Temperature Dependence of Lamellar Thickness in Isothermally

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Temperature Dependence of Lamellar Thickness in Isothermally Crystallized Poly(butylene terephthalate) Takashi Konishi,*,† Waki Sakatsuji,‡ Koji Fukao,§ and Yoshihisa Miyamoto† †

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan Division of Physics and Astronomy, Graduate School of Physics, Kyoto University, Kyoto 606-8502, Japan § Department of Physics, Ritsumeikan University, Noji-Higashi 1-1-1, Kusatsu 525-8577, Japan ‡

ABSTRACT: In order to clarify the crystallization from the melt in poly(butylene terephthalate) (PBT), the heating processes of isothermally crystallized PBT have been investigated by differential scanning calorimetry (DSC), wide-angle X-ray diffraction, and small-angle X-ray scattering. Both the melting temperature and the crystallization temperature dependences of the lamellar thickness show two distinct regions separated by 208 °C, which indicate two equilibrium transition temperatures, 270 and 335 °C. On the basis of the proposed crystallization model, it is interpreted that the crystallization proceeds through the mesophase below 208 °C and directly from the melt above 208 °C. The experimental results also show the constant excess length of lamella over the experimental temperature range.

1. INTRODUCTION When semicrystalline polymers crystallize by quenching from the melt or by heating up from the glass, the morphology is thin crystalline plate, so-called “lamella”,1 or globular crystallite, “nodule”,2 in the nanometer scale separated by an amorphous region due to the entanglements of polymer chains. The melting temperature, Tm, of such crystallites is lower than that of the infinitely large crystal due to the existence of the interface between the amorphous and crystal. The formation mechanism of polymer crystal has been studied for a long time but not fully understood. The lamella is a typical crystalline morphology of polymer, and the stacked lamellae constitute a spherulite in the micrometer scale. The relation between the lamellar thickness, S , and Tm is well-known as the Gibbs−Thomson relation Tm = Tm0 −

2σeTm0 1 ΔHm S

treated as the secondary nucleation process by Lauritzen and Hoffmann (L−H model).3 According to the L−H model, S c obtained at Tc is kinetically determined as Sc =

C1 Tc0

− Tc

(1)

+ C2

(2)

where C1 and C2 are constants and T0c is the equilibrium crystallization temperature, where the infinitely thick lamellae grow. The formation mechanism of the lamellae has been © XXXX American Chemical Society

ΔHm(Tm0 − Tc)

+ δS (3)

where δS is the excess thickness. Comparing eq 2 with eq 3, T0c equals to T0m and δS hardly depends on Tc in the L−H model. Keller and co-workers have studied the formation mechanism of single crystal of polyethylene (PE) under high pressure and high temperature.4,5 They have suggested the crystallization model in which the transient mesophase forms at the growth front of lamella below a temperature, TX, and the transition from the mesophase to the crystal is thermodynamically treated.5 According to Keller’s model, T0c is located above T0m and represents an equilibrium mesophase−crystal transition temperature, T0MC. Strobl and co-workers have also confirmed the existence of T0MC in many polymers and have proposed the crystallization model through the mesophase in which the granular mesophase forms at the growth front of lamella, transforms into the granular crystal, and merges into the lamella.6−8 The excess length δS is not taken into account in Strobl’s model. We have also confirmed the existence of T0MC in isotactic polypropylene (iPP) and poly(butylene terephthalate) (PBT)

where T0m is the equilibrium melting temperature, which is the melting temperature of the infinitely large crystal, σe is the folding surface free energy, and ΔHm is the heat of fusion. The average lamellar thickness, S c, obtained at the crystallization temperature, Tc, is empirically given by Sc =

2σeTm0

Received: January 19, 2016 Revised: February 23, 2016

A

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3. RESULTS AND DISCUSSION 3.1. Melting Behavior of the Isothermally Crystallized PBT. Figures 1a and 1b show the DSC curves of the heating

