Characterization of Dilution Effect of Semidilute Polymer Solution on

Article pubs.acs.org/Macromolecules

Characterization of Dilution Effect of Semidilute Polymer Solution on Intrinsic Nonlinearity Q0 via FT Rheology Hyeong Yong Song,† Seung Joon Park,‡ and Kyu Hyun*,† †

School of Chemical and Biomolecular Engineering, Pusan National University, Busan 46241, South Korea Department of Chemical Engineering and Biotechnology, Korea Polytechnic University, Siheung-Si, Gyeonggi-Do 15073, South Korea

‡

ABSTRACT: Intrinsic nonlinearity Q 0(ω) under medium amplitude oscillatory shear (MAOS) deformation was investigated for monodisperse polystyrene (PS) solutions at various concentrations, which were classified as unentangled or entangled solution in semidilute regime. These two types of PS solutions displayed different shapes when Q0 was plotted as a function of frequency (ω). Unentangled solutions showed increases of Q0 with frequency at the low frequency region and then plateau behavior at the high frequency region. On the contrary, entangled solutions showed an increase of Q0 before the MAOS terminal relaxation time and a subsequent decrease, which is similar to that observed for entangled linear polymer melts. The Q0(ω) curves of each group were superposed in a dimensionless coordinate (Q0/Q0,max vs De), so that transition from the plateau of Q0 to decreasing Q0 at the high frequency region might indicate the onset of entanglement in polymer solution. In particular, all unentangled solutions had the same Q0,max value (0.006) regardless of polymer concentration and molecular weight because Q0 responds to Rouse-like relaxation process only, which is featured as no interchain interaction and chain stretching. However, the Q0,max values of entangled solutions were dependent on the number of entanglements (Z). The master curve of Q0,max as a function of Z showed that Q0,max was constant at low entanglement numbers (few or virtually no entanglements) and then increased with the beginning of entanglement to approach a limiting value at high entanglement numbers, where reptation is the dominant linear relaxation process. In addition, the master curve of Q0,max as a function of Z was used to quantify the degree of tube dilation based on the dynamic tube dilution (DTD) theory. Direct comparison of the Q0,max values of semidilute solutions and melts showed that they followed the same molecular dynamics in MAOS flow like SAOS (small amplitude oscillatory shear) flow. Comparison between static and dynamic dilutions using the Q0,max master curve suggested that this curve could characterize the effective number of entanglements per backbone chain for branched polymers. Because it was confirmed again that Q0(ω) is highly sensitive to various relaxation processes, MAOS tests may provide a new means of investigating molecular dynamics.

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INTRODUCTION The relationship between polymer architectures and their rheological properties is a fascinating subject because rheological properties govern the flow behavior of polymers when they are processed in the melt state.1 To this end, considerable advances have been made in the synthesis of welldefined polymers with various architectures2,3 and theoretical foundations based on the tube model4−6 and simulations.7,8 These advances have enhanced our understanding of the molecular rheology of complex polymers from linear chains to branched polymers, such as star polymers, 9−11 comb polymers,12−14 and H-shaped polymers.15 To determine the effects of architectural variations on rheological behavior, linear viscoelastic (LVE) properties obtained by small amplitude oscillatory shear (SAOS) test have been extensively measured because they are simply obtained and useful for characterizing materials, and their mechanical behaviors are already referenced in many textbooks.1,16,17 However, deformations are quite small © XXXX American Chemical Society

within the LVE regime. In view of the real processing conditions used for polymers in the plastic industry, linear response is not valid anymore. During most processing operations, deformations are large and rapid, and thus, nonlinear material responses are required to evaluate real systems. Various nonlinear deformations, such as nonlinear step shear,12,13 transient start-up shear,18−20 and extensional flow,21 have been used because nonlinear deformations induce molecular chain changes, which are reflected by nonlinear viscoelastic properties. However, subtle differences in architecture may not be detected by even nonlinear measurements mentioned above. For example, Menezes and Graessley19 found linear and star chains behave similarly under time-dependent nonlinear shear flow (start-up shear from rest, relaxation of Received: January 18, 2017 Revised: July 20, 2017

