Rheo-Optical Study of Viscoelastic Relaxation Modes in Block

Article pubs.acs.org/Macromolecules

Rheo-Optical Study of Viscoelastic Relaxation Modes in Block Copolymer Micellar Lattice System Eiko Tamura,‡ Yurika Kawai,‡ Tadashi Inoue,*,‡ and Hiroshi Watanabe†,‡ ‡

Department of Macromolecular Science, Graduate School of Science, Osaka University, Machikaneyama-cho 1-1, Toyonaka Osaka 560-0043, Japan † Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011 Japan ABSTRACT: Dynamic birefringence and viscoelasticity of a diblock copolymer micellar solution were measured in order to clarify the molecular origin of viscoelastic response of the macrolattice structure formed by micelles. The complex strainoptical ratio changed its sign with angular frequency ω and exhibited very complicated ω dependence, suggesting that the stress emerged/relaxed through several mechanisms. With an assumption of the stress-optical rule for each mechanism, the complex shear modulus was separated into four components corresponding, from high to low ω, to the reorientation of corona chains, reorientation of core chains, deformation of core, and deformation of the micellar lattice. Values of the stress optical coefficients for respective components lent support to these assignments.

1. INTRODUCTION When polymeric materials are deformed, they become anisotropic and the stress, σ, and the birefringence, Δn, are observed. The strain-induced birefringence of polymeric materials is intimately related to the stress. For example, for the rubbery materials, the following stress-optical rule (SOR) holds well.1 n(̂ t ) = Cσ (̂ t )

micelles having S cores and B coronas in diene-selective solvents above a certain concentration and at low temperature, as shown in Figure 1. The glass transition temperature, Tg, of the S core could be close to/lower than the room temperature for micellar solutions with the SB volume fraction ϕ < 0.40 and intermediate SB molecular weight of a few tens of thousands, as judged from the glassy window for styrene−isoprene (SI) diblock copolymer in tetradecane.3

(1)

Here, n̂(t) and σ̂(t) are the deviatoric parts of the refractive index tensor and the stress tensor at time t, respectively. The proportionality coefficient, C, is the stress-optical coefficient. The SOR holds for rubbery materials even in the nonlinear viscoelastic region. The proportionality between the stress and the birefringence is not limited for rubbery materials. Assuming that the stress and dielectric tenor are functions of strain, we find the proportionality just from a symmetry argument under small strains irrespective of the origin(s) of stress.2 However, the proportionality coefficient, C, is specific to the molecular/ structural origin of stress and birefringence. Thus, a close investigation of the relationship between stress and birefringence provides us with detailed information on the molecular origin of stress. It is of interest to examine, from this point of view, the origin of the stress of block copolymer systems. Block copolymers exhibit a wide variety of microphaseseparated structures, for example, spheres, cylinders, lamella, and so on. Viscoelastic properties of microphase-separated block copolymers have been extensively studied. In particular, solutions of diblock copolymers dissolved in selective solvent show unique rheology. Styrene−butadiene (SB) diblock copolymers with relatively small S content form spherical © 2012 American Chemical Society

Figure 1. Schematic illustrations of SB micellar solution. Received: April 25, 2012 Revised: July 20, 2012 Published: August 7, 2012 6580

