Stabilization of Nuclei of Lamellar Polymer Crystals: Insights from a

Mar 10, 2016 - ... Lamellar Polymer Crystals: Insights from a. Comparison of the Hoffman−Weeks Line with the Crystallization. Line. Jun Xu,*,†. Ba...
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Stabilization of Nuclei of Lamellar Polymer Crystals: Insights from a Comparison of the Hoffman−Weeks Line with the Crystallization Line Jun Xu,*,† Barbara Heck,‡ Hai-Mu Ye,§ Jing Jiang,∥ Yi-Ren Tang,† Jin Liu,† Bao-Hua Guo,† Renate Reiter,‡ Dong-Shan Zhou,∥ and Günter Reiter‡ †

Advanced Materials Laboratory of Ministry of Education, Department of Chemical Engineering, Tsinghua University, Beijing 100084, China ‡ Institute of Physics and Freiburg Materials Research Center, Albert-Ludwig-University of Freiburg, 79104 Freiburg, Germany § Department of Materials Science and Engineering, China University of Petroleum, Beijing 102249, China ∥ Department of Polymer Science and Engineering, School of Chemistry and Chemical Engineering, Key Laboratory of High Performance Polymer Materials and Technology, Ministry of Education, State Key Laboratory of Coordination Chemistry, Nanjing University, Nanjing 210093, China S Supporting Information *

ABSTRACT: We have studied melting of poly(butylene succinate), isothermally crystallized over a wide temperature range, employing a combination of the Hoffman−Weeks plot and the Gibbs− Thomson crystallization line, determined by small-angle X-ray scattering measurements. A change in the slope α of the Hoffman−Weeks (H−W) line, accompanied by a change of the slope of the crystallization line, was observed for crystallization temperatures higher than 110 °C. α was reaching a value of 1, implying that no intersection point between the H−W line and the Tm = Tc line could be obtained. (Tm is the measured melting temperature and Tc is the temperature at which the sample was crystallized). This observation was corroborated by the crystallization line, which was found to be parallel to the melting line for Tc > 110 °C. We relate these changes in slope to different stabilization mechanisms of the secondary nuclei at the growth front of polymer lamellar crystals. For Tc > 110 °C, secondary nuclei are proposed to be stabilized by coalescence of neighboring nuclei, all having a small width. By contrast, for Tc > 110 °C, the number density of secondary nuclei is low and thus their coalescence is rare. Accordingly, nuclei are stabilized by growing in size, mainly increasing their width.

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temperature (T0m), which is one of the central thermodynamic parameters characterizing crystallization and melting behavior of polymers. For instance, T0m is related to the interfacial free energy between the crystal and the molten environment, which represents a key parameter for controlling isothermal growth kinetics. Traditionally, researchers used two ways to determine T0m. On the one hand, using the Gibbs−Thomson equation, which reflects the dependence on the surface-to-volume ratio of the crystal, one typically presents the melting temperature versus the reciprocal lamellar thickness.6,7 This presentation implicitly assumes that the lateral size of crystalline lamellae is much larger than the lamellar thickness. There, a linear extrapolation to an infinitely thick lamellar crystals yields T0m. On the other hand, the so-called Hoffman−Weeks (H−W) line traces Tm, the melting temperature as a function of Tc, the temperature at which the sample was crystallized. There, a linear extrapolation

INTRODUCTION

A polymer chain consists of a sequence of covalently connected segments (monomers), which have to be integrated in a cooperative manner in a crystal. Thus, when long and flexible polymers crystallize from a quiescent melt or solution at a detectable rate, nanometer-thick lamellar crystals consisting of folded chains are formed.1 The degree of chain folding is significantly affected by the rate at which the polymer crystals grow. In contrast to crystals containing extended chains, foldedchain lamellar crystals are metastable due to the existence of noncrystalline folds and a higher surface-to-volume ratio.2−5 Consequently, the melting point (Tm) of polymer folded-chain lamellar crystals varies with the lamellar thickness (lc). Both, metastability and a melting temperature, which depend on lamellar thickness, affect the operational temperature range for applications. Metastability of polymer folded-chain lamellar crystals can be quantitatively characterized via the two parameters Tm and lc. Ideally, the most stable polymer crystal has an infinitely large size in all the three dimensions, i.e., 1/lc → 0. The melting point of such an ideal polymer crystal defines the equilibrium melting © XXXX American Chemical Society

