Dielectric behavior of Styrene–Isoprene - American Chemical Society

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Dielectric behavior of Styrene−Isoprene (SI) Diblock and SIIS Triblock Copolymers: Global Dynamics of I Blocks in Spherical and Cylindrical Domains Embedded in Glassy S Matrix Yumi Matsumiya, Quan Chen,† Akiko Uno, and Hiroshi Watanabe* Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan

Atsushi Takano, Kohei Matsuoka, and Yushu Matsushita Department of Applied Chemistry, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan ABSTRACT: Dielectric behavior was examined for nanodomain-forming styrene−isoprene (SI) diblock and SIIS triblock copolymers, the former having type-A dipoles aligned along its I block, whereas the latter being the head-tohead dimer of the former and having the dipoles inverted at the midpoint of the middle II block. The slow dielectric relaxation at low temperatures ( cylinder (Cyl) > lamella (La). This order appears to naturally reflect the domain continuity that helps large-scale motion of the I block without violating the osmotic constraint. In the Sph and/or Cyl domains, all II blocks of the SIIS copolymer have their ends fixed on the surface of the same domain and are geometrically classified as loop-type blocks. These loop-type blocks are naively expected to behave similarly, in a dynamic sense, to the tail-type I blocks of SI. Nevertheless, comparison of the dielectric data of the SIIS and SI copolymers suggested that some of the II blocks are classified as dynamically pseudobridge-like blocks behaving differently from the tail-type I blocks. This behavior of the II blocks in the Sph/Cyl domains was speculated to reflect their end-to-end distance: This distance governs the direction of II block tension that acts on the midpoint of this block thereby affecting the dielectrically detected midpoint motion. The II block in the Sph/Cyl domain appears to exhibit the pseudobridge-like behavior when it has a large end-to-end distance and the two halffragments of this block pull the midpoint in the opposite directions. This argument is crude but seems to be physically acceptable, as suggested from an approximate analysis of the distribution of the end-to-end distance in those domains.

1. INTRODUCTION Block copolymers form nanodomains, and the thermodynamics of this domain formation is well understood:1,2 Contacts of chemically different segments should be reduced, each block should randomize its conformation, the block junctions should be located randomly in the boundary between different domains, and the blocks in each domain are required to fill the domain uniformly with their segments. Compromise of these contradicting thermodynamic requirements determines the domain morphology. Specifically, in bulk copolymers, the osmotic requirement of uniform domain-filling is very strong, and the morphology is determined essentially in a way that the other thermodynamic requirements are satisfied as much as possible without violating the osmotic requirement. Obviously, the osmotic requirement also affects the block dynamics:3−5 The spatial distribution of the segments is fully coupled with the block conformation when the blocks are confined in the nanodomain and their ends are fixed at the © 2012 American Chemical Society

domain boundary, and the blocks in each nanodomain are forced to move highly cooperatively so that their motion results in no segment density gradient6 (at a length scale larger than the segment size). This osmotic constraint on the block dynamics broadens/retards the relaxation, as discussed for diblock copolymers composed of polystyrene (S) and cis-polyisoprene (I) blocks:3−5 The I block has so-called type-A dipoles parallel along the chain backbone and the polarization of this block is proportional to its end-to-end vector RI, so that the fluctuation of RI results in slow dielectric relaxation. 5,7,8 At low temperatures T well below the glass transition temperature Tg,PS of the S block, the S block is dielectrically inert and the S− I block junction is fixed at the interface between the glassy S domain and rubbery I domain. At such low T, the fluctuation of Received: July 13, 2012 Revised: August 20, 2012 Published: August 28, 2012 7050

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first subject is related to the nanodomain shape. We expect that the effect of the osmotic constraint on the block dynamics is stronger in the spherical/cylindrical domain than in the lamellar domain, because the lamellar domain has a 2-dimensional continuity and the blocks therein can utilize the unlimited space in the direction parallel to the lamellar surface to move cooperatively rather easily even under the very strong requirement of uniform domain-filling.13 Our recent experiments13 utilized guest homo I chains mixed in the spherical I domain of SIS triblock copolymer having no dipole inversion to probe the effect of the osmotic constraint in this domain through the dielectric behavior of the guest chains. The result suggested that the effect on the guest chain dynamics is larger in the spherical domain than in the lamellar domain (the latter examined in literature22), which lends support to the above expectation. However, it is also desired to focus on the SI diblock copolymers having the dielectrically active, tail-type I block and test the expectation more directly (without utilizing the guest chains). The second subject is related to the loop/bridge population ratio. In the lamellar I domain and in the I matrix embedding spherical S domains, the loop- and bridge-type I blocks of SIS triblock copolymer are clearly distinguished geometrically: Two ends of the loop-type I block are anchored on the same S domain, whereas the ends of the bridge-type I block, on different S domains. Indeed, the loop- and bridge-type II blocks (having the dipole inversion) exhibit different dielectric responses.4,5,9,15 However, the geometrical distinction between the loop and bridge vanishes in the spherical and cylindrical domains surrounded by a single surface (S/I boundary), because any I block therein has the ends on the same surface. It is of interest to examine the dielectric behavior of the II blocks in those domains. Thus, we conducted dielectric measurements for a SI diblock copolymer and its head-to-head dimer, a dipole-inverted SIIS triblock copolymer, both forming cylindrical I domains, as well as for blends of the SIIS and/or SI copolymer with a low-M homo S that form spherical I domains. The results suggested that the effect of the osmotic constraint is larger in the spherical/cylindrical domain than in the lamellar domain, as expected, and that the II block in the spherical/cylindrical domain behaves either as looplike or pseudobridge-like depending on a distance between its ends anchored on the same I domain surface. Details of these results are presented in this article.

