Dynamics of Concentrated Polymer Solutions Revisited

Article pubs.acs.org/Macromolecules

Dynamics of Concentrated Polymer Solutions Revisited: Isomonomeric Friction Adjustment and Its Consequences Zhi-Chao Yan,†,‡ Bao-Qing Zhang,† and Chen-Yang Liu*,† †

Beijing National Laboratory for Molecular Sciences, CAS Key Laboratory of Engineering Plastics, Joint Laboratory of Polymer Science and Materials, Institute of Chemistry, The Chinese Academy of Sciences, Beijing 100190, P. R. China ‡ University of Chinese Academy of Sciences, Beijing 100049, P. R. China S Supporting Information *

ABSTRACT: In concentrated polymer solutions, the concentration (ϕ) dependence of the terminal relaxation time τd reflects ϕ-dependent changes of several factors, the monomeric friction ζ0(ϕ), the entanglement length, a(ϕ), and the correlation length, ξ(ϕ). Usually, the effect of the latter two factors on τd can be cast in a simple power law, τd ∼ ϕv. This power law form is to be examined for τd after correction of the changes of ζ0 with ϕ, but this iso-ζ0 correction is an unsettled problem. The correction based on the concept of “iso-free-volume” has been attempted in literatures. This study focused on four groups of solutions with different local frictional environments to examine universal validity of this correction. The isothermal data of τd were rheologically measured, and then corrected to the isofrictional (iso-ζ0) state. After this correction, τd of most solutions in small molecule solvents showed the power law behavior τd ∼ ϕv with exponent of v = 2.0 ± 0.2, irrespective of the solvent type, either neutral small molecules or an ionic liquid (organic salt), and of the difference of the glass transition temperatures of the solvent and polymer. In contrast, the exponent became smaller (v ≈ 1.3) for the solutions in an oligomeric solvent. These results are discussed within the frame of the two-length scaling theory that considers changes of ξ and a with ϕ.

1. INTRODUCTION The reptation model has achieved great success in describing the terminal dynamics of the entangled homopolymer melt.1 In this model, a linear chain reptates within a 1-D curvilinear tube (that represents the entanglement network). During this reptative motion, the polymer chain experiences the topological hindrance from the tube as well as the inter/intra-molecular monomeric frictions.2 For a linear polymer with molecular weight M, its terminal relaxation time τd can be simply related to these two factors as τd = K (M )ζ0(T )

Although the dynamics of polymer solutions is very complicated, a part of the effects on τd, being insensitive to the friction, can be expressed as a simple power law relationship with the polymer volume fraction ϕ:6−8 τd = K (M )ϕvζ0(ϕ , T )

Here, v is the power-law exponent, and ζ0(ϕ, T) is the monomeric friction factor of polymers in the solution. The two ϕ-dependent factors, ϕ v and ζ0 (ϕ, T), are separable experimentally. The main purpose of this paper is to reduce the ζ0(ϕ, T) factor in a reference state to explore the physics behind the ϕv factor. Recently, Colby6 made a review of the structure and linear viscoelasticity of flexible polymers in solutions from both experimental and theoretical aspects. Features of the ϕv and ζ0(ϕ, T) factors reported in literature (including Colby’s review6) can be summarized as below. Dilution Power Law ϕv. In entangled concentrated polymer solutions, the plateau modulus GN0(ϕ) decreases with decreasing ϕ as3,4

(1)

Here, K(M) is the structure factor representing the global relaxation of the long chain M, and ζ0(T) is the monomeric friction factor that decreases with increasing temperature T. In entangled concentrated polymer solutions3,4 or in miscible blends,5 the chain motion appears to be still described by the reptation model. However, the solvent (or blending partner) changes τd of the chain in different ways. First of all, the solvent tunes a local friction, thereby either accelerating or retarding the local segmental relaxation in a nontrivial way. The solvent also dilutes the entanglement network and hence accelerates the global motion of the chain. Furthermore, the hydrodynamic and excluded volume interactions are important inside the correlation blob in the solution. These effects change with the concentration, solvent quality, and even the size of solvents, thereby strongly affecting the dynamics and rheology of polymer solutions. © 2014 American Chemical Society

(2)

G N0(ϕ) ∼ ϕ2.0 − 2.3

(3)

Received: November 12, 2013 Revised: May 17, 2014 Published: June 19, 2014 4460

dx.doi.org/10.1021/ma402338h | Macromolecules 2014, 47, 4460−4470

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According to an expression G0N(ϕ) = ϕρRT/Me(ϕ) with ρ and R being the density and gas constant,7 the entanglement molecular weight Me(ϕ) in the solution can be expressed as Me(ϕ) ≈

Me(1) ϕ1.0 − 1.3

(4)

The pure reptation model1 utilizes this Me(ϕ) to describe τd in a power-law form, ⎛ M ⎞3 ⎛ N ⎞3 τd ≈ τe(ϕ , T )⎜ ⎟ ≈ τe(ϕ , T )⎜ ⎟ ⎝ Me(ϕ) ⎠ ⎝ Ne(ϕ) ⎠

(5)

Here, τe(ϕ,T) is the relaxation time of the entanglement strand being proportional to ζ0(ϕ, T), N = M/M0, and Ne(ϕ) = Me(ϕ)/M0 with M0 being the monomer molecular weight. Equation 5 is further elaborated in section 6 according to a physical picture of hierarchical dynamics in solutions.6,9 The key in this picture is the existence of two dynamic lengths for entangled polymers in the solution, the entanglement length a and the correlation length ξ. (This makes a contrast from the dynamics in melt governed only by a.) The ϕ dependence of τe emerges not only from the ζ0(ϕ, T) factor but also from the hydrodynamic interaction within the correlation length ξ(ϕ). Thus, the terminal relaxation time τd changes with ϕ through the Ne(ϕ), ζ0(ϕ, T), and ξ(ϕ), as explained in more detail in section 6. Usually, the non-f rictional part of this change (due to Ne(ϕ) and ξ(ϕ)) can be cast in a power law form, τd ∼ ϕv (cf. eq 2), in particular in the concentrated regime. The friction factor ζ0(ϕ, T) is solvent-specific, and its features can be summarized as below. Monomeric Friction Factor ζ0(ϕ, T). The solvent usually provides an extra free volume to accelerate the segmental relaxation of polymers, or, equivalently, reduces the monomeric friction. An example is shown in Figure 1a for 1,4-polybutadiene (PBD) solutions in phenyloctane (PHO) (data taken from Table 10 in ref 3). The ζ0 (ϕ)/ζ0(1) ratio decreases with decreasing ϕ to 0.1 but becomes insensitive to ϕ on further dilution.10 Namely, ζ0 (ϕ) changes with ϕ significantly in the concentrated regime (ϕ > 0.1) but this change becomes negligible in the semidilute regime (ϕ < 0.1). Because of this feature, semidilute solutions have been employed to investigate the dilution effect represented by the ϕv factor explained above.11,12 In contrast, in concentrated solutions, analysis of the ϕv factor requires the isofrictional (iso-ζ0) correction. An example is shown in Figure 1b, where the zero-shear viscosities η0 of PBD/PHO solutions are compared at a fixed temperature (25 °C) and in the iso-ζ0 state: The weaker ϕv dependence of η0 in the iso-ζ0 state demonstrates the importance of the iso-ζ0 correction in the analysis of τd (∼η0/GN). How to obtain ζ0 (ϕ) at a given temperature is a question. ζ0(ϕ, T) depends on both T and ϕ, so that ζ0 (ϕ2, T′) and ζ0 (ϕ1, T′) at different concentrations ϕ2 and ϕ1 but at the same temperature T′ do not coincide with each other; cf. Figure 1a. The principle of the iso-ζ0 correction proposed by Masuda et al.13 is to find a temperature T″ being associated with ϕ2 and satisfying a relationship ζ0(ϕ2 , T ″) = ζ0(ϕ1 , T ′)