crystallized from the glassy states whose morphologies are nodules.9,10 Furthermore, we have studied the crystallization of PBT from the melt and have proposed that the crystallization model for the lamellar formation mechanism through the mesophase.11 According to the model, amorphous PBT transforms into the α-crystal through the mesophase below TX = 208 °C and directly transforms above TX, and the lamellae formed both below and above TX have δS due to the thickening of the stem at the growth front. The relation between S c and Tm are, however, not fully clarified since all Tm’s of S c’s in a previous report are located above TX. PBT has two triclinic crystalline forms, α-form and βform,12−18 and the smectic liquid crystal.19,20 The α-crystalline structure is mainly obtained when PBT is crystallized from the melt or from the glass. The β-crystalline structure forms by stretching the α-form and transforms reversibly to the α-form on removal of the strain. The smectic phase is obtained by stretching the glass below room temperature and transform into the α-form by heating. The c-axis (fiber axis) length of the α-crystalline unit cell, ca. 11.6 Å, is similar to that of the smectic periodicity, ca. 11.7 Å, and is shorter than that of the β-form, ca. 13.0 Å.19,20 The aims of this report are to experimentally obtain the relation among Tm, Tc, and S c in PBT and to confirm whether the relation obeys the proposed model, especially in the lowtemperature region. For these purposes, the melting behavior and the T-dependence of S c formed at Tc during heating were investigated by differential scanning calorimetry (DSC), smallangle X-ray scattering (SAXS), and wide-angle X-ray diffraction (WAXD), and then the results are interpreted by the proposed crystallization model, which takes into account the constant excess length of lamella.

2. EXPERIMENTAL SECTION The polymer used in this study was PBT with Mv = 38 000 purchased from Sigma-Aldrich Co. Ltd. The melting process of PBT isothermally crystallized at Tc between 140 and 225 °C after melting at 280 °C for 2 min was measured by DSC (Shimadzu DSC60), WAXD, and SAXS under nitrogen gas atmosphere. For Tc above 165 °C, PBT was isothermally crystallized after cooling at 70 K/min, and then the crystallized PBT was heated up from Tc (WAXD, SAXS) or from the temperature lower than Tc by 20 K (DSC) to 280 °C at a rate of 10 K/ min. The cooling rate during cooling from Tc to Tc − 20 K was 40 K/ min. The DSC measurements at several heating rates, β, 2−70 K/min, were also carried out for the melting process of PBT isothermally crystallized at 188.2 °C. For Tc below 170 °C, in order to prevent the PBT sample from crystallization on cooling to Tc, the PBT film was quenched on the hot stage at Tc immediately after melting, fully crystallized, and cooled to room temperature. The heating process of the PBT film crystallized below 170 °C was measured from room temperature. The calibrations of temperature and heat flux for DSC measurements were performed using indium and alumina, respectively. Aluminum pans were used for the sample and the reference pan. The SAXS and WAXD measurements were performed using the beamline BL-40B2 at SPring-8, Nishiharima, Japan. The SAXS covers the range of scattering vector, q, from 0.008 to 0.2 Å−1 (q = 4πλ−1 sin θ, λ and θ being X-ray wavelength and scattering angle, respectively). The wavelength λ in SAXS and WAXD was 1.0 Å−1. The temperature of the samples for the SAXS measurements was controlled using a Linkam LK-600PH. The discussion in this study is based on the experimental results at the heating rate 10 K/min except for the DSC results of the various heating rate (Figure 3).

Figure 1. (a) DSC curves of the heating processes of PBT’s crystallized at (a) Tc = 188.2 °C and (c) Tc = 147.3 °C. The curves in (b) and (d) are the enlarged curves between 180 and 240 °C in (a) and between 140 and 240 °C in (c), respectively. The solid and broken lines in (a) and (b) indicate the amorphous and crystalline Cp lines reported by Pyda and co-workers,44 respectively. The filled and open arrows indicate the onset and the top of peaks, respectively. See the text for further explanations.

process of PBT isothermally crystallized at Tc = 188.2 °C. The curve has three endothermic peaks, Tm1, Tm2, and Tm3 peaks in the ascending order, and one exothermic peak, a Texo peak, between Tm2 and Tm3. Similar multiple melting behaviors can be observed in many polymeric materials.21−30,33−43 The multiple melting behavior is ascribed to the existence of two or more distributions of lamellar thickness or perfectness and to the melting and the recrystallization of the lamellae during heating. Two or more distributions of the lamellar thickness in the isothermally crystallized polymer have been actually B