RI is activated only by the motion of the free end of the I block (that is associated with motion of all I segments along the block backbone except at the S−I junction), and the terminal dielectric relaxation of the SI diblock copolymer detecting this motion was found to be much broader and slower than the relaxation of I homopolymer of the same molecular weight.3−5,8,9 This behavior of the I block is mainly attributed to the osmotic constraint on the block dynamics,4,5,8,9 although there could be a secondary effect of spatial confinement of the I blocks in their domain.8,10,11 (The spatial confinement in the absence of the osmotic constraint does not retard the dielectric relaxation of a tethered bead−spring chain (a model for nonentangled I blocks) but enhances the fast relaxation, as suggested from simple bead−spring analysis.8,11) The dynamics of the I block in the SIS triblock copolymer is also osmotically constrained. However, at low T < Tg,PS, both ends of the middle I block are fixed on the glassy S domains to allow no fluctuation RI, and the motion of such end-fixed I block activates no slow dielectric relaxation.12,13 This difficulty in the dielectric observation of the large-scale (global) block motion can be removed if the type-A dipoles of the I block are symmetrically inverted at the midpoint of this block. For such SIIS triblock copolymers, with “II” denoting the dipole-inverted I block throughout this article whenever necessary, the polarization is proportional to a sum of two end-to-center vectors of the II block. Thus, the motion of the midpoint of this block activates the slow dielectric relaxation even at T < Tg,PS.4,5 The SIIS triblock copolymers were actually synthesized through head-to-head coupling of SI− precursor diblock anions, and the slow dielectric relaxation due to the midpoint motion of the II block was found to be broader/retarded compared to the relaxation of I homopolymers, mostly due to the osmotic constraint on the II block dynamics.4,9,14−16 In addition, for the SIS triblock copolymers forming S/I lamellar domain4,9 and/or spherical S domains embedded in the I matrix,14−16 an interesting problem of loop/bridge population ratio of the middle I block has been examined. The middle I block has either bridge- or loop-type conformations, and their population ratio was studied theoretically through the self-consistent field (SCF) approach17−19 and also experimentally with dielectric,4,9,14−16 viscoelastic, 20 and swelling methods.21 Specifically, the dielectric method, applying to the dipole-inverted SIIS copolymer, is based on a molecular argument that the midpoint motion of the loop is similar to the end motion of the tail (=I block of the SI precursor for SIIS) but is slower and more intensive compared to the midpoint motion of the bridge under the strong osmotic constraint.4,5 For lamella-forming I blocks at T < Tg,PS, the bridge fraction ϕbridge dielectrically estimated on the basis of this argument, ϕbridge ≅ 0.4,4,9 is close to the SCF prediction.17−19 In addition, the dielectric similarity of the loop and tail, deduced also in the SCF approach, 17 was experimentally noted for SIIS/SI blends.9 Furthermore, for the I blocks being swollen with n-tetradecane (C14; I-selective solvent) to form the matrix for spherical S domains at T < Tg,PS, the dielectric estimate, ϕbridge = 0.4−0.5 at low C14 content (at low degree of swelling of the I block),14−16 decreased to ϕbridge ≅ 0.2 with increasing C14 content.15 This change of ϕbridge is attributable to stretching/destabilization of the bridge-type I block on swelling. All above results demonstrate the importance of the osmotic constraint for the dynamics of the end-fixed blocks. Those results encourage us to extend the research to two subjects. The

2. EXPERIMENTAL SECTION 2.1. Material. A pair of SI and SIIS samples having a large S content (78.7 wt %) was synthesized with a method reported previously:4 At first, the living SI− anion was polymerized with sbutyllithium in benzene (Bz), and the product was separated in two portions: One portion was terminated with methanol to obtain the SI diblock precursor, and the other was coupled in a head-to-head fashion with 95% equimolar of bifunctional terminator, xylylene dichloride. The product of this anion-coupling reaction was thoroughly fractionated from benzene/methanol mixture to recover the SIIS triblock copolymer sample having the symmetrical dipole inversion in the II block. Molecular characteristics of those SI and SIIS copolymer samples were determined with GPC (CO-8020 and DP-8020; Tosoh) equipped with a refractive index (RI)/low-angle light scattering (LALS) monitor (LS-8000; Tosoh) and an ultraviolet (UV) adsorption monitor (UV-8020; Tosoh). The elution solvent was tetrahydrofuran (THF). The S block molecular weight MS was 7051

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determined from the elution volume calibration for the S precursor (recovered before the copolymerizing step of SI− precursor). Monodisperse polystyrene standards (TSKs; Tosoh) were utilized for this elution volume calibration. The I block molecular weight MI was evaluated from the MS value and the RI and UV signal intensities of the S precursor and the SI (and SIIS) samples with a method explained previously.13,16,23 The molecular characteristics thus determined are summarized in Table 1. The sample code denotes

Table 1. Characteristics of the Copolymer Samples code

10−3MS

10−3MI

Mw/Mna

wIb

SI 48-13 SIIS 48-(13)2-48

47.9 47.9c

13.0 26.0

1.04 1.05

0.213 0.213

a Evaluated from elution volume calibration. bWeight fraction of I block. cEach of the two S blocks of the triblock copolymer has MS = 47.9 × 103.

the block molecular weight in unit of kg mol−1. The targeted molecular weight of the I block was chosen to be close to that of the previously utilized copolymers forming lamellar S/I domains.4 The materials subjected to dielectric and SAXS/TEM measurements were bulk SI and SIIS samples and blends of SI and/or SIIS with a low-M polystyrene (S11; Mw = 10.5 × 103, Mw/Mn = 1.08; supplied from Tosoh). The S11 content in the blends was set to be wS11 = 50 wt %. Films of these materials were obtained by casting from 10 wt % solutions of SIIS and/or SI (plus S11 in the case of blends) dissolved in a benzene/methyl ethyl ketone (Bz/MEK) mixed solvent with Bz/MEK = 2/1 vol/vol for bulk copolymers and 4/1 vol/vol for the blends: The solutions were first cast slowly for 1 week on a Teflon dish under ambient condition, and the resulting films were removed from the dish and placed on an electrode (utilized in dielectric measurements) and thoroughly dried in vacuum at T ≅ 50 °C for 1 week more. Cylindrical I domains were formed in the bulk SIIS and SI films, whereas spherical I domains in the SIIS/S11 and SI/S11 blend films, as shown later in Figures 1 and 2. These I domain shapes were in accord to the volume fraction of the I block in the system,1 ϕI = 0.236

Figure 2. SAXS intensity I(q) (part a) and TEM images (parts b and c) obtained for SI 48-13/S11 and SIIS 48-(13)2-48/S11 blend films (wS11 = 50 wt %) having experienced the dielectric measurements. In part a, the plot of I(q) against the magnitude of scattering vector q is in arbitrary unit (not absolute intensity). The plot for the SI/S11 blend is shifted vertically to avoid heavy overlapping of the data points with those for SIIS/S11 blend. in bulk SIIS and SI systems and ϕI = 0.120 in the blends, and also to the S-selectivitity of the Bz/MEK casting solvent (that allowed the I blocks to segregate in an early stage of casting13,24,25). 2.2. Measurements. For the I-cylinder forming SI and SIIS films as well as for the I-sphere forming SI/S11 and SIIS/S11 blend films, the dielectric measurements were conducted with an impedance analyzer/dielectric interface system (1260 and 1296; Solartron) and a homemade dielectric cell composed of a guarded main electrode, a counter electrode, and a separate shielding jacket (on which these electrodes were firmly attached). The sample on the counter electrode was first annealed and then slowly pressed by the main electrode to be filled in the gap between the two electrodes at 140 °C in Ar atmosphere, and finally cooled to room temperature. The dielectric measurements were made at two temperatures, 0 and 20 °C, where the S blocks were glassy but the I blocks were rubbery. After the dielectric measurements, the sample film was recovered from the dielectric cell and was subjected to small-angle X-ray scattering (SAXS) measurement and transmission electron microscope (TEM) observation. The SAXS measurements were conducted at room temperature with a laboratory goniometer (RINT-2000, Rigaku). The X-ray wavelength was λ = 0.154 nm (Cu Kα line). The sample film was placed in a cell having a mica window, and the scattering intensity I(q) was measured as a function of the magnitude of scattering vector, q = {4π/λ}sin(θ/2) with θ being the scattering angle. For TEM observation, the sample films were stained with 3 wt % OsO4(aq) solution for 6 h at 60 °C, and then embedded in epoxy resin and sliced into ultrathin sections of ∼80 nm thickness with an ultramicrotome (Ultracut UCT; LEICA). These ultrathin sections, sliced at ambient temperature, were further stained with a vapor from 3 wt % OsO4(aq) solution for 30 min at 60 °C, and then subjected to TEM observation with a laboratory TEM system (JEM-1400, JEOL) at ambient temperature. The observation voltage was 120 kV.

