Crystallization Kinetics And Nucleation Mechanisms In The Presence Of Melt Mesophases
Polymer crystallization proceeds by nucleation and growth; for a recent review of homopolymer crystallization, including kinetics and mechanism, see Schultz [2001]. If the polymer is free of contaminants, homogeneous nucleation initiates crystallization. Homogeneous nucleation requires the formation of a critical nucleus; the process is thermally activated and has a large energy barrier. A large undercooling is therefore expected for homogeneous nucleation to occur at a substantial rate. In semicrystalline homopolymers, crystallization is more commonly initiated heterogeneously, by dust particles or other impurities present in the melt. These impurities eliminate the need to form a critical nucleus, so the undercooling associated with crystallization proceeding from heterogen-ous nucleation is typically much smaller than from homogeneous nucleation.
Crystallization in bulk polymers typically follows a sigmoidal time evolution, described by the Avrami equation:
where y is the fraction of the ultimate crystallinity that is achieved at time t; k is the rate constant, and n is the Avrami exponent. In principle, the Avrami exponent provides information about the nucleation mode and crystal growth habit. For example, instantaneous nucleation followed by isothermal three-dimensional growth (as for spherulites) yields n = 3, and indeed, most crystallization data for bulk polymers are adequately described by the Avrami equation with n = 2—4. While spherulites are a specific case, sigmoidal crystallization kinetics (n > 1) are generally expected for any process where the growing crystals "spread" with time: where the amount of material deposited, in successive time intervals, onto the structure formed from a single nucleus increases steadily with time, until the crystallizable material is depleted and the crystallization rate falls. (Again considering the particular case of spherulites, successively deposited spherical shells of the same thickness have progressively greater volume, producing the initial autoacceleration in y(t) reflected in Equation (6.2)).
Restricting crystallization on a nanometer length scale necessarily impacts how crystallization is initiated and how it proceeds. Consider first the case of spheres, where the overall number density of impurities (typically of order 109 cm—3) is many orders of magnitude below the number density of microdomains (typically 1017 cm—3, depending on molecular weight). If crystals are indeed confined to individual spheres, then the overwhelming majority of these spheres must nucleate homogeneously. Moreover, since the diameter of individual microdomains is only a few tens of nanometers, crystal growth over the full spatial extent of the microdomain should be essentially instantaneous once nucleated. In this case, the rate of crystallization will simply be proportional to the fraction of microdomains that have not yet nucleated, yielding an Avrami exponent n = 1 and nonsigmoidal kinetics (rate decreases continuously with time). This idea was first advanced by Lotz and Kovacs [1969], though the techniques available at that time precluded the precise kinetic study coupled with structural characterization needed to confirm this picture. Recently, isothermal crystallizations tracked by synchrotron-based, time-resolved simultaneous SAXS/WAXS, have provided valuable insight into the nucleation modes and growth habits during the crystallization of semicrystalline block copoly-mers, as discussed in the following section.
6.5.1 ISOTHERMAL CRYSTALLIZATION WITHIN SPHERES
Loo et al. applied time-resolved SAXS/WAXS to study crystallization in a range of E-containing semicrystalline-glassy [2001] and semicrystalline-rubbery [2002] block copolymers, where the E block formed spheres or cylinders; many of these were the same polymers whose solid-state structures (confined crystallization vs. breakout) were reviewed earlier in this chapter. Considering first the sphere-forming diblocks, complete confinement could be achieved either through a glassy matrix (E/VCH 5/22) or through strong interblock segregation (E/SEB 9/55). The progress of crystallization could be tracked easily through the increase in the WAXS peak intensity for the (110) reflection of orthorhombic E, or through the growth of the SAXS feature near q = 0.5nm_1, which arises from the formation of crystallites, or through the change in the principal SAXS peak intensity as the electron density difference between spheres and matrix changes when the spheres densify through crystallization. These features can all be seen in the SAXS and WAXS patterns shown as insets in Figure 6.7, which presents the crystallization kinetics for the strongly segregated E/SEB 9/55. The SAXS and WAXS patterns evolve in parallel, and both fit well to first-order kinetics (n = 1), with a half-time that is identical to within measurement error. The observation of first-order kinetics in E/SEB 9/55 simultaneously confirms complete confinement of the growing crystals by the isolated microdomains, and a homogeneous nucleation mechanism.