A

DOI: 10.1021/acs.macromol.7b00119 Macromolecules XXXX, XXX, XXX−XXX

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suggested, Q0(ω) has been frequently used to characterize polymer systems.25,27,29,55 Compared with linear viscoelastic properties of monodisperse homopolymers, Q0(ω) has improved sensitivity for relaxation processes, such as reptation, contour length fluctuation (CLF), and constraint release (CR), by displaying local maxima and minima at characteristic relaxation times.25,30 Hyun and Wilhelm27 reported the time−temperature superposition (TTS) principle is valid for Q0(ω) values measured at different frequencies and temperatures in the MAOS region, as is the case for other nonlinear tests, e.g., steady shear and step shear at larger deformation. The shapes of nonlinear master curves of Q0(ω) depended on polymer architecture. Linear polymers had one peak near the reciprocal value of terminal relaxation time,27,29 whereas comb polymers with many branches had two peaks corresponding to backbone relaxation time at lower frequency and branch relaxation time at higher frequency.25,27 Song et al.30 reported 3-arm star polymers had two weak peaks even though the backbone has only one branch. Since nonlinear step shear and transient start-up tests failed to differentiate linear and star polymers,19,22 it was confirmed that LAOS tests can characterize different topologies. The relaxation processes of different polymer architectures have been explained using dynamic tube dilution (DTD) theory.27 Tube-based theories assume polymer chains are confined by a tube, which consists of neighboring chains. Whereas linear polymers relax by back-and-forth motion through the tube (reptation), branched polymers relax their backbone chains like linear polymers after branches fully relax due to the existence of branching points.12,56 In other words, a hierarchy of motions occurs in the relaxation of branched polymers. When the branches fully relax by CLF, the relaxed branches play a role as effective solvents by diluting the unrelaxed backbone. As a result, the effective tube diameter of the backbone chain increases and then stimulates relaxation of the backbone chain. From the viewpoint of DTD theory, the nonlinear parameter Q0(ω) from MAOS test for branched polymers could be reasonably explained. Because the branches’ relaxation is followed by the backbone’s relaxation for branched polymers, two peaks are observed, that is, one at higher frequency (earlier time) for branches’ relaxation and the other at lower frequency (later time) for backbone’s relaxation. In addition, the maximum value of Q0 (≡Q0,max) corresponding to backbone chain decreases as the molecular weight of branches increases or the amount of effective dynamic solvents increase.27 However, the relation between decreases in Q0,max and the DTD effect has not been quantified, and thus, a fundamental question remains to be addressed: how does Q0,max quantitatively reflect DTD effect induced by fully relaxed branches? To answer this question, we systematically investigated intrinsic nonlinearity Q0(ω) by applying FT rheology to monodisperse linear PS solutions of different concentrations. Linear homopolymer solutions of different concentrations can describe a different degree of tube dilation state of a polymer melt because static dilution (i.e., preparing a polymer solution) and dynamic dilution affect the same final result to tube diameter of backbone chains.57 We dissolved PS in a θ solvent at various concentrations, obtained semidilute unentangled and entangled solutions, and analyzed the nonlinear master curves of Q0(ω) for these solutions. To the best of the authors’ knowledge, no systematic studies have been previously undertaken on unentangled solutions or melts. Therefore,