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Watanabe et al.4−6 focused on a relationship between the structure and rheological properties of micellar solutions. Early experiments7 revealed that moderately concentrated solutions of spherical block copolymer micelles exhibit elasto-plastic behavior attributable to a micellar lattice formed through a balance of the thermodynamic, elastic and osmotic forces, the former randomizing the corona conformation and the latter reducing the corona concentration gradient. The elastic response results from deformation of the lattice under small strain, while plastic response is attributed to large-scale sliding motion of the lattice plane mainly occurring at the grain boundary. In moderately concentrated micelle systems, the intercorona osmotic interaction is effectively screened on addition of homo polybutadiene (hB) chemically identical to but considerably shorter than the corona B block. This screening allows the micelles to be randomly dispersed in the hB matrix and leads to the lattice disordering.6,7 For those randomly dispersed micelles, three linear viscoelastic relaxation processes are observed. The fastest process has been attributed to relaxation of the matrix hB, and the intermediate process, to relaxation of individual corona blocks. The slowest process has been attributed to relaxation of the Brownian stress due to the micelle diffusion.8 Considering the usefulness of the rheo-optical method explained earlier, this paper focuses on the dynamic birefringence of SB micellar lattice system to examine the relaxation mode(s) therein. As described above, the solution of SB block copolymer in a low molecular weight B-selective solvent (without added hB) is expected to show at least two components of stress, the viscoelastic stress due to orientation of individual corona blocks and the elastic stress resulting from deformation of the micellar lattice. We found that the straininduced birefringence actually has four stress components attributable to the orientation of corona blocks, orientation of S blocks in the core, deformation of the core, and the lattice deformation. The estimated values of the stress-optical coefficients lent support to this assignment. Details of these results are summarized in this paper. Those results also demonstrate the usefulness of the rheo-optical method for dynamic analysis of various heterogeneous complex fluids (that include block copolymer solutions).

Δn Z (t ) = 2γ0K 0(ω) sin[ωt + δB(ω)] = 2γ0K ′(ω) sin ωt + 2γ0K ″(ω) cos ωt

(2)

The phase angle δB(ω) appearing in eq 2 may be different from the viscoelastic phase angle. The coefficient of 2 is needed to convert the directly measured ΔnZ to Δnxy (the xy component of refractive index tensor Δn) under small strain. A complex strain-optical coefficient can be defined as K* = K′ + iK″ with K′(ω) = K0(ω) cos δB(ω) and K″(ω) = K0(ω) sin δB(ω), as similar to the expression of the complex modulus. The SOR formulated for K* and G* is expressed as

K * = CG*

(3)

In the actual birefringence measurements, the SB-C14(19.1) sample charged in the apparatus had an area of 10 mm ×10 mm and a thickness of ≅0.5 mm, and the dynamic birefringence and dynamic modulus were simultaneously measured at 30 °C. In order to check the consistency of rheological data, a laboratory rheometer MCR301 (Anton Paar Co.) equipped with a parallel-plate fixture of 12 mm diameter was also used for dynamic viscoelasticity measurements. The measurement was performed at T = 30 °C. After a dynamic strain sweep test, the frequency sweep measurement was conducted in the linear regime to determine G* as a function of angular frequency ω.

3. RESULTS AND DISCUSSION 3.1. Overview. Figure 2 shows the data of the complex shear modulus G* and complex strain-optical coefficient K* at

2. EXPERIMENTAL SECTION 2.1. Materials and Sample Preparation. A commercial available styrene−butadiene (SB) block copolymer (Phillips Co., Solprene1205) was used. This copolymer, being identical to the copolymer examined in the previous studies,4,6 has the molecular weight M = 52000 and styrene content of 29.5 wt %. The solvent ntetradecane (nC14) is highly selective, good for B block, and poor for S block. A SB/nC14 solution was prepared by dissolving prescribed masses of SB and nC14 in excess of methylene chloride (MC) and then allowing MC to thoroughly evaporate at room temperature. The SB concentration, determined after this evaporation, was 19.1 wt % (and the sample solution is coded as SB-C14(19.1)). At this high concentration, the SB micelles were arranged on a macrolattice to exhibit the elasto-plastic behavior, as noted in the previous studies.4,6 2.2. Rheo-Optical Measurement. A homemade apparatus for rheo-optical measurement under oscillatory shear-strain deformation was reported elsewhere.9,10 As in the viscoelastic measurements, a small amplitude oscillatory strain, γ(t) = γ0 sin ωt (with γ0 = 0.06) is applied on the sample sandwiched between two glass plates. Then, the birefringence of the sample is obtained as a sinusoidal function of t:

Figure 2. Frequency dependence of G* and K* obtained for SBC14(19.1) at T = 30 °C. Solid lines represent the G* data measured with MCR301.

