BCC Grain Formation Triggered by Miscibility Jump on Temperature

Mar 2, 2015 - Department of Macromolecular Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Jap...
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BCC Grain Formation Triggered by Miscibility Jump on Temperature Drop Akifumi Matsushita,† Shigeru Okamoto,*,† Eiko Tamura,‡ and Tadashi Inoue‡ †

Department of Materials Science and Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi 466-8555, Japan ‡ Department of Macromolecular Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan S Supporting Information *

ABSTRACT: Formation of a body-centered cubic (BCC) structure and grain growth process triggered by segregationpower jump on a temperature drop was studied by small-angle X-ray scattering (SAXS), rheology, and differential scanning calorimetry (DSC). Polystyrene-block-poly(ethylene-alt-propylene) (SEP) was dissolved in tridecyl-2,2,4-trimethyl hexanoate (salacos913) that was a practically neutral good solvent for both of polystyrene (PS) and PEP block chains at temperature above 84 °C (Tsol), while it was highly selective (good for PEP) below Tsol. Spherical microdomains in a short-range liquid-like order were formed above Tsol; the system was in the so-called “lattice disordered state”, designated as disordered sphere. The solution was annealed at a temperature (130 °C) above Tsol for 10 min and successively subjected to a temperature drop across Tsol. The system stayed in the lattice-disordered state for a certain induction period. During this induction period, stronger segregation power at the lower temperature increased the domain spacing, whereas a storage shear modulus (G′) showed liquid-like behavior (G′∝ ω2) at low frequencies (ω < 0.2 s−1) in a terminal zone and a shoulder at ω ∼ 1 s−1. The shoulder shifted toward the smaller ω region, arising from dissociation of the PS block from the solvent. Once BCC lattice structures of spherical microdomains formed, grains eventually grew in size up to ca. 2.5 μm with a large size-distribution as revealed by the 2d-SAXS with spot-like scatterings, whereas G′ in the terminal region increased, arising from the increase in correlation length of the spherical microdomains. Eventually, the G′ showed plateau at lower frequencies at ω < 0.2 s−1, indicating that the BCC lattice of spheres with long-range order (grain stuructures) was percolated throughout the solution. The number of the grains still continued to increase at the cost of spherical microdomains in the lattice disordered state, which caused the further increase in G′ at the plateau until the end of the ordering process of the BCC structure.

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INTRODUCTION

weight that enhanced nonlinear optical property of gold nanoparticles.17 However, BCPs generally have limited tunability in a structural dimension. For example, it is known that a domain spacing, d, of a neat BCP or in a solution follows the temperature dependence of d = T−1/3: Thus, a BCP shows only less than 10% change in d with the temperature change of 100 °C. Here, T denotes absolute temperature. For more functional devices, such as a tunable laser resonator, an optical switching device, etc., demanded are tunability or drastic changes in the spacing d by environmental stimuli including external fields, such as temperature, pH, magnetic field, electric field and so on. Some researchers,18−21 varied domain spacings by the change in molecular weight of a BCP, blending BCP of different

Self-assembly of block copolymers (BCPs) in a molten state or in a solution allow the formation of microphase-separated structures with a variety of properties and functions; structures are typically on the order of nanometers; the period and morphology are primarily determined by the molecular weight and the volume fraction of constituent block chains, respectively. Recent progresses in BCP synthesis have enhanced potential capability to fabricate materials of customized functions. Applications span many kinds of products from thermoplastic elastomers,1,2 adhesives,3 surfactant,4,5 cosmetics,6,7 etc. to highly functionalized materials, such as a solar cell,8−10 nanolithography,11−13 etc. In these decades, application to photonic crystals (PhCs)14−21 has been also studied to control the flow of light. Yoon et al. applied thin layers of highly ordered lamellar structures to laser resonator.16 Tsuchiya et al. fabricated PhCs in a solution of a BCP with high molecular © 2015 American Chemical Society

Received: January 5, 2015 Revised: February 6, 2015 Published: March 2, 2015 1813

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a solid was seen below TODT. Furthermore, lamellar structure exhibits the lowest moduli, followed by cylinders, perforated layers, and then the cubic phases that appear solid-like over a wide range of frequencies in the terminal region. The lower symmetry phases, such as the BCC spheres and gyroid phases, are classified as solids with regard to their zero-frequency response to stress. Winter et al.31 demonstrated, by small-angle X-ray scattering (SAXS) and rheology, structural evolution after quenching specimens from the temperature above TODT to that below it; they showed microphase-separated cylindrical domains were formed immediately after the quench as detected by SAXS, whereas storage modulus in the terminal region revealed organization of grain network structures required over 13 h. In this paper, we comprehensively investigated time evolution of BCC structure after miscibility jump by quenching from lattice-disordered state to highly ordered state. We used a BCP solution, polystyrene-block-poly(ethylene-alt-propylene) (SEP)/isotridecyl isononanoate (salacos913). Salacos913 is a strongly selective for poly(ethylene-alt-propylene) (PEP) at room temperature; the high selectivity enhances segregation;32 thus, the solution forms highly ordered structure. The specimen formed well-ordered BCC structure and the SAXS profile showed clear seven lattice-scattering peaks that enabled us to conduct a fine analysis on the structural development; we will discuss temporal changes in domain spacing, the degree of order, the size and the number of BCC grains on the basis of SAXS and rheological responses.