Received: September 26, 2015 Revised: March 3, 2016

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Macromolecules to the intersection point of the H−W line with the line of Tm = Tc also is expected to yield T0m.8 This H−W plot requires only the measurement of melting temperatures of samples, which were isothermally crystallized at various temperatures. Thus, it has been frequently adopted for determining T0m. However, most of the times, the values of T0m, which were determined along these two ways, i.e., based on the Gibbs−Thomson equation and derived from the H−W plot, are not the same. Both, the H−W line and the Gibbs−Thomson equation account also for the metastability of polymer lamellar crystals. The Gibbs−Thomson equation links Tm to the surface-tovolume ratio. For lamellar crystals with a width w ≫ lc, the surface-to-volume ratio is characterized by 1/lc. The H−W line reflects metastability indirectly via the deviation of Tm from T0m. In addition, the H−W line reveals the process of crystal growth and stabilization of polymer lamellar crystals via the difference between Tc and Tm. Without any additional growth, the critical nucleus formed at Tc will melt at Tm = Tc, signifying the balance between surface and volume terms. To avoid melting of the nucleus at Tc, its size has to increase. Consequently, the resulting lamellar crystal has a higher melting point than the nucleus initially formed at Tc. Such behavior also applies to crystals of small molecules. There, however, rapid crystal growth usually leads to crystalline structures, which have characteristic dimensions significantly larger than the nucleus. Because of rapid growth, their surface-to-volume ratio approaches zero within short times. Thus, independent of the temperature at which small molecules are isothermally crystallized, large enough crystals are formed, which all have the same melting temperature T0m. By contrast, the surface-to-volume ratio of polymer lamellar crystals never really approaches zero. There, the kinetically controlled process of chain folding causes that lc remains of the order of nanometers. The easily determined H−W line implicitly assumes that the surface-to-volume ratio of the resulting lamellar crystals, characterized by 1/lc, varies linearly with Tc. Any changes in lc with time, or any variation of the surface-to-volume ratio either with time or due to a different thermal history, which is not controlled by Tc only, will cause deviations from a linear variation of Tm with Tc.9,10 Accordingly, deviations from linearity inform about changes in the underlying assumption of a unique relation between lc and Tc. To account for such a stabilization effect, Strobl introduced the crystallization line, namely Tc ∼ 1/lc and proposed that it is the equilibrium transformation line from the initial mesophase to the native lamellar crystal.11,12 However, the kinetically controlled variation of the lamellar thickness (stem length) in a polymer lamellar crystal and how this variation contributes to stabilization still remain highly debated issues. It is generally believed that crystallization of polymer lamellar crystals proceeds by adding a crystalline stem to the nucleus. The length of this stem is determined via the kinetics of secondary nucleation, i.e., selecting the length for which the fastest steady growth of the lamellar crystals can be achieved.2,3,13,14 To avoid instantaneous remelting, this stem length has to be a bit larger than the thermodynamically predicted minimum lamellar thickness of the secondary nucleus. After the initial attachment to the crystalline core, the stem length may increase and can further improve stability, characterized by an increase in melting temperature. This subsequent stabilization of the initially formed crystalline nuclei and the transformation to a more stable crystal phase can proceed via the following three routes: elongation of the

crystalline stem (lamellar thickening), widening of crystalline cluster via laterally attaching additional stems (lamellar widening), or coalescence of neighboring lamellar clusters.15−17 In this work, we chose to examine the metastability of lamellar crystals of poly(butylene succinate) (PBS), formed at various isothermal crystallization temperatures. Within PBS lamellar crystals, chain mobility is low and consequently lamellar thickening after growth is difficult. Moreover, forming a homogeneous nucleus of PBS requires significant undercooling. However, via the addition of a nucleating agent, we could extend the temperature range for isothermal crystallization up to high temperatures. On the basis of the so obtained experimental results, we will discuss the mechanisms of stabilizing secondary nuclei at the growth front of lamellar crystals, based on a combination of the H−W line and the crystallization line.