30 °C measured with the flow birefringence apparatus. Also included is the G* data measured with the conventional rheometer (MCR301). The two sets of the G* data (plots and curves) are in accord with each other. The real part of complex strain-optical coefficient, K′, is negative in the entire range of the angular frequency ω, and |K′| does not monotonically increases with increasing ω. This nonmonotonic behavior of K′ corresponds to changes of the sign of the imaginary part of the coefficient, K″: At low ω (0.1 s−1), K″ decreases 6581

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significantly with increasing ω and changes its sign (becomes negative) at ω ≅ 1 s−1, whereas |K′| is rather insensitive to ω. At higher ω, |K″| first increases and then decreases with increasing ω and K″ again changes its sign at ω ≅ 300 s−1, while |K′| gradually increases with ω. These features strongly suggest the existence of relaxation modes with positive stress-optical coefficients C at ultrahigh ω > 300 s−1 and low ω ≅ 0.1 s−1 and the other modes with negative C at high and ultralow ω (≅100 s−1 and 0 assigned to orientational relaxation of PB corona) are explained later. The above results suggest that there are at least four distinct modes of relaxation contributing to the K* data in our experimental window. Assuming that SOR is valid for each mode, we may split the G* and K* data of SB-C14(19.1) into 4 components, as shown in eqs 4 and 5. G* = G1* + G2* + G3* + G4*

(4)

K * = C1G1* + C2G2* + C3G3* + C4G4*

(5)

Figure 4. Frequency dependence of K* calculated as C*G* for SBC14(19.1) at T = 30 °C. Curves show the results of fitting with the four Maxwellian modes.

Table 1. Characteristic Parameters for Relaxation Mode i for the Stress and Birefringence of the SB-C14(19.1) Micellar System

The subscripts i = 1−4 specify the mechanism of stress generation/relaxation from low to high frequencies. For simple but definite argument, we approximated each component of Gi* as a single Maxwellian component and evaluated the initial modulus Gi, relaxation time τi, and the stress-optical coefficient Ci of the component through fitting of the G* and K* data. (In this analysis, the relaxation of the slowest mode was not clearly observed and hence τ4 was fixed as infinity.) The best-fit results are shown in Figures 3 and 4, and the corresponding parameter

mode index i

Gi/Pa

τi/s

Ci/ 10−9 Pa−1

assignment of origin

1 2 3 4

1100 160 600 2000

∞ 12 0.01 0.001

−1.6 5.0 −13.0 1.5

micellar lattice deformation form birefringence of S core reorientation of S segments reorientation of B segments

can be understood if we consider a model case, two Maxwellian elements, e1 and e2, having stress-optical coefficients Ce1 and Ce2 (≠Ce1), respectively, and being connected in series. These elements reflect different relaxation mechanisms but the connected elements as a whole still satisfy the stress-optical relationship, i.e., the proportionality between the birefringence Δn and stress σ, Δn = (Ce1 + Ce2)σ. Namely, two different relaxation mechanisms can be observed rheo-optically as one relaxation process (one of the terms in eqs 4 and 5) if they are combined in the series fashion, and the rheo-optical data cannot resolve the stress-optical coefficients for individual mechanisms. In what follows, we keep this point in our mind and discuss the four relaxation processes (eqs 4 and 5) with the aid of their C values. 3.2. Assignment of Relaxation Processes. Figure 5 illustrates possible relaxation mechanisms for the SB micellar lattice system. The system at equilibrium (Figure 5a) would be deformed as shown in Figure 5b on application of a small step strain (Figure 5b depicts the tensile strain, but the situation is similar for the shear strain examined in this study.) Specifically, the micellar lattice should be distorted, and the micellar B corona should be deformed as well. Even if the core is glassy,6 the PS blocks therein are connected to the PB corona blocks and thus the tension from the deformed PB corona should force the PS blocks to be oriented a little and the cores should be deformed “a little” accordingly. Deformation of domain and lattice is analogous to that considered in the theory for photoelasticity of inorganic glasses or cubic crystals. For these materials, two effects have been considered: One is the “lattice effect” giving negative birefringence, and the other is positive “atomic effect” (form birefringence of deformed atomic electron cloud).11 For the case of block copolymer, the “lattice effect” corresponds to process L shown in Figure 5, whereas the