Figure 1. SAXS intensity I(q) (part a) and TEM images (parts b and c) obtained for bulk SI 48-13 and SIIS 48-(13)2-48 films having experienced the dielectric measurements. In part a, the plot of I(q) against the magnitude of scattering vector q is in arbitrary unit (not absolute intensity). The plot for SI 48-13 is shifted vertically to avoid heavy overlapping of the data points with those for SIIS 48-(13)2-48.

3. RESULTS AND DISCUSSION 3.1. Nanodomain Structure. Figures 1 and 2, respectively, show the SAXS profiles and TEM images obtained for the bulk SI 48-13 and SIIS 48-(13)2-48 copolymer films and for the SI/ 7052

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S11 and SIIS/S11 blend films. These films were recovered after the dielectric measurements and thus most faithfully retained the domain structures corresponding to the dielectric data discussed later. For bulk SI and SIIS samples, the SAXS intensity I(q) exhibits the first and second order peaks attributed to the interference between the I domains; see Figure 1a. The second peak, seen at q2 ≅ √3q1 with q1 being the magnitude of scattering vector for the first peak, suggests that the I blocks form cylindrical domains aligned on a (hexagonal) lattice,1,2 which is in accord to the ϕI value of the copolymer (=0.236). In fact, in Figure 1, parts b and c, the TEM images confirm the cylinder formation (see the circular cylinder cross sections followed by the side view of cylinders at the grain boundary), although the grains are rather small and the domain alignment therein is not very long-ranged. In addition, this alignment is distorted and looks somehow similar to the simple cubic alignment. However, for our later discussion of the dielectric data, this distortion and the lack of long-ranged order, attributable to the sample preparation method (compression of the cast film at 140 °C followed by no long-time annealing), are not important but the cylinder formation itself is essential. From those TEM images and SAXS data, the cylinder diameter dCyl was estimated as dCyl(TEM) ≅ 18 nm

(1a)

⎛ 2ϕ ⎞1/2 4π dCyl(SAXS) = ⎜ ≅ 19 nm ⎟ ⎝ 3 π ⎠ q1

(1b)

⎛ 3ϕ ⎞1/3 2π dSph(SAXS) = 2⎜ I ⎟ ⎝ 4π ⎠ q1* ≅ 18 nm (for SI/S11), 20 nm (for SIIS/S11) (2b)

The dSph(TEM) value, indistinguishable for the SI/S11 and SIIS/S11 blends, was obtained from a diameter dapp averaged for the dark circles in Figure 2, parts b and c: dSph(TEM) = {4/ π}dapp with the factor {4/π} accounting for a deviation of the slicing plane of the TEM section from the equator of the spherical domains.13 q1* appearing in eq 2b is given by q1* = 0.94ϕI−0.062q1 (for ϕI < 0.4), where the factor of 0.94ϕI−0.062 accounts for a small decrease of q1 on disordering of a bcc lattice into a random dispersion.13,29 (q1* = q1 for a perfect bcc lattice.) The dSph(SAXS) value is a little smaller for the SI/S11 blend than for the SIIS/S11 blend possibly because of a difference of the “randomness” of the domain location in the two blends (note that eq 2b assumes the liquid-like randomness29), but this small difference of the dSph(SAXS) values is not important for our discussion of the dielectric data. We note that the domain radii d/2 (in particular d(TEM)/2 given by eqs 1a and 2a) in the SI systems are close to the unperturbed end-to-end distance of the I block, ⟨R2⟩1/2 ≅ 9.4 nm (evaluated from an empirical equation30) and that the domain diameter d of the SIIS systems is just moderately larger than ⟨R2⟩1/2 (≅ 13 nm) of the II block. These similarities between the domain size and ⟨R2⟩1/2 become a key in our discussion of the dielectric response explained later. 3.2. Dielectric Behavior of SI Systems. 3.2.1. Overview. In this section, we examine the dielectric response of the I blocks having the type-A dipoles to discuss the block dynamics in nanodomains of various shapes. The response of the dipoleinverted SIIS triblock copolymer is affected by the osmotic constraint as well as the loop/bridge population distribution. Thus, for clear comparison of the I block dynamics in the domains of various shapes, we focus on the response of the SI diblock copolymer: All I blocks of SI have the tail-type conformation, and their slow dielectric response at T < Tg,PS (where the S−I junctions are fixed) is unequivocally attributed to motion of their free ends in respective domains, as explained later in more detail in relation to the dielectric relaxation function (eq 3). Figure 3a shows the dielectric loss, ε″, of the SI 48-13 diblock copolymer (filled triangles) and its blend with S11 (filled circles) measured in this study. From the calibration measurement for the empty dielectric cell, the experimental resolution of ε″ in this study was estimated to be ≅10−4. The ε″ data at T = 0 and 20 °C ( 107 s−1 at 0 °C)3,7 to be detected in the range of ω examined in Figure 3. Thus, the ε″/ϕI data seen in Figure 3b are exclusively related to the global motion (end-to-end fluctuation) of the I block. Those data are described by the well-established relationship between εI″ and the component of the end-to-end vector of the I block in the direction of the electric field, RI,E:4,5,7,35

∫0

∞

dΦ(t ) sin ωt dt dt

⟨{RI,E(0)}2 ⟩eq

(3b)