stress from steady state, and start-up shear following resumption of flow), and the same result was obtained using the nonlinear step shear test.22 Furthermore, Neidhöfer et al.23 reported that linear and star-branched polystyrene (PS) solutions behaved similarly under SAOS flow with the same entanglement number. In order to distinguish linear and star polymer solutions, they applied nonlinear deformations. No differences were observed under nonlinear step shear flow, but pronounced differences were monitored under large amplitude oscillatory shear (LAOS) flow. Recently, the LAOS test has received much attention as a useful tool for characterizing various complex fluids, such as polymer solutions and melts,23−30 nanocomposites,31,32 blends,33−36 emulsions,37,38 and suspensions.39 When a sinusoidal strain is applied, stress response is also sinusoidal for small deformations. However, it is no longer sinusoidal for larger deformations and shows a distorted form due to the presence of higher odd harmonics.40 These higher odd harmonics can be quantified by Fourier transform (FT) rheology,27,41,42 which can detect the very weak signals of higher harmonics and quantify them as intensities as functions of harmonic numbers by decomposing distorted stress in the time domain into frequency-dependent spectra.27 However, the presence of many odd harmonics in nonlinear stress complicates data analysis. For this reason, the relative intensity of the third harmonic (I3/1(ω,γ0) ≡ I(3ω)/I(ω), where ω is the excitation frequency), the first normalized nonlinear leading order, has been used as a representative nonlinear parameter. FT rheology has been applied to investigate polymer melt and solutions.23−28,43,44 In particular, many researches have mainly focused on the differentiation of linear and branched polymers. Several researches44−47 have applied FT rheology to characterize long chain branching (LCB) for linear and branched polyethylene (PE) melts. The results obtained confirmed FT rheology is more sensitive for characterizing LCB than linear viscoelastic methodology. Liu et al.48 defined a new LCB index (DLCB) from the observation that branched polymers had higher I3/1 values at large strain amplitude than a linear polymer when I3/1 curves of linear and branched polymers were superposed by vertical shift. Furthermore, usage of LAOS nonlinear factor (NLF ≡ |G1′ /G3′ | at γ0 = 10 and ω/2π = 0.1 Hz) defined by Filipe et al.49 allowed distinguishing different LCB contents and types. Overall, FT rheology has become a representative methodology for detecting LCB and for distinguishing polymer topologies. However, systematic investigations have rarely been conducted on well-defined topologies. Responding to this need, recent experiments have focused on the nonlinear behavior of monodisperse homopolymers with different architectures using a newly defined parameter, intrinsic nonlinearity Q0(ω).25,27,29,30 An asymptotic or intrinsic approach for the nonlinear shear stress allowed for defining theoretically and measuring new intrinsic nonlinear parameters in MAOS region which is an intermediate region between SAOS and LAOS.27,50−52 Hyun and Wilhelm27 introduced new mechanical parameters, that is a nonlinear coefficient Q(ω,γ0) ≡ I3/1/γ02 and an intrinsic nonlinearity Q0(ω) ≡ limγ0→0Q(ω,γ0). Since then, Ewoldt and Bharadwaj50 described full MAOS region using four intrinsic Chebyshev coefficients, [e1](ω), [v1](ω), [e3](ω), and [v3](ω) (two first-harmonic and two third-harmonic nonlinearities).53,54 Especially, Q0(ω) is calculated by combining two thirdharmonic nonlinearities.50,54 Of intrinsic nonlinear parameters B


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Macromolecules Table 1. Characteristics of PS Solutions Used in This Study samples PS100_20 PS100_25 PS100_30 PS100_35 PS100_40 PS100_100 PS183_25 PS183_30 PS183_35 PS183_40 PS300_10 PS300_15 PS300_20 PS300_25 PS300_30 PS300_35 PS300_40 PS300_100

Mw [kg/mol] (PDI) 100.4 (1.02)

183 (1.06)

297.7 (1.07)

ϕa

Zsol

0.187 0.234 0.283 0.331 0.380 1 0.234 0.283 0.331 0.380 0.093 0.140 0.187 0.234 0.283 0.331 0.380 1

0.620 0.840 1.077 1.330 1.600 5.807 1.530 1.961 2.422 2.911 0.722 1.246 1.838 2.489 3.191 3.941 4.736 17.211