The well-ordered microdomain morphologies that block copolymers present allow an unambiguous and precise calculation of the number density of spherical microdomains, so that the overall crystallization rate can be directly trans-
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Figure 6.7 Isothermal crystallization (67 °C) of strongly segregated sphere-forming E/SEB 9/ 55 monitored by time-resolved SAXS/ WAXS. Insets show the SAXS (left) and WAXS (right) patterns for E/SEB 9/55 both at the start (top trace in each panel) and end (bottom trace) of crystallization. E (110) peak intensity is shown as the lower data set (D), integrated SAXS intensity near q = 0.5nm~' is shown as the middle data set (O), both read on the left axis. Both are fit to first-order kinetics (solid curves) with half-times of 3 min. For comparison, the crystallization behavior of a hydrogenated polybutadiene homopolymer (E41) is shown as the top data set (□), read on the right axis. The kinetics are qualitatively different (sigmoidal vs. first-order), and to achieve a similar half-time, a much higher crystallization temperature was required for E41 (95 vs. 67 °C). [Reprinted with permission from Loo et al. 2000a].
Figure 6.7 Isothermal crystallization (67 °C) of strongly segregated sphere-forming E/SEB 9/ 55 monitored by time-resolved SAXS/ WAXS. Insets show the SAXS (left) and WAXS (right) patterns for E/SEB 9/55 both at the start (top trace in each panel) and end (bottom trace) of crystallization. E (110) peak intensity is shown as the lower data set (D), integrated SAXS intensity near q = 0.5nm~' is shown as the middle data set (O), both read on the left axis. Both are fit to first-order kinetics (solid curves) with half-times of 3 min. For comparison, the crystallization behavior of a hydrogenated polybutadiene homopolymer (E41) is shown as the top data set (□), read on the right axis. The kinetics are qualitatively different (sigmoidal vs. first-order), and to achieve a similar half-time, a much higher crystallization temperature was required for E41 (95 vs. 67 °C). [Reprinted with permission from Loo et al. 2000a].
lated into a quantitative value of the homogeneous nucleation rate of the polymer forming the spheres [Loo et al. 2002]. Varying the isothermal crystallization temperature thus provides the temperature dependence of the homogeneous nucleation rate. Figure 6.8 shows the results of such a calculation, including data from the glassy-matrix E/VCH 5/22, the strongly segregated E/ SEB 9/55, and classic data of Koutsky et al. [1967] for the homogeneous nucleation rate in linear polyethylene, measured by dispersing the polymer into micrometer-size droplets in a suspending liquid. The slopes of all three data sets are similar, indicating that the nucleation rate increases by approximately a factor of 3 for each additional 1 °C of undercooling. The three data
Figure 6.8 Homogeneous nucleation rates for polyethylene extracted from measurements on bulk E/VCH 5/22 (•), bulk E/SEB 9/55 (□), and a suspension of linear polyethylene droplets (▼, data of Koutsky et al. [1967]). Slopes S of log(rate) vs. Tc are indicated for each of the three data sets. Inset shows the same data plotted on an expanded rate scale, with the prediction of classical nucleation theory shown as the solid curve (prediction uses the material parameters given by Koutsky et al. [1967]).
Figure 6.8 Homogeneous nucleation rates for polyethylene extracted from measurements on bulk E/VCH 5/22 (•), bulk E/SEB 9/55 (□), and a suspension of linear polyethylene droplets (▼, data of Koutsky et al. [1967]). Slopes S of log(rate) vs. Tc are indicated for each of the three data sets. Inset shows the same data plotted on an expanded rate scale, with the prediction of classical nucleation theory shown as the solid curve (prediction uses the material parameters given by Koutsky et al. [1967]).
sets do not quite collapse onto a single master curve within experimental error, as shown in the inset. This may reflect modest differences in nucleation rate due to chemical microstructure (linear polyethylene vs. hydrogenated polybuta-diene), as well as to the effect that "tethering" the block copolymer chains to the microdomain interfaces might have on the energetics of critical nucleus formation. That said, the agreement between the three sets is remarkable, with the block copolymer data extending the classical rate measurements by over 20 °C in undercooling and nearly seven decades in rate, due to the much smaller volumes of block copolymer microdomains as compared with suspended polymer droplets. This approach to measuring homogeneous nucleation rates should be straightforward to apply to any polymer that can be incorporated into a block copolymer.
