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Strain-Phase-Resolved Dynamic SAXS Studies of BCC-Spherical Domains in Block Copolymers under LAOS: Creation of Twinned BCC-Sphere and Their Dynamic Response Kenji Saijo,* Gakuji Shin,† and Takeji Hashimoto*,‡,∥ Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Katsura, Nishikyo-ku, Kyoto 615-8510, Japan

Yoshiyuki Amemiya and Kazuki Ito§ Department of Advanced Materials Science, Graduated School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan ABSTRACT: We applied a large-amplitude oscillatory shear deformation (LAOS) to polystyrene-block-polyisoprene-blockpolystyrene which had nearly randomly oriented grains of body-center-cubic lattice of spherical microdomains (BCCsphere) before applying LAOS. Mechanical and structural responses of the BCC-sphere were simultaneously measured as a function strain phase ϕ and number of oscillatory strain cycles N by means of shear stress measurements and strainphase-resolved dynamic small-angle X-ray scattering (DSAXS). The DSAXS revealed the following pieces of evidence: (1) LAOS first leads the BCC-sphere to attain two sets of the specific orientations as defined by A- and A′-sphere and by B- and B′-sphere, where both A- and A′-sphere and B- and B′-sphere are in the mirror symmetry with respect to {112} lattice plane which is parallel to the shear plane (the OXZ-plane with the OXand OZ-axis defined as the shear direction and vorticity or neutral axis, respectively) for A- and A′-sphere or to the OXY-plane with the OY-axis defined as the velocity gradient direction for B- and B′-sphere. The two sets commonly have their ⟨111⟩ axis along the OX-axis. (2) Moreover, in parallel to attaining the specific orientations, both A- and A′-sphere and B- and B′-sphere are directed to form twinned BCC-sphere with their {112} lattice planes as the twin plane. While the events (1) and (2) described above were found to occur in the early stage of strain cycles (N ≤ 4), in the late stage of strain cycles (N > 4) the twinned BCClattice itself was found to undergo the oscillatory shear deformation in response to the applied shear strain under the fixed orientation of the {112} lattice plane and the ⟨111⟩ axis as described above. Moreover, we found that the shear deformations of the A- and A′-sphere under those fixed orientations are strikingly unequal due to dominance of the entropy elasticity of coronar block chains over the energy elasticity associated with the strain imposed on the lattice spacings. We elucidated that stress variations with ϕ and N, including nonlinear response with stress hardening, are closely related to the structural responses with ϕ and N, respectively, as described above.

I. INTRODUCTION There have been many reports with respect to effects of the large-amplitude oscillatory shear strain (LAOS) on the microdomain structures of block copolymers (bcp) as studied by means of small-angle X-ray scattering (SAXS) and smallangle neutron scattering (SANS),1,2 since the pioneering works of Keller et al.3 and Hadziioannou et al.4 in the 1970s. However, there have been almost no reports that explored dynamic responses of SAXS and/or SANS patterns to LAOS as a function of strain phase ϕ and a number of the oscillation period, N (designated hereafter as “number of strain cycle”). Thus, in this work we aim to explore dynamic responses of SAXS, designated as DSAXS, as a function of ϕ and N in order to gain new insights into dynamic responses of spherical microdomains of body-centered-cubic lattice (designated hereafter as BCC-sphere) and their effects on dynamic © 2013 American Chemical Society

mechanical responses by simultaneous measurements of DSAXS and shear stress σ. As for small-angle scattering (SAS) experiments of bcps under LAOS, there have been many reports up to now on how lamellar microdomains orient against the shear direction depending on experimental conditions.5−6 As for the study of the spherical microdomains, there have been also many reports by Hamley et al.,12−15 Mortensen et al.,16,17 and Lodge et al.18 on spherical microdomain systems swollen with solvents. However the research on the spherical microdomains of neat bcp melts has not been much performed, except for the studies of Almdal et al.,19 Koppi et al.,20 Okamoto et al.,21 and Shin et Received: December 13, 2012 Revised: January 23, 2013 Published: February 12, 2013 1549

dx.doi.org/10.1021/ma302559w | Macromolecules 2013, 46, 1549−1562

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al.22 Okamoto et al. and Shin et al. carried out the strain-phaseresolved DSAXS experiments simultaneously with stress measurements. Almdal et al.19 studied the influence of LAOS on the orientation of the BCC-sphere in the neat poly(ethylenepropylene)-block-poly(ethylethylene) (PEP−PEE) using SANS where the PEP block was deuterated and had the volume fraction of 0.83. They obtained results that the (110) plane of BCC lattice was aligned parallel to the shear plane (the OXZ-plane defined later in Figure 2) and the ⟨1̅11⟩ axis is oriented parallel to the shear direction (the OX-axis in Figure 2). The scattering patterns showed that the twinned BCC structure (B/B′-twin to be defined later in section IV-1) is formed with its twinning plane (112) vertical to the shear plane. Ackerson et al.23 reported a similar orientation for colloidal suspensions under shear deformation. Koppi et al.20 investigated the shear effects as a function of shear rate and T for the PEP−PEE dibcp having different volume fraction of PEP of 0.25 and molecular weight by using SANS and rheological measurements under LAOS. The dibcp had TOOT = 175 °C and TODT = 220 °C. The direction of the incident neutron beam is parallel to the velocity gradient direction. They observed that the hexagonal cylinder (designated hereafter as hex−cyl) oriented by LAOS at 150 °C transformed to the twinned BCC structure on elevating temperature to 180 °C. They reported the deformation mechanisms of the thus obtained twinned BCC-sphere as a function of shear rate γ̇ at 195 °C. Under a sufficiently slow shear rate (γ̇ ≪ γ̇*; γ̇* represents the inverse relaxation time of the defects), they found not only the same twinned BCC structure as Almdal et al. found (B/B′-twin) but also another type of the twinned BCC structure (A/A′-twin to be defined later in section IV-1) also having the same twinning plane (112) parallel to the shear plane at this slow shear rate. At an intermediate shear rate (γ̇ ≈ γ̇*), the in situ SANS scattering patterns became isotropic independent of azimuthal angle, implying randomly oriented BCC-sphere. However, after cessation of the LAOS, the initial twinned BCC structure of A/A′ and/or B/B′ reappeared. At a high shear rate (γ̇ ≫ γ̇*), an affine and elastic deformation of sphere took place. The intriguing SANS experimental results reported by Koppi et al. and Almdal et al. showed cumulative effects of the LAOS over some strain cycles N. If the change in the scattering pattern could be measured in situ and at real time as a function of the strain phase ϕ and strain cycle N, we anticipate to gain deeper insights into the mechanism and process of orientation and deformation of BCC-sphere and their dynamic responses as well. In this paper we aim to investigate the orientation process and the dynamic response of BCC-sphere in a triblock copolymer of polystyrene-block-polyisoprene-block-polystyrene (SIS) by using simultaneous DSAXS and rheology measurements under LAOS as a function of ϕ and N. The LAOS with γ0 = 0.5 and ω = 0.0944 rad/s was imposed to the sample at T = 210 °C. The choice of this set of T and ω will be clarified later in section II-5. We analyzed two-dimensional DSAXS (2D-DSAXS) patterns obtained by the strain-phase-resolved experiments under LAOS in order to study the deformation and orientation processes of BCC-sphere and by the timeresolved experiments after cessation of LAOS in order to study the relaxation process. We investigated 2D-DSAXS patterns obtained with incident X-ray beam in two directions: parallel to

the vorticity direction (the OZ-axis) and the velocity gradient direction (the OY-axis).