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MATERIALS AND SAMPLE PREPARATION Poly(butylene succinate) (PBS), poly(butylene fumarate) (PBF), poly(butylene succinate-co-5 mol % butylene adipate) (PBSA5), and poly(butylene succinate-co-10 mol % butylene adipate) (PBSA10) were synthesized by esterification and polycondensation in the molten state. The detailed description of chemical synthesis was reported in our previous works.18,19 The molar mass information on the studied samples is listed in Table 1. Table 1. Molar Mass of the Studied Samples samples

Mn (×104)

Mw (×104)

Mw/Mn

PBS PBF PBSA5 PBSA10

3.17 2.95 0.62 3.13

8.04 5.49 1.39 5.05

2.5 1.86 2.22 1.61

PBS samples with 2 wt % of nucleating agent PBF were prepared via solution casting from 1 wt % solution in chloroform. Before isothermal crystallization, all samples were dried in vacuum at 60 °C overnight. Pure PBS samples sandwiched between two glass slides were melted at 160 °C for 3 min to remove the thermal history, and subsequently transferred to a hot stage preset at temperatures ranging from 79 to 101 °C for isothermal crystallization. After adequate crystallization time, impingement of all spherulites was observed under a polarized optical microscope (Weitu, Shanghai). These samples were quenched to room temperature and used for further characterization. Isothermal crystallization of PBS nucleated with 2 wt % of PBF was carried out in a Shimadzu DSC-60 differential scanning calorimeter (DSC) under nitrogen atmosphere. The specimen were sealed in an aluminum pan and heated from room temperature to 160 °C at a heating rate of 20 °C/min. After remaining for 3 min at 160 °C to erase the thermal history, the samples were quenched rapidly by liquid nitrogen in the DSC to the isothermal crystallization temperature, ranging from 101 to 112 °C. The time for isothermal crystallization of nucleated PBS spanned from 30 min at 101 °C to 48 h at 112 °C. Once heat flow of the DSC curve reached back to the baseline, the samples were directly heated from the isothermal crystallization temperature to 160 °C at a heating rate of 10 °C/min. Small-angle X-ray scattering (SAXS) experiments were carried out with the aid of a Kratky camera attached to a B

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Macromolecules conventional Cu Kα X-ray source employing a position sensitive metal wire detector and a temperature controlled sample holder. Samples were isothermally crystallized at different temperatures Tc. For the higher crystallization temperatures (Tc > 94 °C), samples were crystallized in situ after being annealed for at least 4 min at 160 °C, i.e., above the equilibrium melting temperature. For Tc ≤ 94 °C, samples were crystallized in an external oven, cooled to room temperature, then heated again to the respective Tc before the measurements. Subsequent heating across the melting temperature Tm was carried out stepwise, usually in steps of 2 °C, allowing the sample to equilibrate for 2 min at each step. Scattering curves were registered over a period of at least 4 min for the isothermal scattering experiments and within 8 min during the heating experiments. After background correction, a deconvolution algorithm was applied on the slit-smeared scattering data.20 The thicknesses of the crystalline lamellae lc were derived from the interface distribution functions (IDF).21,22

