Viscoelastic Properties of Tightly Entangled Semiflexible Polymer

Oct 11, 2018 - The absolute molar mass of CTC was determined using a gel permeation chromatography (GPC) system with refractive index, light scatterin...
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Viscoelastic Properties of Tightly Entangled Semiflexible Polymer Solutions Yuki Okada, Yuka Goto, Reina Tanaka, Takuya Katashima, Xinyue Jiang, Ken Terao, Takahiro Sato, and Tadashi Inoue* Graduate School of Science, Osaka University, 1-1, Machikaneyama, Toyonaka, Osaka 560-0043, Japan

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S Supporting Information *

ABSTRACT: The dynamic viscoelasticity and birefringence of well-characterized semiflexible polymer solutionscellulose tris(phenyl carbamate)/tricresyl phosphate solutionswere investigated over a wide concentration region ranging from a dilute to a tightly entangled regime where the entanglement spacing is smaller than the Kuhn segment size. The stress optical rule did not hold true in the tightly entangled regime, indicating that the molecular origin of stress is not simply attributed to the orientation of segments but to the bending of chains, which does not contribute to birefringence significantly. At high frequencies, the power law behavior with the exponent 3/4 due to the tension stress was observed for the complex shear moduli at all the concentrations.

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INTRODUCTION Static properties such as conformation and also molecular dynamics of polymers in solution have long been studied. According to previous studies, the conformation of linear polymers can be represented using the wormlike chain model.1 In this model, the radius of gyration can be written as a function of the contour length, L, and the Kuhn segment length, λ−1, which is defined as follows:2 λ −1 =

2ε kBT

A considerable amount of viscoelastic data for dilute solutions of semiflexible polymers was recorded10−13 and analyzed using the hybrid theory, in which the complex shear modulus of the polymer is represented by the sum of the orientational mode of the rod and Zimm modes for flexible polymers.14 Here, the Zimm modes were introduced to represent the internal motion of semiflexible polymers phenomenologically. These Zimm modes were first attributed to the bending motion of semiflexible polymers,15 but the calculated complex modulus using the bending motion showed the much weaker frequency dependence,16,17 ω1/4, than that of the Zimm modes, ω2/3. Morse et al. theoretically investigated the viscoelastic properties of semiflexible polymer solutions and introduced the tension mode, in which frequency dependence of the modulus at high frequencies is ω3/4 in the coil-like regime (L ≫ λ−1) solution in addition to the bending motion.17,18 Later, Morse also developed the theory for rodlike polymers in dilute solutions19 (L ≪ λ−1) and found that frequency dependence of the modulus in the tension mode is ω3/4 at high frequencies and ω5/4 at low frequencies. The frequency dependence of the modulus in the tension mode for intermediate rigidity was not discussed; however, it appeared to be close to ω3/4 as Lλ exceeded 1.19 He also developed the theory for entangled solutions including tightly entangled systems.17,18 However, despite considerable theoretical progess,13,17−20 viscoelastic properties of semiflexible polymers, L ∼ λ−1, are not fully examined experimentally. One of the problems for performing experiments using semiflexible polymers is the limited number of model systems suitable for viscoelastic measurements, which normally requires nonvolatile or less volatile solvents. In previous studies, a large amount of

(1)

Here, kB, ε, and T represent the Boltzmann constant, the bending force constant, and temperature, respectively. There are two limiting cases. One is the Gaussian limit (coil-like limit), Lλ → ∞ at a fixed L, and the other case is the rigid-rodlike limit, Lλ → 0 at a fixed L. Viscoelastic properties of flexible chains have been investigated over a wide concentration range.3 In theoretical studies, the stress supported by flexible chains can be calculated from the orientational anisotropy of segments; this method was experimentally verified as the stress-optical rule (SOR), which says that the molecular origin of stress and birefringence is the orientation of segments.4 Most of the molecular theories pertaining to polymers are based on SOR. Examples are the bead−spring models5,6 for dilute solutions and the reptation theory for entangled systems. These theories use the Gaussian spring to represent entropic forces of chains. For rigid rods, their viscoelastic properties in dilute solutions7 and viscosity over a wide concentration regime including the entangled regime8,9 are also understood. Semiflexible polymers correspond to cases where Lλ ∼ 1. Although physicochemical properties could be described by a function of Lλ, polymer species having large λ−1 values are commonly called semiflexible polymers. Semiflexible polymers include biopolymers such as DNA, polypeptides, and polysaccharides. © XXXX American Chemical Society