II. EXPERIMENTAL METHODS II-1. Sample. The sample studied is a SIS triblock copolymer (Vector4111, Dexco Polymers Co.) having the weight-average molecular weight of Mw = 1.4 × 105, the heterogeneity index Mw/ Mn = 1.11, and the volume fraction of polystyrene f PS = 0.164. The film cast from a 10 wt % toluene solution has a hex−cyl morphology. Figure 1a schematically shows the temperature dependence of the

Figure 1. (a) Temperature dependence of microdomain structure of the sample and (b) the thermomechanical protocol employed in this LAOS experiment. The integer numbers 1−10 designate the following. 1: initial state at γ = 0; 2 to 4: 1st, 5th, and 20th cycle of LAOS; 4 to 10: 0, 5, 10, 30, 100, and 1000 s after cessation of LAOS at γ = 0, all at 210 °C. microdomain structure of the sample. The order−order transition temperature (T OOT) between hex−cyl and BCC-sphere was determined to be 183 °C, while the lattice disordering-ordering transition temperature (TLDOT) was determined to be 215 °C and the dimicellization/micellization transition temperature was estimated as high as 280 °C, using SAXS measurements.24,25 Figure 1b will be described later in section II-5. II-2. Shear Apparatus. Figure 2 shows the shear deformation device of film samples under LAOS together with the definition of the Cartesian coordinate OXYZ, where the OX-, OY-, and OZ-axis are

Figure 2. Schematic diagram of the shear deformation device for film specimens used in this work. The direction of the incident X-ray beam is set parallel to either the OZ-axis (a) and the OY-axis (b). Part (a) shows the cross section normal to the OZ-axis of the shear cell. The OX-, OY-, and OZ-axis are set parallel to shear direction, velocity gradient direction, and neutral (or vorticity) direction. 1550


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taken along the shear direction, the velocity gradient direction, and the neutral (or vorticity) direction, respectively. The 2D-SAXS patterns were taken with the incident X-ray beam along either the OZ-axis (part a) or the OY-axis (part b). The device in Figure 2a shows its cross section normal to the OZ-axis. As-cast films of thickness 0.5 mm, length of ca. 7 mm, and width 5 mm along the Z-axis were stacked to make the total thickness of the sample equal to be 3 mm along the Yaxis as shown in Figure 2. The stacked samples were sandwiched between two metal plates, and LAOS was imposed on the specimen by moving one of the metal plates parallel to the other. We used two beryllium plates fixed in the brass frames for both side of the incident and scattered X-ray beam. The thickness of the beryllium plates used was 0.3 mm. 2D DSAXS patterns with the incident X-ray beam parallel to the OY-axis and the OZ-axis were taken by rotating the sample sandwiched by the shear device by 90° around the OX-axis. The pointfocused incident beam size at the sample was 1.5 mm2. The SAXS intensity distribution was detected with an imaging plate (IP).26 The sample-to-detector distance was 1.6 m. II-3. Strain-Phase-Resolved Data Acquisition of SAXS and Stress during LAOS. The DSAXS system comprises a stage-mounted IP system developed at the Photon Factory27 and a hydraulic sample deformation device developed at Kyoto University.28 Details of the phase-resolved DSAXS system were described elsewhere.26 The Lissajous figures of stress (σ) vs strain (γ) under oscillatory shear deformation were simultaneously obtained with DSAXS. The stress data are expanded to the Fourier series in order to evaluate the nonlinear response of σ against γ, the details of which will be discussed later in section IV-7. II-4. X-ray Source. The phase-resolved DSAXS measurements were performed by using the synchrotron X-ray beam at the beamline 15A of the Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, Japan. The beamline 15A has the demagnifying mirror−monochromator optics.29 The wavelength (λ) of incident X-ray was 0.15 nm, and its spectral distribution was Δλ/λ = 10−3. II-5. Thermomechanical Protocol of LAOS Experiments. Figure 1b shows the protocol used in this work. We raised the temperature from 25 to 210 °C in order to attain BCC-sphere from hex−cyl via the OOT. The initial state of the specimen for the LAOS experiments was attained after holding the specimen at 210 °C for 20 min at zero shear strain (point 1 in the protocol). We conducted in situ the strain-phase-resolved DSAXS experiments under LAOS for 20 strain cycles. We shall present the 2D DSAXS patterns obtained as a function of N and ϕ at the time specified by points 2−4. Then timeresolved SAXS patterns were obtained during the relaxation process after cessation of the LAOS at zero shear strain for 16.7 min at the time specified by points 5−10. The measuring temperature for those data as described above was only 5 °C below TLDOT, and hence the BCC-sphere obtained is expected not to be in a very strong segregation limit. Then we changed temperature from 210 to 188 °C (5 °C above TOOT and 27 °C below TLODT) at the cooling rate of 1 °C/min. After keeping the sample at 188 °C for 20 min, we obtained 2D-SAXS patterns shown in Figure 8 later at the time designated by point 11 in the protocol. 2D-SAXS patterns were obtained with the exposure time of 1 s at the time indicated by points 1−11 in Figure 1b. LAOS with strain amplitude (γ0) of 0.5 and the angular frequency (ω) of 0.0944 rad/s was applied to the sample at 210 °C to obtain the oriented BCC lattice.

γ = γ0 exp(iωt ) = γ0 exp(iϕ)

Figure 3. (a) 2D SAXS pattern before imposing LAOS at the point marked by no. 1 in Figure 1 with the incident X-ray along the OZ-axis, (b) the circular-averaged intensity distribution with q, and (c) azimuthal-angle (μ) dependence of the first-order diffraction maximum for the pattern shown in (a).

b) and azimuthal-angle (μ) dependence of the first-order diffraction intensity maximum at qm and at 210 °C (part c) before applying LAOS, where q is the magnitude of the scattering vector q defined by q = (4π/λ) sin(θ/2), θ being the scattering angle, and qm is the q value at the first-order diffraction intensity maximum. The azimuthal angle μ is defined as the angle which increases in the counterclockwise direction from the OX-axis (see the right part in Figure 3a). The 2D SAXS pattern shows three nearly circular diffraction rings in part a. From the scattering profile shown in part b these rings were found to exist at the peak position of 1, √2, and √3 relative to qm. They have a weak preferential orientation as evidenced by a weak variation of each diffraction intensity with respect to μ in part c. We found a broad scattering peak at qp ≅ 0.6 nm−1, which is highlighted in the inset to part b, reflecting

(1)

where ϕ is the strain phase. It is worth noting that a set of the parameters T and ω selected for the LAOS experiments correspond closely to the terminal flow region in the linear dynamic mechanical experiments.24

III. EXPERIMENTAL RESULTS: SR-DSAXS RESULTS III-1. Initial State of BCC-Spheres. Figure 3 shows the 2D SAXS pattern taken at point 1 in Figure 1b with IP (part a) and circular-averaged intensity distribution with respect to q (part 1551


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patterns at the 5th and 20th cycles is essentially similar to each other but distinctly different from that of the 1st cycle. These trends (i) and (ii) suggest that the dynamic response of BCCsphere at N < 5 is different from that at N ≥ 5, whose difference will be clarified later. Moreover, we note that (iii) the (110) and (200) diffraction spots oriented at particular μ values at phase 2 and 4 in the 5th and 20th strain cycle; (iv) although the ϕ dependence of scattering patterns at the 5th and 20th cycle is similar to each other, the diffraction spots at the 20th strain cycle became sharper than those at the 5th cycle at each strain phase; (v) though the scattering patterns at phase 1 and phase 3 are similar at a first glance, the close inspection reveals that they are in the mirror symmetry with respect to the OYZ- and OXZ-plane, while those at phase 2 and phase 4 are very clearly in the mirror images with respect to the OYZ- and OXZ-plane. The trends (iii) and (iv) may suggest that: LAOS tends to orient particular lattice planes of BCC-sphere with respect to the shear plane and the shear direction with the strain cycle N from N = 1 to 5; then the BCC-sphere with the selected orientation as described above may be subjected to the oscillatory lattice deformation in response to ϕ under the particular lattice plane {112} and axis ⟨111⟩, respectively, being fixed parallel to the shear plane (the OXZ-plane) and the shear direction (the OX-axis) in the strain cycle N > 5. The trend (v) may indicate that the contribution of the oscillatory lattice deformation under the fixed lattice orientation becomes increasingly important than that of the overall orientation of the lattice plane and the lattice axis with N from N = 5 to 20. The detailed discussion including possible lattice structures at phase 1 and phase 3 in Figure 5e will be deferred until sections IV-4 and IV-5. At phases 1 and 3, we discern the four “distorted” diffraction spots at μmax = 47°, 133°, 227°, and 313° and at q’s slightly larger than the qm corresponding to the diffraction ring from {110} plane. Those four distorted diffraction spots appeared to be elongated along the OYdirection. Structural models at phases 1 and 3 shown by a question mark in part e will be discussed in section IV-3, while those in phases 2 and 4 in part e will be discussed in section IV4 III-3. Structural Relaxation after Cessation of LAOS at Zero Strain. Figure 6 shows SAXS patterns obtained at (a) 0, (b) 5, (c) 10, (d) 30, (e) 100, and (f) 1000 s after cessation of LAOS at strain phase 1 of the 21st cycle (corresponding to points 5−10 in Figure 1b) with the incident X-ray along the OZ-axis. Each pattern was taken in situ at 210 °C with an exposure time of 1 s. Four off-meridional and two meridional spots (labeled 1 and 2, respectively) on the first-order diffraction ring in the pattern (a) underwent a remarkable change and became clear, respectively, with an elapse of time after the cessation. Each of the four spots labeled 1 at q slightly larger than qm, which is close to the diffraction ring from the {110} lattice plane, split into two spots: one shifts toward the position q m , corresponding to the peak position from the {110} lattice plane, and the other shifts toward √2qm, corresponding to the peak position of the diffraction from the {200} lattice plane of BCC-sphere (Figure 6f). Thus, the four off-meridional spots in pattern (a) split into the eight off-meridional spots corresponding to the {110} and {200} diffractions during the relaxation process of the strained BCC lattice. The relaxation of the 2D SAXS patterns reveals that the four off-meridional diffraction spots at phases 1 and 3 in Figure 5c are due to the strained