Figure 1. (a) Normalized monomeric friction factor ζ0(ϕ)/ζ0(1) in PBD/PHO solutions at 25 °C as a function of the concentration ϕ. ζ0(ϕ)/ζ0(1) decreases with dilution until approaching a constant value at ϕ ≈ 0.1. (b) Comparison of the zero-shear viscosity η0 at isothermal state (25 °C) and at iso-ζ0 state [ζ0(melt, 25 °C)]. Data obtained from ref 3.

Thus, eq 6 can be converted to the following relationship, f (ϕ2 , T ″)/B2 = f (ϕ1 , T ′)/B1

Here, B is a constant of order unity; B1 and B2 are defined for the solutions having ϕ1 and ϕ2. Thus, the iso-ζ0 correction is equivalent to the iso-free-volume adjustment within the frame of the free volume theory. In practice, the T dependence of τd or η0 can be transformed to that of f(ϕ, T) via the WLF fitting.13 This f(ϕ, T) is utilized to determine T″ (cf. eq 8) and calculate the iso-ζ0 correction factor, ζ0(ϕ2, T″)/ζ0(ϕ2, T′), (cf. eq 7); see also Figure S1 in the Supporting Information. Further details of this calculation are described later in section 4 and Appendix A. As reviewed above, the ζ0(ϕ) and ϕv factors appearing in eq 2 can be separated experimentally through the iso-ζ0 correction. After the first report for the iso-ζ0 correction by Fujita et al.,16 this correction has been widely made in the literature.3,15−18 However, the correction procedure (summarized in Appendix A) is not always well documented in the literature. More importantly, applicability and validity of the correction do not seem to be systematically examined by experiments heretofore. Furthermore, some features after the iso-ζ0 correction, such as the dilution power law for τd (∼ϕv), have not been fully clarified. In this study, we revisit the dynamics in concentrated polymer solutions for two purposes: one is to illustrate the solvent effect on the local friction and test the validity of the iso-ζ0 correction and another is to examine the dilution power law. In order to obtain universal and representative results, we focus on four groups of concentrated polymer solutions with quite different local frictional environments (see Figure 2 for the first two groups). (1) The same polymer (PBD) in three

(6)

According to the free volume theory, ζ0(ϕ, T) and the fractional free volume f(ϕ, T) can be related as 14,15

ζ0(ϕ , T ) ≈ exp[B /f (ϕ , T )]

(8)

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Table 1. Molecular Characteristics of Polymers and Solvents

Figure 2. Model systems used in the present work: (1) PBD in different solvents: OTP, EHB, and PHO; (2) different polymers in EHB: PBD, PS, and PVE; (3) PBD in oBD; (4) PMMA in IL. IL is Bmim[Tf2N]. For parts 3 and 4 not illustrated here, see text. a

different solvents: phenyloctane (PHO) (lower-Tg than PBD), 2-ethylhexyl benzoate (EHB) (iso-Tg with PBD), and o-terphenyl (OTP) (higher-Tg than PBD). The iso-Tg and higher-Tg solvents are rarely found, and thus the solutions in EHB and OTP serve as good model systems for elucidating the solvent effect on the segmental dynamics. (2) The same solvent (EHB) for three different polymers, PBD, 1,2-polybutadiene (PVE), and polystyrene (PS). (3) PBD/oligomeric butadiene (oBD) solutions: the PBD dynamics in oBD is compared with that in ordinary solvents. (4) Poly(methyl methacrylate) (PMMA)/ionic liquid (IL) solutions: IL attracts research interest in recent years19−21 partly because of its complicated interaction with solutes.22,23 Thus, it is highly desired to examine the effect of IL on polymer dynamics. For those solutions, the difference of Tg between the solvent and the polymer (ΔTg) ranges from −200 to +70 °C. Such wide span of ΔTg enables us to examine the universality of both iso-ζ0 correction and the dilution power-law form (ϕv) in a variety of frictional environments. This paper is organized as follows. Samples and experimental conditions are presented in section 2. In section 3, rheological data of some representative solutions are presented, and the solvent effect on the isothermal terminal relaxation time is investigated. In section 4, the principle and procedure of the iso-ζ0 correction are discussed on the basis of experimental data. In section 5, τd in the iso-ζ0 state is compared with τd in the isothermal state, and then the dilution power-law in the iso-ζ0 state, τd ∼ ϕv, is discussed. Finally, in section 6, this power law is discussed in relation to the two dynamic length scales explained earlier, the entanglement length a and the correlation length ξ.

component

Mn (kg/mol)

Mw/Mn

Tg (°C)

ρ (g/cm3, 25 °C)

PBD PVE PS PMMA oBD EHB OTP PHO Bmim[Tf2N]

1100 242 1050 586 0.9

1.13 1.07 1.09 1.70 1.11

−96.7 −18.0 106.0 120.3 −100.5 −98.6 −23.3 −121.0a −87.0

0.90 0.90 1.05 1.20 0.90 0.96 1.10 0.85 1.43

Obtained from ref 3.