Received: July 23, 2018 Revised: October 11, 2018

A

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Macromolecules viscoelastic data for dilute solutions was obtained via bulk rheological measurements,10−13 and data for entangled solutions obtained via microrheological measurements21,22 were reported. However, these data covered the limited frequency region, and some data were not consistent with the frequency dependence of the high-frequency modulus where the internal mode becomes dominant. Recently, Jiang et al. reported that cellulose tris(phenyl carbamate) (CTC), which is known as a semiflexible polymer, with λ−1 = 11.5 nm, can be dissolved into tricresyl phosphate (TCP), and they measured its chain dimensions in TCP via small angle X-ray scattering.23,48 TCP is a glass-forming solvent, and we can change its viscosity by changing the temperature. Thus, solutions of CTC in TCP are anticipated to be an ideal system for viscoelastic measurements over a wide temperature/frequency range. A characteristic feature in the viscoelastic properties of semiflexible polymers is the significant contribution of internal motions (bending/tension modes) to the modulus. In conventional mechanical measurements, quantitative separation of the orientation and internal modes is difficult, and some hypotheses are necessary in this regard. Rheo-optical measurements involving birefringence measurements are effective for exploring the viscoelasticity of such complicated systems because birefringence has different susceptibility depending on the origin of stress. Through simultaneous measurement of birefringence and viscoelasticity, one can divide the viscoelasticity into components according to the origin of stress. Actually, this method has been utilized for mode separation of various polymeric systems such as poly(ionic liquid)s (orientation, local, and glass modes) 2 4 and poly(macromonomer) (main chain, branch, and glass modes)25,26 and also for estimation of partial stresses of matrix polymer and silica particles in silica-filled rubber to clarify reinforcement effects.27 In this study, we separated the modulus of semiflexible polymers into orientational and internal modes using the rheo-optical method. We explored the viscoelastic properties of CTC/TCP solutions over a wide frequency region and wide concentration regime covering the tightly entangled regime via simultaneous measurement of viscoelasticity and birefringence. Our goal was to estimate the internal mode of semiflexible polymers experimentally, particularly for tightly entangled systems.

Figure 1. Concentration regime of semiflexible polymers predicted by Morse.17 Predicted value of exponents, a and b, are equal to 0.5 and 0.4, respectively. L, T, LC, and D represent loosely entangled, tightly entangled, liquid crystal, and dilute regime, respectively. c* and c** respectively represent the concentration where Me is comparable with MK and the critical concentration at the liquid-crystalline transition at L ≫ Lp.

At a certain concentration, Me becomes comparable to MK. The difference between L and T regions comes from the relative ratio of Me and MK. The L region corresponds to ordinary entangled systems of flexible polymers, where Me > MK. Contrary to the L regime, Me is smaller than MK in the T regime. The boundary between L and T for the coil-like case (L > λ−1) can be estimated as c* ∼ δpπ(d/λ−1)2, which is transferred by the critical contour density for the LT boundary, ρ* ∼ 1/Lp2, proposed by Morse.17 Here, δp and d are respectively the density of polymers and steric diameter of the polymer chain. c** is the critical concentration at the liquidcrystalline transition. Now, we discuss the dynamics and viscoelastic properties of the semiflexible polymers. For the case of polymer solutions, the complex shear modulus, G*(ω), can be divided into two contributions. G*(ω) = Gp*(ω) + Gsol*(ω)

(3)

Here subscripts, “p” and “sol” represent respectively the polymeric and solvent contributions. The following approximation is often considered to estimate Gsol*(ω):

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THEORETICAL BACKGROUND Figure 1 shows the theoretical prediction of the viscoelastic diagram for semiflexible polymers. This diagram was initially proposed by Morse.17 Herein, we modified the diagram by changing the vertical and horizontal axes to experimental quantities that can be measured directly. We used the molar mass of the Kuhn segment, MK, for determining the flexibility index instead of the persistent length, Lp ≡ λ−1/2, which is used in the Morse theory. Thus, when the molar mass of the polymer is larger than MK, the polymer shows a coil-like conformation. Abbreviations, D, L, T, and LC mean dilute, loosely entangled, tightly entangled, and liquid crystalline regimes, respectively. The boundary between L and D is characterized by the molar mass between entanglements, Me. When the molar mass of the polymer, M, is larger than Me, entanglements are formed. The concentration dependence of Me is well-known (Me ∼ c−1∼−1.3) for the case of flexible polymers.