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Macromolecules observed by electron microscopy.31,32 The Tm2 peak is absent in the DSC curve of PBT crystallized from the glass whose crystalline morphology is nodule rather than lamella,10 which indicates that the existence of the Tm2 peak depends on the crystalline morphology. The Tm1 and Tm2 peaks should be regarded as the melting peaks of thin and thick lamellae with the lamellar thicknesses, S1 and S2 , respectively, formed at Tc.11,21−29 The Tm1 peak is well-known as an annealing peak. The possibility of assignment of the annealing peak to partial melting30,33−38 or to relaxation of the mobile amorphous fraction39,40 is reported. Recently, DiLorenzo and co-workers have suggested that the annealing peak is ascribed not to a single process, but to simultaneous partial melting and mobilization of the rigid amorphous fraction.41−43 The onset and peak positions of Tm1 peak, Tm1(onset) and Tm1(top), are located ca. 4 and 7 K above Tc, respectively. The solid and broken lines in Figures 1a and 1b indicate the amorphous and crystalline isobaric specific heat, Cp, respectively, reported by Pyda and co-workers.44 The observed DSC curve of the crystallized PBT is located above both the amorphous and the crystalline Cp lines between Tm1 and Tm2 and thus the lamellae thinner than S2 melt during heating between Tm1 and Tm2. Similar results have been reported by some researchers.27,28 The DSC curve below Tm1 is also located above the both Cp lines which may indicate that the melting process of the crystalline lamellae formed during the cooling process down to 168 °C in the DSC experimental procedure. Figures 1c and 1d show the DSC curves of the heating process of PBT isothermally crystallized at Tc = 147.3 °C. The curve also has the Tm1 and Tm3 peaks but the Texo peak instead of the Tm2 peak. The curve in Figure 1d shows two onsets of Texo peak at 188.1 and 197.6 °C represented by Texo(onset1) and Texo(onset2) in the ascending order, respectively. The Tc-dependence of Tm’s and Texo on Tc of PBT crystallized at several Tc’s are represented in Figure 2. The DSC thermograms for several Tc were reported in ref 11. In this study, the DSC curves for Tc lower than 160 °C and the Tcdependence of Texo are examined in detail. Tm1(onset) and Tm1(top) are located ca. 3 and 7 K above Tc, respectively, and the Tm3 peak is independent of Tc and merges into the Tm2 peak above Tc = 215 °C. These results have been reported by numerous researches, and the origin of the Tm3 peak is ascribed to the remelting of the crystal recrystallized during heating.21−29,45 The Tc-dependence of Tm2 slightly changes at Tc = 208 °C, and Tm2 is not observed below 185 °C. Figure 2 shows that two Texo(onset)’s are observed below Tc = 185 °C and that the difference between Texo(onset1) and Texo(onset2) increases with decreasing Tc. In order to clarify the origin of two Texo(onset)’s, we have performed the DSC measurement of PBT crystallized at 188.2 °C at several heating rates, and the results are shown in Figures 3a and 3b. The heating rate dependence of Tm peaks is shown in Figure 3c. The Tm1 and Tm2 peaks shift to a higher temperature with increasing β. The shift of Tm3 peak is smaller than those of the Tm1 and Tm2 peaks, and the Tm3 peak shifts to a slightly higher temperature first decreases and then increases with increasing β around 10 K/min. The Tm2 peak is located in the exothermic peak at a rate of 7 K/min and not observed at a rate of 2 K/ min. Such a complex behavior between the endothermic and exothermic peaks has been reported by Schick and co-workers as the relationship between “the correspondence of the experimental time scale and that of melting−recrystalliza-

Figure 2. Relation among Tm’s, Texo, and Tc. The solid thin line indicates the Tm = Tc line. The open and filled circles indicate the onset and peak temperatures of Tm1 peak, Tm1(onset), and Tm1(top), respectively. The open and filled squares indicate the onset and peak temperatures of Tm2 peak, Tm2(onset), and Tm2(top), respectively. The triangles indicate the peak temperatures of Tm3 peak, Tm3(top). The inversed triangles and the filled diamonds indicate the onset temperatures of Texo peaks, Texo(onset1), and Texo(onset2) in ascending order. The open diamonds indicate the peak temperature of the Texo peaks, Texo(top). The solid thick line indicates the melting line of PBT crystallized from the liquid state given by eqs 8 and 9. The broken line indicates the L−C transition line of PBT crystallization from the mesomorphic phase given by eqs 7 and 9. The chain line indicates the M−C transition line of of PBT crystallized from the mesomorphic phase given by eqs 7 and 10.

tion”.30 The DSC curve at a rate of 2 K/min in Figure 3b shows two onsets of Texo peak: Texo(onset1) and Texo(onset2). The shift of the Tm1 peak dependent on β is ascribed to the superheating behavior.46,47 The degree of the superheating temperature ΔTshift = Tm(β) − Tm(β = 0) increases with heating rate and given as ΔTshift = Aβ z