Tg (≈ −100 °C) is still close to Tg of high-M PBD because of a high 1,2-content of oBD. All solutions were prepared on a weight basis, and the polymer volume fraction ϕ was calculated by assuming no volume change on dissolution. The solutions were made with excess of cosolvent, toluene, except for PBD/PHO (no cosolvent was used) and PMMA/ Bmim[Tf2N] (cosolvent was methylene chloride). Also mixed was 0.2% w/w Ciba IRGANOX B215 antioxidant. After complete dissolution, the solutions were placed under vacuum in dark at room temperature for at least 2 weeks until the cosolvent content decreased below 0.02 wt %. The PMMA/Bmim[Tf2N] solution was further evacuated at 80 °C for 24 h to fully remove the moisture. The polymer concentration was checked selectively by thermal gravimetric analysis (TGA) after this evacuation process. Linear viscoelastic properties of polymer melts and their concentrated solutions were measured by using a TA ARES-G2 rheometer with 8 mm parallel-plate geometry under N2 atmosphere. Oscillatory frequency sweep was performed to measure the storage and loss moduli, G′ and G″, in a range of angular frequency ω between 10−2 and 102 rad/s. The strain amplitude was kept small to ensure the linearity. Further details of measurements have been described before.27,28

3. CONCENTRATION DEPENDENCE OF TERMINAL RELAXATION TIME AT CONSTANT TEMPERATURE The terminal relaxation time τd was determined as τd = 1/ωc (9) where ωc is the angular frequency at the crossing point of the G′ and G″ curves at low frequencies.29 The meaning/validity of this determination is explained in the Supporting Information. For PBD/EHB having Tg(solvent) ≅ Tg(polymer), parts a and b of Figure 3 show the master curves of the moduli at Tref = 25 °C and the corresponding shift factor aT. ζ0 of PBD in EHB hardly changes with ϕ, as suggested from excellent agreement of the aT data of the solutions and bulk PBD in a very wide range of T (between −40 and 60 °C). (Further details of ζ0 are discussed in section 4.) Thus, the decreases of τd and GN0 with ϕ are mainly attributed to dilution of entanglement network. Such solutions requiring no significant ζ0-correction are hereafter referred to as iso-ζ0 (or “iso-Tg”) solutions. For plasticized solutions, PBD/PHO having Tg(solvent) < Tg(polymer), Figure 3c shows the master curves of the moduli. Both τd and GN0 decrease with decreasing ϕ, as similar to the behavior of the PBD/EHB solution. However, τd is shorter in the PBD/PHO solution than in the PBD/EHB solution having the same ϕ (for example, τd = 0.33 and 0.88 s at ϕ = 0.15). This difference is attributed to the plasticizing effect of PHO (that reduces ζ0 of PBD). The other plasticized solutions (PVE/EHB, PS/EHB, and PMMA/Bmim[Tf2N]) exhibited the same features, the decrease of GN0 due to the dilution effect and decrease of τd due to both dilution and plasticizing effects.

2. MATERIALS AND EXPERIMENTS Narrow distribution linear 1,4-polybutadiene (PBD; 1,4-cis:1,4trans:1,2 = 54:36:10), oligomeric butadiene (oBD; 1,4-cis:1,4trans:1,2 = 43:24:33), 1,2-polybutadiene (PVE; 1,2 addition = 83%), and polystyrene (PS) were purchased from Polymer Source, Inc. Relatively wide distribution linear poly(methyl methacrylate) (PMMA) was purchased from Sigma. Small molecule solvents, including phenyloctane (PHO), 2-ethylhexyl benzoate (EHB), and o-terphenyl (OTP), were purchased from Acros Organics, TCI, and Aldrich, respectively. An ionic liquid, 1-butyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide (Bmim[Tf2N]), was purchased from the Center for Green Chemistry and Catalysis, Lanzhou Institute of Chemical Physics, Chinese Academy of Sciences. Molecular characteristics of these polymers and solvents are summarized in Table 1. The molecular weight and polydispersity of polymers were provided by Polymer Source, Inc. Densities were obtained from literatures and handbooks.3,7,24−26 Glass transition temperatures were measured by TA DSC-Q2000 at 10 °C/min heating rate. Microstructure of the PBD, oBD, and PVE samples were determined by 1H NMR measurement (BRUKER DMX-300). Note that oBD has low molecular weight but its 4462


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decrease of τd on dilution is mostly compensated by an increase of ζ0 (antiplasticizing effect) in this case. Comparison of the behavior of the antiplasticized and plasticized solutions demonstrates that the dilution effect and the solvent effect, represented by the ϕv and ζ0(ϕ) factors in eq 2, are clearly distinguished. In Figure 4a, the ϕ dependence of τd at a given T (= 25 °C) is compared for PBD in three different solvents (EHB, PHO,

Figure 4. Concentration dependence of the terminal relaxation time for (a) PBD in different solvents: EHB, PHO, and OTP at 25 °C. (b) Different polymers in EHB: PBD, PVE, and PS at 25 °C. (c) PMMA/ Bmim[Tf2N] at 150 °C.

and OTP). For the iso-Tg PBD/EHB solution, a simple power law relationship, τd ∼ ϕv with v = 2.0, is observed because ζ0(ϕ, T) hardly changes with ϕ. The plasticized PBD/PHO solution and the antiplasticized PBD/OTP solution, respectively, exhibit stronger and weaker ϕ dependence of τd, as easily expected. Figure 4b shows the ϕ dependence of τd at 25 °C for three different polymers (PBD, PVE, and PS) in the same solvent EHB. (The data are unavailable for bulk PS being glassy at 25 °C). Clearly, τd of the PVE/EHB and PS/EHB solutions is not described by a power law of ϕ, and its ϕ dependence is much stronger than that for τd of the PBD/EHB solution. The deviation from the power-law behavior, noted for the first two solutions, can

Figure 3. (a) Master curves of pure PBD and PBD/EHB solutions (Tg,EHB = Tg,PBD). (b) WLF shift factors aT of pure PBD and PBD/ EHB solutions. The good overlap of aT is observed because Tg,EHB = Tg,PBD. (c) Master curves of pure PBD and PBD/PHO solutions (Tg,PHO < Tg,PBD). (d) Master curves of pure PBD and PBD/OTP solutions (Tg,OTP > Tg,PBD). PHO and OTP rather easily crystallized at low T, which did not allow us to measure the data at high ωaT.

For so-called antiplasticized solutions, PBD/OTP having Tg(solvent) > Tg(polymer), the moduli master curves are shown in Figure 3d. As expected, GN0 decreases on dilution. However, τd hardly changes with ϕ, possibly because the 4463


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be related to ΔTg between EHB and polymers (≈ −80 °C for PVE/EHB, and ≈ −200 °C for PS/EHB). Figure 4c shows the ϕ dependence of τd for PMMA/ Bmin[Tf2N] solutions at 150 °C. (Bmin[Tf2N] negligibly evaporates even at 150 °C.30) τd decreases by ∼7 orders of magnitude on a decrease of ϕ from 1 to 0.15 and does not exhibit the power-law dependence on ϕ. This strong ϕ dependence of τd is again attributed to a large ΔTg (≈ −190 °C between Bmin[Tf2N] and PMMA). Similarly strong ϕ dependence has been noted for PMMA/diethyl phthalate (DEP) solutions13 having a similar ΔTg (≈ −170 °C). Thus, the strong, nonpower-law type ϕ dependence of τd (hence of ζ0) is mainly attributable to ΔTg irrespective of the solvent type (neutral small molecule or the ionic liquid).