the form factor for the isolated spherical microdomains as shown by the arrow in part b, from which the average radius of the domains R is estimated to be 9.6 nm (qpR = 5.765). The volumetric considerations indicate that the spheres are arranged in space with a body-centered cubic symmetry (BCC). The first, second, and third peak are the diffractions from the {110}, {200}, and {112} lattice planes of BCC-sphere, respectively. Figure 3c reflects a weak preferential orientation of the BCC lattice as evidenced by a weak μ dependence of the first-order maximum at qm. The two peaks at μ = 90° and 270° are the local maximum with a lower intensity, while the four peaks at μ = 35°, 145°, 215°, and 325° are the local maximum with a higher intensity, which may be a memory reflecting both a weak preferred orientation of hex−cyl in the as-cast film30,31 and the OOT process from hex−cyl to BCC-spheres. The details will be discussed later in section IV-2. III-2. Strain-Phase Resolved SR-DSAXS Studies during LAOS. Figure 4 shows the definition of strain phase ϕ of

Figure 4. Definition of strain phase ϕ of oscillatory shear strain γ (solid line) and rate of shear strain γ̇ (dotted line).

oscillatory strain γ (solid line). The numbers 1−4 in the figure designate the points corresponding to ϕ = 0, π/2, π, and 3π/2 rad, respectively, while γ̇ is the rate of shear strain (dotted line). Figure 5 shows the SR-DSAXS patterns in the parameter space of strain phase ϕ and strain cycle N. The patterns in rows (a), (b), and (c) are obtained at N = 1, 5, and 20, respectively, corresponding to the points 2, 3, and 4 in Figure 1b, respectively. The X-ray exposure time to record each pattern is 1 s, corresponding to the strain-phase interval of 0.0944/(2π) = 0.015 cycle = 5.4° arc in the phase angle. SAXS patterns are noted to change with strain phase ϕ at a given cycle and with strain cycle N at a given phase. The following features are discerned in this figure. (1) N = 1 (Figure 5a): The change in 2D DSAXS pattern with strain phase in the first cycle is not as clear as that in the 5th (Figure 5b) and 20th cycles (Figure 5c). The patterns in the 5th and 20th cycles showed a clear periodic response to the strain phase compared with those in the first cycle. At N = 1, the second- and third-order diffraction maxima at √2qm and √3qm at phase 1, respectively, were found to become weak with increasing ϕ, indicating increasing lattice distortions of BCC-sphere with ϕ. Nevertheless, the first-order diffraction pattern at qm is found to change from nearly circular ring at phase 1, which is almost independent of μ, to the pattern with the clear μ-dependence with increasing ϕ, implying that LAOS tends to orient particular lattices planes of BCC-sphere with respect to the shear direction (the OX-axis) and the shear plane (the OXZ-plane) with ϕ. This trend continued up to N ∼ 5. (2) N = 5 and 20 (Figure 5b,c): We first note that (i) the pattern at phase 1 in the 5th cycle is distinctly different from those in the 1st cycle; (ii) the ϕ dependence of the scattering 1552


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Figure 5. 2D SAXS patterns obtained at (a) 1st, (b) 5th, and (c) 20th cycle of LAOS and at the four representative strain phase as defined in row (d) with the incident X-ray along the OZ-axis. Row (e) shows the BCC-sphere model at phases 2 and 4 as well as the unknown model marked by the question mark ? at phases 1 and 3, which will be later clarified in the text in conjunction with Figures 6 and 12.

of the BCC-sphere, which will be discussed below in section IV1.

BCC lattice, the details of which will be discussed later in section IV-3. As for {112} diffraction, the meridional spots labeled 3 (parallel to the OY-axis) were stationary with time, but the equatorial arcs (along the OX-axis) labeled 4 split along the azimuthal angle into four spots with time. The meridional spots labeled 2 were also stationary with time. III-4. Stress−Strain Behavior under LAOS. Figure 7 shows the Lissajous figures of shear stress (σ) vs shear strain (γ) in (a) the 1st, (b) 5th, (c) 10th, and (d) 20th strain cycles. The maximum stress is gradually increasing with increasing the strain cycle N, indicating the strain hardening. The typical bowtie pattern is built up with time, indicative of the strain hardening and enhanced nonlinear response with increasing N. III-5. Specific Orientation of BCC-Sphere Attained by LAOS. Figure 8 shows the 2D SAXS patterns of the oriented BCC-sphere for the specimens which were first relaxed at zero strain after imposing LAOS over N = 20 at 210 °C (point 10 in Figure 1b) and then cooled down to 188 °C and kept at 188 °C for 20 min (point 11 in Figure 1b). The 2DSAXS patterns were taken with the incident X-ray along (a) the OZ-axis and (b) the OY-axis. Comparisons of the pattern (a) in Figure 8 obtained at 188 °C and the pattern (f) in Figure 6 obtained at 210 °C would indicate effects of changing temperature on orientation

IV. DISCUSSION IV-1. Oriented BCC Structures Attained after Structural Relaxation of Specimens Subjected to LAOS. Figures 9a and 10a show the schematic diagrams of diffraction spots in the real diffraction patterns shown in Figures 8a and 8b, respectively. Those 2D SAXS patterns were taken with the incident X-ray beam along the OZ-axis and the OY-axis, respectively. Figures 9b and 10b present the azimuthal-angle dependence of the first-order diffraction maxima from the {110} lattice plane for each pattern shown in Figures 8a and 8b, respectively. We showed four kinds of orientated BCC-sphere model shown in Figures 9c and 10c to account for those 2D SAXS patterns shown in Figures 9a and 10a, respectively. Moreover, judging from the diffraction patterns, it is important to note that a whole sample space is volume-filled with these Aand A’-sphere and B- and B’-sphere. In Figure 9a, the diffraction spots labeled 1 and 1′ at q = qm are those from {110} plane of A- and A′-sphere, respectively. The spots 2 and 2′ at q = √2qm are those from the {200} plane of A- and A′-sphere. The spots 3 at q = √3qm are the diffraction spots from the {112} plane of A- and A′-sphere. The 1553


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Figure 8. 2D SAXS patterns of the oriented BCC-sphere attained in the relaxed specimen at 188 °C (no. 11 defined in Figure 1) with the incident X-ray along (a) the OZ-axis and (b) the OY-axis.

Figure 6. SAXS patterns taken in situ at 210 °C and at (a) 0, (b) 5, (c) 10, (d) 30, (e) 100, and (f) 1000 s after cessation of 20 cycles of LAOS at zero strain with the incident X-ray along the OZ-axis.