Here, Δε is the dielectric intensity, Φ(t) is the dielectric relaxation function, and ⟨···⟩eq indicates an ensemble average at equilibrium. (The averages ⟨RI,E(t)RI,E(0)⟩eq and ⟨{RI,E(0)}2⟩eq can be replaced by ⟨RI(t)·RI(0)⟩eq and ⟨{RI(0)}2⟩eq if the I blocks have an isotropic orientational distribution in the materials.5,8) The relaxation of Φ(t) is determined by the timeevolution of RI,E(t) (or RI(t)); see eq 3b. For SI copolymers having the S−I junction fixed on the glassy S domain, this time evolution occurs only through the motion of the free end of the I block. Thus, the dielectric relaxation of SI seen in Figure 3 is essentially (and exclusively) related to the free-end motion of the I blocks, although a delicate consideration is necessary for the I block motion in the S/I interphase, as discussed later. In relation to eq 3b, we should note a difference between the I block dynamics and the dielectric response reflecting this dynamics. Obviously, the I block dynamics is specified by the motion of all I segments in the block backbone. In contrast, the dielectric response exclusively detects the end motion, as noted from eq 3b. This end motion is correlated with the motion of the all segments in the I block backbone, but this correlation is reflected in the dielectric Φ(t) just indirectly as the t dependence of Φ(t). In other words, we can fully specify Φ(t) just from the information for the time evolution of RI,E(t) (that includes the correlation between RI,E(t) and RI,E(0)), even if no motional information is available for the segments along the I block. Thus, we cannot fully describe the block dynamics only from the dielectric data.36 Nevertheless, those data are still useful for discussing the effect of the osmotic constraint on the block dynamics, as described in the next section. The dielectric relaxation of homo-PI chains is also described by eq 3 and detects the motion of their free ends (activating the end-to-end vector fluctuation). In this sense, the dielectric feature is similar for homo-PI chains and the I blocks. However, we should note an important difference. For homo-PI, the chain conformation is completely renewed (randomized) and Φ(t) decays to zero at sufficiently long times. In contrast, for the I blocks in their La, Cyl, and Sph domains, Φ(t) does not fully decay to zero because these blocks cannot penetrate the S domains. (This lack of full decay would be easily understood if we consider the I block tethered on a flat, lamellar S domain. RI of this I block tends to be oriented, at any time, in the direction normal to the lamellar surface so that ⟨RI(t)·RI(0)⟩eq does not decay to zero.5,8) Thus, the εI″ data of the I block just detect a relaxable part of the orientational memory (as clearly noted from the dΦ(t)/dt term in eq 3a), which tends to decrease the dielectric intensity Δε of the I block compared to Δε of homoPI. At the same time, we should also note that eq 3b does not include the cross-correlation between different I blocks, ⟨R[α] I,E (t) R[β] I,E (0)⟩eq with α and β (≠α) being the indices of the I blocks, because of the vectorial cancellation in the (huge) ensemble of I blocks:5,8 For a given (βth) I block having R[β] I (0), the ensemble should include the same number of I blocks (αth I [α] blocks) having the opposite orientation, R[α] I (t) and −RI (t), to vanish the cross-correlation. However, this vectorial cancellation does not mean the lack of motional cooperativity. Instead, neighboring I blocks should move highly cooperatively in order to maintain the constant segment density (=osmotic constraint for the block dynamics explained earlier), and Δε of

Figure 3. (a) Dielectric loss ε″ of bulk SI 48-13 copolymer and its blend with S11 reduced at 0 °C. The I blocks formed cylindrical (Cyl) and spherical (Sph) domains in the bulk and blend systems, respectively. For comparison, ε″ data at 0 °C are shown also for bulk PI13 with M = 13.0 × 103 and for the lamella (La) forming SI 1212 copolymer4 (MS = 11.7 × 103, MI = 11.6 × 103). (b) Comparison of the ε″ data shown in part a after normalization by the volume fraction ϕI of the I block.

εI″(ω) = −Δε

⟨RI,E(t )RI,E(0)⟩eq

(3a)

with 7054

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superposition, is sufficient for our second-step argument (molecular argument) summarized below. The retardation/broadening of the I block dynamics is mainly attributable to the osmotic constraint explained earlier. Namely, neighboring I blocks are forced to move highly cooperatively because of the osmotic constraint that allows no density gradient in the domain (at a length scale larger than the segment size), and this cooperativity leads to the retardation/ broadening of the I block relaxation.4,5,9,13 In relation to this point, some comments need to be made for additional mechanisms that may result in the retardation/ broadening. The I block motion is limited within the I domain, and this spatial confinement in the presence of the osmotic constraint could, in principle, result in the retardation/ broadening. (Note that the spatial confinement in the absence of the osmotic constraint changes the entropic tension of the I block thereby just enhancing the fast relaxation.8) However, our previous study13 focusing on guest I chains in the spherical I domain of SIS revealed that the guest chains having various size ⟨R2⟩1/2 (various molecular weights) exhibit nearly the same magnitude of broadening of their relaxation mode distribution. Because the effect of the spatial confinement should be enhanced on an increase of ⟨R2⟩1/2 (up to ∼half of the domain diameter13), this ⟨R2⟩1/2-insensitivity of the mode broadening of the guest chain suggests that the spatial constraint has just a secondary effect for the dynamics in the spherical I domain. The motion of I segments in the S/I interphase is retarded by the S segments therein but not necessarily fully quenched, which could also result in the retardation/broadening of the dielectric response of the I block.38 However, in the guest I/SIS system mentioned above, a fraction of the guest I chain segments in the S/I interphase should decrease with increasing ⟨R2⟩1/2 of the guest chain (because a large chain tends to avoid existing near the domain surface for an entropic reason). Nevertheless, those guest chains exhibit the ⟨R2⟩1/2-insensitive mode broadening.13 In addition, an earlier study3 showed that the dielectric mode distribution of La-forming SI copolymers is insensitive to the block molecular weight MSI despite a decease of the fraction of the I segments in the S/I interphase with increasing MSI. These results suggest that the (nonquenched) retardation of the I segment motion in the S/I interphase is not the main factor resulting in the retarded/broadened dielectric response observed for our SI systems (Figure 3). On the basis of this argument, the retardation/broadening seen for those systems is mainly attributable, with no ambiguity, to the osmotic constraint for the I block dynamics. The ε″/ϕI data in Figure 3b indicate that that the dielectric mode distribution is broader in the order of the Sph (circles) > Cyl (triangles) > La (diamonds) systems. This fact suggests that the effect of the osmotic constraint on the I block dynamics is larger in this order, which is consistent with the behavior of the guest PI chains in the Sph and La domains.13 The effect of the osmotic constraint can be naturally related to the continuity of the domains:13 The La domain is continuous in the two directions parallel to its surface, so that the I blocks can rather easily adjust their conformation to displace their segments over a distance comparable to ⟨R2⟩1/2 in these directions without violating the osmotic constraint. Those displaced segments in turn adjust the local segment density thereby helping the I blocks to change their conformations also in the perpendicular direction without violating the osmotic constraint. In contrast, this large-scale segment displacement could occur only in one direction in the Cyl domain and