molecular weights, blending a BCP with short homopolymers; they successfully ranged over the whole wavelengths of visible light. However, these techniques require many procedures including dissolving different polymers for each desired wavelength as well as synthesis of a BCP with a different molecular weight. Walish et al.18 demonstrated a faster change in domain spacing of a hydrophilic/hydrophobic BCP with a tunability induced by electrochemical stimulation, which showed high tunability of the domain spacing from 350 to 1600 nm. A simple change in solvent concentrations is also a very effective method. Recently, we reported that lamellar domain spacing in solutions varied over a vast range from ca. 300 to 800 nm as a result of drastic change in segregation power by the jump of miscibility between the phases on mixing two immiscible selective solvents with a neutral solvent:22 The selective solvents are separately sequestered in corresponding phases and increase segregation power, while a neutral solvent is uniformly distributed in all the phases and decreases segregation power. Furthermore, the immiscible selective solvents with refractive index difference increased the refractive contrast between the phases that will, in future, provide higher functions to PhCs, such as higher reflectivity, lower threshold of laser resonator, more efficient confinement of light, etc. Nevertheless, this technique also requires some procedures to blend solvents for the tunability. In this study, we control the segregation power by an miscibility jump between a block chain and a solvent on temperature change and study resulting ordering process in which we will discuss grain growth. Here, a “grain” denotes an aggregation of microdomains that are coherently ordered; a grain is one of the best candidates as a highly ordered structure for functional materials, e.g., anisotropic optical materials. BCPs generally form small grains with random orientation. The state of order is affected by chain mobility and segregation power that are strongly dependent on temperature. The order was classically investigated and high order was obtained by simply annealing systems,23−27 especially quenching systems across order−disorder transition temperature (TODT) or latticedisordering transition temperature (TLDT) enabled us to investigate grain growth mechanisms. Sota et al. demonstrated two ordering paths to obtain hexagonally packed cylinders: BCC was formed as an intermediate structure by annealing a system slightly below TLDT (shallow quench), whereas cylinders were directory formed by annealing a system well below TODT (deep quench).23 Symmetry of morphology alters direction or dimensionality of grain growth. Chastek et al.24 induced grains of gyroid structure from disordered state via metastable hexagonally perforated-layer phase; isotropic grain growth was found with a constant velocity of growth front that was dependent on segregation power and relaxation time of polymer chains. Dai et al.28 studied grain growth of hexagonally packed cylinders; growing velocity along the cylinder axis was faster than any other directions; small grains were merged with large grains after the whole sample was filled with grains. Hashimoto and co-workers25 found an isolated lamellar grain of an anisotropic shape nucleated in the matrix of disordered phase by shallow quench from disordered state to temperature below TLDT. Low-frequency viscoelastic measurement is dependent on the state of order and an effective method to identify TODT and TLDT, which was first demonstrated by Gouinlock et al.29 and Chung et al.30 In the disordered state, liquid-like behavior was observed while intermediate response to a Newtonian fluid and

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EXPERIMENTAL SECTION

In this study, we used an SEP, i.e., a hydrogenated polystyrene-blockpolyisoprene (SI), purchased from Kraton Polymer Japan. The number-averaged molecular weight (Mn), polydispersity index (PDI: Mw/Mn) and weight (volume) fraction of polystyrene is 170 000 g/ mol, 1.09 and 28 wt % (25.2 vol %), respectively. Here, Mw is the weight-averaged molecular weight. A selective solvent, salacos913, was kindly supplied by Nisshin OilliO. Salacos913 is highly selective for PEP. The solution of polymer concentration of 10 wt % was prepared at 100 °C, hereafter designated as SEP-S. A cloud point measurement was conducted using a polystyrene (PS) with Mn of 4.06 × 104 g/mol and PDI of 1.03, hereafter designated as PS41. The molecular weight of PS41 is close to that of the PS block of the SEP: Mn = 4.76 × 104 g/ mol. SAXS measurements were performed on Beamline 40B2 at SPring8, Japan. The X-ray wavelength was 0.15 nm. The camera length between the detector and the sample was 4000 mm. The magnitude of scattering vector is defined as q = 4π(sin θ)/λ where 2θ and λ are the scattering angle and the wavelength, respectively. An imaging plate (RAXIS VII, Rigaku) and an II-CCD camera (HAMAMATSU Photonics K.K.) were used as the detector. The beam sizes at the detector position and the sample position were 0.2 × 0.2 and 0.5 × 0.4 mm2, respectively. Additional SAXS measurements were carried out on a Nanoviewer (Rigaku) instrument. The X-ray beam of Cu Kα line (λ = 0.154 nm) was incident on the sample. An imaging plate (R-AXIS IV+ +, Rigaku). A cell of thickness of 0.98 mm with Kapton windows were used for these SAXS measurements. The scattering intensity was corrected for absorption due to the samples, air scattering from an empty cell (also with two pieces of thin Kapton film as windows) and dark current, i.e., electric noise due of the detector. For measurements above room temperature, the temperature was controlled within the error of ±0.1 °C by a TMS 94 controller (Linkam Scientific Instruments Ltd.). Differential scanning calorimetry (DSC) measurements were performed on SEP-S and a reference solvent (Salacos913) filled in a pair of sealed aluminum cells by using Rigaku DSC-8230. 1814

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Macromolecules Viscoelastic measurement was conducted with a rheometer (Anton Paar, Physica MCR 301) using a cone-and-plate geometry (25 or 40 mm diameter, 4° cone angle and 49 μm initial gap). All of the measurements were collected at strain of 1.0%, which is in a linear viscoelastic regime. For the analysis of SAXS profiles to estimate a domain spacing, a radius of a sphere and their distributions, scattering curves were calculated on the basis of the paracrystal theory33 and the Percus− Yevick hard-sphere liquid theory.34 The paracrystal theory was used to calculate scattering for a body-centered cubic lattice (Im3m), taking into account Hosemann’s lattice distortion of the second kind, whereas the Percus−Yevick hard-sphere liquid theory was used to calculate scattering for spheres in a liquid-like lattice-disordered state. In both of the theories, Gaussian distributions of a sphere radius, a domain distance and an interfacial thickness of a sphere were taken into account.

arise from decrease in enthalpy because PS homopolymer is still immiscible with the solvent at Tsol as mentioned below. It is also worth noting that exotherm was not observed in a cooling run, which indicates that it takes some time for the solvent molecules to move out of PS domains. In order to elucidate the above-mentioned conjectural change in the solvent quality for PS, a cloud point of a solution of PS41 in salacos913 at the polymer concentration of 3.0 wt % was measured. This concentration is almost the same as that of the PS block chain in SEP-S. PS was miscible in salacos913 at 130 °C whereas it was immiscible at room temperature, indicating that the solution has a UCST type of phase diagram. Then, the solution was subjected to the turbidity measurement by cooling from 135 °C to room temperature at the rate of ca. −0.3 °C/min. PS41 certainly precipitated at 114 °C. This miscibility jump is the key in this study although the cloud point is much higher than Tsol. We speculate this temperature discrepancy arose from the difference in their molecular architecture: The PS blocks in SEP are connected to the PEP blocks that are readily miscible with salacos913, and hence, the PS blocks are forced to be mixed with the solvent at Tsol at the cost of enthalpy loss that, in turn, is well compensated for by the great entropy gain as mentioned above. Microphase-separated structures formed in SEP-S were clearly affected by the miscibility jump. Figure 2 shows

