function can be obtained as βϕ0(tm) =

1 2π

∞

∫−∞ {F ̅(ω)/F0̅ (ω)}eiωt dω

(20) −1

Here, the dimension of the distribution function is T . Note that the analysis is constructed under the following assumptions: (1) No thermal gradient in the direction of sample thickness during the heating. To overcome the

melting of single lamellae crystals is completed within a narrow temperature range (4−7 °C) despite the fast-heating rate. The results also confirmed that the peak temperature increases with increasing heating rate. D

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shapes of the curve are similar below 120 °C regardless of the heating rate. However, the peak heights and shapes were different in the high-temperature region above 130 °C. Two reasons are cited as a possible cause. First, the parameters of superheating kinetics (i.e., Ac and z) might depend on the size of the lamellae (i.e., the melting temperature). The second reason is thermal lag. The sample was sufficiently small for this study; however, the bare thermal lag might affect the deconvolution. The influence of thermal lag is smallest for the result of 1000 °C s−1. Figure 5b shows the deconvolution and experimental curves at a heating rate of 1000 °C s−1. The heat flow of the deconvolved FSC traces enlarged the normalized ϕ0(tm), as the total heat of fusion became the same as the experimentally observed values. In the temperature region above 110 °C, the temperature shifts by ∼2 °C toward the lower temperature side after deconvolution. 3.5. Determination of Lamellar Thickness. A method used to determine the LTD was also developed in this study. The relation between melting temperature and lamellar thickness is given by the Gibbs−Thomson equation3 as follows:

influence of thermal gradient, the measured sample was obtained by microtoming to a thickness of ∼1 μm. According to Toda,12−14,28 the effect of the thermal lag should be negligible in this study. (2) Physical properties (i.e., density, heat capacity, thermal diffusivity, and thermal conductivity) are uniform during melting. (3) Linear heat flow over the surface of the sample takes place; in other words, the heat radiation and convection are ignored. (4) The parameters of superheating kinetics (i.e., Ac and z) obtained from the heating rate dependence of peak melting temperature Tm are constant during heating. In other words, both Ac and z are independent of lamellar thickness. In previous studies, the heating-rate dependence of the characteristic time (τ) of melting suggested that τ becomes shorter at low temperatures.40,41 It was concluded that the melting process occurs more easily for crystallites with lower melting points. Therefore, Ac and z should also show temperature dependencies. However, it is worth confirming the relation between melting temperature distribution and the parameters of superheating kinetics. The deconvolution method of this study can also be applied to the determination of the excess heat capacity curve excluding the influence of superheating, as caused by the fast-heating rate. The aforementioned assumptions were adopted in this study. Figure 5a shows the normalized melting distribution functions of the linear PE sample at different heating rates. The peak at 125 °C, which corresponds to the TZEP of the sample (see Figure 3), was nearly independent of the heating rate. The

Tm = Tm0 −

2σeTm 0 lΔHm0

(21)

where is the equilibrium melting temperature, σe is the top and bottom fold surface free energy, l is the lamellar thickness, and ΔH0m is the heat of fusion per cubic centimeter of the perfect crystal. The well-known value of σe is 0.09 J m−2 for PE, which was obtained from polymer nucleation theory.42,43 Furthermore, the T0m and ΔH0m of PE are 414.6 K and 293 J cm−3 (we assumed the density is 1 g cm−3 for the perfect crystal of polyethylene), respectively.2 In this study, considering the difference in the heating rate dependence of deconvolution analysis in Figure 5a, the deconvolved FSC curve at a heating rate of 1000 °C s−1 was used to determine the LTD. The conventional DSC curve was also applied for LTD analysis. Figure 6 shows the LTD of the linear PE sample. The lT0m

Figure 6. LTD of a linear PE sample. The black curve was determined from the deconvolved FSC trace, and the blue curve was obtained directly from the conventional DSC trace.

integration of the LTD curve is the crystallinity. The LTD obtained from the conventional DSC trace is broad, and the peak length (lpeak) is larger than that of the LTD obtained from the FSC trace. As mentioned below, the average lamellar thickness estimated from SAXS and Raman spectroscopy is in good agreement with the FSC results. The LTD determined from the conventional DSC curve overestimated the lamellar

Figure 5. (a) Distribution functions of melting temperature for the linear PE sample. Black, red, green, and blue traces are the results for heating rates of 1000, 2000, 3000, and 5000 °C s−1, respectively. (b) DSC traces between before (black curve) and after (pink curve) deconvolution for a heating rate of 1000 °C s−1. E