Figure 3. Frequency dependences of storage and loss moduli, G′ and G″, measured for SB-C14(19.1) at T = 30 °C. Thick curves show the results of fitting with the four Maxwellian modes. Dotted curves represent frequency dependence for each Maxwellian mode.

values are summarized in Table 1. As shown in Figure 4, characteristic features of K* (changes of sign of K″ and nonmonotonic ω dependence of K′) are almost quantitatively reproduced with the combination of four Maxwellian components with the following C values: C4/Pa−1 = 1.5 × 10−9, C3/Pa−1 = −1.3 × 10−8, C2/Pa−1 = 2.5 × 10−9, C1/Pa−1 = −1.6 × 10−9 Pa for the ultrahigh ω, high ω, low ω, and ultralow ω components, respectively. The G* data are also described reasonably well by this combination, as noted in Figure 3. Here, a comment needs to be made for the four processes considered in eqs 4 and 5. One (or more) of these processes may be a complex process that corresponds to two different relaxation mechanisms working in a series fashion. This point 6582

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point, a previous study12 found that the equilibrium modulus Geo measured for micellar lattices is smaller, by a factor of ≅1/ 10, than the modulus expected for entropic elasticity of individual corona blocks, Geo = νcoronakT (νcorona = corona number density, k = Boltzmann constant), and suggested that ≅10 corona blocks cooperatively behave as one entropic stresssustaining unit (under an implicit assumption that the applied strain is affinely transmitted to individual corona blocks). If g blocks behave as the stress-sustaining unit as a whole under the affine deformation, the stress optical coefficient C2 could be g times larger than C4 for the entropic elasticity of individual PB blocks. If we follow this argument based on the C values, g is estimated to be ∼3 for our SB-C14(19.1) system. However, the observed G2 value (160 Pa) is approximately 60 times smaller than Geo = 9100 Pa for the entropic modulus of individual corona B blocks, giving an estimate of g ≅ 60 based on the G2 data. The difference between the two g values (3 and 60) as well as the very small G2 value itself suggests that the B corona layer was not affinely but more weakly deformed during the low-ω process. This process associated with just a small G2 value was not resolved well in the previous rheological studies.4,6 Nevertheless, the positive C2 value suggests existence of this low-ω process related to the form effect of the micellar corona layer. The slowest relaxation process (with τ1 = ∞) could be originated from “the lattice effect” of the micellar lattice because its C1 is negative. As shown in Figure 6, the S cores under the

Figure 5. Schematic illustrations of relaxation processes in micellar lattice systems.