the I block may increase and/or decrease according to the type of cooperative motion. Furthermore, the end-to-end fluctuation of the cooperatively moving I blocks should tend to be suppressed (and the free ends of those blocks tend to be localized) because of the osmotic constraint,4,5,8 which could result in a decrease of Δε. The I blocks are different from homo-PI chains also for these points. 3.2.2. Effect of Osmotic Constraint on I Block Dynamics in La, Cyl, and Sph Domains. Keeping the above points explained for eq 3 in our mind, we now compare the ε″/ϕI data (=εI″) for the I blocks and bulk PI chains. As noted in Figure 3, an average dielectric relaxation time τpeak defined in terms of the ε″-peak frequency ωpeak, τpeak = 1/ωpeak, is not very different for the I block in the La and Cyl domains (filled diamonds and triangles) and for the corresponding bulk PI (dashed curve). From this result, one might conclude that the terminal relaxation time τ1 (corresponding to the slowest mode of end-to-end fluctuation) is nearly the same for those I blocks and bulk PI chains. However, this is not the case at all, as noted from the rigid phenomenological framework of Boltzmann superposition (linear stimulus-response framework specified by eq 3a): As explained thoroughly in previous papers,5,13 this framework indicates that τpeak just corresponds to a group of intensive relaxation modes but does not necessarily coincide with τ1 (see Appendix of ref 13, for example) and that the proportionality, ε″ ∝ ω, should be observed in a range of ω where the full relaxation including the relaxation of the slowest mode is attained, and vice versa. (One can easily confirm this fact by utilizing a general form of the relaxation function, Φ(t) = Φ(∞) + Σp≥1 gpexp(−t/τp), to conduct the integral in eq 3a.) The terminal proportionality ε″∝ω is clearly observed for bulk PI on a decrease of ω just below ωpeak but not for the I blocks in the La and Cyl domains; see Figure 3. Thus, the I blocks in these domains do not fully relax in the range of ω examined (which corresponds to their broad relaxation mode distribution extending to long times) and the terminal relaxation of those I blocks is much slower than that of the bulk PI chain: τ1 ≫ 1/ ωpeak for the I blocks and τ1 ≅ 1/ωpeak for bulk PI. This is the case also for the I block in the Sph domain (filled circles) having an even broader mode distribution and hardly exhibiting the ε″-peak. These facts unequivocally indicate that the end motion (end-to-end fluctuation) of the I block in the La, Cyl, and Sph domains is associated with a broad mode distribution and the slowest mode is highly retarded compared to that of the bulk PI of the same molecular weight. One might argue that the broadening of the ε″ data of the I blocks (Figure 3) corresponds to emergence of a new slow process not observed for bulk PI, and that those data are to be discussed at f irst from a molecular viewpoint for this new process. However, the ε″ data in the range of ω examined are unequivocally attributed to the cooperative end motion (end-toend fluctuation) of the I blocks, and no new process is expected to emerge (unless this end motion is artificially referred to as a new process). More importantly, the molecular argument requires some assumption and is less reliable, as the f irst step, compared to the above argument based on the rigid framework of Boltzmann superposition. In fact, the ε″ data in Figure 3 do not allow us to resolve full details of the end motion because of lack of the knowledge about the eigenmodes of the osmotically constrained I blocks.36,37 Nevertheless, the broadening/ retardation of the end motion of the I block, unequivocally deduced from those data within the framework of Boltzmann 7055

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specified by eq 3b. The corresponding expression of Φ(t) of the II block of SIIS is given by4,5,8

obviously in no direction in the discrete Sph domain. These arguments suggest that the effect of the osmotic constraint is larger in the order of Sph > Cyl > La, as deduced from the ε″/ ϕI data. The effect of the osmotic constraint on the block dynamics can be examined also for the dielectric intensity, Δε (experimentally evaluated as Δε = (2/π)∫ ∞ −∞ε″(ω) dln ω). Figure 3b suggests that the normalized intensity, Δε/ϕI, decreases in the order, Sph < Cyl < La, although we cannot firmly conclude this order because the data are available only in a limited range of ω where the terminal proportionality (ε″ ∝ ω) is not observed. A small Δε/ϕI value is attributable, at least partly, to suppression of the end motion (localization of the ends), as can be noted from eq 3. Thus, the observed order of Δε/ϕI also suggests that the effect of the osmotic constraint is stronger in the order of Sph > Cyl > La. Specifically, the Sph system exhibits very small ε″ not significantly larger than the resolution of ε″ in our experiments (∼O(10−4), as explained earlier for Figure 3a). Thus, the end motion of the I blocks appears to be strongly suppressed by the osmotic constraint in the Sph domain. 3.3. Comparison of Dielectric Behavior of SI and SIIS. 3.3.1. Overview. For La-forming SIS copolymers, the bridgeand loop-type conformations of the middle I block are clearly distinguished geometrically, and the slow dynamics is similar for the loop-type I block and the tail-type I block (of SI copolymers) but is different for the bridge-type I block,4,5 as explained briefly in Introduction (and in more detail later for Figure 4). In contrast, this geometrical distinction between the

Φ(t ) =

[1] [2] [1] [2] ⟨{RII,E (t ) + RII,E (t )}{RII,E (0) + RII,E (0)}⟩eq [1] [2] ⟨{RII,E (0) + RII,E (0)}2 ⟩eq

(4)

The polarization of the II block is proportional to a sum of two [2] end-to-midpoint vectors R[1] II and RII (cf. Figure 5 shown

Figure 5. Schematic illustration of the I block conformation in (a) lamellar I domain of SIIS triblock copolymer, (b) spherical I domain of SI diblock copolymer, and (c) spherical I domain of SIIS triblock copolymer.

later), so that the component of this sum vector in the direction [2] of the electric field, R[1] II,E + RII,E, is included in eq 4. (The vectorial cancellation explained for eq 3b has been accounted in eq 4.) For the II block with its two ends fixed on the surface of the glassy S domain (at T < Tg,PS), the fluctuation of the sum vector and the corresponding relaxation of Φ are activated only by the motion of the dipole-inverted midpoint of the II block. (This midpoint motion is correlated with the motion of all segments in the II block, but this correlation is just indirectly reflected in the t dependence of Φ so that we can specify Φ only from the information for the midpoint motion, as similar to the situation explained for eq 3b.) Thus, comparison of the dielectric data of the SIIS and SI systems allows us to examine the similarity/difference between the midpoint motion of the II block of SIIS and the end motion of the I block of SI.4,5,8 This comparison is made in Figure 4 for the pair of SIIS and SI copolymers forming (a) La, (b) Cyl, and (c) Sph domains of the I blocks. (The data for SI are the same as in Figure 3a.) For each pair, the I block fraction ϕI is the same so that the ε″ data at the same temperature, 0 °C, are directly compared (without the normalization by ϕI). One might argue that the segregation power χN (with χ = interaction parameter between S and I and N = degree of polymerization) is to be set identical for SIIS and SI when their ε″ data are compared. However, the I segment friction is different at such iso-χN temperatures, and a correction is necessary for this difference. Because the ε″ data of the SIIS and SI systems obeyed the time−temperature superposition with the same shift factor (as long as the temperature was kept well below Tg,PS), the comparison at the iso-χN temperatures, after this correction made with the shift factor, is equivalent to the comparison at the same temperature, 0 °C in Figure 4. Thus, Figure 4 allows us to discuss the dielectric behavior of the I and II blocks with no ambiguity.