Gsol*(ω) = (1 − vc)astrainGsol,0*(ω)

(4)

Here, Gsol,0*(ω) is the complex modulus of the neat solvent, and v is the specific volume of the polymer. astrain is the strain amplification factor, which was calculated as (1 − vc)−1 by Einstein28 and more recently by Domurath et al.29 Thus, Gsol*(ω) = Gsol,0*(ω) is often used to estimate the solvent contribution. However, for the case of a glass-forming solvent, Gsol*(ω) = Gsol,0*(ω) does not hold because of the cooperativity of the polymer and solvent dynamics. For polymers having high the glass transition temperatures, the solvent dynamics is retarded by the existence of polymer chain, and this effect is concentration-dependent, resulting that the frequency dependence of Gsol*(ω) varies with polymer concentration. Thus, strictly speaking, the solvent contribution cannot be subtracted by using Gsol,0*(ω). Therefore, we did not consider to subtract the solvent contribution from G*(ω) with Gsol,0*(ω) not to induce any artifact in this study. B

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X-ray scattering measurements.23 The number of Kuhn segments per chain, λL, for the present sample was estimated to be 9.2. Measurements. For the sample solutions and neat TCP, dynamic viscoelastic measurements were conducted under dry air with a conventional rheometer (ARES G2, TA Instruments) utilizing a parallel-plate fixture (diameter = 4.0 mm for the measurement in the glassy and glass-to-rubber transition zones, diameter = 25 mm for the measurement in the flow zone of nonentangled solutions and TCP and diameter = 8 mm for the measurement in the flow and rubbery plateau zones of the entangled sample). The complex modulus was measured at angular frequencies (ω) ranging between 0.01 and 100 s−1. The oscillatory strain amplitude was kept small (from 0.003 to 0.2) to obtain the storage and loss moduli (G′ and G″) in the linear viscoelastic regime. For temperature control, a forced convection oven (TA Instruments) was utilized, and a gas chiller (PGC-152, Polycold Systems Inc.) was also utilized for measurements at temperatures lower than 30 °C. Simultaneous dynamic measurements of flow birefringence and shear stress were also conducted under dry air with a homemade shear birefringence apparatus utilizing a parallel glass plate fixture. Details of the apparatus have been reported elsewhere.26 The contact area of the sample solutions and glass plate fixture was ∼16 mm2 (4 mm × 4 mm) for the tests during the transition from the glassy to the glasstransition zone and ∼225 mm2 (15 mm × 15 mm) for the tests at lower frequencies. The gas chiller was utilized for shear birefringence measurements at temperatures lower than 30 °C. The measurements were conducted in the ω range between 0.03 and 1000 s−1. The oscillatory strain amplitude was kept small (from 0.003 to 0.2) to obtain the storage and loss moduli (G′ and G″) and the real and imaginary components of the stress-optical coefficient (K′ and K″) in the linear viscoelastic regime. The reliability of the modulus obtained using the homemade apparatus was checked by comparing with the data obtained using the conventional rheometer.

Morse used a discretized model of a wormlike chain as a chain of N beads connected by very stiff springs with a preferred distance, a, between neighboring beads, and a threebody bending potential that is a discretized version of the bending energy.17,18 The polymeric mode can be decomposed into three components: Gp*(ω) = G link *(ω) + G bend*(ω) + Gtens*(ω)

(5)

Here, the subscripts “link”, “bend”, and “tens” respectively correspond to stresses arising from “orientational entropy of links”, “bending of chains”, and “tension of chains”. Morse reexpressed the first two terms of this equation for “orientational stress” and “curvature stress”, but we will not use his modulus here.17 According to his result for dilute solutions, G*tens(ω) becomes dominant at high frequencies and G*tens(ω) ∝ ω3/4 for the coil-like conformation (L ≫ λ−1).18 Therefore, the tension stress significantly contributed to the modulus at high frequencies in semiflexible polymers. A similar decomposition into component functions is possible for birefringence: K *(ω) = K p*(ω) + K sol*(ω)

(6)

K p*(ω) = Klink*(ω) + Kbend*(ω) + K tens*(ω)

(7)

Here, the complex strain-optical coefficient, K* = Δn*/γ* is defined as the complex ratio of the oscillatory birefringence, Δn*, to the sinusoidal strain, γ*; similar to the complex shear modulus, G* = σ*/γ* is defined as the complex ratio of the oscillatory stress, σ*, to γ*. In the following discussion, we ignore the form birefringence of the polymer. This approximation works when ∂n/∂c ∼ 0. This condition would be fulfilled for CTC/TCP solutions because the refractive index of CTC is estimated to be 1.57, and this value is close to 1.56 of TCP.30 The ordinary SOR can be described as K* = CG*, where C is the stress-optical coefficient. In the case of semiflexible polymers, the SOR can be modified as follows:

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RESULTS AND DISCUSSION Concentration Regime of CTC. We first estimated the concentration region of CTC solutions with preliminary viscoelastic measurements. Figure 2 shows the concentration regime of CTC/TCP solutions. The figure includes preliminarily data of Me at 0.02 g cm−3 obtained using the Mw = 2.04M sample.23 Blue marks represent the concentrations of our test solutions. The boundary between L and D can be

Klink* + Kbend* + K tens* = C linkG link *(ω) + C bendG bend*(ω) + C tensGtens*(ω)

(8)

Here, Clink, Cbend, and Ctens represent stress-optical coefficient for the orientational, bending, and tension modes, respectively.