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RESULTS AND DISCUSSION Pure PBS was isothermally crystallized within the temperature range from 71 to 101 °C. For Tc in the range from 71 to ca. 89 °C, significant melting−recrystallization occurred during heating at a rate of 10 °C/min.23−25 Thus, the melting peak of the originally formed lamellae was extremely small and difficult to determine by differential scanning calorimetry (DSC). Only for Tc higher than ca. 89 °C, the melting peak of the originally formed lamellae, denoted by Tm,1, was clearly detectable, as shown in Figure 1a. Tm,2 and Tm,3 are the melting peaks of the recrystallized lamellar crystals during heating. Tm,3 hardly varied with the isothermal crystallization temperature. Consequently, only Tm,1 for samples crystallized at Tc > 89 °C were used for the H−W plot. For Tc = 101 °C, complete crystallization of pure PBS required crystallization time (tc) longer than 10 h. When adding the nucleating agent poly(butylene fumarate) (PBF), a highly efficient nucleating agent for PBS,18,26 it became possible to crystallize PBS isothermally even at higher Tc within 1 to 2 days. Figure 1b shows the DSC curves of PBS containing 2 wt % PBF and isothermally crystallized at Tc ranging from 104 to 112 °C. In this temperature range, only one melting peak could be observed clearly. At these high values of Tc, compared to the fast rate of melting, the rate of recrystallization is rather slow. The temperatures of the first melting peak Tm,1 (from Figure 1a) versus crystallization temperature Tc are plotted in Figure 2. Tm,1 of pure PBS, crystallized at 89 to 101 °C without nucleating agent, and those crystallized at 104 to 112 °C with PBF as nucleating agent (from Figure 1b) are shown on the same fitting line. The intersection with the Tm = Tc line yields a value of 127 ± 1 °C. This value is within the range of previously reported values, which varied between 127 and 140 °C and were determined by the same method.27−31 The reason for the uncertainty in determining the exact value of T0m via an extrapolation of the linear Hoffman−Weeks line can probably be attributed to a small range of rather low isothermal crystallization temperatures (80−100 °C) previously used. This limited temperature range and deviations of the data points from a straight line unavoidably introduce large uncertainties in the values of slopes deduced from the variation of Tm with Tc.27−31 In addition, the values of T0m deduced from the H−W plot differ from the value of 146.5 °C, obtained by Gan et al. via an

Figure 1. DSC thermograms of (a) PBS samples isothermally crystallized within the temperature range from 71 to 101 °C; (b) PBS samples with 2 wt % of nucleating agent PBF isothermally crystallized within the temperature range from 104 to 112 °C.

Figure 2. Hoffman−Weeks plot for pure PBS and PBS nucleated with 2 wt % PBF. The three data points at the highest crystallization temperatures (marked by red squares) lie on a line with a slope of 1.0.

extrapolation of the Gibbs−Thomson equation.32 To avoid such problems encountered by employing a linear extrapolation of the H−W plot, a variant of the H−W plot based on a nonlinear extrapolation has been applied for determining T0m.10,33,34 There, the temperature T0m was derived from the intersection of a nonlinear extrapolation of the H−W line with the Tm = Tc line.10,33,34 However, it is highly intriguing that the last three data points in Figure 2, obtained for Tc > 110 °C, C

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1.36 However, the authors did not discuss this observation.35,36 In contrast to our results, in the range of low Tc, the melting point of the copolymer used by Kim et al. did not increase much with increasing crystallization temperature. A H−W line parallel to the Tm = Tc line for data points obtained at the highest values of Tc implies that these two lines have no intersection point. Consequently, the following questions arise: What is the mechanism causing a H−W line with a slope of 1? What is the physical meaning of the temperature Tx determined by a linear extrapolation of the data in the low Tc range of the H−W line? The slope of the H−W line as shown in Figure 4a is defined by α.

suggest a slope of 1.0. Thus, for the highest values of Tc, assuming a linear relation for Tm(Tc), we may obtain a line parallel to the Tm = Tc line. Accordingly, Tm(Tc) cannot intersect the Tm = Tc line. Thus, the here presented results show that it is problematic to obtain Tm0 of PBS via extrapolation of the H−W line. To examine the generality of this trend, we have studied also other polymers. Poly(butylene succinate-co-5 mol % butylene adipate) and poly(butylene succinate-co-10 mol % butylene adipate) random copolymers were chosen. The results, shown in Figure 3, exhibited a similar trend. At the highest