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RESULTS AND DISCUSSION The point to understand the structural change in a solution, SEP-S, described in this report is miscibility jump between PS and the solvent, salacos913 that is a highly selective good solvent for PEP at room temperature. First of all, thermal analysis of SEP-S was conducted by DSC. The concentration of SEP in a solution was fixed at 10 wt % throughout this study. DSC thermogram was recorded at the heating rate of 10 °C/ min and at the cooling rate of −5 °C/min. An endothermic peak was detected at 83.9 °C as shown in Figure 1. It is known

Figure 1. DSC thermogram of SEP-S with the heating rate of 10 °C/ min and the cooling rate of 5 °C/min. Figure 2. SAXS profiles (solid lines) of SEP-S at several temperatures in the heating cycle with superposition of calculated curves (dotted lines).

that ODT and LDT are detectable by DSC measurement for BCPs with symmetric composition,35,36 whereas Kim et al. mentioned that they were undetectable for BCPs with highly asymmetric composition because the DSC signal was too small.37 In this study, the endothermic peak was detected though the SEP used has highly asymmetric composition, suggesting that this endotherm is not due to ODT or LDT. Note that the volume fraction of PS is 25.2 vol %. We assume that miscibility between PS and the solvent gradually increases with increasing temperature and the solvent quality changes from “selectively good for PEP” to “neutral” at 83.9 °C (hereafter designated as Tsol). Thus, the endotherm can presumably be attributed to entropy gain that arises from the solvation of PS in the solvent. However, it is not considered to

circular-averaged 1d SAXS profiles from SEP-S observed as a function of temperature: SEP-S was heated from room temperature to 90 °C across Tsol at the rate of 2 °C/min. The profiles were shifted to avoid overlap. The SAXS profile observed at 26 °C (as-prepared) have seven sharp scattering maxima, arising from lattice symmetry, whose positions relative to the primary one are 1:√2:√3:√4:√5:√6:√7, which is typical of body-centered cubic (BCC) lattice with long-range order. Here, it is considered that PS forms spherical microdomains in the matrix of PEP because of the following 1815

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Macromolecules two reasons; PS has minor fraction in this block copolymer (25.2 vol %); the solvent is selective for PEP at 28 °C and hence it selectively swells PEP. Actually, the volume fraction of the spheres in the solution is 2.75 vol % as mentioned later. As the temperature increased, the scattering intensity decreased with the peak width increased and the higher-order maxima gradually disappeared. It is also noticeable that the scattering maxima shifted toward the higher-q region, i.e., the domain spacing decreased as the temperature increased: e.g., 58.6 nm at 78 °C to 52.6 nm at 90 °C. These results indicate segregation power decreased with the increase of the temperature. Eventually, all of the distinct higher-order maxima vanished at 90 °C slightly above Tsol, where the solvent was more uniformly distributed as a neutral solvent by the change of the solvent quality and shielded the segregation power between PS and PEP. There still remained the broad primary peak with a shoulder peak comprising multiple higher-order broad scattering, arising from interparticle interference without lattice order, which can be attributed to so-called lattice-disordered structure (short-range order in lattice-disordered state). Note that all of these profiles have particle scattering, arising from intraparticle interference, though not observed here because a small-area CCD detector was used for this experiment; particle scattering will be shown later in Figure 4. The SAXS profiles were fitted with theoretical scattering curves (shown by dotted lines in Figure 2) calculated on the basis of the paracrystal theory, from which the volume fractions of PS domain and salacos913 in the PS domain were obtained as summarized in Table 1. The PS volume fraction has

Figure 3. 2d-SAXS patterns obtained from SEP-S: (a) before the thermal treatment at the room temperature, (b) just after the temperature drop from 130 to 28 °C, (c) annealed at 28 °C for 12 h, and (d) 35 h after the temperature drop from 130 °C.

(Figure 3a); just after the temperature drop from 130 to 28 °C (Figure 3b); annealed at 28 °C for 12 h (Figure 3c), and 35 h (Figure 3d). The SAXS from the as-prepared solution in Figure 3a showed the isotropic pattern with multiple sharp scattering peaks, arising from the BCC lattice of spheres with long-range order as described before. That, however, vanished just after experiencing high temperature at 130 °C and only weak broad peak was observed as seen in Figure 3b. The scattering intensity increased without any sign of sharp peaks during the next 12 h (Figure 3c). Eventually, the multiple isotropic sharp peaks appeared after annealing for 35 h, indicating high degree of order. More interestingly, there appeared many scattering spots arising from large grains24 of the BCC lattice of spheres as will be further described later (Figure 3d). The results indicate that the long-range order of the BCC sphere phase can be reproducibly observed at the room temperature; however, it disappears at the temperature higher than Tsol. The time evolution of the long-range order is well discernible in circular-averaged 1d-SAXS profiles. Figure 4a shows circularaveraged SAXS from SEP-S with the same thermal history corresponding to Figure 3, i.e., before the thermal treatment (as-prepared), just after the temperature drop from 130 to 28 °C (0 h), annealed at 28 °C for 4, 12, and 35 h. Each profile was vertically shifted to avoid overlap. Here, a large-area imaging-plate detector was used to collect particle scattering in a high-q region. The profile of “as-prepared” have seven sharp lattice scattering peaks marked by thin arrows whose positions relative to the primary one at q = 0.113 nm −1 are 1:√2:√3:√4:√5:√6:√7, typical of the BCC lattice with long-range order as mentioned before. The broad peak at ca. 0.5 nm−1 marked by the filled thick arrow is particle scattering of the PS spherical microdomains. The domain spacing and the domain size (radius) were estimated to be 55.8 and 12.1 nm, respectively, from the primary peak of the lattice scattering and the particle scattering. Just after the temperature drop from 130 °C (higher than Tsol), the long-range order has not been retrieved yet, which is obvious by the fact that the profile of “0 h” has no sign of the higher-order peaks but only a broad