Figure 9. (a) Schematic diagram of the diffraction spots shown in Figure 8a, (b) azimuthal-angle dependence of the first-order diffraction maxima({110} diffraction) for the pattern shown in Figure 8a, and (c) four types of models (defined by A, A′- and B, B′-sphere) showing orientations of BCC-sphere. The assignment of each diffraction spot in (a) is indicated beneath (a) and (c) where K(hkl) (K = A, A′, B, or B′) denotes the origin of the diffraction spot.

spots 4 and 4′ are also those from the {112} plane of A- and A′sphere, respectively. The spots 5 and 5′ at q = 2qm are the diffraction spots from the {220} plane of A- and A′-sphere, respectively. The spots 6 are the diffraction spots from the {110} plane of the B- and B′-sphere. In Figure 10a, the diffraction spots labeled 1 and 1′ at q = qm are those from the {110} plane of B- and B′-sphere, respectively. The spots 2 and 2′ at q = √2qm are those from the {200} plane of B- and B′-sphere, respectively. The spots 5

Figure 7. Lissajous figures of shear stress vs shear strain in (a) 1st, (b) 5th, (c) 10th, and (d) 20th strain cycles of LAOS at 210 °C.

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Figure 11. Schematic diagram for twinned A- and A′-sphere (A/A′twin) and twinned B- and B′-sphere (B/B′-twin) with the lattice matching [(a) and (b)] and without the lattice matching [(c) and (d)] at the twin boundaries. The parts (a) and (b) represent A/A′-twin, when K and K′ represent A- and A′-sphere, respectively, and the Cartesian coordinate is given by part (e). The parts (a) and (b) represent B/B′-twin also, when K and K′ represent B- and B′-sphere, respectively, and the Cartesian coordinate is given by part (f).

Figure 10. (a) Schematic diagram of diffraction spots shown in Figure 8b, (b) azimuthal-angle dependence of the first-order diffraction maxima({110} diffraction) for the pattern shown in Figure 8(b), and (c) four types of models (defined by A, A′- and B, B′-sphere) showing orientations of BCC-sphere.

the OX-axis as schematically shown in Figure 11a or 11b or independent grains with the disordered twin boundaries as shown in Figure 11c or 11d. The volume occupied by the A-, A′-sphere is definitely larger than the volume occupied by the B, B′-sphere by comparing the area under the respective peaks from the {110} plane shown in Figures 9b and 10b. The ratio of the former to the latter is ∼2.5, though the ratio estimated should be considered as a crudely approximated one, because the estimation excludes the diffraction from other lattice planes. The diffraction pattern obtained from the BCC-sphere relaxed in situ at 210 °C (shown in Figure 6f) is now compared with that obtained in situ at 188 °C shown in Figure 8a or 9a. The two patterns are essentially identical, except for the following minor changes recognized with decreasing temperature: (i) {200} diffraction spots at √2qm (the spots 2 and 2′ in Figure 9a) become weak; (ii) off-meridional {112} diffractions (the spots 4 and 4′ in Figure 9a) become also weak. These facts suggest that A- and A′-sphere tend to lose slightly their perfection with decreasing temperature. IV-2. Initial State of BCC-Spheres. The azinuthal-angle dependence of the first-order diffraction peak at qm shown in Figure 3c is somewhat similar to that in Figure 9b. As shown in Figure 1, we prepared the sample of the BCC-sphere morphology by annealing the as-cast films, which had the hex−cyl morphology, at 210 °C in the quiescent state. The hex−cyls in the as-cast film have usually a partial orientation with the cylindrical axes parallel to the films,30,31 which in turn may end up the partial orientation of the BCC-sphere developed via OOT. Thus, there may exist already memory effects for creating A-, A′- and B-, B′-sphere, which were built

and 6 are those from the {220} and {110} plane of A- and A′sphere, respectively. A- and A′-spheres in Figures 9c and 10c have the {112} plane parallel to the OXZ-plane (the shear plane) and the ⟨111⟩ axis parallel to the OX-axis (the shear direction), and they are in the mirror symmetry with respect to the OXZ or {112} plane. Aand A′-spheres have a relative orientation appropriate for the twinned BCC-sphere. Thus, A- and A′-spheres may be able to form either the twinned BCC structures with the {112} twinning plane parallel to the shear plane (the OXZ-plane) and with the lattice matching at the twin boundaries, as schematically shown in Figure 11a or 11b, or they form independent grains with lattice mismatches at their grain boundaries or with disordered twin boundaries as shown in Figure 11c or 11d. It is worth noting that one cannot distinguish between the model shown in Figures 11a and 11b and that shown in Figures 11c and 11d by the static SAXS alone. This point will be further discussed later in section IV-5. B- and B′-spheres in Figures 9c and 10c have {110} plane parallel to the OXZ-plane and the ⟨111⟩ axis parallel to the OXaxis, and they are in the mirror symmetry with respect to the OXY-plane or {112} plane. They are the structures to be obtained by rotating A- and A′-sphere by 90° around the OXaxis. B- and B′-sphere also have a relative orientation appropriate for the twinned BCC-sphere. Thus, they may be able to form the twinned BCC-sphere with the twinning plane {112} parallel to the OXY-plane and the ⟨111⟩ axis parallel to 1555


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up in the oriented hex−cyl in the solvent-cast films and its subsequent OOT into BCC-sphere, before imposing LAOS. It might be possible that the BCC-sphere structure oriented by LAOS is influenced by these precursory orientation of BCCsphere. However, strong effects of shear on building up the specific orientation of BCC-sphere are clearly discernible by comparing Figure 3a with Figure 8a. IV-3. Mechanism of Structural Relaxation after Cessation of LAOS at Zero Shear Strain. A. Interpretation Based on Equal Lattice Strain for A- and A′-Sphere. We present a schematic model on the lattice deformation of A- and A′-sphere in Figure 12 to explain the changes in the scattering

deformation is applied parallel to the OX-axis and OXZ-plane as shown by the solid arrows so that the shear strain on the lattice, γlattice, exists in the OXY-plane. In parts (a) and (b) as well as parts (d) and (e), A-sphere and A′-sphere are assumed to be subjected to the same shear strain, γlattice = +0.4, smaller than the bulk strain amplitude of γ0 = 0.5, as shown by the arrow. Figure 12a as well as Figures 12d and 12e schematically present the deformation of A-sphere from the “undeformed lattice” (shown by open circles and broken lines) to the “deformed lattice” (shown by filled circles and solid lines). On the other hand, Figure 12b and Figures 12d and 12e also schematically present the corresponding deformation for the A′-sphere. The equally deformed lattice of both A-sphere and A′-sphere with γlattice = +0.4 is expected to give the diffraction patterns as schematically shown in Figure 12c, in which the positions of the set of the four diagonal spots from {110}, {200}, and {112} lattice planes are displaced due to the lattice deformation as follows: The four diffraction spots from the {110} lattice planes are expected to shift along the vectors A11 and A13 for A-sphere and along the vectors A′12 and A′14 for A′-sphere. The four diffraction spots from the {200} lattice planes are expected to shift along the vectors A22 and A24 for A-sphere and along the vectors A′21 and A′23 for A′-sphere. The four diffraction spots from the {112} lattice planes are expected to shift along the vectors A32 and A34 for A-sphere and along the vectors A′31 and A′33. Here in the symbol Kij described above, K designates the type of BCC-sphere (A- or A′-sphere), the subscript i (i = 1−3) means the order of the diffraction, and the subscript j (j = 1−4) means the quadrant where the diffraction spots appear. The two sets of the meridional diffraction spots labeled by nos. 2 and 3 in Figure 12c are those from the {110} plane of B- and B′-sphere and the {112} plane of A- and A′-sphere parallel to the OXZ-plane, respectively, so that they are stationary under the LAOS. The equal deformation of the A/A′-twin (d) and A′/A-twin (e) also are expected to cause the same changes in the diffraction spots as shown in part (c). It is worth noting that the deformation causes the diffraction spots in the first and fourth quadrants to shift downward and those in the second and third quadrants to shift upward. The experimental diffraction patterns for the deformed BCCsphere shown in Figure 6a were taken during the X-ray exposure time over 1 s after the cessation of LAOS, so that the diffraction spots are subjected to some effects of the relaxation on the deformed lattice. Effects of this lattice relaxation as well as those of lattice distortions caused by the imposed LAOS deformation are expected to broaden the diffraction spots along the OY-axis and thereby to give rise to the overlap of the spots along the OY-axis as schematically shown by the broken ellipses in Figure 12c. The pattern shown by the broken ellipses may be able to qualitatively account for the experimental pattern shown in Figure 6a with respect to the distortions of the diffraction spots 1 and 4 as well as the break of symmetry of those with respect to OYZ- and OXZ-plane. B. Analyses of the Relaxation of the Deformed Lattice in Figure 6. When the equally deformed lattice shown by the solid lines in Figure 12 relaxes back to the undeformed lattice of Asphere and A′-sphere as shown by the broken lines, each diffraction spot is expected to shift along the opposite directions specified by the vectors Kij’s shown in Figure 12c. The relaxation reduces the lattice distortions and the variation of effective lattice strain averaged over the exposure time of 1 s, and hence the overlapped spots shown by the broken ellipses tend to split into and shift to the set of four diffraction spots,