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thickness (la < lc).Therefore, the lamellar thickness, lc = llp − la, is determined to be 18 nm from SAXS analysis. Figure 8 shows the low-frequency Raman spectrum for the PE sample. A sharp peak is observed in the low-frequency

thickness due to the influence of cold crystallization and reorganization during the heating process. Here, average lamellar thickness can be estimated as follows:

L1 =

L2 =

∑i ϕili ∑i ϕi

(22)

∑i ϕili 2 ∑i ϕil

(23)

The dispersion of lamellar thickness, L2/L1, is determined as the ratio of L2 to L1. In the linear PE sample, the values of L1, L2, and dispersion were 15 nm, 18 nm, and 1.2, respectively. The lpeak of 15 nm is close to L1. Furthermore, the dispersion of 1.2 suggests that the lamellar thickness of the sample is dispersed with a narrow LTD. Accounting for the wide dispersion of the molecular weight of the sample (Mw/Mn = 2.9), the LTD can be considered to be independent of molecular weight. The relation between lamellar thickness and molecular weight awaits further investigation, including molecular weight dependence of LTD analysis. Figure 7a shows the SAXS pattern of the linear PE sample. The broken line corresponds to the background scattering

Figure 8. Low-frequency Raman spectrum of the linear PE sample.

region at ∼10 cm−1, which is assigned to the longitudinal acoustic vibration mode (LAM). Using this peak, the average lamellar thickness can be determined as follows:45 m lLAM = 0.5 E (2c Δv) ρ

()

(24)

where Δv is the wavenumber of LAM, lLAM is the straight chain length, m of 1 is odd order, c is the velocity of light, E (240 GPa) is the elastic modulus of a PE crystal, and ρ (1000 kg m−3) is the density of a PE crystal. The average lamellar thickness of 22 nm was determined from the analysis of lowfrequency Raman spectra. The average lamellar thicknesses obtained from SAXS (lc = 18 nm) and low-frequency Raman spectroscopy (lLAM = 22 nm) are in good agreement with L2 of 18 nm from FSC.

4. CONCLUSIONS FSC was used to examine the melting behavior of wellcrystallized linear PE sample. By taking sufficiently small sample mass and thickness, the influence of thermal lag during fastheating is thought to be negligible. Furthermore, by fastheating, reorganization and cold crystallization are fully suppressed. Second, based on the experimental dependence of melting peak temperature on fast-heating rate, melting kinetics of single-sized lamellae without cold crystallization and reorganization were calculated for several heating rates. The results were then applied for deconvolution analysis, and the deconvolved FSC traces, which correct for the influence of superheating, cold crystallization, and reorganization, were determined. Under such precautions, the resulting FSC heating curve reflects the melting point distribution of metastable-butnot-reorganized folded-chain crystals. The developed method will be applied to other semicrystalline polymers in the future. Finally, the Gibbs−Thomson equation was applied to calculate the LTD from the melting point distribution. An L1 of 15 nm, L2 of 18 nm, and dispersion (L2/L1) of 1.2 were determined. The lpeak of 15 nm is close to L1. The dispersion of 1.2 suggests that the lamellar thickness of the sample is dispersed with a narrow LTD. The average lamellar thicknesses

Figure 7. (a) SAXS pattern of the linear PE sample. (b) Correlation function calculated from the SAXS pattern.

pattern. Figure 7b shows the correlation function determined from the SAXS pattern using a Fourier transform technique. The peak at 24.1 nm corresponds to the long period length, llp, of the linear PE sample.44 In this study, the intersection at 6 nm was assumed to be the thickness of the amorphous layer, la. Here, if the crystallinity of the sample is higher than 50%, the length of the amorphous layer is shorter than the lamellar F

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obtained from SAXS (lc = 18 nm) and low-frequency Raman spectroscopy (lLAM = 22 nm) are in good agreement with the value of L2 of 18 nm calculated from FSC.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected], [email protected]; Tel +81(77)533-8603 (Y.F.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors offer special thanks to Dr. Hideaki Takahashi, Mr. Kazuyuki Okada, Dr. Takeshi Nakagawa, and Dr. Kazuhiko Ishikiriyama for their meaningful advice. The authors express their appreciation to Ms. Tomoko Kinoshita for her invaluable assistance. The synchrotron radiation experiment was performed at the FSBL BL03XU in SPring-8. This work was financially supported by Toray Research Center, Inc., Japan.

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REFERENCES

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DOI: 10.1021/acs.macromol.5b02278 Macromolecules XXXX, XXX, XXX−XXX