Table 1. Volume Fractions of PS Domain and Solvent in PS Domain Estimated by Fitting SAXS Profiles of SEP-S with Theoretical Curves Based on the Paracrystal Theory temp, °C 90 88 86 80 78 70 26

volume fraction of PS domain, % 4.70 4.53 4.40 3.67 3.52 2.84 2.75

± ± ± ± ± ± ±

1.53 1.21 0.977 0.710 0.519 0.429 0.384

volume (weight) fraction of salacos913 in PS domain, % 50.8 48.9 47.4 37.0 34.2 18.4 15.7

± ± ± ± ± ± ±

12.1 10.8 9.54 10.2 8.46 10.7 10.4

(45.8 (43.9 (42.5 (32.5 (29.9 (15.6 (13.2

± ± ± ± ± ± ±

12.3) 10.9) 9.53) 9.81) 8.00) 9.58) 9.16)

negligibly small difference between 26 and 78 °C where the PS domain contains a quite small amount of the solvent, whereas both of the PS and solvent volume fractions significantly increased in the vicinity of Tsol. This miscibility jump, i.e., the solvent quality change at Tsol caused the order−order transition from BCC to disordered spheres structure. Temperature dependence of SAXS from a solution of SEP-S was studied by two-step annealing. First, SEP-S was annealed at 130 °C for 10 min to bring solutions into lattice disordered state, where the selectivity of the solvent for PEP is much less pronounced as described before; salacos913 is practically a neutral solvent for both of PS and PEP. Second, temperature was dropped to lower temperatures (25−82 °C) where solutions were annealed for decades of hours; the solvent is highly selectively good for PEP. Cooling speed was moderate: e.g., it took 2 min from 130 °C down to 60 °C; 5 min to 28 °C. Figure 3 shows 2d-SAXS patterns from a SEP-S solution before and during annealing process; at room temperature before the temperature jump to 130 °C, i.e., as-prepared 1816

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Figure 4. (a, b) Circular-averaged SAXS profiles of SEP-S. From the bottom, (a) as-prepared, annealed at 28 °C for 0, 4, 12, and 35 h after the temperature drop from 130 °C, and (b) annealed at 60 °C for 0 to 80 min of annealing after the temperature drop from 130 °C.

primary peak, a shoulder peak marked by the open thick arrow and a particle scattering marked by the filled thick arrow (typical of disordered micelle structure as mentioned above).23,37 It is worthy to note that the existence of the particle scattering proves the system is in phase-separated state. The lattice and particle scattering peaks still locate at slightly higher q region than those of “as-prepared”; the domain spacings and the particle sizes are estimated to be 52.2 and 11.4 nm. The profiles obtained after annealing for 4 and 12 h at 28 °C show the same feature as seen in that of “0 h”. Therefore, it is considered that during annealing for 12 h, SEP-S was kept in the lattice-disordered state. In this period, the lattice and particle scattering peaks shifted slightly toward lower q region: i.e., the domain spacing and the particle size increased; the domain spacings and the particle sizes were estimated to be 54.2 and 11.7 nm from the profiles of 4 and 12 h, respectively. This indicates that the segregation power gradually increased by the above-mentioned slow dissociation of the solvent from PS domains. Sharp multiple lattice scattering peaks were retrieved by annealing for 35 h and the peak positions shifted back to the q values almost identical with those in the profile of “asprepared”: i.e., the domain spacing and the particle size fully recovered and the regular BCC lattice was formed again after such long annealing period. Note that the profile of SEP-S annealed for 35 h shows sharper peaks than that of the asprepared, indicating that the degree of order was increased; hence, it is considered that large grains were formed by annealing, as the spot-like scatterings were observed as shown in Figure 3d. Figure 4b shows time evolution of circular averaged SAXS profiles of SEP-S annealed at 60 °C after the temperature drop from 130 °C. The profiles for the first 10 min are quite similar to those observed for the first 12 h in Figure 4a; hence, the system is also considered to be in the lattice-disordered state. During the further annealing, the scattering intensity gradually increased and higher-order maxima appeared; the √2 and the √3 peaks were observed 15 min after the drop, when spot-like scattering as shown in Figure 3d started to be observed in 2dSAXS though not shown here; eventually, four lattice scattering peaks were observed 40 min after the drop, which indicates that BCC structure was actually formed also at 60 °C. Further

annealing up to 80 min made no more difference in the structure. The comparison of the result obtained at 60 °C with that at 28 °C gives us a perspective on structure formation: the BCC lattice with long-range order is formed in shorter time at higher temperature, which may arise from the fact that SEP has higher mobility in the solution; the degree of structural order is higher at lower temperature, as evidenced by the fact that SEPS annealed at 28 °C for 35 h showed a larger number of sharper scattering peaks than the one annealed at 60 °C for 80 min. Here, it is worthy to note that the intensity ratio of the √2 peak to the √3 peak of the profiles in Figures 4b is not identical to that in Figure 2. The √2 peak is well suppressed in Figure 4b, whereas the √2 peak in Figure 2 is strong and clearly seen. See, for example, the profile at 70 °C in Figure 2: This is the profile we measured at the temperature closest to 60 °C in Figure 4b. The difference in the intensity ratio is presumably ascribed to difference in the volume fractions of the constituent phases originating in the thermal history. The profiles in Figure 2 were obtained in the heating process at the rate of +2 °C/min, while those in Figure 4b were isothermally measured after the temperature drop. Reorganization of the BCC lattice, accompanied by the redistribution of the solvent molecules, may take relatively long time with the temperature change. One should be reminded that an exothermic peak was not observed at Tsol in the cooling process, presumably because the diffusion of the solvent to go out of the PS phase was so slow. Below Tsol, the diffusion in the PS phase may be even slower because the PS chain mobility is lower at the lower temperature at the higher polymer concentration. Therefore, in Figure 2, the intensity ratio above 70 °C is very close to that of the virgin sample at the room temperature (26 °C). In other words, the structures above 70 °C were far from the equilibrium and close to that at 26 °C. In contrast, the structures shown in Figure 4b are considered to be much more equilibrated by annealing for such a long time. In addition, and more profoundly, it is worth while to note that all of the multiple lattice scattering peaks are very sharp: See for example the profiles at 40, 60, and 80 min in Figure 4b, whereas the intensity decreases without broadening the width as time goes back from 80 to 60 min. Moreover, the peaks do not have a shape of a usual brag scattering: The widths at the 1817

