Dielectric Behavior of Guest cis-Polyisoprene Confined in Spherical

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Dielectric Behavior of Guest cis-Polyisoprene Confined in Spherical Microdomain of Triblock Copolymer. Quan Chen, Yumi Matsumiya, Tatsuya Iwamoto, Koji Nishida, Toshiji Kanaya, 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 ternary blends of guest cis-polyisoprene (PI) chains, coguest PS chains, and host PS−PI−PS triblock copolymers. The guest PI chains had the molecular weight Mg‑PI well below Mb‑PI of the PI blocks and were mixed in the spherical PI domains formed in the copolymers. PI has the type-A dipoles parallel along the chain backbone and its end-to-end vector fluctuation activates dielectric relaxation. The dielectric test was conducted at temperatures well below the glass transition temperature of the PS blocks, so that the PI blocks had their ends fixed on the glassy PS matrix and exhibited no end-to-end vector fluctuation. Thus, the slow dielectric response of the systems exclusively detected the global motion of the guest PI chains confined in the spherical PI domain. The dielectric mode distribution of the guest PI chains thus observed was broader than that of the same chains in bulk systems, reflecting the osmotic and spatial constraints on the chain dynamics in the spherical domains. The guest PI chains of different Mg‑PI exhibited nearly the same extent of mode broadening, despite their average coil size approached the domain size with increasing Mg‑PI. This result suggested that the osmotic constraint was the primary factor affecting (broadening and retarding) the guest chain dynamics. The osmotic constraint appeared to be stronger in the spherical domain than in the lamellar domain, the latter being examined in literature.

1. INTRODUCTION Block copolymers exhibit microphase separated domain structures, and the thermodynamic driving force for this structure formation is rather well understood:1,2 Namely, the covalently connected but chemically different blocks segregate in space as much as they can, while satisfying the thermodynamic requirements of randomizing the location of the block junctions, randomizing the block conformation, and uniformly filling the domains with the constituent segments. These requirements, in particular the last two, are contradicting to each other because the uniform domain-filling requires neighboring blocks to have correlated (and not fully randomized) conformations, and compromise of those requirements determines the size and shape of the microdomains.1−3 The requirements explained above result in conformational correlation of the blocks in the microdomain and should affect (retard) the large-scale dynamics of the blocks. Dielectric measurements for copolymers consisting of cis-polyisoprene (PI) and polystyrene (PS) blocks have been proven to be very useful for examination of this effect(s). PI has the so-called type-A dipole parallel along the chain backbone, while PS does not.4,5 Thus, the slow dielectric responses experimentally detected for the copolymer can be exclusively related to the end-to-end vector fluctuation of the PI block (having no inversion of type-A © 2012 American Chemical Society

dipoles), in particular at low temperatures (T) well below the glass transition temperature Tg,PS of the PS blocks where the PS segmental motion is quenched (undetectably slow). For monodisperse linear PI, extensive studies5−7 revealed that the dielectric mode distribution in bulk systems hardly changes with the molecular weight M (although slight mode narrowing is observed for high-M PI8) and that the mode distribution narrows moderately in semidilute and dilute solutions. For all these cases, the linear PI chains exhibit narrow distribution of the terminal dielectric relaxation modes characterized by a sharp peak of the dielectric loss ε″ immediately followed by a lowfrequency (low-ω) part exhibiting the proportionality, ε″ ∝ ω. In contrast, in bulk PS−PI diblock copolymer forming lamellar domains, the PI block does not exhibit this low-ω proportionality but shows the mode distribution much broader compared to linear PI chains of the same M,9 which indicates that the terminal dielectric relaxation time ⟨τ⟩w (=second-moment average relaxation time defined in terms of the relaxation spectrum5,6) is much longer for the PI block than for the linear PI chains. This enormous retardation/broadening of the PI block dynamics can be Received: January 22, 2012 Revised: March 3, 2012 Published: March 16, 2012 2809