(4)

where A and z are constants. The thermal lag effects from the equipment can be ignored when z is smaller than 0.5.46 Generally, the value of Tm(β = 0) for Tm1 is regard as Tc.46,47 The dependences of ΔTshift for Tm1(onset) and Tm1(top) on β with Tm1(β = 0) = Tc = 188.2 °C are shown in Figure 3d and can be fitted by eq 4 with A = 0.94 and z = 0.42 and with A = 2.1 and z = 0.44, respectively. Here the superheating analysis is also applied to the dependences of ΔTshift for Tm2 and Texo with Tm2(β = 0) = Texo(β = 0) = 205.5 °C (Figure 3d). The dependence of ΔTshift for Tm2(onset) and Tm2(top) falls onto that for Texo(onset1) at β < 10 K/min and Texo(onset2) at β = 2 K/min, respectively. Thus, Texo(onset1) and Texo(onset2) should correspond to Tm2(onset) and Tm2(top), respectively. ΔTshift for Tm2(onset) at β ≥ 10 K/min and Texo(onset1) at β < 10 K/min and that for Tm2(top) at β ≥ 7 K/min and Texo(onset2) at β = 2 K/min can be fitted by eq 4 with A = 1.0 and z = 0.40 and with A = 3.2 and z = 0.25, respectively. ΔTshift for Tm2(onset) and Texo(onset1) coincides with that for Tm1(onset). From the above result Texo(onset1) should be regarded as Tm2(onset), that is, Tm of S2 formed at Tc, and the relation can be applied to Texo(onset1) for Tc < 185 °C in Figure 2. The origin of Texo(onset2) will be discussed again later. In order to examine the melting behavior from the C

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Figure 3. (a) DSC curves in the heating process of PBT crystallized at 188.2 °C at several heating rates, 2−70 K/min, (b) the enlarged DSC curves for 2, 7, and 10 K/min in (a), (c) the heating rate dependence of Tm’s and Texo, and (d) the heating rate dependence of the shift temperature ΔTshift for Tm’s and Texo. The circles, squares, and triangles indicate Tm1, Tm2, and Tm3, and the filled and open symbols indicate the onset and the top of peaks, respectively. The inversed triangles indicate Texo(onset1), and diamonds represent Texo(onset2). The thick solid and broken lines in (c) and (d) are the fitting lines of ΔTshift for Tm1(onset) and Tm1(peak), respectively, by eq 4. The thin solid and broken lines are those of ΔTshift for Tm2(onset) at β ≥ 10 K/min and Texo(onset1) at β < 10 K/min and for Tm2(top) at β ≥ 7 K/min and Texo(onset2) at β = 2 K/min, respectively.

viewpoint of the lamellar thickness, the SAXS and WAXD results for the heating process of isothermally crystallized PBT will be discussed in the next section. 3.2. Lamellar Thickness of Crystallized PBT during Heating. Figures 4a and 4b show the WAXD profiles of PBT’s quenched to 188.2 °C from the melt and isothermally crystallized for 35 min, respectively. The crystallized PBT has the α-form because the Bragg peaks of α-form at q = 0.66, 1.15, 1.25, 1.46, 1.50, 1.61, 1.65, 1.76, 1.78, and 2.05 Å−1 are observed. The Bragg peaks in the WAXD profile were fitted by the Lorentzian functions. A broad peak at q = 1.65 Å−1 is required to reproduce the observed WAXD intensity, the origin of which is unclear at the moment. The crystallinity of PBT is calculated as 37.5% from Figure 4b. Figures 5a and 5b show the WAXD profiles in the heating process of PBT crystallized at 188.2 °C and the T-dependence of crystallinity calculated from the WAXD result in (a), respectively. Figure 5b shows that the crystallinity keeps constant up to 192 °C, gradually decreases up to 209 °C, further decreases up to 212 °C, slightly increases up to 215 °C, and drastically decreases to 225 °C. The temperatures 192, 209, 212, and 215 °C in Figure 5b correspond to T m1 (onset), T m2 (onset), T m2 (top), and Texo(top), respectively, of the DSC curves in Figure 1. The WAXD result shows that the crystalline phase melts gradually between Tm1 and Tm2, which corresponds to the DSC results in Figure 1. Figure 6a shows the Lorentz factor corrected SAXS curve, Iq2, of PBT crystallized at 188.2 °C for 35 min. The lamellar thickness is estimated from the SAXS profile by the Fourier transformation of the correlation function of the interfaces