Tx − Tm = α(Tx − Tc), Tm = Tc + ΔT ,

α=1

α 110 °C, the crystallization line was found to be parallel to the melting line. Consequently, our SAXS results are thus consistent with the conclusions drawn from the DSC measurements. We notice that Strobl et al. also reported for syndio-tactic poly(propyleneco-octene) a decrease in slope of the crystallization line for the highest crystallization temperatures where they employed the “self-seeding” technique.38 The observation that the crystallization line becomes parallel to the melting line at high Tc thus deserves further studies in order to test the generality of such a behavior. Our results unambiguously show that the intersection of the extrapolated H−W line with the Tm = Tc line does not yield the equilibrium melting point. The increase in slope of the H−W line is closely related to the decrease in slope of the crystallization line. This change in slope implies a distinct change in the selection mechanism for the lamellar thickness chosen during isothermal crystallization of PBS for large and low supercooling, respectively. The initial lamellar thickness is determined by secondary nucleation.2,13,39 There, the entropic barrier increases with the thickness of the lamella. For a given thickness, the width of a lamellar cluster (here and in the following, the term “cluster” indicates a nucleus large enough to allow for further growth) can be increased without having to overcome such a barrier. Thus, it is reasonable to assume that the lamellar thickness selected in the course of the growth process is deviating NOT much from the minimum lamellar thickness determined by thermodynamic parameters, i.e., secondary nucleation. However, changes in the width of these lamellar clusters are possible. During growth, the width of secondary lamellar clusters will increase in time. This widening process has to be taken into account here.

(3)

where K is the slope of the melting line in the plot of melting temperature versus the reciprocal of lamellar thickness lc. Combining eq 1 and 3, we get: Tx − Tc =

K⎛1 1⎞ ⎜ − ⎟, α ⎝ lc lx ⎠

Tm0 − Tx = K ×

α 104 °C, lamellar thickening during a DSC heating run was not considerable, resulting in DSC curves with only one melting peak. By contrast, melting of PBS in the SAXS measurement with a low heating rate was accompanied by significant melting−recrystallization and all the samples melted almost at the same temperature of 120 °C (Figure S1 in the Supporting Information). Furthermore, the thickness of lamellar PBS crystals, isothermally crystallized at 100 °C, varied little with time up to 6000 s (Figure S2 in Supporting Information). This observation revealed that lamellar thickening did not happen during the studied time range. In addition, we prolonged the duration of isothermal crystallization of pure PBS and nucleated PBS up to 48 h. Even after such a long duration, the observed increase in melting peak temperature was less than 2 °C (Figure S3 in Supporting Information). Consequently, the time for isothermal crystallization has only limited influence on lamellar thickening of PBS lamellar E

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Figure 6. Scheme showing the formation process of polymer lamellar crystals and the effect of lamellar width (w) on melting. (a) Secondary nucleation and growth of individual lamellar polymer cluster at the growth front, (b) coalescence of neighboring lamellar clusters, initiated by multiple nuclei. Coalescence most probably happens at high supercooling and provides another stabilization method of the newly formed lamellar clusters. (c) Scheme showing the effect of width of crystalline clusters on the melting line. The red and blue lines indicate the melting line of the secondary nuclei with minimum width and that of the lamellar crystals with infinite large width, respectively. For w → ∞, the blue melting line will be established. Figure 5. Melting line (ML) and the crystallization line (CL) showing the DSC melting point and the isothermal crystallization temperature, respectively, versus the reciprocal of initial lamellar thickness determined by SAXS measurement. (a) The melting line and crystallization line of both pure PBS and PBS nucleated with 2 wt % of PBF, represented by red and blue squares (CL) and red and blue crosses (ML), respectively. (b) Magnified view of the dashed square in (a) showing only data points of nucleated PBS with 2 wt % of PBF. The initial lamellar thickness of the isothermally crystallized samples was used to draw the solid crystallization line. However, the real crystallization line at high temperatures may deviate from the solid line at lower temperatures, which can be reflected by the fact that the four melting points for samples crystallized at the highest temperatures show deviations from the melting line, i.e., the reciprocal values of the initial lamellar thickness cannot be presented with respect to the solid crystallization line. To bring the four melting points back to the melting line, we have to shift also the crystallization line accordingly (data points are shifted to larger lamellar thickness values, represented by gray triangles). This procedure led to the dashed line, which is the real crystallization line at high temperatures and which is parallel to the melting line. The data points of PBS nucleated with PBF indicated by blue open squares demonstrated the same trend.