“domain effect” (process FB in Figure 5) is analogous to the “atomic effect” of the inorganic glasses or cubic crystals. The fastest, ultrahigh ω relaxation process (Gi = 4) must be related to the reorientation of PB corona blocks shown as the process RB in Figure 5, as judged from the large mobility of those blocks. This assignment is supported from the C4 value (=1.5 × 10−9 Pa−1) for this process being close to the C value of bulk PB (=2.0 × 10−9 Pa−1). At the same time, we note that the conformation of the PB corona bocks in our micellar system cannot be fully randomized/disoriented because of the osmotic interaction among them. The second fastest, high ω process (process RS illustrated in Figure 5) could be attributed to reorientation of PS blocks in the core, as suggested from the negative C3 value (=−1.3 × 10−8 Pa−1) for this process. This negative C3 value is about two times larger, in magnitude, than the negative coefficient (=−7.0 × 10−9 Pa−1) for ordinary polystyrene solutions (in the liquid state at room temperature). Although this difference is not trivial, we note that the PS blocks are confined in the spherical domain and this confinement could change the chain statistics and may affect the stress-optical coefficient. In addition, the S cores are connected to the B corona phase in a series fashion, which suggests that G3 is an apparent modulus (because the deformation of the core is much smaller than that of the soft B corona) and its |C3| can be larger than |CPS| of the PS solution. The third, low-ω relaxation process (process FB in Figure 5) could originate from the “form birefringence” of deformed micelles (mostly attributed to the deformed corona, as explained later), as judged from its positive C2 value (=5.0 × 10−9 Pa−1). The deformation of the corona layer should relax through cooperative motion of all PB blocks in a given micelle and thus this relaxation would be slower than the reorientational relaxation of individual PB blocks (ultra high-ω processes). Although the estimation of C2 value is not so acculate because of small amplitude of G2, we note that the C2 value is about three times larger than the C4 value (=1.5 × 10−9 Pa−1) for the relaxation of individual B corona blocks, possibly because the origin of the stress for this mode is mainly the “cooperative reorientation” of the corona B blocks: The conformation of the corona blocks cannot be fully randomized/disoriented in the fastest mode, because of the osmotic interaction of those blocks explained earlier. Concerning this

Figure 6. Concentration profile of core (a) and core and corona (b).

applied strain are anisotropically distributed in space. The origin of the stress for this slowest process has been attributed to “the orientation of the B corona blocks” under the osmotic interaction. In this context, the birefringence for that process could be a sum of the intrinsic birefringence of B blocks due to remaining orientation and the birefringence originated from the deformation of the lattice of the S cores. The G1 value (1100 Pa) is approximately 10 times smaller than the entropic modulus of individual B blocks, Geo = 9100 Pa. As described earlier, this difference could be attributed to the osmotic coupling of PB blocks, and the C2 value suggests g ≅ 3 corona blocks may behave as one entropic stress-sustaining unit. If we follow this argument, the intrinsic contribution by PB blocks to the mode 1 is estimated to be gC4G1 ∼ C2G1 ∼ 5.5 × 10−6. On 6583

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The parameters, a and d, should change with the deformation. If the spherical domain and lattice commonly exhibit affine deformation on application of a small tensile strain, ξ, a (of the domain) and d (of the lattice) change by the same factor, (1 + ξ) in the tensile direction (x direction) and by the factor (1 + ξ)−1/2 ≅ (1 − ξ/2) in the other two (y and z) directions under the incompressible condition. Then, the strain affects the factor q0a (∼2πa/d) in none of the three directions, thereby resulting in no significant birefringence Δn ∼ εxx − εyy; cf. eq 9. However, if only the lattice deforms affinely but the domain does not deform (pseudoaffine deformation), the optical anisotropy emerges. This anisotropy due to the tensile strain can be estimated as

the other hand, the actual birefringence for the slowest mode 1 is C1G1 = −1.8 × 10−6. These gC4G1 and C1G1 values are quite different (even for their signs), suggesting that the birefringence C1G1 for the slowest process is determined by a subtle balance between the intrinsic contribution of the B corona blocks and the neat contribution from the lattice deformation. This point is further examined below with the aid of a theory of form birefringence for the deformed lattice. 3.3. Analysis of Form Birefringence. The form birefringence of “deformed spherical domain” and “deformed lattice” may be described with the theory of Onuki and Doi (OD).13 They derived a relationship between fluctuations in polymer solution concentration and the macroscopic dielectric tensor, the latter connecting the macroscopic electric displacement of the material with the macroscopic electric field. Anisotropies in this dielectric tensor determine the conservative dichroism and birefringence of the material. If the spatial anisotropies in the concentration arise from an imposed shear, then the theory predicts the optical properties of the material under deformation. Starting with Maxwell’s equation, the OD theory relates the local electric field to the local dielectric tensor for a charge free system. (The anisotropic part of the dielectric tensor due to the form and intrinsic contributions is obtained after making the ensemble average of a perturbation solution of the field equation.) Later, Onuki and Fukuda applied this theory to the form birefringence of the microdomain structure of block copolymer.14 Their expression for the dielectric tensor εij is εij = −