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EXPERIMENTAL SECTION

Sample. The synthesis of CTC was reported in a previous study.30 A fraction with Mw = 110 kg mol−1 (Mw/Mn = 1.08) was used here. The absolute molar mass of CTC was determined using a gel permeation chromatography (GPC) system with refractive index, light scattering, and viscosity detectors (Malvern Panalytical). Details of GPC measurements are presented in the Supporting Information. The solvent, TCP, was purchased from Kishida Chemical Co. and used without further purification. Dichloromethane (DCM), purchased from Wako, was used as a cosolvent without further purification to prepare a high-concentration test solution. The CTC sample used in a previous study was dried in a vacuum oven at 25 °C overnight and dissolved in TCP at room temperature to prepare 0.01 and 0.05 g cm−3 test solutions. The solution with the highest concentration, 0.23 g cm−3, was prepared using the cosolvent method. Weighed CTC was first dissolved in a mixture of TCP and DCM. The concentration of the resulting solution was ∼0.05 g cm−3. Then, the solution was kept under atmospheric pressure for 2 days to allow evaporation of DCM and then in a vacuum oven at 25 °C overnight to remove DCM completely. The Kuhn segment length and molar mass of the Kuhn segment in TCP were estimated to 11.5 nm and 12 kg mol−1, respectively, in a previous experiment using with small-angle

Figure 2. Concentration dependence of the molar mass between entanglements for CTC/TCP solutions. The molar masses of a viscoelastic segment and Kuhn segment are shown as broken lines. The same abbreviations as in Figure 1 are used. c* and c** for M > MK are also presented. C

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Macromolecules considered as certain concentrations where M = Me. Thus, the sample with the lowest concentration corresponds to the dilute solution. The boundary between L and T is a controversial issue depending on choice of flexibility index. Several indexes such as λ−1 and LP have been used to characterize the stiffness of the chain. The viscoelastic segment (Rouse segment) defined as the minimum structural unit that holds the orientational stress also can be used as flexibility index. In contrast to λ−1 and LP which can be determined from the chain dimension, the viscoelastic segment is determined by the limiting modulus at high frequencies for the orientational stress determined by the rheo-optical measurement (see the Supporting Information for details of the determination of the viscoelastic segment). For the case of polymer melts, the viscoelastic segment size is very close to the Kuhn segment size.31 On the other hand, the viscoelastic segment size in solution becomes larger than the Kuhn segment size.32 Larson explained this larger viscoelastic segment size in solution by the consideration of the energetic barriers to bond rotation.33 In this study, we considered that the viscoelastic segment size is suitable for the boundary between T and L because we concern molecular dynamics and viscoelastic properties of polymer solution. This choice will be verified later in this study. The Me value of the sample with the highest concentration is comparable to the molar mass of the viscoelastic segment, Ms, as shown in the figure. This sample is located on the boundary of L and T entangled regimes, where c* ∼ 0.23 g cm−3. This viscoelastic diagram is a guide for further discussion; therefore, the categorization based on this diagram, particularly for the sample with the highest concentration, will be reexamined based on the result of the rheo-optical measurements. The phase boundary of the liquid crystalline region was estimated from the reported result for the CTC/THF solution, with c** ∼ 0.4 g cm−3.34 The critical concentration of the liquid-crystalline transition, c**, depends slightly on temperature. Here, we discuss only the estimation at 25 °C to avoid complications. We noted that the concentrations of all our samples were below c**. Dilute Solution. We examined the rheological properties of dilute solutions to characterize the CTC polymer. Figure 3 shows the storage and loss moduli (G′ and G″) for the 0.01 g cm−3 CTC solution in TCP, together with the viscoelastic data for neat TCP. Here, the method of reduced valuables was used to construct the composite curves.35 The reference temperature, Tr, is −10 °C. The time−temperature shift factors for the 0.01 g cm−3 CTC/TCP solution and neat TCP are presented in Figure S1 of the Supporting Information. At high frequencies, G′ and G″ of the CTC solution are close to those of the solvent showing typical glass transition behavior. The values for the CTC solution are slightly shifted to lower frequencies compared to those for neat TCP. This can be attributed to the slightly higher glass transition temperature of the solution than the solvent. At the middle frequencies, G′ showed power law behavior, G′ ∝ ω2/3, over a wide frequency region, 2 < log(ωaT/s−1) < 6. Compared to G′ of TCP, a remarkable increase of G′ in these frequencies was observed. The exponent of the power law, 2/3, agrees with previous experimental studies and the exponent used in the hybrid theory10−13 but smaller than the theoretical value of 3/4 for the tension mode of semiflexible chains with Lλ−1 ≫ 1.18 We noted that this power law behavior should not be attributed to the Zimm mode, which can be observed after relaxation of the tension mode, because such power law behavior is not