Finally, Reiter et al. [2001] recently presented exciting atomic force microscopy (AFM) experiments on thin films (one microdomain thick) of an EO/aB diblock (EO/aB 4/21, where aB is atactic polybutene) supported on a Si wafer substrate. The elasticity differences between amorphous aB, amorphous EO, and crystalline EO permitted the clear resolution and identification of amorphous and crystallized EO spheres within the aB matrix, as shown in Figure 6.9. By imaging the array of spheres after various times of isothermal crystallization and simply counting the crystallized and uncrystallized spheres to determine the fraction remaining uncrystallized, the authors showed - in real space - that each spherical microdomain crystallizes independently, and generally follows firstorder kinetics. Some deviations from simple first-order kinetics were observed towards the end of the crystallization process (last 10 % of the domains), where the rate slowed substantially; similar deviations from precise first-order kinetics are seen in the isothermal crystallization kinetics of bulk EO/aB 4/21 measured by DSC [Rottele et al. 2003]. These deviations indicate that there is some variation in the nucleation rate across the population of EO spheres, with some spheres having a distinctly lower rate of nucleation than the average, though the origin of this variation remains unclear.
6.5.2 ISOTHERMAL CRYSTALLIZATION WITHIN CYLINDERS
Confined crystallization in cylindrical microdomains is also expected to result in first-order crystallization kinetics, provided the cylinders are not "connected" at their ends through grain boundaries between regions of cylinders with different orientation. Though the length of the cylinders - typically of order 1 micrometer - is much greater than their diameter, the number density of microdomains still vastly exceeds the typical density of homogeneous nuclei, and even a 1-mm growth distance can be covered rapidly at the deep undercooling at which homogeneous nucleation is effective. Both glassy-matrix E/ VCH [Loo et al. 2001] and strongly segregated E/SEB diblocks [Loo et al. 2002] forming E cylinders indeed exhibited first-order crystallization kinetics, with
time [min]
Figure 6.9 AFM phase images of a single-microdomain array of sphere-forming EO/aB 4/ 21. White circles represent crystallized EO spheres; dark circles are amorphous EO spheres. Panel A shows the array after a 5-min isothermal hold at —23 °C, while panel B shows the array after 15 min at —23 °C. By counting the uncrystallized spheres that remain after various crystallization intervals, the kinetic plot shown in panel C was created; the straight line represents the fit to first-order kinetics (Avrami n = 1). [Reprinted with permission from Reiter et al. 2001].
time [min]
Figure 6.9 AFM phase images of a single-microdomain array of sphere-forming EO/aB 4/ 21. White circles represent crystallized EO spheres; dark circles are amorphous EO spheres. Panel A shows the array after a 5-min isothermal hold at —23 °C, while panel B shows the array after 15 min at —23 °C. By counting the uncrystallized spheres that remain after various crystallization intervals, the kinetic plot shown in panel C was created; the straight line represents the fit to first-order kinetics (Avrami n = 1). [Reprinted with permission from Reiter et al. 2001].
rates approximately a factor of 30 faster than for their sphere-forming counterparts, reflecting the correspondingly larger volume per microdomain (cylinder vs. sphere).