τL [s] at 22 °C −3

3.48 × 10 1.02 × 10−2 2.91 × 10−2 8.70 × 10−2 3.66 × 10−1 1.74 × 10−2 b 9.75 × 10−2 3.04 × 10−1 9.33 × 10−1 3.20 × 100 6.00 × 10−3 3.00 × 10−2 9.26 × 10−2 3.02 × 10−1 1.17 × 100 4.15 × 100 1.44 × 101 8.11 × 10−1 b

entangled or notc unentangled

entangled

entangled

unentangled entangled

ϕ = (mPS/ρPS)/[(mPS/ρPS) + (mDOP/ρDOP)] where ρPS = 1.07 g/cm3 and ρDOP = 0.98 g/cm3. bTerminal relaxation time of melt state at 190 °C. Based on the existence of a crossover point near the terminal region.

a c

Figure 1. Linear master curves of semidilute PS solutions at Tref = 22 °C. (a) PS100 series, (b) PS183 series, and (c) PS300 series. Moduli were shifted by multiplying 10−4 (10 wt %), 10−2 (15 wt %), 100 (20 wt %), 102 (25 wt %), 104 (30 wt %), 106 (35 wt %), and 108 (40 wt %) to enable changes in curve shape to be easily visualized. Note that solutions with the same concentration were shifted to the same extents. The solid lines indicate reciprocal values of terminal relaxation time obtained from linear viscoelastic properties (τL).

Q0(ω) of unentangled solutions was measured and compared with several constitutive models. The solution state instead of

melt state was selected in order to investigate two aspects. First, semidilute linear polymer solutions dissolved in θ solvents C


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were recorded in “transient” mode and then automatically transformed to stress intensities using the “FT to Frequency Spectrum” function in TRIOS software. Data obtained at different frequencies and temperatures were superposed onto nonlinear master curves using the same shift factors (aT) used for linear master curves because the applicability of TTS principle to LAOS tests was experimentally verified for a thermo-rheologically simple fluid.64

follow the same molecular dynamics as linear polymer melts because a chain in a semidilute solution can be considered as a melt of correlation blobs acting as a rescaled monomer.58 As a result, their linear viscoelastic properties coincide at the same entanglement number.59,60 Here, we investigated whether this superposition is also valid for Q0(ω) by comparing solution data with previously published melt data.25,29,30 Second, because DTD is distinct from static dilution (dissolution of a polymer into a solvent) but has the same final result (tube dilation of the backbone chain),57 we tried to compare static dilution with dynamic dilution directly using Q0,max of linear polymer solutions and backbone chains of branched polymers. Furthermore, we calculated the effective number of entanglements per backbone chain in branched polymers by considering effective degree of backbone dilution according to the concept suggested by Kapnistos et al.12,13 As branched polymers, comb polymers with many branches,25 recently published in another paper, were selected for comparison purposes.

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RESULTS AND DISCUSSION SAOS Experiments. Linear Master Curves for Semidilute PS Solutions. Linear master curves of G′(ω) and G″(ω) are plotted in Figure 1. With increasing concentration, the terminal region was shifted toward lower frequencies and the width of the G′(ω) and G″(ω) crossover region increases. This corresponds to the development of a plateau zone due to the onset of entanglement.16 Terminal relaxation times were calculated from linear viscoelastic data as follows: τL = [G′/ (ωG″)]ω→0.61 In the next section, another terminal relaxation time will be defined from nonlinear master curves of Q0(ω). Subdivision into Unentangled and Entangled Solutions. The solutions in Figure 1 can be classified into two groups with or without a crossover point of G′ and G″ near the terminal region. Solutions with no crossover point display a power law behavior of moduli (G′ and G″ ∝ ωn) at the high frequency region. In the case of n = 0.5, linear rheological behavior corresponds to an effective Rouse-like behavior.61 The Rouse model, which describes Rouse behavior, is based on the assumption that a flexible polymer chain is represented as a chain of N Gaussian submolecules connected by springs,61 and it is well-known that semidilute unentangled solutions or unentangled melts follow the Rouse model. Figure 2 compares