ΔFnuclei 2σ 2σ = + e − Δgnuclei V w lc

(7)

where ΔF indicates the change of total free energy of the secondary nuclei, and σ and σe are the interfacial free energy of the lateral and fold surface, respectively. Δgnuclei is the change of bulk Gibbs free energy per unit volume of the nuclei. V is the volume of the secondary crystalline polymer nucleus and equals to b0wlc, where b0 is the stem width normal to the growth direction. At the early stage of their formation, lamellar thickness lc and width w of secondary nuclei are correlated and cannot change independently. The correlation is determined by the minimization of the total interfacial free energy at a fixed volume4,40 or based on Wulff’s construction.41 From eq 7, we have ΔFnuclei 2σ 2σ ≥2 × e − Δgnuclei V w lc

2

We consider a secondary lamellar nucleus with a width w formed at the growth front built up of stems having a uniform length lc (see Figure 6a). The free energy change of the formation of the secondary nuclei per volume:

2σ σσeb0 2σ × e =4 w lc V

(9)

where eq 8 gets the minimum value only when σ σ = e w lc F

(8)

(10)

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and thus may proceed at a faster rate than lamellar widening. For the third route of a transformation of a less stable mesophase into a more stable crystalline phase, we could not yet preclude such possibility since we observed a crystallization line extrapolated to a temperature higher than T0m. In the report by Marand et al.,56 the change of the amorphous phase was reflected by a small DSC endotherm at low temperatures, caused by secondary crystallization. This change was accompanied by a large endotherm at higher temperatures caused by primary crystallization. The two populations of crystals were formed consecutively during the isothermal crystallization.56 In the present work, where PBS was isothermally crystallized at 100 °C, there were two endotherms on the DSC heating curve, with the endotherm at higher temperatures decreasing with increasing heating rate (as shown in Figure S4a in the Supporting Information). This indicates that the second endotherm corresponded to recrystallizationmelting occurring during heating. In addition, when the crystallization temperature was higher than 104 °C, there was only one observable endotherm in the DSC curve. Thus, the melting points we used for H−W plot correspond to the primary crystals. The possible change of amorphous phase during secondary crystallization had only limited influence on the melting point of the primary crystals. On the basis of the above analysis, we propose that lamellar widening is the most efficient and thus most probable route for increasing the stability of the crystalline lamellar clusters. We should point out that the red line in Figure 6c, the melting line of the secondary nuclei with finite width, may not intercept with the vertical axis at Tm0 when the lamellar thickness is extrapolated to infinity. The enthalpy change per unit volume of the secondary nuclei having a finite width, Δhnuclei, may be only a fraction of that of the lamellar crystals having infinite width, Δhcrystal. Similarly, it is also reasonable to assume that the entropy change per unit volume of the secondary nuclei having a finite width, Δsnuclei, may be only a fraction of that of the lamellar crystals with infinite width, Δscrystal. This is reasonable since nuclei of very small size may have an intermediate structure (e.g., packing density) between the totally amorphous melt and the crystal bulk, similar to what has been observed in the melting of small molecular crystal57 and the crystallization of colloids.58 Thus, we obtain

Therefore, the preferred shape of secondary nuclei follows: lc σ = e w σ

(11)

For a secondary nucleus being stable against remelting, the minimum lamellar thickness lc,min and the minimum lamellar width wmin lead to a zero total free energy change of the nuclei. They are lc ,min =

4σe Δgnuclei

and

wmin =

4σ Δgnuclei

(12)