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temperature dependence of chain mobility and driving force of the BCC formation originating in segregation power. Namely, the driving force is stronger/weaker at lower/higher temperature, where conversely, higher/lower viscosity slows down/ assists the formation of BCC, respectively.24,27,38−40 Here, we elucidate the ordering kinetics over the temperature range from 40 to 70 °C by the Avrami plot along with the Arrhenius plot for the SAXS. We selected the above temperature region because the induction period had minimum at 70 °C and increased with temperature as seen in Figure 5. The Avrami equation was fitted to integrated scattering intensity Q(t), where Q(t) is defined as

bottoms are clearly wider. This can be attributed to a possible excess scattering at the bottom. It is considered that the SAXS profiles comprise two structure factors: one arises from the regular BCC lattice, the other from disordered spheres. We superposed scattering profiles (red dots) obtained in the azimuthal angles, in which no spot-like scattering was observed, of the corresponding 2d-SAXS patterns with many scattering spots: See the profiles at 35 h in Figure 4a and at 80 min in Figure 4b. The profiles (red dotted lines) well fit the bottom curve of those with multiple sharp scatterings, indicating that the structure formed in SEP-S consists of the regular BCC lattice and disordered spheres. Therefore, we decomposed the scattering into that from the BCC lattice and that from disordered spheres: We define the percentage of the BCC lattice by the percentage of the intensity from the BCC lattice in the total intensity of the primary peak (IBCC(t)). The values of IBCC(t) were, respectively, 89 and 43% for 35 h annealing time at 28 °C and 80 min at 60 °C, which means that SEP-S annealed at the lower temperature contained larger amount of the regular BCC lattice than at the higher temperature. Time evolution of IBCC(t) after the temperature drop from 130 to 60 °C will be discussed later. As mentioned above, the regular BCC lattice was not formed in the early stage of annealing; namely, there was an induction period after the temperature drop as revealed by the late appearance of the spot-like scattering with the multiple sharp peaks. The induction period was strongly dependent on annealing temperature as seen in Figure 4. Here, we define induction period by the emergence of the second-order scattering maximum at √2q*, where q* is the magnitude of the scattering vector at the primary peak position. Figure 5

Q (t ) =

∫q

q2

I ( q , t )q 2 d q (1)

1

The values of q1 and q2 were selected as all the lattice peaks are covered by the q region of q1 < q < q2. The values of Q(t)/ Q(∞) measured at 40, 60, and 70 °C followed the Avrami equation as shown in Figure S1a of the Supporting Information: Q(t)/Q(∞) ∼ 1 − exp(−Kta). The deviation between the experimental data and the Avrami equation was seen at ca. 12, 25, and 100 min that are in good agreement with the induction period in Figure 5. The exponent, a, was, respectively, 2.79, 3.47 and 2.74, whereas the Avrami rate constant K was 7.34 × 10−7, 5.05 × 10−6 and 2.16 × 10−4 at 40, 60, and 70 °C, respectively. These values of K was fitted by the Arrhenius equation as seen in Figure S1b, yielding the activation energy of Ea = 155 kcal/mol. As a function of annealing time, we estimated the grain size from SAXS according to the method presented by Hosemann, i.e., from the so-called Hosemann plot, in which a grain size is estimated from scattering peak width of multiple peaks of higher-order harmonics of the same lattice planes.41,42 For this evaluation, the scattering from the disordered sphere was subtracted so as to extract the scattering arising from the regular BCC lattices. A grain size can be more precisely estimated by using a larger number of higher-order harmonics unless they violate the criterion mentioned below; however, only scattering peaks of {110} lattice planes and second-order harmonic of {110}, i.e., {220} lattice planes were available in this study because we had only seven lattice scattering peaks, i.e., six higher-order maximum from √2 to √7 relative to the primary peak. A grain size was estimated by the following equation: (Δqn /2π )2 = 1/D2 + π 4g 4 n 4 /d 2

(2)

under the criterion of 2π g n ≪ 1, where Δqn is a full width at half-maximum of a nth harmonic of the {110} lattice planes, d is a distance of the {110} lattice planes (domain spacing), g is Hosemann’s lattice distortion of the second kind, D is a grain size along the normal vector of the {110} lattice planes. Note that D is considered to be a diameter of the grains because BCC grains generally grow spherically. Note also that the values of n are 1 and 2 for {110} and {220}, respectively. At each annealing time, a grain size was estimated at least 10 times using different profiles randomly selected from a SAXS pattern, though the plots are not shown here. Figure 6 shows time evolution of the grain size estimated from the SAXS profiles obtained during annealing at 60 °C after the temperature drop, corresponding to the measurement shown in Figure 4b. The onset of the BCC formation was 15 min as mentioned above. In the time region from this onset time to 25 min, the fourth2 2 2

Figure 5. Induction period of the BCC formation process in SEP-S as a function of the annealing temperature.

shows induction periods thus obtained for various annealing temperatures with error bars arising from inaccuracy to determine the emergence of sharp peaks and experimental variability. The BCC lattice was not formed above Tsol, whereas BCC lattices formation was not observed below room temperature within accessible experimental time though expected to be observed; thus induction period was determined in the temperature range between 25 and 82 °C. The induction period drastically decreased with increasing temperature in the range below 70 °C, whereas it slightly increased above 70 °C as seen in the inset. This opposite tendency is attributed to 1818