4. ISOFRICTIONAL CORRECTION As noted in the previous section, the solvent significantly affects the polymer dynamics in concentrated solutions through dilution of the entanglement (common for all solvents), and also through changes of ζ0(ϕ, T) that are highly dependent on ΔTg between the solvent and the polymer. For ΔTg ≈ 0, ζ0(ϕ, T) hardly changes with ϕ. In contrast, ζ0(ϕ, T) decreases and increases, respectively, for the cases of ΔTg < 0 and ΔTg > 0. Thus, changes of τd due to the dilution effect (represented by the ϕv factor in eq 2) in several different solvents can be analyzed only after the iso-ζ0 correction. The principle and procedure of this correction based on the WLF fitting are rather well-known, but they are graphically summarized in Figure 5 (for data of the PVE/EHB solutions as an example) so as to demonstrate that the data presented in this study were obtained in considerably wide ranges of ϕ and T. For each ϕ, the fractional free volume f(ϕ, T) was obtained via the standard WLF fitting of the T dependence of τd (or the shift factor aT); cf. Appendix A. For the PVE/EHB solution, the ϕ and T dependence of f(ϕ, T)/B (with B being a constant close to unity) is presented as the 3-D plot in Figure 5a. For a given ϕ, the increase of f(ϕ, T)/B due to an increase of T represents an extra free volume acquired by enhanced thermal motion. The increase of f(ϕ, T)/B on dilution at a fixed T reflects the free volume introduced by the plasticizing solvent, EHB. The f(ϕ, T)/B curves for respective solutions and melt penetrate an isothermal plane (at T = 298 K) at different f(ϕ, T)/B values, as noted for the red points in Figure 5b. This f(ϕ, T)/B value increases (and ζ0(ϕ, T) decreases) with decreasing ϕ. In contrast, the data for a constant f/B value specify the iso-ζ0 temperatures T″ for respective solutions that are to be compared with the melt at a given temperature T′. For example, Figure 5c shows that the solution with ϕ = 0.3 at T″ = 242 K and the melt at T′ = 298 K are in the iso-freevolume state (f/B = 0.06) to have the same ζ0. Once T″ is known, the friction correction factor for the solutions, ζ0(ϕ, T″)/ζ0(ϕ, T′), is calculated easily. The algebraic equation finding T″ for a given T′ is shown in Appendix A. As the other examples, parts a and b of Figure 6 show the f(ϕ, T)/B vs T plots for the antiplasticized PBD/OTP solutions and the iso-ζ0 (iso-Tg) PBD/EHB solutions, respectively. For both solutions, f(ϕ, T)/B increases with increasing T. However, the difference is noted for the changes of f(ϕ, T)/B with ϕ: On dilution, the PBD/OTP solution shows considerable decreases of f(ϕ, T)/B, whereas the PBD/EHB solution exhibits no “significant” change of f(ϕ, T)/B (that corresponds to the friction correction factor between 1.2 and 1.4 explained below). This difference is a natural consequence of the solvent nature

Figure 5. (a) Temperature dependence of f(T)/B for PVE/EHB at concentrations indicated. Fractional free volume surface can be defined by four lines. (b) Four lines of f(T)/B cross the “isothermal” plane (T = 298 K) at different f/B. (c) Four lines of f(T) cross the “iso-freevolume” plane ( f/B = 0.06 for the melt at 298 K) at different temperatures which correspond to the “iso-free-volume” (or iso-ζ0) reference temperature T″.

(antiplasticizing vs iso-Tg). The iso-ζ0 temperatures, T′ for melt and T″ for solutions, can be easily determined as temperatures where the f(ϕ, T)/B factor has a given, constant 4464


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in the v value of the other solutions. In the next section, we examine the ϕ dependence of τd of those solutions in the iso-ζ0 state, keeping this uncertainty in our mind.

5. CONCENTRATION DEPENDENCE OF TERMINAL RELAXATION TIME IN ISOFRICTIONAL STATE Figure 7a summarizes the ϕ dependence of the terminal relaxation time τd of PBD solutions in three solvents (EHB,

Figure 6. Fractional free volume as a function of temperature for (a) PBD/OTP and (b) PBD/EHB. The horizontal lines represent the fractional free volume f/B of undiluted PBD at T′ = 25 °C, which cross the “iso-free-volume” (or iso-ζ0) reference temperature T″ for solutions.

value. The f(ϕ, T)/B vs T plots for other solutions are shown in the Supporting Information. For all solutions, the iso-ζ0 temperatures and the corresponding ζ0(ϕ,T″)/ζ0(ϕ,T′) ratio (friction correction factor) were thus determined. The results are summarized in Appendix A (Table 4) together with the related WLF parameters and the free volume parameters. Here, a comment needs to be made for the values of the friction correction factor, ζ0(ϕ,T″)/ζ0(ϕ,T′), for the PBD/ EHB solutions. For these solutions having ϕ = 0.15−0.5, T′ = 25 °C, and T″ = 20.8−17.8 °C, the correction factor has a value between 1.2 and 1.4, as shown in Table 4 in Appendix A. This change in the friction factor is comparable, in magnitude, with the uncertainty in the rheological evaluation of τd, ∼ 0.1 log units (1.26; see Supporting Information). Thus, the free volume correction is not “significant” for PBD/EHB solutions, and these solutions can be approximately treated as the iso-ζ0 solutions. This was the case for PBD/oBD solutions. The dilution exponent v for these two solutions, listed in Table 2,

Figure 7. Concentration dependence of the terminal relaxation time in the iso-ζ0 state with the reference temperature for melts as indicated. (a) PBD in different solvents: EHB, PHO, and OTP. (b) Different polymers in EHB: PBD, PVE, and PS. (c) PMMA/Bmim[Tf2N] and PMMA/DEP. Data of PMMA/DEP obtained from ref 13.