which gives the red melting line in Figure 6c. To ensure that crystal growth can be initiated by a secondary nucleus, its size should be somewhat larger than the minimum size. Consequently, because nuclei are stabilized additionally by growth beyond the minimum size, the melting point of such clusters can be higher than the isothermal crystallization temperature. Growth of the clusters leads to a decrease of their surface-to-volume ratio. This stabilization of the initially formed clusters can proceed via the following three routes: First, increase of the thickness (length) of a crystalline stem at the growth front.11,12,15,38,42 Via chain sliding within the crystalline lamella for polymers having high chain mobility in the crystal lattice, such lamellar thickening also can occur for chains buried in the bulk of the crystal.42−53 As proposed by Marand and Huang,54 lamellar thickening may also take place via reeling in amorphous segments. The influence of lamellar thickening on the melting point and the corresponding H−W plot has been systematically addressed by Marand et al.10,55 Second, an increase of the width of a quasi-two-dimensional set of crystalline stems at the growth front of a lamella can also increase the stability of a lamellar cluster. Widening may occur by attachment of more crystalline stems or by coalescence of neighboring clusters, as indicated in Figure 6b. The two widening routes have different consequences. When attaching more stems at sides of a cluster, its volume increases and thus the required minimum stem length for a cluster to be stable against melting will decrease as the cluster widens. The cluster, however, always stays close to its minimum size (volume). By contrast, when two such lamellar clusters coalesce, the width (and the volume) of the resulting cluster increases significantly and abruptly, resulting in a cluster which (for this now larger width) has a thickness above the required lc,min. Third, stabilization may be achieved also via a change of the bulk Gibbs free energy per volume through transformation of a less stable meso-phase into a more stable crystalline phase11,12,38 or, in the course of secondary crystallization, via a decrease of the chain configurational entropy of the remaining amorphous phase constrained by the surrounded crystals.56 These three stabilization routes have different probabilities: After attaching a crystalline stem that has the required minimum length lc,min, the length lc can increase with the logarithm of time.47−54 As lc increases, the energy barrier also increases. In contrast to the increasing energetic penalty for increasing lc, an increase of the lamellar width w is proportional to the time needed for attaching additional crystalline stems, which, in turn, depends on the process of lateral diffusion of polymers on the crystals growth surface. During primary crystallization, the energy barrier for lateral diffusion is always the same and does not vary in time. The coalescence of neighboring clusters can be considered as a percolation process

Δgnuclei = Δhnuclei − Tc Δsnuclei = f1 Δhcrystal − Tfc 2 Δscrystal = f1 Tm0Δscrystal − Tfc 2 Δscrystal ⎛ f T0 ⎞ = f2 Δscrystal ⎜⎜ 1 m − Tc ⎟⎟ ⎝ f2 ⎠ ⎛ f T0 ⎞ m = Δsnuclei⎜⎜ 1 − Tc ⎟⎟ ⎝ f2 ⎠

(13)

where Tc is the isothermal crystallization temperature and T0m is the equilibrium melting point of the crystal with infinite 3dimensional size. We define f1 Tm0 f2 G

=

Δhnuclei ≡ Tc∞ Δsnuclei

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Macromolecules where T∞ c is a virtual temperature, reflecting the extrapolated melting point of an infinitely large crystal which has the same structure as the nuclei. However, when the nuclei grow to a sufficient size, their structure will change to that of the bulk crystal. Substitution of eq 14 into eq 12 leads to lc ,min = wmin

4σe , nuclei Δsnuclei(Tc∞ − Tc) 4σnuclei = Δsnuclei(Tc∞ − Tc)

Tm0 − Tm 1 lc

2σe , crystal Δgcrystal

=

⎡ ⎢ 1 α=⎢ ⎢γ 1 + ⎣

and

2σe , crystal Δscrystal(Tm0 − Tc)

(

(15)

,

when w → ∞

Equation 16 gives the blue melting line drawn in Figure 6c. The scheme drawn in Figure 6, parts a and b, outlines the two stages of secondary nucleation and growth in width (lateral spread of the lamellar clusters and coalescence of nearby clusters) on the crystal growth face. Our scheme is different from the multistage scheme proposed by Strobl, where a sequential change from an initial meso-phase, to a native crystal and finally to a stabilized crystal happens in the radial direction of the spherulite, from the growth front to the interior of a bulk lamellar crystal.11,12,38 In addition, in our scheme, the increase of stem length (lamellar thickening) mainly occurs only during the initial first nucleation stage. During the subsequent growth in width of the clusters and their coalescence, lamellar thickening hardly appears. In contrast, in Strobl’s scheme, an increase of stem length occurs in both stages: formation of the meso-phase and further thickening of the meso-phase until it transforms to the native crystal. More generally, if coalescence and reorganization of lamellar clusters at the growth front occurs, the width of lamella just before coalescence is β times the minimum width given in eq 12, the minimum thickness of a stable lamella is ⎛ 1⎞ ⎜1 + ⎟ , − Tc) ⎝ β⎠ 4βσnuclei = Δsnuclei(Tc∞ − Tc)

■

when w = βwmin

(17)