EXPERIMENTAL SECTION

Materials. The polymer solutions used in this study were monodisperse linear PS dissolved in dioctyl phthalate (DOP) at different concentrations (Table 1). The sample nomenclature used was as follows: polymer type, weight-average molecular weight in kg/ mol, and concentration based on weight percent. PS100 and PS300 were provided by BASF (Ludwigshafen, Germany) and characterized by gel permeation chromatography (GPC). PS183 was purchased from and characterized by Polymer Source Inc. (Quebec, Canada). PS was dissolved in DOP, which is a θ solvent for PS at the θ temperature of 22 °C.24 DOP was selected due to its low volatility (1.6 hPa at 93 °C). Tetrahydrofuran (THF), a good solvent for PS, was added as a cosolvent to facilitate mixing and ensure homogeneity of solutions. Mixtures were stirred at room temperature for 1 day, and afterward THF was gradually removed under vacuum conditions without substantial DOP loss. Finally, solutions were filtered through Teflon membrane (30−60 μm pore size, Savillex) to remove possible impurities. Polymer solutions are classified as dilute or semidilute based on overlap volume fraction (ϕ*).61 The volume fractions (ϕ) of samples are listed in Table 1. When the volume fraction of a solution is below the overlap volume fraction, the solution is called dilute (ϕ < ϕ*). Otherwise, it is called semidilute (ϕ > ϕ*). The overlap concentration can be calculated approximately using ϕ* = 2.33/[η],62 where [η] is the intrinsic viscosity in mL/g ([η] = 0.84(Mw/104)0.5 at 22 °C).63 Estimated overlap concentrations were 0.098 for PS100, 0.073 for PS183, and 0.057 for PS300. Therefore, all solutions are semidilute. Semidilute solutions are further subdivided as semidilute unentangled or entangled depending on the entanglement volume fraction (ϕe).61 Classification based on entanglement will be discussed in the Results and Discussion section. Rheological Measurements. Linear and nonlinear rheological properties were measured using a commercial strain-controlled rheometer (ARES-G2, TA Instruments). For solutions, rheological properties were measured using an advanced Peltier system (APS) with 25, 40, and 50 mm parallel plate geometries. Melt states were also measured in order to check volume fraction of 1.0. For melts, forced convection oven (FCO) with 25 mm parallel plate was used in a nitrogen environment to prevent oxidative degradation of samples. Frequency sweep tests (SAOS tests) were conducted at different temperatures within the SAOS region to obtain linear master curves using the time−temperature superposition (TTS) principle. Data shifting was performed using TRIOS software provided by TA Instruments. Reference temperatures were fixed at 22 °C for solutions and 190 °C for melts. Strain sweep tests (LAOS tests) were also carried out within the MAOS region. Experiments were stopped before approaching the LAOS region to calculate intrinsic nonlinearity, which is defined in the MAOS region. Stress curves as a function of time

Figure 2. A log−log plot of reduced moduli against aTωτR for the Rouse model. The plots of all solutions with no crossover point are superposed and follow the model qualitatively.

experimental data with the Rouse model. The predicted curves of the Rouse model are expressed by the following equations:61 G′(ω)M w = cRT

ω 2τp 2

N

∑ p=1

1 + ω 2τp 2

(G″(ω) − ωηs)M w cRT

τp =

τR p

2

=

N

=

∑ p=1

(1)

ωτp 1 + ω 2τp 2

(2)

6M w η0 π 2p2 cRT

(3)

where c is polymer concentration, R is the gas constant, T is absolute temperature, ηs is solvent viscosity, η0 is the zero-shear D


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Macromolecules viscosity of the solution, p is the mode number, and τp is the Rouse relaxation time at the pth mode. The plot of reduced moduli versus aTωτR enables all data with no crossover point to coincide.61 Experimental data were in accord with model prediction qualitatively. In particular, the power law index of ∼0.5 at high frequency for experimental data indicates that they follow the effective Rouse-like behavior. Therefore, it does not matter that the solutions with no crossover point in our systems are named semidilute unentangled solutions. The onset of entanglement in polymer solutions begins at a critical value, that is, at the entanglement volume fraction (ϕe). If the volume fraction of a solution is above the entanglement volume fraction, the solution is entangled (ϕ > ϕe). Otherwise, it is not entangled (ϕ* < ϕ < ϕe). Entanglement volume fraction has been defined as follows:16 ϕe =

Mcmelt Mw

Figure 3. Plateau modulus against the volume fraction of the polymer solution used. Two theoretical slopes of 2 (dashed line) and 7/3 (solid line) are indicated. The dilution exponent of 4/3 is deemed to be more appropriate than one of 1.