Figure 3. Dynamic storage and loss moduli (G′ and G″) and real and imaginary components of complex strain-optical coefficient (K′ and K″) for the CTC110k 0.01 g cm−3 solution. The reference temperature is −10 °C. The gray broken line represents storage and loss moduli for TCP (G′sol and G″sol). The blue continuous line represents the power law G′ ∝ ω2/3.

observed in K′ which we describe later. At low frequencies, log(ωaT/s−1) < 1.0, the terminal flow behaviors G′ ∝ ω2 and G″ ∝ ω were observed. In addition, the real and imaginary components of K* are presented in Figure 3. In constructing the composite curves for them, we used the same horizontal shift factor, aT, for the complex modulus and also a small temperature-dependent vertical shift factor, bT. This is mainly because the optical anisotropy of the repeating units of CTC would depend on temperature. The averaged direction of three benzene rings in side chains of CTC would vary with temperature. A similar effect involving the side chains is observed for poly(methyl methacrylate).36 At high frequencies, K′ and K″ are positive, and this can be attributed to the contribution of the solvent, Ksol*, similar to the case of G*. At low frequencies, birefringence originates in the link orientation mode of CTC, Klink*. The sign of orientational birefringence was negative. This is reasonable because CTC has three phenyl rings in the side chains of its repeating units like polystyrene (PS), which shows negative orientational birefringence.32 The frequency dependence of the negative components of K′ and K″ at low frequencies is rather close to the single Maxwellian model than the Zimm model, reflecting the small number of Kuhn segments per chain, n = 9.2. Thus, CTC is optically like a rigid-rod-like polymer while its viscoelasticity is very similar to that of flexible polymers. As described in the previous section, Kp* can include the three components, namely link, bend, and tension modes in principle. However, the stress-optical coefficients of the bending and tension modes are expected to be much smaller than that of the orientation mode. This is because Glink*(ω) has an entropic origin while the others are basically energetic, which means that the compliance of Gbend*(ω) and Gtens*(ω) is much smaller than Glink*(ω). A smaller compliance means smaller internal strain per unit stress. As birefringence originates from internal strain, the above argument leads to a lower stress-optical coefficient for the bending and tension modes. Therefore, the following inequality holds: Klink*(ω) ≫ Kbend*(ω) ∼ K tens*(ω) D

(9)

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model. The obtained G′link is shown in Figure 4 as the black broken line. Figure 4 also indicates that the power law behavior of G′ at approximately 3 < log(ω/s−1) < 6 should not be attributed to the orientation mode but to the tension mode. The theory proposed by Morse et al. predicts that the contribution of the bending stress is small and negligible when compared to the tension stress; Gtens′ > Gbend′. In such cases, G* − G*link can be regarded as G*sol + G*tens, and the modulus at middle frequencies is mainly supported by the tension mode. Thus, we can obtain G*tens experimentally, and the real component of G*tens is shown in Figure 4 as the green broken line. Details of the analysis method are presented in the Supporting Information (S3). Our result is qualitatively consistent with the prediction by Morse, Gtens′ ∝ ω3/4 (Gtens″ also should be Gtens″ ∝ ω3/4), although the total modulus showed Zimm-like behavior with G′ ∝ ω2/3 (G″ also should be G″ ∝ ω2/3) at middle frequencies. Thus, the separation into modes is extremely important and necessary to understand the molecular origin of stress. The above argument verifies that CTC can be regarded as a semiflexible polymer. The viscoelastic segment size, MS, can be estimated from the limiting value of C−1K′(∞) ∼ C−1Klink′(∞) ∼ 1300 Pa at high frequencies. Details of the estimation method of MS were presented in our previous studies, and its application to the CTC solution is explained in the Supporting Information (S4) briefly.37 The estimated value of MS is 15 kg mol−1, which is close to the Kuhn segment size, MK = 12 kg mol−1. The obtained MS values are summarized in Table 1. Contrary to PS