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related to constraints for the PI block dynamics,5 the so-called spatial constraint due to conformational restriction within the microdomain (that reduces possible dynamic paths for conformational changes) and the osmotic constraint due to the very strong requirement of uniform domain-filling (that forces the blocks to move cooperatively). Although the spatial constraint has some effect on the mode broadening,10−12 the osmotic constraint appears to dominate the retardation/broadening, as suggested from experimental observation that the mode distribution of PS−PI copolymers narrows on moderate dilution with a PI-selective solvent13 and that the mode distribution is narrower for guest PI chains in the lamellar domain than for the PI block therein.5,9 The osmotic and spatial constraints, affecting the chain dynamics in the block copolymer domain, would change their magnitudes with the domain shape: These constraints are expected to affect the chain dynamics significantly in the discrete, spherical domain than in the continuous, lamellar domain. We may attempt to examine this expectation through dielectric tests. Specifically, we can focus on PS−PI−PS triblock copolymers having guest PI chains in the PI domain thereby distinguishing the effects of the two constraints on the chain dynamics: At low T ( RPI from the surface of the spherical PI domain (boundary to the glassy PS matrix), suggesting that a large fraction of those chains are spatially constrained. In contrast, for low-M guest PI chains having the RPI/dPI ratio as small as 0.1, this constraint would be very weak.

where ϕcore is the volume fraction of the core. Since the spatial distribution of the PI domains in our PI/ SIS/PS blends is highly disordered (cf. Figures 1 and 2), we utilized eq 2 to estimate dPI (=dcore in eq 2) from the q1 data (Figure 3) and the known volume fraction Φ of the SIS copolymer plus guest-PI after the minor correction with eq 1. (The guest PI is included in the PI domains thereby contributing to Φ.) The dPI in the blends thus estimated from the SAXS data is shown with the squares in Figure 4a. For comparison, dPI estimated from the SAXS data for the neat copolymers are shown with the horizontal solid arrows. (For the neat copolymer, the q*/q1 ratio was set to be unity because Φ = 1.) As seen in Figure 4a, the estimates of dPI obtained from the TEM images and SAXS data are in reasonable agreement within experimental uncertainties. These dPI values decrease 2813

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not covered in our experimental window. For these reasons, the neat copolymer exhibits negligible dielectric relaxation in the experimental window examined in Figure 6. In contrast, the blend clearly exhibits slow dielectric relaxation as noted from the ε″ data shown with squares in Figure 6b, although its ε′ data (Figure 6a) are close to the data for the neat copolymer (because the majority component in the blend and copolymer is the dielectrically inert PS). This relaxation, observed for all ternary blends examined, is exclusively attributable to the end-to-end vector fluctuation of the guest PI chains having the type-A dipoles and moving in the spherical PI domain. These guest chains have just a small volume fraction in the blend, ϕg‑PI = 0.056, but their dielectric loss can be clearly resolved because of the lack of the loss of the S blocks/coguest PS chains and the I blocks. Here, we attempt to examine the dielectric behavior of the PI chains (I block and guest PI) in the spherical PI domain. For this purpose, we can utilize the Maxwell−Wagner (MW) equation24,25 for the complex dielectric permittivity ε*(ω) (= ε′(ω)−iε″(ω)) of dilute dispersions of spheres,

Thus, in summary, the spatial constraint for the guest PI chains examined in this study should be considerably enhanced on an increase of Mg‑PI that results in an increase of the RPI/dPI ratio from 0.1 to 0.4 (and more; see Figure 4b). If the spatial constraint is the main factor governing the global dynamics of the guest PI chains, their dielectric relaxation should be more significantly broadened and retarded (compared to the relaxation in the bulk state) with increasing Mg‑PI. We keep this point in our mind to examine the dielectric behavior of the guest PI chains in the next section. 3.3.1. Dielectric Behavior. Overview. Figure 6 shows the dynamic dielectric constant ε′(ω) and dielectric loss ε″(ω)

ε*(ω) = ε*m (ω)

{2 − 2ϕ}ε*m (ω) + {1 + 2ϕ}ε*d (ω) {2 + ϕ}ε*m (ω) + {1 − ϕ}ε*d (ω)

for small ϕ (≤0.1)