Figure 12. Model for explaining the changes in the diffraction patterns shown in Figure 6 during the relaxation process after cessation of the LAOS at 20th strain cycle and at zero shear strain (phase 4).

patterns shown in Figure 6 during the structural relaxation process after the cessation of the LAOS at ϕ = 0 in the 21st strain cycle. Figure 12 schematically presents the equal shear deformation of the lattice for A-sphere (part a) and A′-sphere (part b) as well as those in the A/A′-(d) and A′/A-twin boundary (e) under the particular lattice orientation in which the {112} lattice plane and the ⟨111⟩ axis for A-sphere and A′sphere are fixed parallel to the OXZ-plane (the shear plane) and the OX-axis (the shear direction), respectively. The LAOS 1556


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axis. It is striking to note that the deformation and its relaxation behavior of A-sphere and A′-sphere are quite different. For γlattice > 0, the A′-sphere undergoes the large lattice strain of ∼+0.4, which relaxes slowly as evidenced by the time changes in the azimuthal angles from ∼45° to ∼55° for r*200,A′,d and from ∼48° to ∼35° for r*110,A′,d (cf. Figure 12b), while Asphere undergoes only a small deformation and/or a rapid relaxation (at the time scale of less than ∼10 s). If the uniform deformation were the case, the azimuthal angle should change from 61° to 55° for r*200,A,d and from 17° to 35° for r*110,A,d in Figure 12a. Figure 13b show the lattice strains imposed on the {110} plane and the {200} plane for A-sphere, designated respectively as A{110} and A{200}, and those for A′-sphere, designated respectively as A′{110} and A′{200}. The lattice strain ε for each plane in A- or A′-sphere was calculated by using the relationship

each set of which arises from the {110}, {200}, and {112} lattice planes as shown by the series of the patterns given in Figures 6b−f. A series of the patterns (a) to (f) shown in Figure 6 reveals themselves the splitting and shifting of the diffraction spots 1 and 4 in the pattern (a) toward {110} and {200} diffraction positions and toward {112} diffraction position, respectively, during the relaxation process. The pattern in Figure 6a taken during the exposure time of 1 s after the shear cessation reflects already a shifting of the diffraction spots of nos. 1 and 4 to some degree so that the spots are extended along the OY-axis. The extended spots 1 and 4 in Figure 6a hence reveal the relaxation of the deformed lattice as well as the relaxation of the lattice distortions. In order to check validity of the assumption concerning the shear deformation with the equal lattice strain on the lattice of A- and A′-sphere, we analyzed the time changes in the diffraction spots during the structural relaxation process in Figure 6. Figure 13a shows the time changes in the azimuthal angle of the reciprocal lattice vectors of the {110} plane for Asphere, r*110,A,d, and {200} plane for A′-sphere, r*200,A′,d, both of which were measured counterclockwisely from the equator (the OX-axis) and those of the {110} plane for A′-sphere, r*110,A′,d, and {200} plane for A-sphere, r*200,A,d, both of which were measured clockwisely from the opposite direction of the OX-

ε = [(1/qm,d) − (1/qm,0)]/(1/qm,0)

where qm,d and qm,0 are the magnitude of scattering vector at the maximum diffraction intensity for each diffraction plane in A- or A′-sphere after and before the shear deformation, respectively. It is striking to note again that the results reveal by themselves the surprisingly large difference in the lattice deformation and relaxation behavior: a large compliance and slow relaxation for A′-sphere vs a small compliance and rapid relaxation for Asphere for the positive γlattice. Of course, it should be noted that this unequal lattice deformation for A- and A’-sphere can account for the 2D SR-DSAXS pattern in Figure 6a and its relaxation from part a to part f more rigorously than the equal lattice deformation model adopted in Figure 12. IV-4. Interpretation of Strain-Phase-Resolved DSAXS Patterns. The analyses in the preceding section definitely provide useful information to understand the phase-resolved SR-DSAXS patterns shown in Figure 5. In each phase at the first strain cycle (Figure 5a), the specific orientation of the BCC-sphere as clarified in conjunction with Figures 8−10 is not yet well developed, so that the strain-phase resolved 2D SRDSAXS patterns will not give clear-cut information. However after the 5th strain cycle (Figure 5b) or more, e.g., 20th cycle (Figure 5c), BCC-spheres are well oriented into the volumefilling A- and A’-sphere and B- and B’-sphere so that the clear diffraction spots from these BCC-spheres subjected to the LAOS should appear at the specific azimuthal angles. These diffraction spots change with strain phase as will be described below. In strain phase 2 with a maximum strain γ or zero strain rate γ̇, the diffraction spots observed are essentially equivalent to those of the relaxed A-sphere shown in Figure 9c, as shown in row (e) in Figure 5. However, in this strain phase, the diffraction spots corresponding to the relaxed A′-sphere, a counterpart of A-sphere in Figure 9c, did not appear in situ under LAOS for some reasons which are not well understood up to this stage. In the strain phase 4 with a maximum absolute value of strain |γ|, in the direction opposite to the strain direction in phase 2, and zero strain rate γ̇, the diffraction spots corresponding to the relaxed A′-sphere as shown in Figures 9c are observed. Again the diffraction spots from the counterpart of the relaxed A-sphere in Figure 9c did not appear in situ under LAOS. The diffraction patterns at phases 1 and 3 in Figure 5c are approximately identical. More precisely, the pattern at phase 1

Figure 13. Relaxation process of deformed A/A′-sphere; time dependence of (a) the azimuthal angle of the reciprocal lattice vectors of the {110} and {200} plane and (b) the lattice strain on the {110} and {200} plane. Note that the way how the azimuthal angle is specified depends on the reciprocal lattice vector: it was taken counterclockwisely for r*110,A,d, and r*200,A′,d with respect to the OXaxis and clockwisely for r*110,A′,d and r*200,A,d with respect to the opposite axis to the OX-axis, as defined in the text (section IV-3.B). 1557