Curiously, however, the same high molecular weight E/MB diblocks (E/MB 17/45 and E/MB 23/63) that appeared to retain the melt morphology (cylinders of E) upon crystallization [Quiram et al. 1997a] showed sigmoidal crystallization kinetics (n = 1.7—3.4 [Quiram et al. 1997b]), rather than the expected first-order kinetics. Similar findings are evident in the data of Shiomi et al. [2002] for cylinder-forming EO/B 5/10, where SAXS demonstrated that the hexagonal macrolattice is clearly retained even for relatively slow crystallizations, yet Avrami exponents n = 2.2—3.1 were measured. These puzzling results were explained by Loo et al. [2002], after obtaining TEM images of E/ MB 17/45 that showed occasional crystals traversing from one cylinder to another. Thus, a single nucleus can crystallize the material initially in several cylinders, and this "spreading" habit - created through infrequent "poke through'' events - produces the observed n > 1. Loo et al. [2002] termed this regime "templated" crystallization: the overall morphology of hexagonally packed cylinders is retained; the individual cylinders are effective in guiding the growing crystals (fast growth axis aligned with cylinder axis); but the crystallization kinetics are sigmoidal, and the overall crystallization rate is faster - often by orders of magnitude - than for analogous polymers (E/VCH and E/SEB) where E crystallization is wholly confined, and each cylinder must be separately nucleated.
Indeed, this strong dependence of rate on the connectivity of the crystalliz-able domains also manifests itself in the freezing point (Tf) measured by DSC at a constant rate of cooling. Typically, it is found that Tf decreases in the order lamellae > cylinders > spheres for polymers in the same chemical family, including EO/B [Chen et al. 2001a], E/VCH [Loo et al. 2001], and EO/BO [Xu et al. 2002a]. Frequently, a substantial depression of Tf from its value for the homopolymer, or a lamellar block copolymer, is taken as evidence of confined crystallization. However, this interpretation is likely oversimplified, especially for systems with rubbery matrices [Muller et al. 2002]. For example, lower molecular weight E/SEB sphere-forming diblocks show a depression of Tf by some 25 °C from the value for E homopolymers and E/VCH lamellar diblocks, but also exhibit sigmoidal crystallization kinetics and extensive "breakout" during isothermal crystallization [Loo et al. 2002]. Indeed, the Tf values for lower molecular weight E/SEB sphere-forming diblocks are actually slightly lower than for their higher molecular weight analogs, where confined crystallization is observed for all crystallization conditions. In the lower molecular weight diblocks, crystallization is still nucleated homogeneously, so during dynamic cooling, more time is required to nucleate each of the smaller spheres formed at lower molecular weights. But at the slower rates characteristic of isothermal crystallization (vs. dynamic cooling), each crystal can grow to span the material originally contained within several spheres, producing extensive breakout and sigmoidal kinetics. Two points should be drawn from these observations: first, that freezing points measured during dynamic cooling are only loosely related to the state of confinement during isothermal crystallization, and second, that direct structural measurements (e.g., by in-situ SAXS or post-facto TEM) are essential to properly confirm the state of confinement imposed on the crystals.
Based on the results from these E/MB and E/SEB block copolymers forming spheres or cylinders of E, Loo et al. [2002] were able to compile a "classification map'' where the normalized interblock segregation strength (the ratio of interblock segregation at the crystallization temperature to that at its order-disorder transition temperature) is plotted against the volume fraction of the crystalliz-able component, as shown in Figure 6.10. In the case of sphere-formers, crystallization is effectively confined within the microdomains when the normalized interblock segregation strength is high. Below the threshold segregation strength of 3 (normalized), dramatic structural rearrangement is observed on
Figure 6.10 Classification map of crystallization modes in E-based semicrystalline diblocks with rubbery matrices. Segregation strength at the crystallization temperature, normalized to that at the ODT, is indicated on the y axis. Volume fraction of ethylene block (vE) in each diblock is shown on the x axis; polymers with vE < 0.19 form spheres of E (circles), those with vE > 0.19 form cylinders of E (squares). Open symbols denote complete destruction of the melt mesophase upon crystallization ("breakout''); filled symbols denote complete confinement, as evidenced through first-order kinetics (Avrami n = 1); symbols with a vertical hatch denote templated crystallization, where SAXS indicates a general retention of the cylindrical melt morphology but sigmoidal crystallization kinetics (n > 1) indicate a ''spreading'' growth habit. [Reprinted with permission from Loo et al., Macromolecules (2002), 35, 2365-2374. Copyright (2002) American Chemical Society.]