(4)

where ε m *(ω) and ε d *(ω), respectively, indicate the permittivity of the medium and dispersed spheres at the angular frequency ω. For our blends, ε*(ω), εd*(ω), and εm*(ω) are equivalent to εb*(ω) of the blend as a whole, εI*(ω) of the PI components in the spherical domain, and εS*(ω) (=εglassy‑S′ = 2.64) of the glassy S chains in the matrix, respectively, and ϕ = ϕtotal‑PI = 0.112. Thus, eq 4 allowed us to evaluate εI*(ω) from the εb*(ω) and εglassy‑S′ data. The results are shown in Figure 6 with the circles. (We also utilized the Hanai equation26 for concentrated dispersions, but the results were very close to those shown in Figure 6 because our blend had just small ϕtotal‑PI.) For the εb* data shown in Figure 6, one might assume, instead of the MW equation (eq 4), a simple blending law for the parallel model: Figure 6. Dielectric data of I-9/SIS 131−57−133/S-11 blends (square) and neat SIS 131−57−133 copolymer (plus symbol) measured at 0 °C. The εI* of the spherical PI domain obtained from the Maxwell−Wagner analysis are shown with the circles. For further details, see text.

ε b′(ω) = ϕtotal ‐ PIεI′(ω) + (1 − ϕtotal ‐ PI)εglassy ‐ S′

(5a)

ε b″(ω) = ϕtotal ‐ PIε I″(ω)

(5b)

(εglassy ‐ S″ = 0)

We calculated εb* from eq 5, with εI′(ω) and εI″(ω) therein being obtained from eq 4 (circles in Figure 6). The calculated εb*, shown in Figure 6 with the triangles, are close to the εb* data (squares), indicating an approximate but satisfactory validity of the simple parallel model for our blends. In fact, eq 5 can describe εb* within acceptably small errors for copolymers systems having either spherical or nonspherical PI domains, given that the PS phase is glassy and exhibits no dielectric loss. Nevertheless, at high T where the PS matrix exhibits considerable dielectric loss due to the segmental relaxation, we confirmed that eqs 4 and 5 give considerably different εb* values for the same set of εI* and εS*. Thus, in general, the use of eq 4 is desired for evaluation of εI*(ω) of the PI components in the spherical domain. For this reason, we utilized eq 4 to evaluate εI*(ω) for all blends examined. Those εI*(ω) data are examined below to discuss the guest chain dynamics in the spherical PI domains.

measured for the I-9/SIS 131−57−133/S-11 ternary blend at 0 °C (squares). These data are double−logarithmically plotted against the angular frequency ω (squares). For comparison, the data obtained for the neat SIS 131−57−133 copolymer are also shown (plus symbols). This neat copolymer exhibits negligibly small ε″ and ω-insensitive ε′, which reflects the microdomain structure illustrated in Figure 5: Namely, the S blocks and coguest PS chains are in the glassy state at 0 °C to exhibit no dielectric (segmental) relaxation. In contrast, the I block is in the rubbery state at 0 °C and exhibits active thermal motion (at a length scale smaller than the end-to-end distance). However, both ends of the I block are fixed at the boundary to the glassy PS matrix, so that the dielectric relaxation due to the noninverted type-A dipoles (being activated only when the end-toend vector fluctuates) is quenched.23 In addition, the segmental relaxation of the I blocks occurs at high ω (>107 s−1 at 0 °C4) 2814

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3.3.2. Guest PI Dynamics in Spherical PI Domain. The top panel of Figure 7 compares the εI″(ω) data obtained for all ternary blends (containing various guest PI chains; cf. Table 2)

In the spherical domain, the guest PI chains are subjected to a very strong requirement of keeping the constant segmental density in the domain. This osmotic requirement forces the guest PI chains to move cooperatively with the PI blocks therein, which leads to the retardation/broadening of the dielectric relaxation of the guest PI chains explained above.5,9,10 In addition, the guest PI chains are not allowed to escape the spherical domain, and this spatial constraint could also affect the relaxation of the guest PI chains.5,10 Effects of these osmotic and spatial constraints on the guest PI motion/dielectric relaxation are further examine below. Since the guest PI chains have the volume fraction υI = ϕg‑PI/ ϕtotal‑PI = 0.5 in the spherical domain, the dielectric loss of those chains normalized to unit volume is given by εI″/υI. In Figure 8,

Figure 7. Top panel: εI″ data of the guest PI chains in spherical PI domains in the blends as indicated. Bottom panel: εI″ data of the guest PI chains in respective bulk state.