Figure 4. Comparison of dielectric loss ε″ data at 0 °C obtained for SI diblock copolymer and its dimer SIIS, the latter having symmetrically inverted type-A dipoles in its II block. Parts a−c show the results of comparison in systems forming the La, Cyl, and Sph I domains, respectively. The solid curves show the ε″ data of SI diblock systems multiplied by factors as indicated. For further details, see text.

bridge- and loop-type I blocks vanishes in the Cyl and/or Sph domain surrounded by a single surface (S/I boundary). It is of interest to examine whether all I blocks in the Cyl/Sph domains are dynamically similar, or, some blocks behave differently. This similarity/difference can be examined dielectrically for the dipole-inverted SIIS triblock copolymer and its half fragment (precursor) SI diblock copolymer at T < Tg,PS. The dielectric relaxation function Φ(t) of the I block of SI has been 7056

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(illustrated in Figure 5b) and in the Cyl domain (not illustrated), as similar to the conformation in the La domain. However, differing from the situation in the La domain, all II blocks of SIIS 48-(13)2-48 have their ends fixed on the same I domain surface and are geometrically classified as the loop-type blocks. From the similarity between the tail and loop in the La domain, one may naively expect coincidence of εSIIS″(ω) and εSI″(ω) for the Sph/Cyl-forming systems. Nevertheless, parts b and c of Figure 4 demonstrate differences of magnitudes of the εSIIS″(ω) and εSI″(ω) data in the Cyl and Sph domains, in particular in Cyl. We also note that those data exhibit similar mode distributions at low ω (associated with no terminal proportionality between ε″ and ω): εSIIS″(ω)≅0.65εSI″(ω) and εSIIS/S11″(ω) ≅ 0.85εSI/S11″(ω) in the Cyl and Sph domains at low ω < 104 s−1, as shown with the solid curves in Figure 4, parts b and c. These results lead us to interpret/speculate that some of the geometrically loop-type II blocks in the Sph/Cyl domain behave dynamically as pseudobridge-type blocks (as similar to the bridge-type block in the La domain). This interpretation/speculation is further examined below. 3.3.2. Loop/Pseudobridge Hypothesis in Sph and Cyl Domains. In the Sph domain, the end-to-end distance r of the II block in the Sph domain can never exceed the domain diameter dSph and has a distribution in a range of r ≤ dSph. Two half-fragments of the II block of SIIS having r well below dSph exert the tensions to the midpoint in (nearly) the same direction and would dielectrically respond similarly to the tailtype I block of SI. In contrast, for r ≅ dSph, the half-fragments should pull the midpoint in (almost) opposite directions, thereby possibly behaving as the pseudobridge to exhibit the dielectric relaxation faster and weaker compared to the tail relaxation. (The dSph value is not significantly larger than ⟨R2⟩1/2 of the II block, as explained earlier. Thus, the conformation with r ≅ dSph can be realized without difficulty.) The situation is similar for the II block in the Cyl domain. The II block would behave similarly to the tail-type I block if its end-to-end distance rφ projected on the Cyl cross-section is well below the cylinder diameter dCyl, whereas the II block with rφ ≅ dCyl would dynamically behave as the pseudobridge-type block. Thus, all II blocks in the Sph and Cyl domains are geometrically classified to be loop-type, but we can interpret/ speculate that some of them having r ≅ d and/or rφ ≅ d may be dynamically classified as pseudobridge-type. On the basis of this molecular interpretation, we propose a hypothesis that the fraction ϕps‑bridge of the dynamically pseudobridge-type II blocks is estimated as

For convenience of our discussion for the Cyl- and Sphforming copolymers, we start with the comparison of previously examined La-forming SI 12-12 and SIIS 12-(12)2-12 copolymers;4 see Figure 4a. The ω dependence of ε″ at low ω, showing no terminal proportionality even at ω well below ωpeak because of the osmotic constraint, is very similar for these SI and SIIS copolymers but the magnitude of ε″ is smaller for the latter: εSIIS″(ω) ≅ 0.6εSI″(ω) at ω < 103 s−1, as shown with the solid curve. The dipole inversion itself does not affect the magnitude of ε″ and the dielectric intensity Δε, as deduced from bead−spring analysis39 and also confirmed experimentally for a series of dipole-inverted homo PI.36b−e Thus, the coincidence of the ω dependence of ε″ and the difference in the magnitude of ε″ seen for the SI and SIIS copolymers has been related to the loop/bridge distribution for the II block of SIIS,4,5 as explained below. The dielectric relaxation of the II block detects its midpoint motion (cf. eq 4). For the loop-type II block illustrated in Figure 5a, two half-fragments, each corresponding to the I block of SI (precursor of SIIS), always exert the thermal tension to the midpoint in the same direction and thus the elastic force acting on the midpoint is twice of the force acting on the end of the tail-type I block of SI. The friction against the motion driven by this tension is also larger, by the factor of 2, for the loop-type II block than for the tail-type I block (because each half-segment of the II block is frictionally equivalent to the tailtype I block). Thus, the midpoint motion of the loop-type II block would be similar to the end motion of the tail-type I block even under the very strong osmotic constraint. (This similarity has been confirmed for the dielectric behavior of blends of SIIS and SI copolymers.9) The friction for the bridgetype II block is identical to that for the loop-type II block. However, the midpoint of the bridge-type II block is pulled by the two fragments in the opposite directions (cf. Figure 5a) and tends to be localized at around the middle plane in the lamellar domain because of the osmotic constraint. These molecular arguments suggest that the dielectric response of the loop-type II block is similar to the response of two tail-type I blocks but the response of the bridge-type II block is faster and weaker compared to the tail response. Then, at low ω, we expect that εSIIS″(ω) and εSI″(ω) exhibit the same relaxation mode distribution (as observed) and that the εSIIS″(ω)/εSI″(ω) ratio is equal to the loop fraction for the II block thereby giving the bridge fraction as ϕbridge = 1 − [εSIIS″(ω)/εSI″(ω)]low ω. The data in Figure 4a suggest ϕbridge ≅ 0.4 for SIIS 12-(12)2-12, which is close to ϕbridge deduced from the self-consistent field calculation.18,19 In relation to the above argument, we should emphasize that the dipole inversion in the II block allowed the SIIS copolymer to exhibit a strong dielectric response at low T ( φ*specifying the dynamically pseudobridge-type blocks) as ϕps‑bridge = ∫ πφ*Φφ(φ) dφ = ∫ πφ*exp(α cos φ) dφ/∫ π0exp(α cos φ) dφ. Utilizing the values of ϕps‑bridge (=0.35 in Cyl domain; eq 7), dCyl (≅18 nm evaluated from TEM images; eq 1a), and ⟨R2⟩1/2 (≅13 nm) in this relationship and making a numerical calculation, we find φ* ≅ 0.93 rad. This φ* value seems to be reasonable as the threshold, as suggested from an argument for f1·f2 similar to that for Sph (f1·f2 > 0 for φ < φ*). Thus, our crude interpretation/speculation appears to be acceptable also for the Cyl domain, although it does not consider orientational distribution of the Cyl grains and a refinement is to be made first for this point. Here, we should emphasize that the above arguments for θ* and φ* are based on the crude approximation of the elastic free energy Fel(r). Obviously, a further quantitative study is desired for better understanding of the loop/bridge problem. Nevertheless, the hypothesis about the dynamically pseudobridgetype II blocks, proposed in this work for the first time, is believed to be physically acceptable, as suggested from those arguments. We also emphasize that the bridge fraction ϕbridge (in La) and the pseudobridge fraction ϕps‑bridge (in Sph/Cyl) were estimated on the basis of the molecular argument of the dynamic similarity between the tail-type and loop-type blocks.4,5 This argument, though sounds reasonable as suggested from experiments for SI/SIIS blends,9 needs to be further examined for better understanding of the block dynamics.40