Figure 4. WAXD profiles of PBT’s isothermally crystallized at 188.2 °C from the melt for (a) 0 min (just after quenching) and (b) 35 min. The circles in (a) and (b) show the observed intensity. The gray solid line in (b) is the fitting curve reproduced by summing up the component of crystalline (thin solid line) and amorphous phase (broken line).

between amorphous and crystal with Gaussian distributions the “Gaussian correlation model”.48 The SAXS profile is fitted in the q range from 0.015 to 0.1 Å−1 by the model, and the D

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at 188.2 °C in the heating process at a rate of 10 K/min. The peak positions of the Iq2 curves, qmax, shift to a low q-value on heating. The invariant, Q, is defined by1 Q=

∫0

∞

4πq2I(q) dq

= (ρc − ρa )2 ϕc(1 − ϕc)

(5)

where ϕc is the crystallinity, and ρc and ρa are the densities of crystal and amorphous, respectively. Figures 7a−c show the T-

Figure 5. (a) WAXD profiles in the heating process of PBT crystallized at 188.2 °C from the melt at a rate of 10 K/min and (b) Tdependence of crystallinity calculated from the WAXD result in (a). The vertical dotted line indicates the onset temperature of the Tm2 peak in the DSC curves of PBT crystallized at 188.2 °C (Figure 1).

Figure 7. T-dependences of (a) the invariant Q, (b) qmax, and (c) the lamellar thickness S calculated from the results of the SAXS profiles in Figure 6. The vertical dotted lines in (a)−(c) indicate Tm2(onset) for Tc = 188.2 °C.

dependences of Q, qmax, and S , respectively, in the heating process from Tc = 188.2 °C. The values of S are obtained from the SAXS profiles in Figure 6c fitted in the q range from 0.015 to 0.1 Å−1 using the Gaussian correlation model. The invariant Q during heating keeps constant up to 209 °C in Figure 7a. Since the term (ρc − ρa)2 is a monotonic increasing function of T and ϕc is lower than 0.5 from the WAXD results, the constant Q up to 209 °C indicates the decrease in ϕc, i.e., melting behavior, which corresponds to the DSC (Figure 1) and WAXD (Figure 5) results. The decrease and increase in Q from 209 to 225 °C also correspond to those of the crystallinity from the WAXD results, and the temperature 209 °C corresponds to Tm2(onset). The T-dependence of qmax, which approximately equals to 2π/Λ, where Λ is the average long period, in Figure 7b shows that qmax remains constant up to 192 °C, gradually decreases up to 209 °C and decreases much more from 209 °C. The temperatures 192 and 209 °C correspond to Tm1(onset) and Tm2(onset), respectively. The T-dependence of S in Figure 7c shows that S gradually increases with T up to Tm2(onset) and becomes still larger from Tm2(onset). In order to examine the behavior of S for PBT crystallized at a lower Tc, the heating process from room temperature of PBT crystallized at 147.3 °C for 20 min is shown in Figure 8. Figure

Figure 6. (a) Lorentz factor corrected SAXS curve, Iq2, of PBT crystallized at 188.2 °C for 35 min and (b) in the heating process of PBT crystallized at 188.2 °C from the melt at a rate of 10 K/min. The solid curve in (a) is a fitting curve calculated using the “Gaussian correlation model”48 (see text).

following parameters are obtained: the average, S , and the standard deviation of lamellar thickness are 52 and 13 Å, respectively, those of amorphous thickness are 70 and 32 Å, respectively, and the standard deviation of long period is 40 Å (Figure 6a). Figure 6b shows the Iq2 curves of PBT crystallized E