with the aid of eq 4. These εI″(ω) data are exclusively attributed to the end-to-end vector fluctuation of the guest chains, because the I block having the fixed ends exhibits no dielectric relaxation, as explained earlier. As a reference, the εI″(ω) data of the guest PI chains in respective bulk state are shown in the bottom panel. The data were measured and compared at 0 °C, except for the two high-M guest chains; the data in the blend and bulk for I-19 and I-35 were measured at Texp = 40 and 60 °C, respectively, and compared at 0 °C after the timefrequency shift from those Texp to 0 °C: The ω-shift factor aT evaluated for the guest PI chains in bulk was applied also to those chains in the PI domain. The εI″(ω) data of those high-M guest PI chains were also subjected to a minor correction for the relaxation intensity (∝1/T with T being the absolute temperature). Figure 7 clearly indicates that the dielectric relaxation of the guest PI chain, either in the spherical PI domain or in bulk, becomes slower with increasing Mg‑PI because the relaxation reflects the global motion at the end-to-end length scale. We also note that the angular frequency ωpeak at the εI″-peak of each guest PI is a little higher in the spherical domain than in bulk. From this small difference of ωpeak, one might consider that the global motion of the guest PI chain is slightly accelerated in the spherical domain. However, this is not the case at all: All PI chains in bulk clearly exhibit the proportionality between εI″ and ω at low ω that characterizes the full relaxation in our experimental window, whereas this proportionality is observed for none of those chains in the spherical domain. Thus, the global motion of the guest PI chains is highly retarded (and its mode distribution is broadened) in the spherical domains compared to respective bulk systems, as explained in more detail in Appendix A in relation to the general, phenomenological framework of dielectric relaxation: For example, the I-3 chain has the second-moment average relaxation time (terminal relaxation time) ⟨τ⟩w = 4.0 × 10−5 s and ⟨τ⟩w > 10−3 s in the bulk state and blend, respectively, and thus its relaxation is much slower in the blend; see Appendix A.

Figure 8. Comparison of dielectric mode distribution for guest PI chains in spherical domains in PI/SIS/S-11 blends, guest PI chains in a lamellar domain, and bulk PI.

the εI″/υI data of the guest PI chains in the spherical domains are double−logarithmically plotted against a frequency reduced at the εI″ peak, ω/ωpeak; see symbols. Those plots are indistinguishable for the guest PI chains having various Mg‑PI (and being subjected to the spatial constraint of various magnitudes, as explained earlier). For comparison, the plots are shown also for those chains in respective bulk state (where υI = 1); see the dashed curve. These bulk chains have quite M-insensitive dielectric mode distribution, as explained in more detail in Appendix B. In literature,9 the εI″/υI data at 0 °C are available for guest PI chains (with 10−3Mg‑PI = 4.5 and 9.8) confined in lamellar polybutadiene (B) domains formed by a styrene−butadiene (SB) copolymer having 10−3MS = 8.6 and 10−3MB = 8.5, with MB being comparable to Mg‑PI. Those εI″/υI data, insensitive to Mg‑PI and ϕg‑PI (in the range of ϕg‑PI ≤ 0.1), are also shown in Figure 8; see the thick solid curve. In Figure 8, we first confirm that the slow dielectric relaxation of the guest PI chains is much broader in the spherical domain than in bulk and no terminal proportionality (ε″ ∝ ω) is observed in the spherical domain. This broad mode distribution is characterized with an empirical fractional powerlaw, ε″ ∝ ωα with α ≅ 1/2, seen at ω < ωpeak. We also note that the dielectric mode distribution of the guest PI in the spherical domains is insensitive to Mg‑PI. (This Mg‑PI-insensitivity is noted also in lamellar domains examined in the previous study.9) Since the spatial constraint for those chains should be enhanced considerably on an increase of Mg‑PI (i.e., on an increase of the chain/domain size ratio, RPI/dPI, from 0.1 to 0.4 and more, as discussed for Figure 4b), the Mg‑PI-insensitive mode distribution suggests that the global motion of the guest PI chains is dominantly affected by the osmotic constraint and the 2815