This expression is a crude approximation because it considers neither the spatial confinement nor the osmotic constraint, but it still serves as a good starting point of our argument. In the polar coordinate system with the azimuthal angle θ being measured from the position of one end of the II block (cf. Figure 5c), we find r2 = d2Sph(1 − cosθ)/2. The corresponding distribution function of θ, ψSph ∝ exp(−Fel/kBT), is expressed in a form including the normalization constant: α exp(α cos θ ) for 0 ≤ θ ≤ π ΨSph(θ ) = α (8a) e − e −α with α=

3dSph 2 4⟨R2⟩

(8b)

Following the above molecular interpretation/speculation, we may regard the II blocks having θ larger than a threshold value θ* (giving r close to dSph) to be dynamically pseudobridge-type. Then, the fraction ϕps‑bridge of those blocks can be related to ΨSph as π

ϕps ‐ bride =

α cos θ *

−α

∫θ* ΨSph(θ) sin θ dθ = e eα − −e−eα

(9)

Utilizing the values of ϕps‑bridge (≅ 0.15 in Sph; eq 6), dSph (≅ 17 nm evaluated from TEM images; eq 2a), and 1/2 (≅ 13 nm) in eq 9, we find θ* ≅ 1.8 rad. The midpoint of the II block would be mostly located near the Sph center, and θ is related to the tensions f1 and f2 from the half fragments as cos θ ≅ f1·f2/f1 f 2 with f i = |fi|. We note that f1·f2 > 0 (midpoint pulled by those fragments in nearly the same direction) for θ well below θ* ≅ 1.8 rad (≅ 1.1π/2) and f1·f2 < 0 (midpoint pulled in nearly the opposite directions) for θ > θ*. Thus, θ* appears to reasonably separate the II block conformations into the loop- and pseudobridge-type conformations. Analysis for Cyl. For the Cyl domain, we may make a similar analysis on the basis of the same approximation for Fel(r). In a cylindrical coordinate system (z,φ) with the height z and the longitudinal angle φ being measured from the position of one end of the II block, we find r2 = z2 + rφ2 with rφ = dCyl{(1 − cos φ)/2}1/2 being the end-to-end distance projected on the Cyl cross-section. Correspondingly, the distribution function of (z,φ) can be expressed as ΨCyl(z , φ) = Φz (z)Φφ(φ)

4. CONCLUDING REMARKS We have examined the dielectric response of SI diblock and SIIS triblock copolymers, the latter being the head-to-head dimer of the former and having the dipole inversion at the midpoint of the II block. The observed response reflects the end motion of the I block (for SI) and the midpoint motion of the II block (for SIIS). These copolymers formed nanodomain structures, and their slow dielectric relaxation was found to be broader and retarded compared to bulk PI, mainly because of the osmotic constraint on the block motion. Comparison of the dielectric mode distribution of the I blocks in the La, Cyl, and Sph nanodomains suggested that the effect of the osmotic constraint on the block dynamics is larger in the order of Sph > Cyl > La. This order is in harmony with the domain continuity that helps the large-scale motion of the I block without violating the osmotic constraint. In the spherical and/or cylindrical domain, all II blocks of the SIIS copolymer have their ends fixed on the surface of the same domain and are geometrically classified to be loop-type. Nevertheless, comparison of the dielectric data of the SIIS and SI copolymers suggested that some fraction of the II blocks dynamically behaves as pseudobridge-type blocks. This behavior of the II blocks was interpreted to be related to their end-toend distance that governs the direction of II block tension acting on the midpoint of this block thereby affecting the dielectrically detected midpoint motion. The II block appears to exhibit the pseudobridge-type behavior when it has a large end-to-end distance and its two half fragments pull the midpoint in the opposite directions. This argument is crude/ speculative but seems to be still acceptable in a physical sense, as suggested from an approximate analysis of the distribution of the end-to-end distance.

(10a)

with ⎛ π ⟨R2⟩ ⎞−1/2 ⎛ 3z 2 ⎞ exp⎜ − Φz (z) = ⎜ ⎟ ⎟ for 0 < z < ∞ ⎝ 2⟨R2⟩ ⎠ ⎝ 6 ⎠ (10b)

and Φφ(φ) =

exp(α cos φ) π

∫0 exp(α cos φ) dφ

for 0 ≤ φ ≤ π (10c)

α is given by eq 8b with dSph therein being replaced by dCyl of the Cyl domain. Because ΨCyl(z,φ) is factorized into Φz(z) and Φφ(φ), with the former being identical to the distribution function in a free space, we may focus on Φφ(φ), as the crudest approximation, to examine the fraction ϕps‑bridge of the dynamically pseudobridge-type II blocks. Specifically, this 7058