This inequality is widely accepted, and only the link orientation is considered for the origin of birefringence in the Morse theory. Thus, the assumptions Cbend = 0 and Ctens = 0 are quite reasonable. In addition, we noted that the relaxation time distribution of K* for the shorter CTC sample was very close to the single-exponential relaxation, as shown in the Supporting Information (S6). This result supports our assumption that the relaxation of birefringence is governed only by the link orientation mode. Consequently, eq 7 can be rewritten as follows: K p*(ω) ≅ Klink*(ω) = CG link *

(10)

Here, C corresponds to Clink, and we treat Clink as C in the following discussion. Thus, the SOR between the total modulus and total strain-optical coefficient, Kp* = CGp* (total SOR in short), is not expected to hold for semiflexible polymers. However, in the flow region where the orientation mode is dominant and K′ ≈ CG′ holds true, the comparison between G* and K* is helpful for analyzing the internal modes. Figure 4 compares the real and imaginary components of G* and K′C−1. Reflecting the above difference between G* and

Table 1. Viscoelastic Segment Size of CTC

Figure 4. G′, G″, and K′ divided by stress-optical coefficient (K′C−1) for CTC110k 0.01 g cm−3 solution. The reference temperature is −10 °C. Three broken lines represent G′sol (blue), G′link obtained by fitting K′C−1 (gray), and G′tens obtained from analysis using SOR (see Supporting Information (S3)).

(11)

K ′ ≅ CG link ′

(12)

C−1K′link(∞) (kPa)

MS (kg mol−1)

0.01 0.05 0.23

1.3 7.2 28

15 13 16.5

or polyisoprene (PIP) solutions, the concentration dependence of MS was not clearly observed.32,38 This result is reasonable because a larger viscoelastic segment size of PS or PIP in dilute solutions is attributed to the faster solvent dynamics and is therefore expected to be observed only for flexible polymers having small MK.33 A detailed discussion on this issue is provided in a previous study.39 The viscosity of the solution, η, can be calculated from the terminal behavior, G″ = ηω at low frequencies. G″ is close to C−1K″ at low frequencies. This means that the bending and tension modes contribute to η; ηTB is small, and therefore η is mainly determined by the link orientation mode. The concentration dependence of the link orientation mode for semiflexbile polymers was discussed by Sato et al. using the fuzzy cylinder model.40 In their theory, the tension modes were not considered. For a precise theoretical estimation of η, the contribution of the bending and tension modes should be considered, but the present results indicate that the contribution of both the modes are not significant. Tightly Entangled Solution. Figure 5 displays the real and imaginary components of G* of the CTC solution at the highest concentration, c = 0.23 g cm−3, which is lower than the critical isotropic−cholesteric liquid-crystalline transition concentration.34 The rubbery plateau region where G′ > G″ is clearly observed in the range −3 < log(ωaT/s−1) < 0. The estimated rubbery plateau modulus is approximately 29000 Pa,

K*, the total SOR, K* = CG*, does not hold as predicted. In the limited frequency region, log(ωaT/s−1) < 2, the relationship K′ = CG′ holds true with C = −3.0 × 10−8 Pa−1. This value is ∼6 times larger than PS and is in accordance with the larger Kuhn segment sizes. The reason for the validity of K′ = CG′ at low frequencies is because other components have shorter relaxation times than the orientation mode and relax at low frequencies. In this region, G* and K* may be simplified as follows: G* = G*link + iηTBω + iηsω

c (g cm−3)

Here, ηTB and ηs represent respectively the viscosity of the tensile and bending contributions and that of the solvent. Equations 11 and 12 are valid only at low frequencies, and eq 12 is essentially the same as eq 10. These results are consistent with theoretical predictions and the SOR for semiflexible polymers holds true only at long times where the link orientation mode is dominant. We determined G′link from K′pC−1 and represented it using the generalized Maxwell E

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weak, indicating that the orientation of the chain is almost constant while the modulus significantly decreases from G′ ∼ 107−104 Pa. As a result, the SOR does not hold in this frequency region. The contribution of the bending modes to the modulus was predicted by Morse,18 but his theory using a fixed tube predicts the simultaneous relaxation of the orientation and bending modes. In this case, the SOR holds true with a smaller stress-optical coefficient in the rubbery plateau zone. The breakdown of the SOR in the rubbery plateau zone indicates that the bending stress has a shorter relaxation time. We will discuss this issue in the section on mode separation of modulus. The existence of such a mode justifies that the 0.23 g cm−3 CTC/TCP solution is a tightly entangled system. Analysis of Link Orientation Mode. As discussed above, the envelope of K′ and K″ likely corresponds to the link orientation mode. Here, we compare G*link to G* of other flexible polymers. For this purpose, we determined G′link and G″link of the CTC solution from the envelope of K′C−1 and K″C−1. Figure 7 compares the determined G′link and G″link with