Table 2. Dilution Power Law Exponents v of Terminal Relaxation Time τd in Concentrated Solutions polymer

solvent

v (before adjustment)

v (after adjustment)

PBD PBD PBD PBD PVE PS PMMA

EHB oBD PHO OTP EHB EHB Bmim[Tf2N]

2.0 1.2 − − − − −

1.9 1.4 1.3 2.1 2.1 2.2 1.8

PHO, and OTP) reduced at the iso-ζ0 state defined with respect to the PBD melt at 25 °C (f/B = 0.14). (The friction correction factor is summarized in Appendix A.) The solid line shows the least-squares fit with the slope of 2.0 obtained for the double-logarithmic plots of τd against ϕ for the PBD/EHB and PBD/OTP solutions. The PBD/PHO solution shows much weaker ϕ dependence of τd that can be characterized with the slope of 1.3. In Figure 7b, the τd data at the iso-ζ0 state are shown for three polymers (PBD, PVE, and PS) in the same solvent EHB.

decreases just a little (by a factor of 0.1−0.2) on the iso-ζ0 correction. This small decrease may be regarded as uncertainty 4465


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Melts of PBD at 25 °C (f/B = 0.14), PVE at 25 °C (f/B = 0.06), PS at 120 °C (f/B = 0.04) are chosen as the references. (Minor iso-ζ0 correction was made for the PBD/EHB solutions, as explained in the previous section.) The dilution exponent is 2.2, 2.1, and 1.9 for the PS/EHB, PVE/EHB, and PBD/EHB solutions, all being close to 2.0 (within the uncertainty explained in the previous section.) Figure 7c shows the τd data for the PMMA/Bmim[Tf2N] solutions at the iso-ζ0 state defined with respect to the PMMA melt at 150 °C (f/B = 0.04). For comparison, literature data for PMMA/DEP solutions13 are also shown. For both solutions, the dilution exponent is again close to 2.0. In summary, for all solutions in low-M solvents, τd in the iso-ζ0 state exhibits power-law type ϕ dependence, irrespective of the solvent type (neutral molecules or ionic liquid) and the ΔTg value. This result suggests validity of the iso-ζ0 correction for those solutions (even for the solutions in the ionic liquid that interacts with the polymer in a complicated way.) From a quantitative aspect, almost all solutions have a similar power law (dilution) exponent, v = 2.0 ± 0.2; see Table 2. The exception is found for the PBD/PHO solution. Indeed, the exponent for this solution, v = 1.3 < 2.0, is close to the result in ref 3, as demonstrated in Figure 8.

Table 3. Dilution Power Law Exponents v of Terminal Relaxation Time τd in Concentrated Solutions, Obtained by Two Groups in the Literaturea group polymerb 1

2

PVAc PMMA PBD PBD PBD hPB PBD PBD PBD PI

10−3Mw (g/mol)

solventb

ϕ range

v

ref

240 191 350 925 925 53.2−440 155 411.5 230 153

DEP DEP Flexon 391 DOP PHO alkane oBD (Mw = 1800) oBD (Mw = 2000) oBD (Mw = 4700) oPI (Mw = 3100)

0.4−1 0.2−1 0.1−1 0.1−1 0.1−1 0.2−1 0.1−1 0.1−1 0.2−1 0.1−1

2.5 1.9c 1.8c 2.2c,d 1.3c 1.8c 1.3c 1.1 1.1c 1.7

17 13 18 3 3 32 31 29 18 33

a

1. Concentrated polymer solutions adjusted to iso-monomericfriction state. 2. Iso-Tg concentrated solutions without adjustment. b Polymer: PVAc, poly(vinyl acetate); PMMA, poly(methyl methacrylate); PBD, 1,4-polybutadiene; PS, polystyrene; hPB, hydrogenated polybutadiene; PI, 1,4-polyisoprene. Solvent: DEP, diethyl phthalate; Flexon 391, the trade name of a kind of oil; PHO, phenyloctane; DOP, dioctyl phthalate; oBD, oligomer PBD; oPI, oligomer PI. cτd−ϕ exponents were recalculated from concentration dependence of η0/GN0, η0Je0 or η0Je0N in the literature, where η0 is the zero shear viscosity, GN0 is the entanglement plateau modulus, Je0 is the steady-state compliance, and Je0N is the entanglement compliance. d Colby et al.3 assumed that the isomonomeric friction adjustment is unnecessary for PBD/DOP solutions, though Tg of DOP is 11 °C higher than Tg of PBD. However, Table 9 and Figure 7 in ref 3 explicitly show that the isothermal (25 °C) fractionnal free volume of PBD/DOP solution is slightly reduced from 0.125 at ϕ = 1 to 0.105 at ϕ = 0.157, and the adjustment factor is calculated as 0.22 by utilizing above data and eq 16 in ref 3. Then the zero-shear viscosity η0 of PBD/DOP 15.7% solution was adjusted, and it was found that the exponent v for η0/GN0 is 2.2 when ϕ > 0.1.

exponent v for PMMA (1.8 in ionic liquid and 1.9 in DEP) is insensitive to the solvent type, as explained for Figure 7c. Most of the iso-Tg solutions reported in literature (“group 2” in Table 3) are mixtures of polymers and their oligomers,18,29,31−34 because the chemically different small molecules rarely behave as the iso-Tg solvent. The v values of those solutions are smaller than those of ordinary solutions (“group 1” in Table 3). This fact can be confirmed for PBD in two different iso-Tg solvents, oBD and EHB, examined in this study: As shown in Figure 9, v = 1.2 (from four sets of the PBD/oBD data) and 2.0 in oBD and EHB. In fact, the small v value in oBD is consistent with the literature data18,29,31 shown therein. This difference of v in the two solvents may be attributed to considerable screening of the excluded volume and hydrodynamic interactions in the oligomers that could allow the long polymer chain to exhibit the local Rouse dynamics even at a considerably small length scales. (In relation to this point, we note that the density change on mixing is negligible for both PBD/oBD and PBD/EHB solutions and thus irrelevant to the difference of their v values.) Consequently, we expect a critical molecular weight for the oligomer to behave as an ordinary solvent. This conjecture can be tested in our future work for solutions in a series of oBDs with different molecular weights.

Figure 8. Concentration dependence of normalized terminal relaxation times for PBD/PHO before (squares) and after (circles) iso-monomeric-friction adjustment. Filled symbols represent data in this study, while open symbols represent data in ref3. Moderate deviation of the two results of ζ0-correction could be attributed to the difference of polymer’s microstructure, temperature control, and WLF fitting of the data.