(4)

solutions in the θ solvent. A recent systematic study supports an α value of 4/3.68 Shahid et al.68 showed that for polymer solutions in small-molecule solvents the plateau modulus was better fitted by ϕ7/3 than ϕ2, and thus, we used α = 4/3 consistently in the present study. Calculated values are listed in Table 1. In particular, entanglements cannot exist in unentangled solutions, which means entanglement number cannot be defined for these solutions. However, to obtain a master curve of maximum Q0 versus number of entanglements, it was calculated using eq 5 by assuming that the theoretical slope of 7/3 in Figure 3 is asymptotically valid for lower volume fractions. MAOS Experiments. Nonlinear Master Curves for Semidilute PS Solutions. Nonlinear master curves of Q0(ω) are plotted in Figure 4. All solutions displayed the typical terminal law (Q0 ∝ ω2) in the terminal region,27,54 and this region shifted toward lower frequency with increasing concentration. Entangled solution data followed the typical curve shape of entangled linear polymer melts, which has already been reported in several publications.25,27,29 For entangled solutions, Q0 increased as a quadratic function of frequency (Q0 ∝ ω2) until the peak, which is a characteristic feature of monodisperse linear homopolymer melts, was reached and then started to decrease with constant slope in a log−log plot. In the present study, many points were obtained at frequencies after the peak value for entangled solutions (De > 1). Recently, Cziep et al.29 pointed out difficulties associated with measurements at De > 1 after maximum Q0 due to low measurement temperatures close to the glass transition temperature, wall slip, sample failures, torque overload, and other phenomena. Solvent in polymer solutions reduces interactions between polymer chains and thus decreases melt stiffness. As a result, chains in the solution follow the oscillation motion at high De better than in the melt. MAOS measurements taken in the solution state might be useful for investigating high De regions of intrinsic nonlinear master curves. Interestingly, unentangled solutions displayed plateau behavior (constant Q0) rather than decreases. As a sample phase changes from unentangled to entangled, intrinsic nonlinear master curve exhibits different behavior after the peak, that is, a constant Q0 for unentangled solutions and a decreasing Q0 for entangled solutions. To compare unentangled and entangled solutions, linear and nonlinear master curves of PS100_30 (unentangled) and

Mcmelt

where is the critical molecular weight, which is melt approximately 2Mmelt value of 17 300 g/mol was e . A Me obtained from experimental plateau modulus using the relation Mmelt = (4/5)ρRT/Gmelt e N . The plateau modulus of entangled solutions was determined at the frequency where tan δ (= G″(ω)/G′(ω)) is minimal. The calculated values of entanglement volume fractions were 0.344 for PS100, 0.189 for PS183, and 0.116 for PS300. Only PS100_35 displayed little difference (0.013). According to these criteria, solutions displaying a crossover seem to be entangled (see details in Appendix 1). In addition, the Doi−Edwards model, based on the tube theory, gives terminal relaxation time corresponding to the frequency of the crossover point of G′(ω) and G″(ω). Taken together, it seems reasonable that solutions with a crossover point are called entangled solutions. For simple discussion, we will call solutions with a crossover entangled solutions and solutions with no crossover unentangled solutions in the remainder of this article (Table 1). Calculation of the Number of Entanglements per Chain. Numbers of entanglements per chain (Zsol = Mw/Msol e ) were calculated by taking into account the solvent-mediated (static) dilution effect. When a polymer melt is diluted, the tube constraining the polymer chain is widened, and this has an effective impact on the entanglement molecular weight. The entanglement molecular weight for a polymer solution is expressed by56 Mesol(ϕ) = Memeltϕ−α

(5)