2 1 ⎛ dε0 ⎞ ⎜ ⎟ ε0q0 2 ⎝ dϕ ̅ ⎠

∂ϕ(r) ∂ϕ(r) ∂xi ∂xj

εxx − εyy = −

⎛ q 0 a ⎞⎫ − sin 2⎜ ⎟⎬ ⎝ 1 − ξ /2 ⎠⎭ ≅

ΔnT =

εxx − εyy 2n

∼

2

2

6A2 q0a ⎛ dn ⎞2 ⎜ ⎟ (sin 2q0a)ξ n ⎝ dϕ ̅ ⎠

(11)

From this Δn , the birefringence coefficient for the shear deformation is obtained as

(6)

2 1 ΔnT 2A2 ⎛ dn ⎞ Δn shear = ∼ K= q a⎜ ⎟ sin 2q0a 3 ξ γ n 0 ⎝ dϕ ̅ ⎠

(12)

As can be noted from eq 12, the birefringence of pseudoaffinely deformed lattice, which corresponds to the neat lattice effect, is proportional to square of the difference of refractive index of two domains, δn, and inverse-square of the domain volume fraction, δϕ. We note eq 12 gives negative birefringence for a > d/4 while positive birefringence for a < d/4. This can be explained as follows. According to eq 6, εij is determined by magnitude of concentration fluctuation, dϕ(r)/dxi. The concentration fluctuation for the lattice is a function of a/d, i.e., dϕ/dx ∼ sin(2πa/d), and this shows a maximum for a = d/ 4. Therefore, for the case of a < d/4, tensile strain elongates the lattice size, d, along the tensile axis and therefore the concentration fluctuation is suppressed along the tensile direction and enhanced in the other two directions. This results in positive birefringence. On the other hand, for the case of a > d/4, the concentration fluctuation is enhanced along the tensile direction to give negative birefringence. For our SB-C14(19.1) system, a/d ≅ 0.18 < 1/4 (as estimated from the a and d data for a SB-C14(20) system5 having a slightly larger concentration, 20 wt %). Thus, the above argument just considering the spherical S core suggests a positive birefringence. However, the slowest mode is actually associated with negative birefringence (C1G1 < 0), suggesting that the deformation of the B corona blocks largely contributes to the observed birefringence. Therefore, we need to refine the concentration profile as shown in Figure 6b to consider contributions from both core S and corona B blocks. In order

(7)

(8)

Here, a is the radius of spherical domain (core), and A is the amplitude of fluctuation being related to the volume fraction of S core, ϕS as A∼ ϕS sin(q0a)/(3q0a). From eqs 6 and 8, the principle values of the dielectric tensor is obtained as 2 2A2 ⎛ dε0 ⎞ 2 ⎜ ⎟ sin q0a ε0 ⎝ dϕ ̅ ⎠

2

T

Δϕ(r) = 2A sin q0a[cos(q0 e1r) + cos(q0 e2r)

εii = −

(10)

Since n = ε0 and [dε0/dϕ̅ ] = 4n [dn/dϕ̅ ] , the corresponding birefringence due only to the lattice deformation is given by 2

In eq 7, ei is the unit vector along axis i of the lattice. For Δϕ(r), we start with an assumption of the stepwise concentration profile considering only for the spherical core, as shown in Figure 6. The corresponding lowest-order Fourier component, being equivalent to that specified by eq 6, is given by

+ cos(q0 e3r)]