In literature, the WLF analysis is usually made for noniso-Tg concentrated solutions3,13,17,18 to reduce their data at the iso-ζ0 state, whereas no iso-ζ0 correction is made for the iso-Tg solutions18,29,31−33 irrespective of the solvent type. Table 3 lists the v data for those solutions found in literature. It should be noted that the exponent is not necessarily the same in concentrated and semidilute solutions because of the excluded volume/hydrodynamic interactions in the latter, as discussed by Adam and Delsanti.11,12 However, all solutions examined in this work (ϕ ≥ 0.15) and those listed in Table 3 are in the concentrated regime. For the solutions in small molecule solvents classified as “group 1” in Table 3, the exponent v is in a range of 2.1 ± 0.3 (except for PBD/PHO). Moderate scatter in the v value is still acceptable, if we consider diverse sources of literature data. In this study, we obtained a similar exponent (v = 2.0 ± 0.2; cf. Table 2) but with smaller scatters, because our solutions were subjected to identical rheological measurements and their data were analyzed in the same way. We should also note that the

6. FURTHER DISCUSSION OF DILUTION EXPONENT The current theory of the polymer solutions, built on the key concept of the correlation length proposed by de Gennes,35,36 is favorably compared with experiments.6,7,37 This theory, fully described in Colby’s review6 and the textbook,9 is briefly 4466


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Rouse chain confined inside the tube. An entanglement strand, taken as the structural unit of this Rouse chain, has the relaxation time τe specified by

⎛ N ⎞2 τe ≈ τξ⎜ e ⎟ ⎝g⎠

Here, Ne and g is the number of monomers per entanglement strand and correlation blob, respectively. From eqs 5, 10, and 11, the terminal relaxation time τd deduced from the pure reptation model (considering no contour fluctuation and constraint release) is given by

Figure 9. Isothermal concentration dependence of normalized terminal relaxation times τd(ϕ)/τd(1) for PBD/EHB (blue symbols) and PBD/ oBD (red symbols). Data of PBD/oBD were obtained from this study (solid symbols; at T = 25 °C) and literatures (refs 18, 29, and 31 at T = 25, 40, and 27 °C, respectively; open symbols). Note that the normalized τd(ϕ)/τd(1) in these iso-ξ0 (iso-Tg) solutions is insensitive to T.

2 3 ⎛ N ⎞3 ⎛ ξ(ϕ) ⎞3⎛ Ne(ϕ) ⎞ ⎛ N ⎞ τd ≈ τe⎜ ⎟⎜ ⎟⎜ ⎟ ≈ τ0⎜ ⎟ ⎝ b ⎠ ⎝ g (ϕ) ⎠ ⎝ Ne(ϕ) ⎠ ⎝ Ne(ϕ) ⎠

(12)

According to eq 12, the concentration dependence of τd emerges not only from Ne(ϕ), but also from ξ(ϕ) and g(ϕ). The scaling predictions for solution structure, the entanglement length, and the terminal dynamics, pioneered by de Gennes, are summarized in Table 2 of Colby’s review.6 In semidilute solutions in good solvents, Ne(ϕ) ∼ ϕ−1.3, ξ(ϕ) ∼ ϕ−0.76, and g(ϕ) ∼ ϕ−1.3, so that the predicted dilution exponent is v = 1.6. The correlation length ξ decreases with increasing ϕ, but the size of the thermal blob ξT remains independent of ϕ.9 Therefore, at some concentration ϕ**, ξ coincides with ξT. This is the crossover between the semidilute and concentrated regimes. For ϕ ≥ ϕ**, chains are ideal on all length scales, and then Ne(ϕ) ∼ ϕ−4/3, ξ(ϕ) ∼ ϕ−1, and g(ϕ) ∼ ϕ−2 so that v = 7/3. It is worth noting that the upper concentration ϕ** of the semidilute regime is solventdependent according to the strict definition, although most of researchers assumed that ϕ** is close to 0.1−0.15.38 Theoretically, there are two limiting cases: the athermal solvent and the Θ solvent. In the athermal limit, the “semidilute” regime persists to high concentrations, ϕ** ≅ 1, since ξ = ξT = b at ϕ = 1. The dilution exponent v of 1.6 is expected to be observed when ϕ > ϕe. In the Θ limit, the “concentrated” regime extends to very small ϕ, since the size of the thermal blob ξT is larger than the size of chain R, and the correlation length ξ cannot reach ξT even at the overlap concentration ϕ*. The dilution exponent v of 2.3 is expected in this case. Discussion of v Data. If we adopt the scaling theory explained above, we expect that the experimental v for entangled solutions has a value between 1.6 and 2.3 (that changes with the solvent quality).6,9 Indeed, most of the data in Tables 2 and 3 locate between 1.6 and 2.3. However, we also note important exceptions for the PBD/PHO and PBD/oBD solutions; cf. Figures 8 and 9. The exponent for these solutions (v ∼ 1.3) is smaller even compared to the athermal asymptote (v = 1.6). This small exponent could be still qualitatively consistent with the expectation, because the oligomer is regarded as a near-athermal solvent for the polymer of identical chemical structure,37 and PBD in PHO is reported to have a large coil expansion exponent in dilute solutions.3 Nevertheless, problems still remain, as explained below. For a solution in a good solvent, the current scaling theory predicts that the dilution exponent v in the semidilute and concentrated regimes is 1.6 and 2.3, respectively. However, PBD/PHO solutions exhibit weak ϕ dependence of the viscosity η (η ∼ ϕ3.6) which corresponds to τd ∼ ϕ v with v = 1.3 for 0.1 < ϕ < 1, and strong dependence (η ∼ ϕ4.7) which corresponds to v = 2.4 for 0.01 < ϕ < 0.1, as reported by

explained below for convenience of further discussion of the dilution exponent obtained in this work. Scaling Theory. Entangled solutions in ordinary solvents have two dynamic lengths, the entanglement length a, and the correlation length ξ. The ϕ dependence of a, ξ, and the average chain size (end-to-end distance), R, in a good solvent is schematically shown in Figure 10 as the double-logarithmic

Figure 10. Chain size R, tube diameter a, and correlation length ξ in a good solvent. The semidilute unentangled regime is ϕ* < ϕ < ϕe; the semidilute entangled regime is ϕe < ϕ < ϕ**; the concentrated regime is ϕ** < ϕ < 1. Here, ϕ*, ϕe, and ϕ** represent overlap concentration, entanglement concentration, and the upper concentration of the semidilute regime, respectively. (Redrawn according to Figure 9.7 of ref 9.)

plots. For ϕ larger than the entanglement threshold ϕe, the length scale is divided in three regimes, below ξ, between ξ and a, and above a. Therefore, the relaxation time of the chain is calculated as a hierarchy of length scales. In the first regime, the hydrodynamic interaction is not screened so that the Zimm dynamics is expected to dominate the relaxation. ⎛ ξ ⎞3 τξ ≈ τ0⎜ ⎟ ⎝b⎠

(11)

(10)

Here, τξ is the relaxation time of the correlation blob, τ0 is the segmental relaxation time, and b is the statistical length of the monomer. The sequence of correlation blobs behaves as the 4467