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g factor by fitting theoretical curves based on the paracrystal theory to the experimental profiles. The comparison found that the g factors obtained from the Hosemann plot (gH) were smaller than those obtained by the fitting method (gf) as shown in Figure S3 of the Supporting Information. This discrepancy is possibly attributed to smearing effect on a scattering peak width by the distribution of the incident X-ray. Although the beam size at the detector position was as small as 0.2 × 0.2 mm2, the peak widths of the scattering spots were also so small; actually, the diameters of the primary peak ({110}) and the second harmonics ({220}) were less than 1 mm (typically, ca. 0.5 mm). Such a sharp peak is strongly expected to be affected by the smearing effect. Moreover, the smearing effect, generally, is greater on a sharper peak at a lower q region. Thus, the effect is greater on the {110} peak than on the {220} peak, causing the intercept and the slope in the Hosemann plot to increase and decrease, respectively. As a result, the value of gH becomes smaller. To the contrary, the broadening of peak width by the same smearing effect will increase the gf factor determined by the fitting method. Therefore, it is conceivable that the value of gH was evaluated to be smaller than that of gf. Finally, SEP-S was subjected to the dynamic viscoelastic measurement to shed light on time evolution of mechanical properties, coupled with the state of order, during the BCC formation under the same thermal history corresponding to the SAXS experiments shown in Figure 4. Measurements were conducted on the sample annealed at 60 °C every 5 min by a scan from 6.28 to 0.135 s−1. It takes ca. 4.3 min per a scan that is too long for such a fast structural evolution: the last data in one scan is taken ca. 3.4 min after the start and ca. 1.6 min before the start of the next scan. From thus obtained data, therefore, we reconstructed quasi-isochronal graphs with the exact interval of 5 min by extrapolation or interpolation as demonstrated in Figure S4 of the Supporting Information. Figure 7b shows the thus reconstructed quasi-isochronal storage modulus (G′) of SEP-S annealed at 60 °C as a function of angular frequency (ω) of oscillatory shear. Figure 7a shows G′ obtained during annealing at 28 °C as a function of ω without the above-mentioned reconstruction because temporal change of G′ was measured for decades of hours with an interval of several hours though a scan takes ca. 45 min. For all the annealing time in Figure 7a, the fast mode in the high frequency region at ω > 20 s−1 is dominated by single-

Figure 6. Evolution of grain of BCC formed in SEP-S at 60 °C.

order maximum (the second-order harmonic of {110}) was not observed. Thus, grain size was not estimated by eq 2 but by another method43 using Δq/2π = 1/D, where Δq is a peak width of a primary peak: Grain sizes obtained by the latter method are slightly smaller than that obtained by the former one because the second term of eq 2 is omitted in the latter method. It is, however, worthy to add these data here, which enable us to grasp an overall tendency. The lattice distortion factor, g, was obtained from the Hosemann plot according to eq 2 to confirm the validity of the grain size obtained below. The value scattered over a wide range from 0.005 to 0.039 as shown in Figure S2 of the Supporting Information: These values satisfied the above criterion. The grain size at the onset of the BCC formation was ca. 700 nm; it increased until ca. 30 min and leveled off at ca. 2 μm, where the grain size had such a wide distribution. It should be noted that for the estimation of the grain size, we excluded profiles in which the scattering peaks were overlapped; by this way, small grains, intrinsically showing broad weak scattering intensity, were eliminated from the evaluation with relatively high probability. This means that the real grain size should cover also a smaller value region. In other words, the real grain size presumably has a huge distribution from 700 to 2500 nm, indicating that formation of the BCC lattice continuously may take place via homogeneous nucleation. It is also interesting to compare the above g factor with the one evaluated by a different method. We evaluated the

Figure 7. Time evolution of storage modulus (G′) of SEP-S as a function of angular frequency (ω) annealed at 28 °C (a), and 60 °C (b) after the temperature drop from 130 °C. Bold lines represent a slope of 2, i.e., G′∝ ω2. 1819

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modulus Ge is given by kT, where k is Boltzmann constant and T is absolute temperature.

chain dynamics and entanglement of PEP corona chains. The slow mode at ω < 0.2 s−1 is strongly dependent on the annealing time: In the beginning, it was typical of liquid or molten homopolymer (G′∝ ω2) until 14 h after the temperature drop; then, it became independent of ω; i.e., the plateau region was seen after 14 h, which is typical of threedimensional morphologies of BCPs including BCC, gyroid, etc.44,45 In the intermediate region at ω ∼ 1, there existed a shoulder of additional elastic stress-and-dissipation, and the shoulder shifted toward smaller ω value until 14 h after the temperature drop. We also measured time-evolution of the elastic moduli at 60 °C as shown in Figure 7b. The result was essentially the same as that obtained at 28 °C, excepting that the fast mode at ω > 20 s−1 was not observed due to the experimental limitation. The shift of the shoulder was also detected until 5 or 10 min and after that the modulus seemed to continuously increase until the plateau was observed. These periods in which the shoulders were observed at 28 and 60 °C were in good agreement with those in which spheres in the lattice disordered state were observed as shown by SAXS in Figure 4, a and b, respectively. It is conceivable that the shoulders arise from the lattice-disordered state. Similar shifts of shoulders were observed in time-resolved rheological measurements on a BCP melt after quenching from disordered state to the temperature below order−disorder transition temperature,37,46 where the shoulder and the shift were, respectively, attributed to “metastable lattice-disordered micelles” and the increase in their number or stability; the shoulder shifted toward smaller ω with decreasing temperature. In this study, however, the rheological measurement was performed as a function of time at the constant temperature. The shifts were observed in the intermediate state until the regular BCC lattice was formed after the temperature drop; it is conceivable that, in this intermediate state, the solvent slowly went out of the PS domains, resulting in the increase of the segregation power and the stability of the spherical microdomains. Note that the endothermic peak was not observed in the cooling run of the DSC measurement, indicating that it takes a long time for the solvent to go out of the PS domains. Miwa et al.47 reported that a polymer chain connected to ionic aggregates having lower mobility shows lower Tg than a corresponding free polymer, which indicates the motion of the polymer is restricted by the connectivity. It is considered that dissociation of PS chains in SEP-S from the solvent reduced the chain mobility of PS, and hence, the mobility of PEP chains was more limited by the restriction of PS chains. This dynamic restriction slowed reorientation of micelles through rearrangement of PEP chains, resulting in the shift of the shoulder in G′. The G′ curves eventually showed plateau at lower frequencies at ω < 0.2 s−1 after annealing for a long time, indicating that well-ordered BCC lattice was formed.44 Watanabe et al.48 investigated a BCP in a solvent selectively good for a constituent block chain. They explained that at a moderate concentration, the BCP/solvent formed a BCC lattice by the thermodynamic constraint: Corona chains are required to randomize their conformation in order to maximize the conformational entropy and to simultaneously have mutually correlated conformation with neighboring chains in order to minimize concentration variation or osmotic free energy. The osmotically constrained corona chains entropically sustain equilibrium elasticity arising from the BCC lattice. They mentioned that each corona chain behaves as an independent stress-sustaining unit: Its contribution to the equilibrium