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spherical domain irrespective of their molecular weights Mg‑PI. These results suggest that the global motion of the guest PI chains in the spherical domains is dominantly affected by the osmotic constraint (requiring uniform filling of the domain with the segments) so that the guest chains are forced to move highly cooperatively with the I blocks: The spatial confinement of the guest chains in the spherical domain gives just a secondary effect for the guest chain dynamics. Furthermore, comparison of the dielectric mode distribution of the guest chains in the spherical and lamellar domains suggested that the osmotic constraint is stronger in the spherical domain: In the lamellar domain, the guest PI chain would move rather freely over a large length scale in the direction parallel to the lamellar surface, and this motion results in adjustment of the local segment density on motion of the other guest chains in the direction normal to the lamellar surface, thereby possibly reducing the osmotic constraint for the motion also in the normal direction. The lack of this mechanisms in the discontinuous spherical domain possibly led to the stronger constraint and the broader dielectric mode distribution.

spatial constraint has just a secondary effect (at least in the range of RPI/dPI examined). In relation to this osmotic constraint, it should be noted that even the lowest-M and highest-M guest chains, I-3 and I-35, exhibit similarly broad dielectric mode distribution: Since the I3 molecular weight (= 3.0 × 103) is well below the entanglement molecular weight (Me ≅ 5.0 × 103 for PI22), the I-3 chains in the bulk state essentially obey the Rouse dynamics6 free from the entanglement effect. In contrast, the I-35 chains are rather well entangled to exhibit the reptation-like dynamics6 in the bulk state. Thus, the coincidence of the broad dielectric mode distribution for the I-3 and I-35 (and all other) chains, characterized with the power-law type behavior (ε″ ∝ ωα with α ≅ 1/2) at low ω, suggests that the osmotic constraint in the spherical domain affects the motion of the guest chains at a length scale ξ well below the entanglement length a (=5.8 nm for PI22) and this effect is accumulated in a self-similar way (irrespective of the entanglement) up to the chain size RPI. Detail of this accumulation is an interesting subject of future work. Finally, we note that the dielectric mode distribution at high ω > ωpeak is nearly the same for the guest PI chains confined in the spherical and lamellar domains (and for the chains in bulk), but the distribution at low ω < ωpeak is a little broader in the spherical domain; compare the symbols and thick solid curve Figure 8. This result suggests that the osmotic constraint is stronger in the spherical domain than in the lamellar domain, possibly because the lamellar domain is continuous and has no wall (domain boundary) perpendicular to the lamellar surface. This lack of the wall would allow the guest PI chain to move and transfer its segments over a large length scale in the direction parallel to the lamellar surface without feeling a strong osmotic constraint. Furthermore, this segment transfer should help adjustment of the local segment density on motion of the other guest chains in the direction normal to the lamellar surface, thereby possibly reducing the osmotic constraint for the motion also in the normal direction. In contrast, in the spherical domains, the guest chain feels the wall in all directions at a distance within the domain diameter, and the segment transfer is always subjected to the strong osmotic constraint. This difference between the lamellar and spherical domains could have resulted in the observed difference of the slow dielectric mode distribution of the guest chains in these domains. As explained above, we may naturally explain the difference of the guest dynamics in the lamellar and spherical domains in relation to the continuity of the domain. However, this explanation is just qualitative, and no quantitative analysis of the difference has been made. This quantitative analysis deserves further studies.

■

APPENDIX A. PHENOMENOLOGICAL FRAMEWORK OF DIELECTRIC RELAXATION A reviewer for this paper suspected that the dielectric relaxation of the guest PI chains is not significantly retarded in the spherical I domain because the angular frequency of the ε″ peak, ωpeak, is not very different for the PI chains in those domains and in bulk state. However, we can unequivocally conclude the significant retardation on the basis of the general, phenomenological framework of dielectric relaxation (in the linear stimulus-response regime), as explained below. It is well-known that the decrease of dynamic dielectric constant from its static value, Δε′(ω)  ε′(0) − ε′(ω), and the dielectric loss, ε″(ω), are related to the normalized dielectric relaxation function Φ(t) (=1 at time t = 0) and the dielectric relaxation intensity Δε as5−7 Δε′(ω) = ωΔε

ε″(ω) = ωΔε

∫0

∫0

∞

∞

Φ(t ) sin ωt dt

Φ(t ) cos ωt dt

(A1)

(A2)

For the slow dielectric relaxation of the guest PI chains having noninverted type-A dipoles, Φ(t) is expressed as the autocorrelation function of the end-to-end vector R:5−7 Φ(t ) =