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ζ0(ϕ, T) can be fitted by the William−Landel−Ferry (WLF) equation for the solution with concentration ϕ.14,15

Colby and coauthors (see Figure 8 and eqs 17 and 18 in ref 3). This trend is opposite to, i.e., qualitatively different from, the theoretical expectation. It is worth noting that the change in temperature, being necessary to make the time−temperature superposition for the iso-ζ0 correction, could change the solvent quality thereby affecting the correlation length ξ (eq 12) and the structure factor K (eq 2). For this case, K is dependent on both M and T, and the strategy of reduction in the iso-ζ0 state with the use of the shift factor does not work on a sound basis. Nevertheless, this delicate problem hardly affects our analysis as long as the solvent remains good in the range of temperature examined.39 This seems to be the case for our five noniso-Tg solutions that exhibited the simple power law, τd ∼ ϕv in the iso-ζ0 state, which suggests that the iso-friction correction is generally valid. In relation to this point, we also note that the Flory−Huggins interaction parameter χ specifying the solvent quality changes with T and ϕ.37 Then, the current treatment utilizing the same v value in the entire range of ϕ could result in some inconsistency. Problems remain also for the dynamics combined with the scaling argument. For example, the Zimm model does not rigorously describe the dynamics of dilute polymers in good solvents.6 It remains unclear how the coupling of the excluded volume and the hydrodynamic interactions affects the scaling behavior. Furthermore, eq 12 is deduced from the scaling theory combined with the pure reptation model considering no contour length fluctuation/constraint release. Obviously, τd scales as M3.4 in experiments. Colby et al.3 listed all possibilities of the exponent for the cases of one dynamic length or two dynamic lengths, and also for the cases of pure reptation and experiments. In fact, the use of the experimental scaling M3.4 in eq 12 affects the v value for τd (∼ϕv) discussed in this study. This issue is an important subject of further study.

τd[ϕ , ζ(ϕ , T )] ζ (ϕ , T ) = log 0 τd[ϕ , ζ(ϕ , Tref )] ζ0(ϕ , Tref )

log a T = log =

−C1(T − Tref ) C2 + T − Tref

(A1)

Here, aT is the temperature shift factor; τd[ϕ, ζ0(ϕ, Tref)] and ζ0(ϕ, Tref) are the terminal relaxation time and the monomeric friction factor at arbitrary reference temperature Tref; C1 and C2 are WLF fitting parameters. According to eq 7, the WLF parameters and the free volume parameters can be related by C1 = B/[2.303f(ϕ, Tref)] and C2 = f(ϕ, Tref)/α.3,14,15 Here, f(ϕ,Tref) is the fractional free volume at Tref, α is the thermal expansion factor for the free volume, and B is a constant of order unity. The expression of f(ϕ, T) can be written as a linear equation8,13 with the aid of these experimentally fitted WLF parameters. f (ϕ , T ) = f (ϕ , Tref ) + α(T − Tref ) = B(2.303C1)−1 + B(2.303C1C2)−1(T − Tref )

(A2)

Therefore, for polymer solutions ϕ1 and ϕ2, the fractional free volume can be obtained from rheological measurements. f (ϕ1 , T ) = f (ϕ1 , Tref 1) + α1(T − Tref 1)

(A3)

f (ϕ2 , T ) = f (ϕ2 , Tref 2) + α2(T − Tref 2)

(A4)

At a given dynamic reference temperature T′ f (ϕ2 , T ′)/B2 ≠ f (ϕ1 , T ′)/B1

(A5)

In order to be adjusted to the same frictional state as the ϕ1 solution, the ϕ2 solution requires a new temperature T″. f (ϕ2 , T ″)/B2 = f (ϕ1 , T ′)/B1

7. CONCLUSION Systematic experiments have been performed to examine the validity of the isofrictional (iso-ζ0) correction for the terminal relaxation time τd of entangled polymer solutions. This correction, widely utilized as an effective approach to reduce the monomeric friction ζ0(ϕ, T) to a reference state,3,13,15−18 was found to lead to consistent results for a variety of solutions. The terminal relaxation time τd thus reduced at the iso-ζ0 state exhibited power-law dependence on the polymer volume fraction ϕ, τd ∼ ϕv with the exponent v being in a range of 1.3−2.2. This result lends support to the current scaling theory combined with the pure reptation model: The theory assumes two dynamic lengths (entanglement and correlation lengths) to give v = 1.6 and 2.3 for the solutions in the athermal and Θ solvents. Nevertheless, the pure reptation model does not describe the experimental relationship, τd ∼ M 3.4, and the use of this relationship in the scaling theory affects the v value deduced from the theory. This problem deserves further study.

(A6)

For a given temperature T′, the corresponding T″ can be easily solved from eqs A3, A4, and A6. Substituting T′ and T″ into eq A1, the friction adjustment factor ζ0(ϕ2, T″)/ ζ0(ϕ2, T′) can be calculated. All WLF parameters and adjustment factors are listed in Table 4. Here, the polymer melt (ϕ1 = 1) at a given temperature is set as the reference state. It should be noted that our “iso-Tg” solvent is a solvent requiring minor ζ0-correction comparable, in magnitude, with the experimental uncertainty of the evaluation of τd (cf. Supporting Information). To our knowledge, the iso-monomeric-friction adjustment was first proposed by Fujita et al.16 in 1958 soon after the famous “WLF” paper.14 An correction factor at isothermal condition was defined as log

ζ0(ϕ1) ζ0(ϕ2)

=

B ⎛1 1⎞ ⎜⎜ − ⎟⎟ f2 ⎠ 2.303 ⎝ f1

(A7)

where f 2 and f1 are the isothermal fractional free volumes at concentrations ϕ2 and ϕ1, respectively. The temperature is the WLF reference temperature Tref1 (to construct the master curve). The concentration ϕ1 usually represent the undiluted state. Two underlying prerequisites exist in Fujita’s approach. (1) The dynamic reference temperature T′ equals the WLF reference temperature of the undiluted polymer Tref1. (2) The experimental temperatures for the undiluted polymer and the

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APPENDIX A. ISOMONOMERIC FRICTION ADJUSTMENT The procedure of the isomonomeric friction adjustment is reviewed as follows. The temperature dependence of the terminal relaxation time τd[ϕ, ζ0(ϕ, T)] or the monomeric friction factor 4468


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Table 4. WLF Parameters, Free Volume Parameters, and Isomonomeric Friction Adjustment Results for Pure Polymers and Their Solutions in This Study solution (Tg/°C) PBD/EHB (−97 vs −99)

PBD/oBD (−97 vs −100)

PBD/PHO (−97 vs −121)

PBD/OTP (−97 vs −23)