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(15) Watanabe, H.; Sato, T.; Osaki, K. Macromolecules 2000, 33, 2545. (16) Watanabe, H.; Matsumiya, Y.; Sawada, T.; Iwamoto, T. Macromolecules 2007, 40, 6885. (17) Zhulina, E. B.; Halperin, A. Macromolecules 1992, 25, 5730. (18) Matsen, M. W. J. Chem. Phys. 1995, 102, 3884. (19) Matsen, M. W.; Schick, M. Macromolecules 1994, 27, 187. (20) Takano, A.; Kamaya, I.; Takahashi, Y.; Matsushita, Y. Macromolecules 2005, 38, 9718. (21) Li, B. Q.; Ruckenstein, E. Macromol. Theory Simul. 1998, 7, 333. (22) Yao, M.-L.; Watanabe, H.; Adachi, K.; Kotaka, T. Macromolecules 1991, 24, 2955. (23) Chen, Q.; Matsumiya, Y.; Masubuchi, Y.; Watanabe, H.; Inoue, T. Macromolecules 2011, 44, 1585. (24) Inoue, T.; Soen, T.; Hashimoto, T.; Kawai, H. J. Polym. Sci., Part A-2 1969, 7, 1283. (25) Arai, K.; UedaMashima, C.; Kotaka, T.; Yoshimura, K.; Murayama, K. Polymer 1984, 25, 230. (26) Hashimoto, T.; Fujimura, M.; Kawai, H. Macromolecules 1980, 13, 1660. (27) Hashimoto, T., In Polymer Alloys, 1st ed.; Kotaka, T., Ide, F., Ogino, K., Eds.; Kagaku Dojin: Tokyo, 1981. (28) Watanabe, H. Acta Polym. 1997, 48, 215. (29) Hashimoto, T.; Shibayama, M.; Kawai, H.; Watanabe, H.; Kotaka, T. Macromolecules 1983, 16, 361. (30) Fetters, L. J.; Lohse, D. J.; Colby, R. H. Chain Dimensions and Entanglement Spacings. In Physical Properties of Polymers Handbook, 2nd ed.; Mark, J. E., Ed.; Springer: New York, 2007; Chapter 25. (31) Chen, Q.; Matsumiya, Y.; Masubuchi, Y.; Watanabe, H.; Inoue, T. Macromolecules 2008, 41, 8694. (32) Wagner, K. W. Arch. Elektrotech. 1914, 2, 371. (33) Sillars, R. W. J. Inst. Electr. Eng. 1937, 80, 378. (34) Steeman, P. A. M.; van Turnhout, J. Dielectric Properties of Inhomogeneous Media. In Broadband Dielectric Spectroscopy; Kremer, F., Schönhals, A., Eds.; Springer: Berlin, 2003; Chapter 13. (35) Schönhals, A.; Kremer, F. Theory of Dielectric Relaxation. In Broadband Dielectric Spectroscopy; Kremer, F., Schönhals, A., Eds.; Springer: Berlin, 2003; Chapter 1. (36) (a) For dielectrically resolving the chain dynamics (motion of all segments in the chain), we need to have some sort of dielectric label. Inversion of the type-A dipoles serves as this dielectric label, as demonstrated for a series of homo-PI chains having either symmetrically or asymmetrically inverted dipoles.36b−e Thus, it is possible to resolve the chain dynamics with the dielectric method. However, this is not the case if we just focus on the dielectric data of dipole noninverted chains (that includes the I block of SI). (b) Watanabe, H.; Urakawa, O.; Kotaka, T. Macromolecules 1993, 26, 5073. (c) Watanabe, H.; Urakawa, O.; Kotaka, T. Macromolecules 1994, 27, 3525. (d) Watanabe, H.; Yamada, H.; Urakawa, O. Macromolecules 1995, 28, 6443. (e) Watanabe, H. Polym. J. 2009, 41, 929. (37) (a) It might look tempting to analyze the dielectric behavior of the I blocks on the basis of the primitive relaxation modes of beadspring chains,7,37b,c Φbs(t) ∼ Σp=oddp−2 exp(−tp2/τ1) for free linear chains and tethered chains. These relaxation modes result from the sinusoidal eigenfunctions associating to the equation of motion of the bead-spring chains.5,37c However, experiments utilizing dipole-inverted linear homo-PI chains36b−e demonstrated that the actual eigenfunctions of those chains are not sinusoidal (unless at infinite dilution) and thus Φbs(t) does not apply even to non-entangled bulk PI chains. More importantly, the osmotic constraint on the block dynamics is not incorporated in the bead-spring model giving the Φbs(t) shown above, which rules out the use of Φbs(t) in the analysis of the I block dynamics. The eigenmodes of such osmotically constrained I block have not been derived by now, but the simple observation described in the text (broadening/retardation due to the osmotic constraint) is sufficient for our discussion about the difference of the I block dynamics in La, Cyl, and Sph domains. (b) Boese, D.; Kremer, F. Macromolecules 1990, 23, 829. (c) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address †

Department of Materials Science & Engineering, The Pennsylvania State University, University Park, PA 16802. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors appreciate kind help for SAXS measurements by Professor K. Nishida and Professor T. Kanaya at ICR, Kyoto University. This work was supported by the Grant-in-Aid for Scientific Research (B) (Grant No. 21350063), Grant-in-Aid for Scientific Research (A) (Grant No. 24245045), and Grantin-Aid for Young Scientists (B) from MEXT (Grant No. 22750204) and partly by Collaborative Research Program of ICR, Kyoto University (Grant No. 2012-35).

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REFERENCES

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(38) (a) Alig, I.; Floudas, G.; Avgeropoulos, A.; Hadjichristidis, N. Macromolecules 1997, 30, 5004. (b) Floudas, G.; Alig, I.; Avgeropoulos, A.; Hadjichristidis, N. J. Non-Cryst. Solids 1988, 235−237, 485. (39) For barely entangled I and II blocks, the former having the fixed and free ends and the latter having two fixed ends and symmetrical dipole inversion, the bead-spring model gives exactly the same ε″ that exhibits the terminal proportionality (ε″ ∝ ω) at ω immediately below ωpeak.8,11 Incorporation of the spatial confinement in this model accelerates the fast relaxation but hardly affect the terminal relaxation time (≅1/ωpeak).8 Thus, the weak (nonterminal) ω dependence of ε″ observed for SI and SIIS is mainly attributable to the osmotic constraint, and the difference of the magnitudes of the εSI″ and εSIIS″ data at low ω data can be related to the difference of the loop and bridge behavior under this constraint.4,5,8 (40) (a) ϕbridge and ϕps‑bridge were estimated from the dielectric data at ω where the terminal proportionality (ε″ ∝ ω) has not been attained. Thus, the real slowest mode of motion, which should be affected by knots between the loops, is not fully reflected in the estimates of ϕbridge and ϕps‑bridge and thus these estimates include the knotted loops, as discussed previously.4,8 It is an interesting subject of future work to estimate the fractions of the dangling and knotted loops as well as knotted bridges separately and examine properties of respective species.40b,c (b) Hong, L.; Jin, F.; Li, J.; Lu, Y.; Wu, C. Macromolecules 2008, 41, 8220. (c) Tan, W. S.; Zhu, Z.; Sukhishvili, S. A.; Rubner, M.; Cohen, R. E. Macromolecules 2011, 44, 7767.

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