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thin lamellae with S 1 starts to melt and recrystallize, while the thick lamellae with S 2 remain. Above T m2 (onset) or Texo(onset1) the thick lamellae with S 2 start to melt and recrystallize. Thus, the average lamellar thicknesses S formed at Tc slightly thickens below the melting temperature of the thick lamellar thickness S 2, which corresponds to Tm2(onset) or Texo(onset1), and S increases above Tm2(onset) or Texo(onset1). Hereinafter, we regard S 2 as the main lamellar thickness S c formed at Tc. In order to examine the behavior of S above Tm2(onset) or Texo(onset1), the T-dependence of S −1 is shown in Figure 9. Figure 9a shows the relation between S −1 and T during heating in PBT’s crystallized at 188.2 °C (Figure 7c), 169.3 and 147.3 °C (Figure 8b). The relation S −1(T) for Tc = 188.2 °C can be fitted by linear correlation (a thin solid line) above 210 °C, which corresponds to Tm2(onset) in Figure 1b. For Tc = 147.3 °C the relation S −1(T) above 208 °C can be also fitted on the same line as that for Tc = 188.2 °C, and the relation S −1(T) between 190 and 208 °C can be fitted by another linear correlation (a thick solid line). The lower crossover temperature 190 °C for Tc = 147 °C corresponds to Texo(onset1) in Figure 1d. For Tc = 169.3 °C the relation S −1(T) can be similarly fitted by the same lines above 208 °C, and between Texo(onset1) and 208 °C, respectively Figure 9b shows the S c−1 dependences of Tc, Tm2(onset), and Texo(onset1). Two solid lines in Figure 9b are the same as those in Figure 9a. According to the Gibbs−Thomson relation (eq 1), the extrapolation to S −1 → 0 gives T0m. The DSC and SAXS results indicate that Tm2(onset) and Texo(onset1) are the melting temperatures of S c (S 2) for isothermally crystallized PBT. Thus, the T-dependences of S c−1 during heating in Figure 9 show interesting results. Two fitting lines below and above 208 °C give T0m = 335 and 270 °C, respectively. When Tm of the lamellar thickness is higher than 208 °C, the lamella thickens along the fitting line giving T0m = 270 °C with T above Tm2(onset). When that is lower than 208 °C, the lamella thickens along the fitting line giving T0m = 335 °C between Texo(onset1) and 208 °C and then along the line leading to T0m = 270 °C above 208 °C. We will discuss about the two T0m’s and the relation between Tc and S c in the next section.

Figure 8. (a) Iq2’s in the heating process of PBT crystallized at 147.3 °C from the melt at a rate of 10 K/min. (b) T-dependences of S obtained from the results of the SAXS profiles in (a). The broken and chain lines in (b) represent Texo(onset1) and Texo(onset2) decided by the DSC result in Figure 1d, respectively.

8a shows the Iq2 in the heating process at a rate of 10 K/min of PBT crystallized at 147.3 °C from the melt. Figures 8b shows the T-dependences of S fitted by the Gaussian correlation model. The value of S gradually increases with T up to Texo(onset1) = 188 °C determined by DSC results in Figure 1d and becomes still larger from Texo(onset1). The change of the T-dependence of S at Texo(onset1) for Tc = 147.3 °C is similar to that at Tm2(onset) for Tc = 188.2 °C. The DSC results also indicate that Texo(onset1) corresponds to Tm2(onset). The DSC and SAXS results indicate as follows. In the heating process from Tm1(onset) to Tm2(onset) or to Texo(onset1) the

Figure 9. (a) T-dependences of S −1 of PBT crystallized at several Tc and (b) the S c−1 dependences of Tc, Tm2(onset), and Texo(onset1). The circles, triangles, and squares in (a) indicate the T-dependences of S −1 of PBT crystallized at 188.2 (Figure 7c), 169.3, and 147.3 °C (Figure 8b), respectively. The open and filled squares in (b) indicate the relations between S c and Tc and between S c and Texo(onset1), respectively, for the PBT isothermally crystallized at Tc = 147.3, 158.3, and 169.3 °C. The open and filled triangles in (b) indicate the relations between S c and Tc and between S c and Tm2(onset), respectively, for the PBT isothermally crystallized at Tc.50 The thick and thin solid lines indicate the M−C and L−C transition lines from eqs 10 and 9, respectively.11 The thick and thin broken lines indicate the crystallization lines obtained from the eqs 8 and 7, respectively. F

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where δS LC is the excess thickness for Tc > TX. Equation 6 gives the transition temperatures from crystal to liquid, TLC(S ), and to mesophase, TMC(S ), as