PVE/EHB (−18 vs −99)

PS/EHBa (106 vs −99)

PMMA/Bmim[Tf2N] (120 vs −87)

ϕ

Tref (°C)

T′ (°C)

C1

C2 (K)

f (ϕ, Tref)/B

104α/B (K−1)

T″ (°C)

1 0.5 0.3 0.15 1 0.5 0.3 0.15 1 0.5 0.3 0.15 1 0.5 0.3 0.15 1 0.5 0.3 0.15 1 0.5 0.3 0.15 1 0.5 0.3 0.15

25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 120 25 25 25 150 150 150 150

25

3.04 2.96 2.93 2.90 3.04 3.15 3.21 3.23 3.04 2.74 2.41 2.06 3.04 3.57 4.18 4.44 6.80 4.30 4.00 3.74 10.0 7.28 5.20 4.07 10.3 7.63 6.46 5.06

158 155 155 154 158 160 164 164 158 172 170 164 158 124 119 108 91.9 123 136 147 70.2 94.1 133 144 207 291 354 437

0.143 0.147 0.148 0.150 0.143 0.138 0.135 0.134 0.143 0.158 0.180 0.211 0.143 0.122 0.104 0.098 0.064 0.101 0.108 0.116 0.043 0.060 0.084 0.107 0.042 0.057 0.067 0.086

9.0 9.5 9.6 9.7 9.0 8.6 8.3 8.1 9.0 9.2 10.6 12.8 9.0 9.8 8.7 9.0 7.0 7.8 8.0 7.9 6.2 6.3 6.3 7.4 2.0 2.0 1.9 2.0

25.0 20.8 19.8 17.8 25.0 30.8 34.7 36.1 25.0 7.90 −10.3 −27.9 25.0 46.5 69.6 74.7 25.0 −22.7 −31.1 −40.9 120 −0.6 −39.0 −60.6 150 74.3 17.7 −72.6

25

25

25

25

120

150

ζ0(ϕ, T″)/ζ0(ϕ, T′) 1.0 1.2 1.3 1.4 1.0 0.78 0.66 0.62 1.0 2.0 4.3 9.6 1.0 0.30 0.072 0.040 1.0 3.2 × 6.3 × 1.1 × 1.0 5.2 × 6.3 × 8.5 × 1.0 4.8 × 7.1 × 1.8 ×

102 102 103 102 104 105 102 103 105

a Since 120 °C is not within the test temperature range of the three solutions, ζ0 (ϕ, T″) /ζ0 (ϕ, T′) is calculated by extrapolating the experimental data to higher temperature in order to cover 120 °C. Details are explained in Text.

solution must have an overlap, so that the same WLF reference temperature can be used in both cases (Tref,2 = Tref,1). Then the fractional free volumes f1 and f 2 at isothermal condition can be obtained. When ΔTg between the solvent and the polymer is not too large, or the volatility of solvent is negligible, these prerequisites can be satisfied in most cases. Such case includes PBD/PHO, PBD/OTP, PVE/EHB, and PMMA/Bmim[Tf2N], as shown in Table 4. However, when ΔTg is quite large and meanwhile the solvent is volatile at high temperatures, the experimental temperatures for the polymer melt (ϕ1) and solutions (ϕ2) do not overlap so that Tref,2 < Tref,1. Such case only includes PS/EHB in Table 4. Therefore, in the latter case, since the dynamic reference temperature T′ does not locate in the test range of the ϕ2 solution, the state ζ0(ϕ2, T′) and thus the correction factor ζ0(ϕ2, T″)/ζ0(ϕ2, T′) cannot be reached without the aid of extrapolation, as displayed by PS/EHB solution (ϕ = 30%) in Figure 11. Data of other PS/EHB solutions in Table 4 also come from the extrapolation. The friction correction factor ζ0(ϕ2, T″)/ζ0(ϕ2, T′) is usually employed to calculate the isofriction τd(ϕ2, T″): τd(ϕ2,T ″) = τd(ϕ2,T ′) × ζ0(ϕ2,T ″)/ζ0(ϕ2,T ′)

Figure 11. Fractional free volume as a function of temperature for PS melt and PS/EHB solution (ϕ2 = 30%). PS melt at 120 °C and 30% PS/EHB at −39 °C are located on the iso free volume line. Friction adjustment factor ζ0(ϕ2, T″)/ζ0(ϕ2, T′) can be obtained only by extrapolating data of the 30% solution to T′ = 120 °C.

extended to the case that two sets of data even have not an overlap, e.g. PS/EHB solutions. One of merits of Masuda’s approach is the broader application range.

■

(A8)

ASSOCIATED CONTENT

S Supporting Information *

In fact, τd(ϕ2, T″) can be directly calculated by using eq A1, without going through the friction adjustment factor ζ0(ϕ2, T″)/ζ0(ϕ2, T′), once T″ is determined.13 This is Masuda’s approach for the iso-monomeric-friction adjustment. Therefore, the outcome of iso-monomeric-friction adjustment can be

Graphical representation for the iso-free-volume (iso-ζ0) temperature T″, evaluation of τd and experimental uncertainty, and f/B vs T plots for some solutions.This material is available free of charge via the Internet at http://pubs.acs.org. 4469


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(30) Mok, M. M.; Liu, X. C.; Bai, Z. F.; Lei, Y.; Lodge, T. P. Macromolecules 2011, 44, 1016−1025. (31) Roovers, J. Polym. J. 1986, 18, 153−162. (32) Tao, H.; Lodge, T. P.; von Meerwall, E. D. Macromolecules 2000, 33, 1747−1758. (33) Nemoto, N.; Odani, H.; Ogawa, T.; Kurata, M. Macromolecules 1972, 5, 641−644. (34) Note that if the oligomer has a lower Tg than the polymer, the isomonomeric friction adjustment is still required. (35) Daoud, M.; Cotton, J. P.; Farnoux, B.; Jannink, G.; Sarma, G.; Benoit, H.; Duplessix, R.; Picot, C.; de Gennes, P. G. Macromolecules 1975, 8, 804−818. (36) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (37) Graessley, W. W. Polymeric Liquids and Networks: Structure and Properties; Garland Science: New York, 2003. (38) Graessley, W. W. Polymer 1980, 21, 258−262. (39) According to Figure 3.16 in ref 9, the temperature dependence of chain size is very weak in good solvent. However, near the Θ temperature, the reduction of chain size is remarkable.

AUTHOR INFORMATION

Corresponding Author

*E-mail: (C.-Y.L.) [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research work was supported by National Natural Science Foundation of China (Grant Nos. 21174153 and 21374127). We thank the reviewers for their very constructive comments.

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REFERENCES

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