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is almost equivalent to the pattern shown in Figure 6a, while the pattern at phase 3 is almost identical to the pattern which is in mirror symmetry to the pattern in Figure 6a with respect to the OYZ-plane. Thus, the pattern at phase 1 reflects the lattice subjected to a large lattice strain, γlattice, as indicated by the solid lines and filled circles and squares shown in Figures 12a and 12b, respectively, and the pattern at phase 3 reflects the lattice subjected to −γlattice. It should be noted here that at phase 1 the lattice strain on A-sphere is much less than that on A’-sphere in Figure 12a and 12b, as revealed in the preceding section. This is consistent with a large stress level shown by the points P1 and P3 in Figure 7d. Consequently, the microscopic lattice strain γlattice is accompanied by a very large phase difference against the applied macroscopic strain γ. Surprisingly the oscillatory lattice deformation for A- and A′-sphere under the fixed orientation of the {112} lattice plane and the ⟨111⟩ axis parallel to the OXZ-plane and the OX-axis, respectively, responds approximately in phase to the periodic strain rate γ̇, revealing a quite viscous response of lattice deformation at this given LAOS condition. This lattice response with the applied strain is consistent with a large value of the mechanical loss tangent, tan δ1, of the linear component of shear stress σ, as will be discussed later in section IV-7. IV-5. Proposed Model for Dynamic Response of BCCSphere to LAOS: Creation and Response of Twinned BCC-Sphere. In section IV-4, we elucidated the piece of evidence that the LAOS first develops the specific orientation of BCC-sphere as defined by A- and A′-sphere with their {112} lattice plane parallel to the OXZ-plane (shear plane) and the ⟨111⟩ axis parallel to the OX-axis as well as B- and B′-sphere with the {112} lattice plane parallel to the OXY-plane and the ⟨111⟩ axis parallel to the OX-axis in the strain cycle of N < 5. Then the LAOS at N ≥ 5 was found to oscillatorily deform BCC-sphere under the fixed orientation as described above. Here it is interesting to note that the DSAXS patterns in strain phases 1 and 3 in Figures 5b and 5c may be explained in the case when the shear deformation of A-sphere cooperatively occurs with the shear deformation of A′-sphere, as illustrated in Figures 12a and 12b, except for the unequal lattice strain for Aand A’-sphere. The cooperative deformation in turn must suggest that A- and A′-sphere essentially form twinned BCCsphere with the lattices of A- and A′-sphere being matched on the twin plane of the {112} lattice plane, as illustrated in Figures 11a and 11b together with the coordinate shown in Figure 11e. The B- and B′-spheres also are expected to form the twinned BCC-sphere, which is also shown by Figures 11a and 11b together with the coordinate shown in Figure 11f. The twinned BCC-sphere with the distorted twin boundaries as shown in Figures 11c and 11d may give γlattice much smaller than γ for bulk so that it may not be able to account for the observed dynamic lattice response shown in Figures 5c and 6. Figure 14 illustrates a set of the twinned BCC-sphere composed of A- and A′-sphere in Figure 11 under the LAOS deformation at strain phase 1 to 4, where parts (a-1) to (a-4) sketch the deformation of the twinned BCC in Figure 11a at strain phase 1 to 4, respectively, while parts (b-1) to (b-4) sketch that of the twinned BCC in Figure 11b at strain phase 1 to 4, respectively. In the figure, the undeformed twins are shown by the broken lines, while the deformed twins are shown by the solid lines. At strain phase 1, both A-sphere and A′sphere are deformed with a fixed orientation of the twin plane and the ⟨111⟩ axis parallel to the OXZ-plane and the OX-axis, respectively, as shown in parts (a-1) and (b-1).

Figure 14. Schematic diagrams showing the response of the twinned A- and A′-sphere to the strain phase of LAOS. A-sphere has a small compliance than A′-sphere in phase 1, which becomes opposite in phase 3. The difference of A- and A′-sphere with respect to the compliance is schematically illustrated by the difference in the length of the arrow in parts (a-1), (b-1), (a-3), and (b-3). The hatching in A′sphere in phase 2 and that in A-sphere in phase 4 imply that the lattices in these spheres are still under stress and bear larger distortions, hence being invisible with the DSAXS patterns in Figure 5.

As clarified in section IV-3 (see Figure 13), the shear deformation of A-sphere and that of A′-sphere are remarkably different: A′-sphere is subjected to a larger γ lattice than A-sphere as schematically illustrated by the difference in the length of the arrow in parts (a-1) and (b-1). These unequally deformed BCC-spheres are expected to give a primary reason to account for the distorted DSAXS patterns at phase 1 shown in Figures 5b and 5c. This is because, in light of the discussion given in conjunction with Figure 12c, (i) the shifts of the diffraction spots given by |A11| and |A13| are much smaller than those given by |A′12| and |A′14|; (ii) those given by |A22| and |A24| are much smaller than those given by |A′21| and |A′23|; (iii) those given by |A32| and |A34| also are much smaller than those given by |A′31| and |A′33|. These asymmetric shifts of the diffraction spots as characterized by (i) to (iii) described above will give a primary reason why the diffraction spots from the {110} and {200} and those from {112} plane are overlapped in the deformed state. The similar arguments can be applied to strain phase 3 shown in Figures 14a-3 and 14b-3 in order to account for the DSAXS patterns at phase 3 in Figures 5b and 5c. At strain phase 2, A-sphere in the twinned BCC in parts (a2) and (b-2) in Figure 14 are relaxed from the shear deformation and thereby should give rise to the diffraction 1558


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spots from the relaxed A-sphere as shown in Figures 5b and 5c and by the model shown in Figure 5e. This is because the lattice of A-sphere is less deformed than that of A′-sphere at phase 1, and thereby it is relaxed faster than A′-sphere and its residual lattice strain may be smaller than A′-sphere. On the other hand A′-sphere is subjected to a larger lattice deformation and thereby a larger degree of lattice distortions than A-sphere, which in turn may cause the diffraction spots from A′-sphere weak in their intensities at phase 2. Thus, we expect that Asphere is relaxed, but A′-sphere is still under the stress. This asymmetric relaxation is expected to account for such experimental observation that A-sphere is visible, which is illustrated by the unhatched A-sphere, but A′-sphere is invisible at phase 2, which is illustrated by hatched A′-sphere. It also accounts for a relatively large stress level shown by the point P2 in Figure 7d and elucidates that the deformation of A′-sphere is primarily responsible for the stress level shown by P2 in Figure 7d. At strain phase 3, γlattice is negative so that in this case Asphere can be more deformed than A′-sphere, as illustrated in the sketch shown in parts (a-3) and (b-3) with respect to the length of the arrow. This unequal lattice deformation causes the asymmetric relaxation of the deformed lattice at strain phase 4, which in turn causes A′-sphere be relaxed and visible, as shown by the unhatched A′-sphere in parts (a-4) and (b-4), but Asphere be under a stressed state with some lattice distortions and invisible as schematically shown by the hatched A-sphere. This accounts for the large stress level shown by the point P4 in Figure 7d. IV-6. Physical Factors Controlling the Unequal Lattice Deformation for A-Sphere and A′-Sphere in A/A′-Twin. We can estimate the lattice strain imposed on {110} and {200} lattice spacing, ε110 and ε200, respectively, when A- and A′sphere are subjected to the γlattice = +0.4, as shown in Figure 15, where

Figure 15. Schematic diagram showing the sheared BCC lattice with the lattice strain γlattice = +0.4 for A-sphere (part a) and A′-sphere (part b), where blue dotted lines and red solid lines represent the lattice before and after the shear deformation. The parts (c) and (d) present simplified lattice for A- and A′-sphere, respectively, and schematic illustration of the conformations of the bcp chains in the matrix subjected to the shear deformation. The shear deformation of the spheres are ignored for the sake of simplicity.