We assume that the most probable lamellar thickness formed at Tc is γ times of the minimum thickness described in eq 17. Then, the crystallization line is given by Tc∞ − Tc 1 lc

=

⎛ 1⎞ ⎜1 + ⎟ − Tc) ⎝ β⎠

2γσe , nulcei Δsnuclei(Tc∞

(19)

1 β

)

⎤ ⎥ σe , crystal Δsnuclei ⎥ ⎥ σe , nucleiΔscrystal ⎦

(20)

CONCLUSION

To summarize, we have measured the peak melting temperature of PBS, which was isothermally crystallized at low supercooling, taking advantage of the added nucleating agent. We observed that at low supercooling the Hoffman−Weeks line (Tm − Tc line) exhibited a slope of approximately 1.0. Assuming that lamellar thickening during the DSC heating scan was negligible, we deduced a quantitative relationship between Hoffman−Weeks line, crystallization line and melting line. A Hoffman−Weeks line with slope of 1 observed at the low supercooling indicates that in this temperature range the crystallization line is parallel to the melting line. In addition, the intersection temperature of Hoffman−Weeks line with the Tm = Tc line of PBS isothermally crystallized at large supercooling corresponds to the temperature of the intersection of crystallization line and melting line, rather than the actual equilibrium melting point, T0m. Furthermore, we discuss three complementary mechanisms for the stabilization of secondary nuclei: thickening, widening, and impingement of neighboring nuclei. At high supercooling, secondary lamellar nuclei are most likely stabilized by coalescence of neighboring nuclei. The higher lamellar thickness of the crystallization line at high supercooling is because the small width of the initial nuclei (which eventually will coalesce) required a much longer stem length within these small nuclei in order to reach the critical surface-to-volume ratio needed for stability. At low supercooling, coalescence of secondary nuclei occurs rather rarely, thus the lamellar thickness is only slightly larger than the minimum lamellar thickness determined by the melting line. Consequently, when evaluating the stability of polymer lamellar crystals, besides the thickness of lamellar clusters also their width should be taken into account.

2σe , nuclei Δsnuclei(Tc∞

Δscrystal

In the case of PBS, the lamellar thickening ratio γ after crystallization is approximately 1, which means negligible lamellar thickening after formation of lamellar crystals. As a result of varying supercooling, the alteration of the widening ratio (the ratio of the lamellar width just before coalescence to the minimum width described by eq 12), β, will change the slope of H−W line, α. At large supercooling, secondary nucleation is relatively easy and a multitude of secondary nuclei will form at the growth front. Thus, coalescence of lamellar clusters will occur frequently and β ≈ 1, so α is around 0.5, provided that σe,crystalΔsnuclei ≈ σe,nucleiΔscrystal. At low supercooling, very few secondary nuclei can form at the growth front, thus impingement of neighboring nuclei hardly happens and lamellar width will be very large comparing to the lamellar thickness. Consequently, β ≫ 1 and α equals 1.

(16)

lc ,min =

2σe , crystal

Dividing eq 19 by eq 18, the ratio of the slopes of the melting line to that of the crystallization line, α equals:

eq 15 indicates that the melting line for the secondary nuclei, formed at large supercooling with a threshold size required to survive, will cross the vertical axis at the virtual temperature T∞ c . 0 The experimental results show that T∞ c > Tm, indicating that f1 > f 2. In eq 15, the interfacial free energy of the secondary nuclei may also be different from that of the lamellar crystal with infinite width, however, it will not change the extrapolated temperature T∞ c . When the lamellar width is large (w → ∞), the minimum lamellar thickness is lc ,min =

=

(18)

From eq 16, the slope of the melting line is H

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02123. Variation of the reciprocal of lamellar thickness, variation of the interface distribution function with time, variation of the melting peak temperatures, DSC curves, and variation of the melting processes (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(J.X.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the National Natural Science Foundation of China (Grant No. 21374054, 51473085), the Sino-German Center for Research Promotion (Grant No. GZ 833), and the National High-tech R&D Program of China (863 Program) (Grant No. 2011AA02A203). J.X. is deeply grateful to the Alexander von Humboldt Foundation for supporting his research fellowship in 2012.

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