3A2 q0a ⎛ dε0 ⎞2 ⎜ ⎟ (sin 2q0a)ξ ε0 ⎝ dϕ ̅ ⎠ (up to the first order ofξ)

where q0 is the peak wavenumber (which corresponds to the radius of gyration of the constituent blocks not significantly different from lattice spacing d; q0 ∼2π/d), ε0 is the dielectric constant at the optical frequency, ϕ̅ is the average local concentration, and ϕ(r) is the concentration including fluctuation. For the simple cubic lattice near the order− disorder transition, the local deviation of the concentration from the average, Δϕ =ϕ(r) − ϕ̅ would vary sinusoidally in space as15 Δϕ(r) ∝ [cos(q0 e1r) + cos(q0 e2r) + cos(q0 e3r)]

2 2A2 ⎛ dε0 ⎞ ⎧ 2⎛ q0a ⎞ ⎟ ⎜ ⎟ ⎨sin ⎜ ε0 ⎝ dϕ ̅ ⎠ ⎩ ⎝ 1 + ξ ⎠

(9)

We note that εii given by eq 9 depends on the ratio of the domain size to the lattice spacing, a/d (∼ q0a/2π). 6584

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that if the domains are slightly deformed, the magnitude of observed birefringence would decrease, and it should increase with the relaxation of domain. This process could be observed as the mode 2. As noted earlier, the actual strain of corona estimated from the G3 value is 1/60 of macroscopic strain. For such a small stain, the form contribution is estimated Ktheo/60 ∼ 7 × 10−8 and it could be negligible compared to C2G2 = 8 × 10−7. One of the reviewers made an important comment that the BCC structure should have structurally series connection of B and S chain and that we should consider the series coupling (stress coincidence) of intrinsic relaxation of B and S chains in the rheo-optical analysis. We carefully examined this comment and came to a conclusion that the method of analysis explained in this paper is close to the best achievable at this moment, as explained below. As a starting example for discussing the stress coincidence in the series connection, we may focus on dilute polymer solutions. Normally, no series connection is considered for the solvent molecules and polymer chain despite a fact that the polymer chains are embedded in the solvent, and the stress of the solution is calculated just as a sum of stress of individual chains. Furthermore, the segments at the chain ends and middle sustain different stresses (smaller for the end segment, as easily noted from the bead−spring model), despite the fact that all segments in the chain are connected. (This difference between the end and middle segments can be found also for entangled chains, as well.) Thus, the stresses of the connected units do not always coincide on the molecular level, and the local stress can be inhomogeneous within the same chain during the relaxation process. Namely, the series model is not always valid on the molecular level. We may also consider another example, a rubber containing liquid droplets chemically identical to the rubber matrix. The droplets do not contribute to the equilibrium modulus irrespective of their viscosity, because the viscous droplets fully relax at equilibrium. This example strongly suggests that the stress of PS core in our copolymer solution would relax at long times because the PS chains have much faster dynamics as assigned to mode 3. More importantly, in the calculation of birefringence originated by mode 1, we ignored deformation of core. Thus, we believe that S core does not sustain detectable stress at long times (as considered in our analysis), despite the direct series connection of the PS cores and PB corona on the molecular level. In conclusion, the above theoretical consideration with the OD theory and pseudoaffine deformation of lattice structure revealed that the strain-induce birefringence due to the lattice structure reflects the structural details such as size of core and corona, volume fraction, and optical contrast, δni, and therefore dynamic birefringence is a useful method to analyze the deformation/relaxation processes of the block copolymer micellar solutions. The theory predicts that a positive birefringence due to core will be observed if the difference in refractive index between corona chain and solvent is small. Thus, the lattice effect for block copolymer macro-lattice systems shows more complex behavior than that for inorganic glasses or cubic crystals, for which only positive “lattice effect” and negative “atomic effect” (form birefringence of deformed atom) have been considered.11

to take into account the difference of the dielectric constants of the S and B blocks, we introduce the ratio of (∂ε0/∂ϕ) for components. ⎛ ∂ε ⎞ ⎛ ∂ε ⎞ r = ⎜⎜ 0 ⎟⎟ /⎜⎜ 0 ⎟⎟ ⎝ ∂ϕB ⎠ ⎝ ∂ϕS ⎠