Table 1. Strain ε110 and ε200 Imposed on (110) and (200) Lattice Spacings, Respectively, and Strain ε1 and ε2 Imposed on Interdomain Distances along the Direction 1 and Direction 2 Defined in Figure 15, Respectively, When A- and A′-Sphere Are Subjected to the Shear Strain on the Lattice, γlattice = +0.4, As Shown in Figure 15. K is A-Sphere or A’Sphere

ε110 = (d110,d − d110,u)/d110,u ε200 = (d 200,d − d 200,u)/d 200,u

and dhkl,u and dhkl,d are the lattice spacing of (hkl) plane before and after imposing the given lattice shear strain γlattice. Figures 15a and 15b show the BCC unit cells for A and A′-sphere, respectively, before (blue broken lines) and after the given shear strain on the respective lattice (red solid lines). The estimated values for ε110 and ε200 are shown in Table 1, where the positive and negative values indicate the expansion and contraction of the lattice spacing. We can estimate also the strain ε1 and ε2 imposed on the chains which bridges between the interfaces of the neighboring spheres along the direction 1 and the direction 2 defined in Figures 15a and 15b. In the calculation of ε1 and ε2, we ignored deformation of the spheres for the sake of simplicity. The results are also shown in Table 1 where the positive and negative values indicate the expansion and contraction, respectively. The values ε1 and ε2 intimately involve the deformation of the corona PI block chains emanating from the PS spheres and hence are related to the conformational entropy loss and entropy elasticity of the corona chains as shown schematically in the simplified A-sphere lattice (Figure 15c) and A′-sphere lattice (Figure 15d). The results in Table 1 elucidate the following pieces of important information. Under the given lattice shear strain, γlattice = +0.4, imposed on the lattice for A- and A′-sphere, A-

strain

A-sphere

A′-sphere

ε110.K ε200.K ε1.K ε2.K

+0.17 −0.16 +0.42 −0.11

−0.18 +0.22 +0.18 −0.13

sphere is subjected to smaller strains on ε110 and ε200 than A′sphere but larger strains on ε1 and ε2. Thus, in the case when γlattice > 0, A-sphere is favored from the viewpoint of the elastic energy contribution to the lattice deformation. However, Asphere is favored from the viewpoint of the entropic contribution to the deformation. In other words, if the energy elasticity outweighs the entropy elasticity, A-sphere is expected to have a higher compliance than A′-sphere; if the entropy elasticity outweighs the energy elasticity, A′-sphere has a higher compliance than A-sphere.32 In the case when γlattice < 0, the lattice compliance for A- and A′-sphere is opposite to the case of γlattice > 0. The conclusion drawn in the previous section IV-5 in conjunction with the model sketched in Figure 14 obviously supports the dominance of the entropy elasticity against the energy elasticity in our bcp system. IV-7. Analyses of Nonlinear Mechanical Responses under LAOS. The nonlinear response of shear stress was analyzed by expanding it into the Fourier series, given by 1559


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Figure 16. (a) Absolute value of the first- (σ1), third- (σ3), and fifthorder Fourier components (σ5) of shear stress σ and (b) loss tangent of the linear part of the shear stress (tan δ1) as a function of strain cycle N at 210 °C.

In the late stage, the stress σk (k = 1−3) start to increase more markedly with N. The response of σ starts to become increasingly nonlinear. If the decay rate of tan δ1 with N is given by a power law of tan δ1 ∼ N−α, the rate is slowed down as evidenced by the decrease of the exponent from α = 2/5 to 1/3 and eventually to 1/7 in late stages 1 and 2 to be defined below, respectively. Further the DSAXS responses show the oscillatory shear deformation of the twinned BCC-sphere under the fixed orientation of the twin plane parallel to the shear plane (OXZplane) for A/A′-twin or the OXY-plane for B/B′-sphere and the ⟨111⟩ axis parallel to the shear direction (the OX-axis). More precisely, the late stage is further divided into the following two stages. In late stage 1 (4 ≤ N ≤ 10), the nonlinear stress terms σ3 and σ5 strikingly increase relative to σ1; σ3/σ1 and σ5/σ1 almost double with N, increasing from 0.14 to 0.25 and 0.03 to 0.06, respectively. In late stage 2 (N > 10), the increase of the nonlinear contributions is slow down to reach the steady state deformation. We may anticipate that the late stage 1 is the stage where the plastic deformation occurs on A- and A′-sphere to create increasing number of the twin boundaries and that the late stage 2 is the stage where the number of the twin boundaries roughly reach a steady state and the lattice undergoes more elastic oscillatory deformation. IV-8. Comparison with Other Works. Our results reveal that two types of oriented twinned BCC-sphere structures are formed in the sphere-forming SIS block copolymer melt subjected to LAOS. One is twinned A- and A′-sphere shown in Figures 8−11 having the twinning plane of the {112} plane parallel to the shear plane of OXZ-plane and the ⟨111⟩ axis parallel to the shear direction OX. Hereafter, we call this Atwin. Another is twinned B- and B′-sphere having the twinning plane vertical to the shear plane as shown also in Figures 8−11. We call this B-twin. Almdal et al.19 observed only B-twin. Koppi et al.20 pointed out the existence of A-twin in addition to Btwin. Because Koppi et al. carried out the SANS experiments with the incident beam from only one direction parallel to a shear gradient direction, the investigation of A-twin was not as complete as that of B-twin. According to our results, the volume fraction of A-twin is larger than that of B-twin. It seems that the results of Koppi et al. are opposite to ours. At this stage we do not know the reasons why the A-twin is dominant over its counterpart structure in our case. As described in section IV2, it might be possible that the structure developed by LAOS is affected by the precursors already presented in the initial state.

stress σ, while Figure 16b shows the loss tangent (tan δ1 = σ1″/ σ1′) of the linear component of σ, both as a function of strain cycle N. These data were obtained in situ at 210 °C simultaneously with the SR-DSAXS. Overall trends observed on the dynamic mechanical response shown in Figures 7 and 16 and on the dynamic response of DSAXS pattern with N shown in Figure 5 are consistently classified into the early stage where N < 4 and the late stage where N ≥ 4. In the early stage, the linear component of the stress, σ1, remains nearly independent of N and dominates the nonlinear components σ3 and σ5, so that the response of σ is roughly linear (σ3/σ1 ≅ 0.06−0.14 and σ5/σ1 ≅ 0.01−0.03). The value tan δ1 (= σ1″/σ1′) linearly decreases with log N, and the DSAXS responses show that the specific orientation of the BCC-sphere toward A- and A′-sphere. In more precisely, even in the early stage, the nonlinear stress term σ3 slightly increased with N, indicative of the nonlinear lattice deformation of BCCsphere to create twinned BCC-sphere.

V. CONCLUSIONS Polystyrene-block-polyisoprene-block-polystyrene (SIS) bulk having almost randomly oriented grains composed of spherical domains in BCC lattice (BCC-sphere) were subjected to a large amplitude oscillatory shear deformation (LAOS) in order to study shear-induced orientation of BCC-sphere and the dynamical response of the oriented BCC-sphere to LAOS by means of strain-phase-resolved synchrotron dynamic smallangle X-ray scattering method (SR-DSAXS) with a twodimensional (2D) imaging detector. The LAOS with strain amplitude of 0.5 and angular frequency 0.0944 rad/s was applied to the specimens at 210 °C in the BCC-sphere phase, 27 °C above OOT between hex−cyl and BCC-sphere, and 2D SR-DSAXS patterns were measured simultaneously with shear stress σ as a function of strain phase ϕ and strain cycle N up to N = 20. The orientation of BCC-sphere was found to advance progressively with N. After N > 4 the distinct orientations of

σ (t ) =

∑ (σk′ + iσk″) exp(ikωt ) k

(2)

Here the absolute value of kth-order Fourier component of the stress, σk, is defined by σk = |σk*| =

(σk′)2 + (σk″)2

(3)

where σk* = σk′ + iσk″,

k = 1, 2, 3, ...