4. CONCLUDING REMARKS For ternary blends of PS−PI−PS triblock copolymer, guest PI chains, and coguest PS chains, we have examined the dielectric behavior at low T ( τ2 > τ3 ... (A4)

Here, gp and τp are the intensity and relaxation time of pth relaxation mode, with the first (slowest) mode corresponding to the motion of the backbone of the chain as a whole. (gp satisfies a relationship, ∑p≥1 gp = 1, which corresponds to the initial condition, Φ(0) =1.) 2816

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As clearly noted from eqs A1, A2, and A4, Δε′(ω) and ε″(ω) are expressed in terms of the relaxation times and intensities of the modes:

∑

Δε′(ω) = Δε

gp

p≥1

ε″(ω) = Δε

∑

ω2τp2 1 + ω2 τp2 ωτp

gp

p≥1

Monodisperse linear PI chains in bulk state have a narrow distribution of the relaxation modes (cf. Appendix B), so that their dielectric ⟨τ⟩n and ⟨τ⟩w are rather close to each other (⟨τ⟩w ≅ 2⟨τ⟩n) and a relaxation time defined in terms of the ε″peak frequency, τpeak = 1/ωpeak, has a value between the ⟨τ⟩n and ⟨τ⟩w values. 5−7 Thus, for the PI chains utilized in this study, we may safely utilize τpeak evaluated in Figure 7 (instead of ⟨τ⟩w) as the dielectric terminal relaxation time in the bulk state. In contrast, for the guest PI chains confined in the spherical domains, the terminal proportionality, ε″(ω) ∝ ω (eq A6), is not observed in our experimental window. This fact enables us to conclude a relationship for the chains in those domains, ⟨τ⟩w > ⟨τ⟩n > 1/ωlowest with ωlowest being the lowest angular frequency examined in Figure 7. (The ε″(ω)/ω ratio increases with decreasing ω and thus [ε″(ω)/ω]ω→0>ε″(ωlowest)/ωlowest, which allows us to deduce this relationship.) Thus, the dielectric terminal relaxation time of the PI chains is significantly longer in the spherical domains than in the bulk state: For example, ⟨τ⟩w > ⟨τ⟩n > 1/ωlowest = 10−3 s for I-3 in the spherical domain and ⟨τ⟩w ≅τpeak = 4.0 × 10−5 s for bulk I-3; see Figure 7. In relation to the above argument, a comment needs to be made for τpeak for a relaxation process with a broad mode distribution. Since the phenomenological framework is the same for G″ and ε″, we here focus on G″ of Rouse chains (bead−spring chains) as a well-known example of loss factor associated with the broad mode distribution. For the Rouse chain composed of N + 1 beads and N springs, G″(ω) is expressed as29

,

1 + ω2τp2

(A5)

This expression is formally identical to the expression of the storage and loss moduli, G′ and G″, in terms of the viscoelastic mode intensities and relaxation times.5,6 Thus, all relationships established for the moduli are applicable to Δε′(ω) and ε″(ω). Specifically, completion of the relaxation is characterized by the relationships that hold only at low ω well below 1/τ1 (with τ1 being the longest relaxation time): Δε′(ω) = Δεω2

∑

gpτp2 ∝ ω2 ,

p≥1

ε″(ω) = Δεω

∑

gpτp ∝ ω

p≥1

(A6)

(These relationships are analogous to the relationships wellknown for the moduli, G′ ∝ ω2 and G″ ∝ ω at low ω where the relaxation is completed.) Consequently, we can unequivocally conclude that the relaxation is not completed in a range of ω where the proportionality between ε″(ω) and ω (and between Δε′(ω) and ω2) is not observed. This is the case for the guest PI chains in the spherical I domains, as noted in Figures 7 and 8. Namely, in those domains, the slowest mode of end-to-end fluctuation of the guest PI chains has not completed in the range of ω examined. In contrast, the proportionality between ε″(ω) and ω is clearly observed for the same PI chains in respective bulk sate; see Figure 7. Thus, the end-to-end fluctuation of the PI chains is significantly retarded in the spherical domain compared to the bulk system. We can also utilize an average relaxation time to quantitatively compare the PI dynamics in the spherical domains and in bulk systems. The first- and second-moment average relaxation times ⟨τ⟩n and ⟨τ⟩w, being defined in terms of the mode relaxation times τp and intensities gp, can be directly evaluated from the dielectric data at low ω (cf. eq A6).5,6