Figure 6.10 Classification map of crystallization modes in E-based semicrystalline diblocks with rubbery matrices. Segregation strength at the crystallization temperature, normalized to that at the ODT, is indicated on the y axis. Volume fraction of ethylene block (vE) in each diblock is shown on the x axis; polymers with vE < 0.19 form spheres of E (circles), those with vE > 0.19 form cylinders of E (squares). Open symbols denote complete destruction of the melt mesophase upon crystallization ("breakout''); filled symbols denote complete confinement, as evidenced through first-order kinetics (Avrami n = 1); symbols with a vertical hatch denote templated crystallization, where SAXS indicates a general retention of the cylindrical melt morphology but sigmoidal crystallization kinetics (n > 1) indicate a ''spreading'' growth habit. [Reprinted with permission from Loo et al., Macromolecules (2002), 35, 2365-2374. Copyright (2002) American Chemical Society.]
crystallization. For cylinder-formers, structural rearrangement is again observed at weak interblock segregation (< 1.5, normalized) and confined crystallization is again observed at strong interblock segregation (> 4). However, ''templated'' crystallization occurs between these two limits; here, crystallization generally occurs within the microdomains determined by microphase separation, but crystallization produces local distortions to the regular microdomains established in the melt, extending even to occasional interconnections between cylinders that permit a single nucleus to crystallize the material originally in many cylinders. Xu et al. [2002b] also found that the EO/BO system hewed closely to this same classification map, including the positions of the dividing lines (especially the critical value of 3 for the normalized segregation strength needed to confine crystallization within spheres). Given the substantial chemical and physical differences between the EO/BO, E/SEB, and E/MB systems, the classification map in Figure 6.10 should be a useful guide for determining the conditions needed to confine crystallization in any semicrystal-line-rubbery block copolymer.
6.5.3 ISOTHERMAL CRYSTALLIZATION IN INTERCONNECTED MICRODOMAINS
Both the "templated" crystallization in cylinders described by Loo et al. [2002], and the orientationally registered crystals in thin films of the lamellar EO/B 6/5 described by Hong et al. [2001a, 2001b] point out the strong impact that interconnections between microdomains can have on the crystallization process. When growing crystals can percolate through all the E-rich microdomains present in the melt, then the crystallization kinetics are not expected to differ qualitatively from those of homopolymers - even when the microphase-separ-ated morphology established in the melt is preserved into the solid. For example, Loo et al. [2001] investigated crystallization within the gyroid channels in a semicrystalline-glassy diblock (E/VCH 8/13), and found unremarkable sigmoidal kinetics (n = 1.7) despite faithful preservation of the gyroid structure by the glassy matrix.
As noted by Hong et al. [2001a, 2001b], lamellae present a particularly interesting case. Like cylinders, the lamellae in an idealized block copolymer grain are totally unconnected. However, the practical difficulty in isolating lamellae from each other is even greater than for cylinders, because of the larger volume per lamella (vs. cylinder); an extremely low defect density would be required to observe isolated crystallization within lamellae. Consequently, lamellar block copolymers are generally reported to exhibit sigmoidal crystallization kinetics, even when the other block is vitreous; for example, Hamley et al. [1998b] reported an Avrami n = 3 for E/VCH 8/7. Still, this result might be expected to depend strongly on the defect density in the mesophase structure, which is difficult to characterize independently. By contrast, Loo et al. [2001] observed a two-step crystallization process in a lamellar semicrystalline-glassy diblock (E/VCH 12/8) similar to the E/VCH 8/7 studied by Hamley et al. [1998b]; the (minority) higher-temperature crystallization process followed sigmoidal crystallization kinetics, while the (majority) lower-temperature process followed first-order kinetics. This two-step process reveals the presence of two distinct populations of E lamellae: a minority interconnected population (perhaps through grain boundaries or screw dislocations), and a majority population of isolated E lamellae, each of which must be independently and homogeneously nucleated.
These results demonstrate the exquisite sensitivity of crystallization kinetics to microdomain topology, particularly interconnection of the domains formed by the crystallizable component. Compared with conventional methods such as mechanical testing or gas-transport measurements, the crystallization kinetics -both Avrami exponent and undercooling - can reveal even low levels of microdomain-connecting defects, at the level of the percolation threshold (two connecting defects per microdomain).
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