(13)

Then the effective concentration fluctuation may be written as Δϕeff (r) = 2[(AS − rAB) sin q0aS + rAB sin q0aB] 3

∑ cos(q0ejr) j=1

(14)

Here, the subscripts S and B stand for S and B block, respectively. From eq 14, the principle values of the dielectric tensor is obtained as εii = −

2 2 ⎛ dε0 ⎞ ⎜⎜ ⎟⎟ [(AS − rAB) sin q0aS + rAB sin q0aB]2 ε0 ⎝ dφS ⎠

(15)

The corresponding shear strain-optical coefficient is calculated as K=

2 4 ⎛ dn ⎞ ⎜⎜ ⎟⎟ [(AS − rAB) sin q0aS + rAB sin q0aB] n ̅ ⎝ dϕS ⎠

[(AS − rAB)q0aS cos q0aS + rABq0aB cos q0aB]

(16)

The above calculation was made for simple cubic lattice, and the same results can be obtained for BCC lattice. Since the micellar lattice is formed due to the osmotic interaction of the overlapping corona blocks of neighboring micelles, we can naturally assume aB/d = (aS + rPB)/d = 0.433 for BCC. (Because the concentration profile of the B block is not perfectly uniform in the B matrix phase, we adopt the aB/d value of 0.433, rather than 0.5, when cast in the step-like profile shown in Figure 6b.) The other parameters appearing in eq 16 were estimated from data,16 (∂n/∂ϕS) =nPS − nC14 = 0.171, ϕS = 0.051, aPS/d = 0.18, (∂n/∂ϕB) = nPB − nC14 = 0.091, and ϕB = 0.12. From these parameter values, the strain-optical coefficient was evaluated as Ktheo = −1.0 × 10−5, being approximately 5 times larger than the experimental value, C1G1 = −1.8 × 10−6. As explained earlier, the actual C1G1 value is determined by the subtle balance of the intrinsic contribution and the neat deformed lattice contribution, Ktheo. We note that the difference between Ktheo and C1G1, ∼ 8.2 × 10−6, is fairly close to the intrinsic contribution by PB blocks, gC4G1 ∼ C2G1 = 5.5 × 10−6, where g ∼ 3. Thus, the same value of osmotic coupling factor for the corona chains, g ∼ 3 can explain the intrinsic contribution of PB chain for both the modes 1 and 2. The difference between this value and larger estimation of g = G1/ Geo ∼ 10 in the earlier study can be attributed to the effective strain of corona chains supporting the lattice plateau modulus. In the earlier study, the strain of B block chains was assumed to be the same as macroscopic strain (affine deformation), whereas the present study revealed that there exist four kinds of relaxation modes and the second mode possibly reduces the effective strain of B block chain at ultra low ω (in the equilibrium plateau region) to ∼1/3 of macroscopic strain. As noted before, if both the domain and lattice are deformed affinely, the birefringence will not be observed. In other words, the lattice effect is canceled by the domain effect. This means

4. SUMMARY We have examined rheo-optical response of the microdomain structure of a model SB block copolymer micellar lattice system 6585

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Macromolecules

Article

in tetradecane. Frequency dependence of K* suggested that the system exhibits four relaxation processes. These processes were attributable to (1) reorientation of B corona blocks, (2) reorientation of S blocks in the micellar core, (3) deformation of the core, and (4) lattice deformation, occurring from high to low frequencies in this order. Thus, the rheo-optical method is useful for characterizing the molecular origin of rheological behavior of block copolymers. The Onuki−Doi theory predicting the form birefringence of microdomain sutructures was extended to calculate strain-induced birefringence of micellar lattices. The theory reasonably explained the magnitude of strain-induced birefringence of SB block copolymers solution having micellar lattice. Measurements on other systems will be reported in our feature work.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partly supported by Grant-in-Aid for Scientific Research on Priority Area “Soft Matter Physics” from the Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 18068009 and 20340112).

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REFERENCES

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