N

G″(ω) = νkBT

ωτ[R,G] p

2 [R,G] 2 } p = 1 1 + ω {τp

(A8)

where ν, kB, and T denote the number density of the chain, Boltzmann constant, and absolute temperature, respectively, and τp[R,G] is the viscoelastic relaxation time of pth Rouse mode: τ[R,G] = p

⎫ ⎧ pπ ζb2 ⎬ sin−2⎨ 24kBT ⎩ 2(N + 1) ⎭

(A9)

In eq A9, ζ and b represent the friction coefficient and step length of the Rouse segment. The intensity factor appearing in eq A8, νkBT, is common for all Rouse modes. The first- and second-moment average viscoelastic Rouse relaxation times, ⟨τ⟩n[R,G] and ⟨τ⟩w[R,G], are given by30 ⟨τ ⟩[R,G] ≡ n

∑p ≥ 1 gpτp

1 ⎡ ε″(ω) ⎤ ⟨τ ⟩n ≡ = , ⎢ ⎥ ∑p ≥ 1 gp Δε ⎣ ω ⎦ω → 0

[R,G] ∑N p = 1 νkBT τ p

∑N p = 1 νkBT

=

ζb2 (N + 2) 36kBT

(∝ N for large N )

∑p ≥ 1 gpτp2

⎡ Δε′(ω) ⎤ ⟨τ ⟩w ≡ =⎢ ⎥ ⎣ ωε″(ω) ⎦ω → 0 ∑p ≥ 1 gpτp

∑

⟨τ ⟩[R,G] ≡ w

(A7)

(These ⟨τ⟩n and ⟨τ⟩w are analogous to the well-known viscoelastic relaxation times,28,29 η0/G∞′ and Jeη0 with η0, G∞′, and Je being the zero-shear viscosity, high-frequency storage modulus, and the steady-state recoverable compliance, respectively.) ⟨τ⟩w is longer than ⟨τ⟩n for any relaxation mode distribution, except for the case of single Maxwellian relaxation for which ⟨τ⟩w = ⟨τ⟩n. It should be noted that ⟨τ⟩w heavily weighs on slow relaxation modes and can be utilize as the terminal dielectric relaxation time,5,6 which is analogous to the use of Jeη0 as the viscoelastic terminal relaxation time.28,29

(A10)

[R,G] 2 ∑N } p = 1 νkBT {τ p

=

[R,G] ∑N p = 1 νkBT τ p

ζb2 (2N 2 + 4N + 9) 180kBT

(∝ N 2 for large N )

⟨τ⟩n[R,G]

(A11)

⟨τ⟩w[R,G]

τN[R,G]

Obviously, these and are longer than for the fastest (Nth) Rouse mode. For several values of N, Figure 9 shows plots of a reduced loss modulus G″(ω)/νNkBT against a reduced frequency ωτN[R,G]. 2817

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The arrows indicate the relaxation frequency of the slowest Rouse mode, 1/τ1[R,G], and the thin dotted curve indicates a modulus contribution from the fastest mode (with p = N) for the case of N = 5. In Figure 9, one might attempt to utilize the peak relaxation time τpeak (= 1/ωpeak) as an average relaxation time. However, this τpeak is insensitive to N and much shorter than τ1[R,G] (cf. arrow), ⟨τ⟩n[R,G], and ⟨τ⟩w[R,G]. This fact demonstrates that τpeak cannot be utilized for discussion of the slow dynamics of the chain given that the chain exhibits a broad mode distribution and its loss peak is not immediately followed by the low-ω proportionality between the loss factor and ω. This conclusion is not limited to the Rouse modulus but applies to any broad relaxation process, i.e., to the broad dielectric relaxation process of the guest PI chains in the spherical domains (seen in Figures 7 and 8). Finally, it is informative to compare the modulus contribution from the fastest Rouse mode for N = 5 (thin dotted curve shown in Figure 9) and the modulus curve (plots for N = 5).