Msol e

where is the effective entanglement molecular weight in a solution, Mmelt is the entanglement molecular weight of the e melt state (ϕ = 1), and α is the dilution exponent. Two possible values of the dilution exponent, 1 and 4/3, have been suggested.56,65 However, several researches have experienced difficulties in determining an exact value for the dilution exponent between 1 and 4/3.65−67 Figure 3 shows the relationship between plateau modulus and volume fraction. The plateau modulus is given as a function of volume fraction by GNsol(ϕ) = GNmeltϕ1 + α

(6)

The value obtained from melt state at 190 °C was vertically shifted to 22 °C by considering changes of density with temperature as follows: ρ(T) = 1.2503 − 6.05 × 10−4T.12 The dilution exponent of 4/3 seems to be more adequate for our PS E


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Figure 4. Nonlinear master curves of semidilute PS solutions at Tref = 22 °C: (a) PS100 series, (b) PS183 series, and (c) PS300 series. The numbers in parentheses indicate the effective entanglement numbers (Zsol) shown in Table 1. The typical scaling law (Q0 ∝ ω2) is observed at low frequency.

Figure 5. Linear and nonlinear master curves of PS100_30 (unentangled) and PS300_30 (entangled). The reciprocal value of the terminal relaxation time is indicated as solid line. Both linear and nonlinear master curves well characterize the different sample phases.

PS300_30 (entangled) are plotted (Figure 5). The figure shows obvious differences between unentangled and entangled solutions, which suggests transition from a constant Q0 to a decrease of Q0 in a nonlinear master curve might be used as a criterion for distinguishing unentangled and entangled states likewise the existence of a crossover point in a linear master curve. Unentangled solutions have more intriguing intrinsic

nonlinear behaviors. This is discussed in detail in the next section. For each group (unentangled and entangled solutions), normalized intrinsic nonlinearity (Q0/Q0,max) was plotted as a function of a dimensionless number (De = aTωτL) defined by terminal relaxation times (Figure 6). Results (Q0/Q0,max vs De) were almost superposed for each group (see Figure 6a,b). As was explained above, unentangled solutions display plateau F


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Figure 6. Dimensionless plots of normalized intrinsic nonlinearity as a function of De (= aTωτL). (a) Unentangled solutions show plateau behavior at De > 0.4, whereas (b) entangled solutions show a decrease of Q0 with a constant average slope of −0.23 at De > 1.

behavior at De > 0.4, whereas entangled solutions show a decreasing in Q0 with a constant “average” slope of −0.23 at De > 1. In Figure 6a, evidence for the plateau behavior of Q0 seems to be rather weak. The data points do not go much above De = 1. There is a possibility that the plateau region is followed by power law behavior with an increasing or decreasing slope. However, this plateau behavior can be regarded as a characteristic feature of unentangled solutions because entangled solutions definitely display a maximum point followed by thinning of Q0. In Figure 6b, the power law value of −0.23 differs from the −0.35 mentioned in a recent report on entangled polymer melts.29 This difference is induced from poor data reproducibility at De > 1, especially for polymer melts. Cziep et al.29 suggested an average scaling exponent after Q0,max of −0.35 with a standard deviation of up to 15%. Nevertheless, the decreasing slope of −0.23 appears more reliable. It was reported for comb PS with many branches, slope after Q0,max corresponding to backbone chain increases as branch length increases and that comb PS with unentangled branches which has Q0 behavior similar to linear one has decreasing slope of −0.22.27 At the current level of knowledge, assertions regarding actual slope are ill-founded due to the experimental difficulties mentioned above. Nevertheless, the important point is that intrinsic nonlinear master curves consistently show a specific shape. As the plateau zone is developed by increased entanglement, linear master curves change and the superposition of linear viscoelastic properties (G′, G″, and tan δ) become impossible even in dimensionless coordinates (Figure 1). It is concluded that nonlinear master curve shapes depend on the onset of entanglement and on polymer topologies. MAOS Behavior of Unentangled Solutions. Intrinsic nonlinear master curves for unentangled solutions are presented as a function of De defined using terminal relaxation time (Figure 7). Interestingly, all curves of unentangled solutions were superposed although Q0-axis was not normalized by Q0,max. Namely, all semidilute unentangled solutions had the same Q0,max. This result provides evidence that intrinsic nonlinearity Q0(ω) responds more sensitively to terminal relaxation processes because intrinsic nonlinearity has a constant value level irrespective of concentration whereas linear viscoelastic moduli increase continuously with increasing concentration. By definition, the Q0 parameter is calculated by normalizing intrinsic third-harmonic complex modulus by linear viscoelastic counterpart. Figure 8 shows two complex moduli as a function of De. In Figure 2, the unentangled solutions followed the