Figure 5. G′, G″, K′, and K″ for CTC110k 0.23 g cm−3 solution. Reference temperature is −10 °C. Plots for K′ and K″ at each temperature are denoted as follows: red circles (80 °C), brown circles (70 °C), green circles (60 °C), turquoise circles (50 °C), blue circles (40 °C), purple circles (30 °C), red squares (20 °C), brown squares (10 °C), green squares (0 °C), turquoise squares (−10 °C), blue squares (−20 °C), and purple squares (−30 °C).

and this value gives Me = 16 kg mol−1, which is comparable to its viscoelastic segment size, 16.5 kg mol−1, and also the Kuhn segment size, 12 kg mol−1. Thus, the CTC solution at the highest concentration is expected to show the characteristics of a tightly entangled system. At higher frequencies, G′ and G″ show a power low behavior, G′ ∝ G″ ∝ ω3/5, over a wide frequency region corresponding to 0 < log(ωaT/s−1) < 3. The apparent powerlaw exponent, 3/5, is smaller than the theoretical value of 3/4 for tightly entangled systems or semiflexible polymer networks and experimental values;41−43 G′ ∝ ω3/4. The composite curves for K′ and K″ are shown in Figure 5. In constructing composites curve for them, we used the same vertical shift factor obtained at low concentrations. The superposition of K″ is not perfect in the rubbery region. However, the data at low frequencies for each temperature formed an envelope, and systematic small deviations were observed only for the data at high frequencies for each temperature. These deviations could be related to the small contribution of the bending modes having a small stress-optical coefficient; we ignored the contribution of the bending modes to birefringence in the present discussion for simplicity. The relaxation at high frequencies, log(ωaT/s−1) > 2, with positive birefringence can be attributed to the solvent contribution, Ksol*. Figure 6 compares G′, G″, K′/C−1, and K″/C−1 for the CTC110k solution at c = 0.23 g cm−3. The relaxation of K′C−1 around the frequency region of −2 < log(ωaT/s−1) < 4 is very

Figure 7. Comparison of orientation modes for the entangled system. G′link and G″link respectively represent the storage and loss moduli of link orientation mode in the tightly entangled system with Mw/Mn ∼ 1. G′PIP and G″PIP represent those of PIP, which is a loosely entangled system with Mw/Mn ∼ 1.13.44 G′PS and G″PS represent those of PS, which is a loosely entangled system with Mw/Mn ∼ 1.92.45

previously reported experimental data of G′ and G″ PIP44 (Mw = 1.1M) with a narrow molar mass distribution, Mw/Mn = 1.13, and PS (Mw = 250K) with the most probable molar mass distribution, Mw/Mn = 1.92.45 These data were shifted by certain horizontal shift factors so that G″ values overlap with each other in the terminal flow zone of G″ (G″ ∝ ω). It is known that G′ and G″ spectra show a universal frequency dependence irrespective of the molar mass when M/Me > 10. M/Me values of PIP and PS are considerably high; MPIP/Me,PIP ∼ 280 and MPS/Me,PS ∼ 14. The contribution of the Rouse mode was not recognized in the data for PIP. For PS, the contribution of the Rouse mode was carefully subtracted for an easier comparison. Therefore, these data can be regarded as pure reptation modes of the loosely entangled system with narrow and broad molar mass distributions. It is clear that the CTC solution has a much broader relaxation time distribution in the terminal distribution than PIP and is rather close to PS. This result indicates that G*link is the orientation mode for the entangled system and M/Me of CTC would have a broader relaxation time distribution. In the case of ordinary polymers, the distribution of M/Me comes from the distribution of M. On the other hand, the molar mass distribution of CTC is narrow. One possible explanation for this broad relaxation time distribution in CTC is as follows. For tightly entangled systems, the entangled point may be not so stable because the

Figure 6. G′, G″, K′C−1, and K″C−1 for CTC110k 0.23 g cm−3 solution. The reference temperature is −10 °C. The gray broken line represents K′link(∞) C−1, which is equal to G′link(∞). F

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Macromolecules chains cannot entangle deeply owing their rigidity. These unstable entanglements may cause a broader distribution of Me. Another possible explanation is the large concentration fluctuations of the CTC solution because the concentration is close to the liquid-crystalline transition concentration, ∼40 wt %.34

Figure 9. Hypothetical relaxation process of bending stress constrained by entanglements in a short time scale, Δt ≪ τlink (top figure). Solid blue lines and open circles represent polymers and obstacles presented by entanglements, respectively. The local small conformation change in polymer orientation can occur on removing the obstacle. This process releases the bending stress significantly but does not contribute to overall orientation of polymers (bottom figure).