Figure 10. Comparison of dielectric mode distribution of bulk PI chains of various molecular weights.

distribution of the PI chains is insensitive to M in the nonentangled and moderately entangled regimes (Me < M < 10Me).5−7,27 These M-insensitive plots of ε″(ω) against ω/ωpeak are shown in Figure 8 with the dashed curve. In relation to the above feature of the nonentangled to moderately entangled PI chains, we note that highly entangled PI chains (M > 20Me) exhibit a slightly narrower terminal mode distribution characterized with ω−1/2 dependence of ε″ at ω just above ωpeak.8 However, in the plots of ε″ against ω/ωpeak, this narrowing for highly entangled PI corresponds to just a moderate decrease of ε″ at ω ≫ ωpeak, by a factor of 20−30%, compared to ε″ of moderately entangled PI. Thus, in the entire range of M, the terminal dielectric mode distribution of bulk PI is essentially insensitive to M, unless we focus on the ω−1/2 dependence of ε″ of high-M PI at ω just above ωpeak. This M-insensitivity of the dielectric mode distribution differs from a considerable M-dependence seen for the terminal viscoelastic mode distribution.6−8 This difference reflects a difference of the relationships between the chain conformation and the measured quantities, the orientational correlation at two separate times reflected in the dielectric quantity and the isochronal orientational anisotropy, in the viscoelastic quantity.5−7,27

Figure 9. Plots of reduced viscoelastic loss modulus of Rouse chains composed of N + 1 beads and N springs (N = 5, 10, 20, and 40) against reduced frequency. The thin dotted curve indicates a contribution to the modulus from the fastest mode for the case of N = 5.

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The peak for this fastest mode contribution emerges at ωτN[R,G] = 1, whereas the peak for the modulus is observed at ωτN[R,G] ≅ 0.6. The contributions from the second and third fastest modes have the peaks at ωτN[R,G] = 0.804 and 0.536, respectively (cf. eq A9). Thus, none of these modes exhibits the peak at ωτN[R,G] ≅ 0.6 observed for the modulus.31 This fact demonstrates that a peak of G″ data (and of ε″ data) does not always correspond to a particular mode of molecular relaxation: For the broad mode distribution, the peak frequency ωpeak has, at best, just a semiquantitative meaning as an average relaxation frequency for a group of relaxation modes having large intensities. This fact also suggests that τpeak of the broad ε″ data of the guest PI chains in the spherical domains cannot be utilized for discussion of the global motion of those chains.

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

This work was supported by the Grant-in-Aid for Scientific Research (B) (Grant No. 21350063), by Grant-in-Aid for Young Scientists (B) from MEXT (Grant No. 22750204), and partly by the Collaborative Research Program of Institute for Chemical Research, Kyoto University (Grant No. 2012-35).

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(1) Bates, F. S.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990, 41, 525. (2) Hamley, I. W. The Physics of Block Copolymers: Oxford Press: Oxford, U.K., 1998. (3) Matsushita, Y. Macromolecules 2007, 40, 771. (4) Adachi, K.; Kotaka, T. Prog. Polym. Sci. 1993, 18, 585. (5) Watanabe, H. Macromol. Rapid Commun. 2001, 22, 127. (6) Watanabe, H. Prog. Polym. Sci. 1999, 24, 1253. (7) Watanabe, H. Polym. J. 2009, 41, 929. (8) Glomann, T.; Schneider, G. J.; Bras, A. R.; Pyckhout-Hintzen, W.; Wischnewski, A.; Zorn, R.; Allgaier, A.; Richter, D. Macromolecules 2011, 44, 7430.

APPENDIX B: DIELECTRIC BEHAVIOR OF BULK PI The dielectric mode distribution is characterized with the ω dependence of the dielectric loss, ε″(ω), and the decrease of the dynamic dielectric constant, Δε′(ω) ≡ ε′(0) − ε′(ω),5−7,27 as explained in Appendix A. In Figure 10, the ε″(ω) and Δε′(ω) data measured for the guest PI chains in respective bulk systems are plotted against the reduced frequency ω/ωpeak (with ωpeak =ε″-peak frequency). The data at low ω(τ), are summed to give just a broad single peak of the loss factor (G″ and/or ε″) at ω between 1/τ and 1/τ′, i.e., at ω not corresponding to the relaxation frequencies of the two modes, given that the τ′/τ ratio is smaller than 5.8.

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dx.doi.org/10.1021/ma3001687 | Macromolecules 2012, 45, 2809−2819