Ge = νkT

(3)

where ν is the number density of the corona chains in the solution. They measured Ge for a solution of a polystyrene-bpolybutadiene in n-tetradecane at the polymer concentration of 10−35 wt % at 25 °C. The value of Ge was successfully proportional to kT (see Figure 1 in their article). In this study, the value of Ge/kT for the SEP-S solution at 28 °C was 23.0 m−3 at ν = 23.5 m−3, whereas the value of Ge/kT for their system was 22.7 m−3 at ν = 23.5 m−3. The value for the SEP-S solution was in good agreement with eq 3 and their value (rigorously, our value was between their value and the one on eq 3), indicating that our system is quite close to the model they proposed, except the fact that the solution of SEP-S is not fully occupied by the BCC lattice but partially by the disordered spheres as discussed above. It is conceivable that the grains are percolated, i.e., three-dimensionally connected by commensuration through the grain boundaries: The percolation is considered as the origin of the plateau in the terminal zone. Moreover, the slight increase in G′ at the plateau (after 30 h at 28 °C and after 35 min at 60 °C) is attributed to the increase in the number ratio of the BCC lattice at the cost of disordered spheres23 as explained later in Figure 8 until the cessation of the increase.

Figure 8. G′ under oscillatory shear at 0.292 s−1 (G′0292(t)), d spacing, full width at half-maximum of the primary SAXS peak reduced by the primary peak position (σq/q*), and the fraction of BCC (IBCC(t)) evaluated by the SAXS intensity of the primary peak are plotted as a function of the annealing time after the temperature drop from 130 to 60 °C.

Finally, we compare the results of the SAXS and the rheology measurements. Figure 8 shows G′ under oscillatory shear at ω = 0.292 s−1 in the terminal zone (G′0292(t)), domain spacing (d), full width at half-maximum (σq) of the primary SAXS peak reduced by the primary peak position, q*, (σq/q*) and the percentage of the BCC lattice (IBCC(t)) as a function of the annealing time after the temperature drop from 130 to 60 °C. Solid lines were drawn as eye-guides. The structural evolution of the BCC lattice is separated into two stages: stage I from 0 to 15 min; stage II from 15 to 60 min after the temperature drop from 130 to 60 °C. At stage I, the d spacing increased and σq/ q* decreased by the abrupt increase of the segregation power arising from the miscibility jump by the temperature drop, which indicates the degree of order in disordered spheres increased. Here, we employ theoretical calculation of SAXS in order to quantitatively describe disordered sphere in stage I. A 1820

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Table 2. Avrami Exponent (a), the Avrami Rate Constant (K), Activation Energy (Ea) Obtained from the Arrhenius Plot at temperature Q(t) φ(t) IBCC(t)

a K a K a K

28 °C

40 °C

60 °C

70 °C

Ea (kcal/mol)

− − 3.07 2.25 × 10−11 − −

2.79 7.34 × 10−7 − − − −

3.47 5.05 × 10−6 4.52 1.04 × 10−8 2.93 1.45 × 10−5

2.74 2.16 × 10−4 − − − −

1.55 × 102 1.60 × 102 −

formation and the growth rate of r showed the second increase. The formation and the increase in the number of the regular BCC lattice also drastically increased G′0292(t). The biggest slopes for these three changes were observed around time = 30 min, when the plateau of the storage modulus in the terminal zone started to be seen as already shown in Figure 7b. At the same time, the grain size increased up to 2500 nm with a large distribution as shown in Figure 6. It is worthy to note that these results clearly show the above-mentioned features: The SEP-S solution was not fully occupied by the BCC lattice and the amount of the BCC lattice was still increasing at the cost of disordered spheres; hence, the plateau presumably arises from the three-dimensional grain networks in which BCC lattice planes are connected by commensuration at the grain boundaries. The increases in G′0292(t) and IBCC(t) and the decrease in σq/q* continued until 60 min after the temperature drop. Here, the Avrami equation was fitted to the temporal increase in SAXS and rheology data so as to give an insight into the kinetics of the grain growth. The temporal increase rate in the storage modulus at 0.292 s−1(G′0292(t)) is given as follows;

theoretical scattering profile was calculated on the basis of the paracrystal theory33 (designated PC) and the Percus−Yevick hard-sphere liquid theory34 (designated PY). The detail of the calculation and the results is given in Figures S5 and S6 of the Supporting Information. Both of the PC equation and the PY equation were well fitted to the experimentally obtained SAXS profiles (Figure S5a,b), except that the PY equation was not well fitted to scattering around the primary peak (Figure S5a). In the beginning of stage I, the radius of the sphere (r) was evaluated to be ca. 10 nm and gradually increased by a few percentage during the induction period (Figure S6a,d). The size distribution (σr) was 1.75 and 1.40 nm at t = 0 and gradually decreased down to 1.46 and 1.35 nm by the PC and PY analysis, respectively. Half of the minimum distance (rHS) between nearest neighboring spheres was 30.3 nm at t = 0 and gradually increased up to 32.0 nm at 10 min, whereas the standard deviation of rHS (σHS) was almost constant at 3.0 nm (Figure S6b). The domain spacing (d) evaluated by PC increased from 52.5 to 54.4 nm, whereas the g factor decreased from 0.130 to 0.115 (Figure S6e). Here, g is given as standard deviation of d (σd) divided by d. The distance between nearest spheres is √2d (= ca. 75 nm) that is in good agreement with double the value of rHS. √2σd/2 = g*d/√2 ≈ 4−5 is also in good agreement with σHS. The rate of increase in d is larger than that in r, thus the volume fraction (φsphere) decreases at first (Figure S6f). However, r continues to grow until 60 min, resulting in the increase in φsphere (Figure S6h). The size and the distribution of the disordered sphere were well characterized by both of the methods as above. In other words, both of the PC and PY equations were well fitted to the experimentally obtained profile from the disordered sphere. It is worthy to note that this fact means that “spheres in the disordered lattice” can be described in two ways: condensed spheres with the disordered arrangement (PY) and spheres on the highly distorted BCC lattice without regularity. The change in the sphere size is explained as follows: After the temperature drop across Tsol, the dissociation of PS from the solvent takes place. The dissociation causes two effects: shrinkage of the PS domain and enhancement of the segregation power. More interestingly, Figure S6h shows gradual increase in its size up to 60 min in stage II, which will be discussed later. The value of d leveled off around the turn of the stages (time ≈ 20 min). In contrast, the size of the PS domains (r) still continued to increase as already shown above. We added the temporal change of r in Figure 8. The continuous increase is presumably ascribed to the slow dissociation of PS from the solvent as mentioned before. The growth rate of r also seems to have slowing down around 20 min. At stage II after the induction period of 15 min, the regular BCC lattice started to form as indicated by the emergence of the scattering spots and the successive increase of IBCC(t); simultaneously, σq/q* showed the second drastic decrease caused by the BCC