(4)

Figure 16a shows the absolute value of the first- (σ1), third(σ3), and fifth-order Fourier components (σ5) of the shear

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(2) Hadjichristidis, N.; Pispas, S.; Floudas, G. In Block Copolymers; John Wiley & Sons: New York, 2003. (3) Keller, A.; Pedemonte, E.; Willmouth, F. M. Kolloid Z. Z. Polym. 1970, 238, 385−389. (4) Hadziioannou, G.; Mathis, A.; Skoulios, A. Colloid Polym. Sci. 1979, 257, 136−139. (5) Koppi, K. A.; Tirrell, M.; Bates, F. S.; Almdal, K.; Colby, R. H. J. Phys. II 1992, 2, 1941−1959. (6) Okamoto, S.; Saijo, K.; Hashimoto, T. Macromolecules 1994, 27, 5547−5555. (7) Zhang, Y.; Wiesner, U. J. Chem. Phys. 1995, 103, 4784−4793. (8) Schulz, M. F.; Khandpur, A. K.; Bates, F. S.; Almdal, K.; Mortensen, K.; Hajduk, D. A.; Gruner, S. M. Macromolecules 1996, 29, 2857−2867. (9) Zhang, Y.; Wiesner, U. J. Chem. Phys. 1997, 106, 2961−2969. (10) Wiesner, U. Macromol. Chem. Phys. 1997, 198, 3319−3352. (11) Chen, Z.-R.; Kornfield, J. A. Polymer 1998, 39, 4679−4699. (12) Hamley, I. W.; Pople, J. A.; Fairclough, J. P. A.; Ryan, A. J.; Booth, C.; Yang, Y.-W. Macromolecules 1998, 31, 3906−3911. (13) Hamley, I. W.; Mortensen, K.; Yu, G.-E.; Booth, C. Macromolecules 1998, 31, 6958−6963. (14) Hamley, I. W.; Pople, J. A.; Booth, C.; Derici, L.; Imperor-Clerc, M.; Davidson, P. Phys. Rev. E 1998, 58, 7620−7628. (15) Hamley, I. W.; Pople, J. A.; Fairclough, J. P. A.; Terrill, N. J.; Ryan, A. J.; Booth, C.; Yu, G.-E.; Diat, O.; Almdal, K.; Mortensen, K.; Vigild, M. J. Chem. Phys. 1998, 108, 6929−6936. (16) Mortensen, K.; Theunissen, E.; Kleppinger, R.; Almdal, K.; Reynaers, H. Macromolecules 2002, 35, 7773−7781. (17) Mortensen, K. J. Polym. Sci., Part B 2004, 42, 3095−3101. (18) Torija, M. A.; Choi, S.-H.; Lodge, T. P.; Bates, F. S. J. Phys. Chem. B 2011, 115, 5840−5848. (19) Almdal, K.; Koppi, K. A.; Bates, F. S. Macromolecules 1993, 26, 4058−4060. (20) Koppi, K. A.; Tirrell, M.; Bates, F. S.; Almdal, K.; Mortensen, K. J. Rheol. 1994, 38, 999−1027. (21) Okamoto, S.; Saijo, K.; Hashimoto, T. Macromolecules 1994, 27, 3753−3758. (22) Shin, G.; Sakamoto, N.; Saijo, K.; Suehiro, S.; Hashimoto, T.; Ito, K.; Amemiya, Y. Macromolecules 2000, 33, 9002−9014. (23) Ackerson, B. J.; Clark, N. A. Phys. Rev. A 1984, 30, 906−918. (24) Sakamoto, N.; Hashimoto, T.; Han, C. D.; Kim, D.; Vaidya, N. Y. Macromolecules 1997, 30, 1621−1632. (25) Han, C. D.; Vaidya, N. Y.; Kim, D.; Shin, G.; Yamaguchi, D.; Hashimoto, T. Macromolecules 2000, 33, 3767−3780. (26) Suehiro, S.; Saijo, K.; Seto, T.; Sakamoto, N.; Hashimoto, T.; Ito, K.; Amemiya, Y. J. Synchrotron Radiat. 1996, 3, 225−230. (27) Amemiya, Y.; Kishimoto, S.; Matsushita, T.; Satow, Y.; Ando, M. Rev. Sci. Instrum. 1989, 60, 1552−1556. (28) Suehiro, S.; Saijo, K.; Ohta, Y.; Hashimoto, T.; Kawai, H. Anal. Chim. Acta 1986, 189, 41−56. (29) Amemiya, Y.; Wakabayashi, K.; Hamanaka, T.; Wakabayashi, T.; Matsushita, T.; Hashizume, H. Nucl. Instrum. Methods 1983, 208, 471− 477. (30) Hashimoto, T.; Shibayama, M.; Fujimura, M.; Kawai, H. In Block Copolymer Science and Technology; Meier, D. J., Ed.; MMI Press Symposium Series; Harwood Academic Pub.: London, 1983; Vol. 3, pp 63−108. (31) Sakurai, S.; Momii, T.; Taie, K.; Shibayama, M.; Nomura, S.; Hashimoto, T. Macromolecules 1993, 26, 485−491. (32) Free energy (F0) of a single bridge chain of PI block emanated from the interfaces of the nearest-neighbor PS spheres is given by F0 ∝ H02, where H0 is the end-to-end distance of the bridge chain. When H0 changes to H by the shear strain on the lattice, γlattice, as shown in Figure 15, the corresponding free energy changes from F0 to F, where F ∝ H2. Thus, the relative free energy change due to γlattice defined by ΔF/F0 ≡ (F − F0)/F0 is given by (H2 − H02)/H02 = |ε|(|ε| + 2), where ε ≡ (H − H0)/H0. The value of ε along the direction 1 (defined by ε1,K) and that along the direction 2 (defined by ε2,K) for K-sphere (K = A or A′) is given by Table 1, where the directions 1 and 2 are defined

BCC-sphere, as characterized by A- and A′-sphere, and B- and B′-sphere in the text, which fill a whole sample space, were observed to be attained, and an oscillatory lattice deformation with ϕ was clearly discerned under the fixed orientation of the twin plane of {112} lattice plane parallel to the shear plane (the OXZ-plane) for A- and A′-sphere and the ⟨111⟩ axis parallel to the shear direction (the OX-axis). This lattice deformation was found to have a large phase difference to the applied LAOS strain, γ, showing the maximum lattice strain γlattice approximately at the strain phases 1 and 3 where γ = 0 or |dγ/dt| becomes maximum. The response of σ with γ is quite nonlinear, exhibiting a typical bow-tie Lissajous pattern where the maximum value of σ in each strain cycle increases with N and hence a trend for the strain-hardening behavior. A nonlinear analysis of σ indicates that both linear component σ1 and nonlinear components σ3 and σ5 increased with N; σ1 increased from about 200 to 300 Pa, and σ3 increased from 10 to 70 Pa. Correspondingly loss tangent, tan δ1, of the linear component decreased from 8 to 2. The increase of σ and σi (i = 1, 3, and 5) is also consistent with increased orientation and ordering of BCC-sphere with N. After the cessation of LAOS at phase 1 (γ = 0) and N = 21, the relaxation process of the deformed lattice was explored by means of the time-resolved SR-DSAXS experiment with 1 s exposure time for X-ray with incident beam along the vorticity direction, and the results unveiled are summarized in Figures 12 and 13. It is striking to discover the facts that the shear deformations of A-sphere and A′-sphere in A-twin are quite unequal; A-sphere has a smaller compliance than A′-sphere for positive shear strain imposed on the lattice, while the opposite is observed for the case of the negative shear strain; and this unequal deformation is proposed to be primarily due to the entropy elasticity of BCC-sphere associated with the conformational entropy change in coronar block chains with the shear deformation.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; [email protected]. jp. Present Addresses †

Mitsubishi Chemical Performance Polymers, Inc., 2001 Hood Rd, Greer, SC 29650. ‡ Visiting Researcher, Quantum Beam Science Directorate, Japan, Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan. § Rigaku Corporation, 3-9-12, Matsubara-cho, Akishima-shi, Tokyo 196-8666, Japan. Notes

The authors declare no competing financial interest. ∥ Professor Emeritus, Kyoto University, Kyoto 606-8501, Japan & Honorary Chair Professor, National Tsing Hua University, Hsinchu 30013, Taiwan.

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ACKNOWLEDGMENTS The SAXS measurements in this work were conducted under approval of the Photon Factory Advisory Committee (Proposal Nos. 97G245 and 99G266).

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REFERENCES

(1) Hamley, I. W. In The Physics of Block Copolymers; Oxford University Press: Oxford, 1998. 1561


Macromolecules

Article

in Figure 15. Thus, the ratio of the relative free energy change for Asphere [defined by (ΔF/F0)A] and A′-sphere [defined by (ΔF/F0)A’] with respect to the lattice deformation is given by (ΔF/F0)A/(ΔF/ F0)A’ = [2|ε2,A|(|ε2,A| + 2) + |ε1,A|(|ε1,A| + 2)]/[2|ε1,A′|(|ε1,A′| + 2) + |ε2,A′| (|ε2,A′| + 2)] = 1.4. Thus, the free energy change of the bridging chains due to the lattice deformation for A′-sphere is smaller than that for the A-sphere.

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