3.3. Interpretation of Lamellar Thickness Behavior Using Possible Crystallization Model. In this section, we discuss the experimental results on the basis of the proposed crystallization model11 of crystalline lamella formation through a mesophase at the growth front. The model, which is based on the Keller’s model,5 takes into account the constant excess length δS forming immediately after the transition from the mesomorphic stem into the crystalline one. The formation mechanism of the constant δS has been reported by Rault during crystallization directly from the melt.49 Rault has suggested that the lamellar thickening by the isothermal thickening at the growth front gives the constant δS . We extend the idea of constant δS in the case of the formation process of lamellae with the constant δS through the mesomorphic stems. The model is based on the prerequisites that the mesophase is thermodynamically metastable for sufficiently thick crystals and that the mesomorphic stem is always mobile, while a crystalline stem hardly thickens when surrounded by the crystalline ones. The monomeric free energy differences among the liquid (L), the crystal (C), and the mesophase (M) are given as ΔGi(T , S) =

2σi ΔHi(Ti0 − T ) − S Ti0

0 TLC = TLC −

0 − TMC = TMC

(i = LM, LC, MC)

where ΔHLM, ΔHLC, and ΔHMC are the heats of fusion of L− M, L−C, and M−C transitions, respectively, T0LM, T0LC, and T0MC are the equilibrium L−M, L−C, and M−C transition temperatures, respectively, σLM and σLC are the surface free energies of L−M and L−C interfaces, respectively, and σMC = σLC − σLM. Under the above assumptions, ΔHLM < ΔHLC, T0LM < T0LC < T0MC are given, and σLM < σLC is expected. A temperature TX and a lamellar thickness S X satisfy ΔGLM(TX, S X) = ΔGLC(TX, S X) = ΔGMC(TX, S X) = 0.11 When a polymeric material crystallizes below TX, the crystallization of the lamella proceeds from the melt as follows: (i) The mesomorphic stem with a length S m forms at the growth front of lamella and then thickens due to the decrease in ΔGLM(S ). (ii) When S reaches the length S * satisfying ΔG LM ( S *) = ΔG LC ( S *), the mesomorphic stem transforms into a crystalline one. (iii) Since the crystalline stem contacts with the mobile mesomorphic stem on the growing surface, the crystalline stem just after the transition from the mesophase into crystal can still thicken; the thickening from S * to S c due to the isothermal annealing occurs in the crystalline phase. (iv) When the crystalline stem is surrounded by other crystalline stems, the stem hardly thickens. When the excess thickness in the process (iii) is denoted as δS MC, the determinate lamellar thickness S c is given by 0 2σMCTMC 0 ΔHMC(TMC − Tc)

+ δ SMC

(Tc < TX)

(7)

For Tc higher than TX, the crystalline lamella directly forms from the molten state and the processes (i) and (ii) are absent. The thickening process suggested by Rault,49 however, occurs in a way similar to the process (iii) of the crystallization below TX, process (iii)′, because the growth front of the lamella always contacts with the liquid state. The thickness S c is given by Sc =

0 2σLCTLC 0 ΔHLC(TLC − Tc)

+ δ SLC

(Tc > TX)

0 2σMCTMC 1 ΔHMC S

(TLC > TX)

(TMC < TX)

(9)

(10)

Thus, the lamella with S > S X should melt above TLC. In the case of S < S X the lamella might transform into the mesomorphic lamella above TMC and immediately transform into the crystalline lamella with thickening. We interpret the experimental results using the above model through the mesophase. The solid lines above and below 208 °C in Figure 9b correspond to the L−C and M−C transition lines by eqs 9 and 10, respectively. The parameters in eqs 9 and 10 can be estimated as T0LC = 270 °C, T0MC = 335 °C, σLC/ΔHLC = 3.05 Å, σMC/ΔHMC = 5.58 Å, and TX = 208 °C. These values are almost identical to those given in a previous study.11 When we give both δS MC and δS LC as 9 Å in eqs 7 and 8,11 the fitting curves by eqs 7 and 8 become thin and thick broken lines in Figure 9b, respectively; the Tc-dependence of S c is reproduced.11,50 For Tc = 188.2 °C in Figure 9a the S −1−T relation shows the gradual increase in S during heating up to Tm2(onset) (Figure 7c), and overlaps the L−C transition line above Tm2. For Tc = 147.3 and 169.3 °C the S −1−T relation also shows the gradual increase in S during heating up to Texo(onset1) and overlaps the M−C transition line up to TX = 208 °C and the L−C transition line above TX. It should be noted that the “melting” behavior below TX observed in this study is the transition from the crystal to the mesophase. The evidence of this suggestion is not, however, confirmed by WAXD results. The Tc-dependence of the Tm2 and Texo’s in the DSC results (Figure 2) are evaluated using the parameters obtained by the SAXS results (Figure 9). In Figure 2 the solid thick curve indicates the L−C line of PBT crystallized from the liquid state given by eqs 8 and 9. The broken line indicates the L−C transition line of PBT crystallized from the mesomorphic phase given by eqs 7 and 9. The chain line indicates the M−C transition line of PBT crystallized from the mesomorphic phase given by eqs 7 and 10. The dependence of Tm2(onset) on Tc above 208 °C lies on the solid line and that between 185 and 208 °C on the broken line. The dependence of the Texo(onset1) on Tc below 185 °C lies on the chain line. The proposed model can explain the SAXS and DSC results. We comment on the origin of Texo(onset1) and Texo(onset2). In Figure 3 we have pointed out that Texo(onset1) and Texo(onset2) indicate Tm2(onset) and Tm2(top) for Tc < 185 °C, respectively. The Tc-dependence of Texo(onset2), however, seems to lie on the M−C transition line of the lamella crystallized through the mesophase given by eqs 7 and 10 in Figure 2. Figure 10 shows the schematic relationship between ΔGLC and ΔGLM and T for the heating process of the PBT with S 2(Tc) crystallized at Tc below 185 °C, which is the intersection of broken and chain lines in Figure 2. The free energy of PBT crystallized at Tc is expressed by the filled circle and passes through two routes, 1 and 2, during heating in Figure 10. The route 1 is the normal route in the proposed model and shows that the free energy of the crystallized PBT proceeding along the ΔGLC line transfers to the ΔGLM line with the endotherm at