Figure 8. Component functions of complex modulus for CTC110k 0.23 g cm−3 solution. The reference temperature is −10 °C. Broken and continuous lines respectively represent the loss and storage moduli of each mode.

Separation of the Complex Modulus into Three Modes. Figure 8 shows the component functions of G′ and G″ for the CTC 0.23 g cm−3 solution. Here, G′link and G″link were determined from the envelope of K′C−1 and K″C−1 as explained earlier. G*tension was obtained by fitting for G*− G*link (see Supporting Information (S5)). The resultant part of the modulus was attributed to G*bend. The frequency dependence of G*bend is rather close to the complex modulus of the Rouse theory. Therefore, we used the Rouse-type spectra for the fitting G*bend (see Supporting Information (S5)). The characteristic relaxation times of each mode are also shown in Figure 8. τlink is the second-moment average viscoelastic relaxation time of G*link. τbend was obtained by fitting with the Rouse spectra, and τtens was estimated using G′tens(τtens−1) = cRT/Ms (see Supporting Information (S5)). As mentioned earlier, Figure 8 indicates that G*bend has a shorter relaxation time than G*link. G*bend is a local stress supported by a portion of the bent viscoelastic segment with a shorter spatial scale than λ−1. Therefore, the bended conformation of segments constrained by entanglements can relax with release of constrain accordingly. This consideration indicates that the bending stress of segments can relax not only through reptation but also via constraint release. Figure 9 shows a schematic illustration of the hypothetical relaxation process of the bending stress. The total orientation degree of segments does not change with local conformation changes depicted in the bottom of this figure. Concentration Dependence. Finally, we recap the above discussion to explain concentration dependence. Figure 10 compares G′ and K′pC−1 (G′link) of CTC at various concentrations. K′pC−1 is represented with a function obtained by fitting for the positive component of K* using the generalized Maxwell model. The relaxation spectra around the glass transition region are broadened with an increasing concentration, which may be attributed to separated glassy modes or dynamic heterogeneity,46,47 often observed in polymer blends or solutions composed of components having largely distinct glass transition temperatures.32 On the other hand, in the low-frequency side of the glass-to-rubber

Figure 10. Concentration dependence of G′ and K′pC−1 in CTC/ TCP solutions.

transition zone, the power law behavior, G′ ∝ω3/5, is systematically observed for all the examined samples except for dilute solutions. The exponent for G″ was rather close to 3/ 5 for the sample with the highest concentration. The characteristic exponent of 3/4 for the tension stress cannot be clearly observed without mode separation. In the rubbery plateau zone, the bending mode constrained by entanglements is observed for the sample solution with the highest concentration. In addition, the relaxation of birefringence is rather insensitive to polymer concentration. We also note that for all the samples K′ = CG′ holds true in the terminal flow zone.

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CONCLUSION In this study, we first determined the viscoelastic properties of well-characterized semiflexible polymers over a wide frequency region ranging from the glassy to the terminal flow region. The viscoelastic data included the complex modulus and strainoptical coefficient for the tightly entangled regime. The full relaxation spectra for tightly entangled regime have not been reported. The main conclusion is that the ordinary SOR is not G

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Macromolecules

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valid for tightly entangled systems in the rubbery plateau zone although the viscoelastic spectra are very similar to those of ordinary flexible polymer solutions. The breakdown of the SOR can be attributed to the large contribution of the bending and tensile stresses to the modulus, while the strain-induced birefringence is mainly determined by the orientation mode. We believe that our systematic observations will contribute to the progress in the study of semiflexible polymers. The nonlinear viscoelasticity of tightly entangled systems is an interesting concept, and this will be the subject of our future studies.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01582. The estimation method of absolute molar mass by GPC, time−temperature shift factors for the composite curves, the estimation of G*tens of CTC/TCP 0.01 g cm−3, the evaluation of viscoelastic segment, the estimation of G*tens and G*bend of CTC/TCP 0.23 g cm−3, and the real and imaginary components of G* and K* data for 0.0123 g cm−3 CTC25k solution in TCP (PDF)

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AUTHOR INFORMATION

Corresponding Author

*(T.I.) E-mail: [email protected]. ORCID

Yuki Okada: 0000-0002-2596-2623 Takuya Katashima: 0000-0001-6184-3835 Ken Terao: 0000-0001-7363-4491 Takahiro Sato: 0000-0002-8213-7531 Tadashi Inoue: 0000-0002-9934-1299 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was partially supported by JSPS KAKENHI Grant JP16H04204 and ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan). The authors thank Prof. Urakawa at Osaka University for the helpful discussions and valuable advice of this study.

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REFERENCES

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