′ (t ) − G0292 ′ (0))/(G0292 ′ (∞) − G0292 ′ (0)) φ(t ) ∼ (G0292 (4)

which followed the Avrami equation, φ(t) ∼ 1 − exp(−Kta). φ(t)’s at 28 and 60 °C are hereafter designated φ28(t) and φ60(t). The exponent (a) for φ28(t) and φ60(t) was evaluated, respectively, to be 3.07 and 4.52, and the Avrami rate constants (K) for φ28(t) and φ60(t) were, respectively, 2.25 × 10−11 and 1.04 × 10−8 as shown in Figure S7, parts a and b, of the Supporting Information. Note that the onset of the deviation between the experimental data and the Avrami fit was, respectively, observed at 13.8 min and 21 h at 60 and 28 °C that are in good agreement with the induction period presented in Figure 5. The Avrami indices obtained above increased from 3.07 to 4.52 with temperature. The values indicate that the BCC structure was formed by three-dimensional nucleation and growth mechanism at both of the temperatures. However, the nucleation mechanism was different: heterogeneous nucleation at 28 °C and homogeneous one at 60 °C. Note that the homogeneous nucleation and growth mechanism observed at 60 °C is consistent with the fact that the homogeneous nucleation was predicted by the time evolution of the grain size at 60 °C in Figure 6. Next, the Avrami equation was fitted to IBCC(t)/IBCC(∞) at 60 °C. The values of IBCC(t)/IBCC(∞) well followed the Avrami equation as shown in Figure S8 of the Support Information: IBCC(t)/IBCC(∞) ∼ 1 − exp(−Kta). The onset of the deviation between the experimental data and the Avrami equation was seen at ca. 20 min that is in good agreement with the induction period in Figure 5. The Avrami exponent (a) was evaluated to be 2.93 that is consistent with the exponent (a = 3.47) obtained 1821

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Macromolecules for 60 °C in Figure S1. The Avrami exponent, the Avrami rate constant, activation energy obtained from the Arrhenius plot were summarized in Table 2. The Avrami exponent, a, evaluated by Q(t) or IBCC(t), was almost constant at ca. 3 regardless of temperature. In contrast, the exponent evaluated by (φ(t)), was ca. 4 at the higher temperature (60 °C) and ca. 3 at the lower temperature (28 °C). The discrepancy in the Avrami exponents obtained from φ60(t) and IBCC(t)/IBCC(∞) is presumably ascribed to the difference in the length scale accessible by these two techniques. SAXS (IBCC(t)) observed the formation of the BCC lattice and the grains: regularity of the lattice, size and the number of the grains are reflected. In contrast, rheology (φ(t)) observed the time evolution of the network percolation of the BCC grains in the matrix of the disordered sphere. In other words, at 60 °C, the grain network was formed by homogeneous nucleation, whereas the grains were formed heterogeneously. At 28 °C, the grain network growth slowed down as depicted by the time evolution of G′ in Figure 7 and by the above Avrami rate constant (k), which is attributed to the dominancy of viscosity. Here, the heterogeneous nucleation of the network was observed by rheology. The BCC lattice was heterogeneously nucleated regardless of temperature. Although only two data points are available, the Arrhenius fit to the Avrami rate constant k at 28 and 60 °C yielded the activation energy: Ea = 1.60 × 102 kcal/mol (rheology): this value was in good agreement with that obtained by the fit of Q(t) in Figure S1b.

structural transition induced by the miscibility jump with temperature will give high tunability to a photonic BCP material in the future.

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ASSOCIATED CONTENT

S Supporting Information *

Avrami plots, time evolution of the grain size, evaluation of the g factor, process to reconstruct the quasi-isochronal graph, fit of the theoretical curves to the experimental SAXS profiles, and radii of spheres r, hard-shells rHS, and volume fraction η of hardshells. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*(S.O.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS A.M. greatly thanks Prof. Hiroshi Watanabe for his valuable advice. The synchrotron radiation experiments were performed at BL40B2 of SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposal Nos. 2011B1502, 2012B1476, 2013B1689, 2014A1423), and at BL8S3 of Aichi Synchrotron Radiation Center, Aichi Science & Technology Foundation, Aichi, Japan (Proposal No. 201402054 and No. 201405071). This work was supported by Japan Society for the Promotion of Science KAKENHI Grant Nos. 21550208 and 24550250.

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CONCLUSION In this study, we investigated the ordering process of the BCC lattice structure in a diblock copolymer solution of SEP/ salacos913 by SAXS, rheology and DSC. Highly ordered BCC lattice was formed by the miscibility jump on quenching the solution from the temperature much above the cloud point of PS (114 °C) to that lower than Tsol (ca. 84 °C). The structural evolution was relatively slow because the dissociation of PS from the solution took long time. The time evolution of the BCC lattice structure was divided into two stages. At stage I (the induction period), the solution was in the latticedisordered state. Disordered sphere was studied by SAXS on the basis of the paracrystal theory and the Percus−Yevick theory. The domain spacing increased with its distribution decreased and reached equilibrium without long-range order. The particle size also increased with their distribution decreased. Around the turn of the stages, the volume fraction showed a minimum and the domain size distribution leveled off. At stage II, the BCC lattice structure started to emerge in the matrix of the disordered spheres. The number of the lattices increased and, eventually, grains were formed. The size of grains was evaluated by the Hosemann’s plot: They had a large size distribution (700−2500 nm). The solution showed terminal behavior in the beginning. The value of G′ in the terminal region gradually increased and finally leveled off at a plateau modulus. It is conceivable that the plateau modulus was sustained by percolation of the grain network structure. We found that grain formation kinetics of the BCC lattice was dependent on temperature and length scale of observation. Formation of BCC grains, as revealed by SAXS, followed heterogeneous nucleation mechanism regardless of temperature. In contrast, formation of grain network structure followed homogeneous (heterogeneous) nucleation mechanism at higher (lower) temperature as shown by rheology. The

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REFERENCES

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