(6)

Sc =

0 2σLCTLC 1 ΔHLC S

(8) G

DOI: 10.1021/acs.macromol.6b00126 Macromolecules XXXX, XXX, XXX−XXX

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ACKNOWLEDGMENTS

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REFERENCES

This work was partially supported by KAKENHI(Grant-in-Aid for Young Scientists (B) (No. 21740311, 25800236)) from the Ministry of Education, Culture, Sports, Science and Technology, Japan. The synchrotron radiation experiments were performed at the BL-40B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal No. 2010A1291, 2010A1200, 2010B1479, 2010B1482, 2011B1398, 2012A1393, 2012B1310, 2013A1549, 2014A1457, and 2014B1509).

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Figure 10. Schematic relationship between the monomeric free energy differences ΔGLC, ΔGLM, and temperature T for the heating process of the PBT crystallized at a crystallization temperature Tc below 185 °C without thickening and recrystallizing. The filled circle indicates the free energy of PBT crystallized at Tc.

TMC on heating. Thus, the lamella through the route 1 might start to show the endothermic behavior from TMC (an open square in Figure 10). On the other hand, the route 2 shows the free energy proceeds along ΔGLC line up to TLC without the transformation. The lamella through the route 2 might start to show the endothermic behavior from TLC. The state between TMC and TLC in the route 2 is metastable, so-called “superheating”. Texo(onset1) and Texo(onset2) in the DSC results (Figure 2) indicate the transition from the crystal into the mesophase and one from the crystal into the liquid with superheating, respectively. The evidence of these transitions has not been confirmed yet. The melting process through the mesophase will be studied in the near future.

4. CONCLUSION The heating process of the isothermally crystallized PBT has been investigated by DSC, WAXD, and SAXS. The DSC and SAXS results show that the Tm1 and Tm2 peaks are melting peaks of thin and thick lamellae formed at Tc and that Texo(onset1) obtained by PBT crystallized below 185 °C corresponds to Tm2(onset) above 185 °C. The WAXD and SAXS results also show the melting during heating between Tm1 and Tm2. The lamellar thickness S formed at Tc above 185 °C thickens along the L−C transition line during heating above Tm2(onset). For Tc below 185 °C S thickens along the M−C transition line between Texo(onset) and 208 °C and then along the L−C transition line during heating above 208 °C. According to the crystallization model through the mesophase, the T-dependences of the lamellar thickness S obey the L−C and M−C transition lines above and below TX = 208 °C, respectively. The relation between S c and Tc gives the constant δS = 9 Å in the full range of Tc. The T-dependences of S taking account of the constant δS can quantitatively explain the crystallization temperature dependences of Tm2(onset) and Texo(onset1) in the DSC results.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; tel +81-75-753-6775; fax +81-75-753-6805 (T.K.). Notes

The authors declare no competing financial interest. H

DOI: 10.1021/acs.macromol.6b00126 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.6b00126 Macromolecules XXXX, XXX, XXX−XXX