Figure 7. Intrinsic nonlinearity of unentangled solutions at Tref = 22 °C. Lines are of analytical solutions of different constitutive equations.

effective Rouse-like behavior with increasing slope of 0.5 in log−log scale. The intrinsic nonlinear third-harmonic complex modulus also increases with 0.5 power-law slope. Two moduli displays different terminal scaling but the same power-law slope at De > 0.4. This explains the plateau behavior of the unentangled solutions observed at De > 0.4. In addition, the unentangled solutions had the same Q0 values at the same De, which signifies that intrinsic third-harmonic nonlinearity under MAOS flow appears in proportion to linear viscoelastic properties under SAOS flow. At low De, rotations of polymer chains and all possible configurational rearrangements occur within the period of an oscillating cycle. However, with increasing De, the orientation of polymer chains cannot keep pace with the alternating oscillation, and as a result, the response of the polymer chains will be limited to stretching chemical bonds.16,69 The Rouselike behavior with a constant slope of 0.5 in Figure 8a shows the transition from random configurational changes to stretch of chains. Within this transition region, the ratio of energy dissipated to energy stored in oscillatory deformation becomes constant, which is a characteristic relaxation process of unentangled linear chains. The same slope of 0.5 shown in Figure 8b does not mean the same physical interpretation as linear viscoelasticity but at least might be used as indication of the Rouse relaxation process. This characteristic feature of Rouse relaxation results in not only 0.5 power-law slope of linear viscoelastic moduli as shown in Figures 2 and 8 but also a constant thinning slope of −0.5 in steady shear viscosity. Colby et al.69 showed that shear thinning of unentangled chains originates from strongly stretched chains most of the time along the flow direction in shear although they also tumble. G


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Figure 8. (a) Linear viscoelastic complex modulus and (b) intrinsic nonlinear third-harmonic complex modulus as a function of De at Tref = 22 °C.

Therefore, the effect of chain stretch becomes important at a time scale shorter than the longest Rouse relaxation time. In Figure 7, unentangled solutions displayed plateau behavior in nonlinear master curves at De > 0.4, and their Q0,max values were the same as about 0.006, regardless of concentration. These unique characteristics of intrinsic nonlinearity in unentangled solutions are expected to originate from no interchain entanglement, chain stretch, and alignment to flow direction. In particular, the fact that unentangled solutions had the same Q0 at the same De regardless of molecular weight and concentration confirms that intrinsic nonlinearity is highly sensitive to relaxation processes. The graphs of nonlinear master curves of the unentangled solutions are similar to analytical solutions of several singlemode constitutive equations including Pom-Pom,29 molecular stress function (MSF),29,30,70 corotational Maxwell,71 dilute rigid dumbbell,72 and fourth-order fluid expansion.54 Comparison between experimental data and analytical solutions is presented in Figure 7. Because measured Q0 values are apparent values under inhomogeneous shear flow, they should be converted to true values using the correction factor.70,73 Q 0,true(ω) =

3 Q (ω) 2 0,app

rigid dumbbell: Q 0(ω) =

× {[(1 + De 2)(625 + 9250De 2 + 26649De 4 + 214488De 6 + 11664De8)] /[(25 + 4De 2)(1 + 4De 2)2 (1 + 9De 2)2 1/2

(25 + 9De 2)2 ]}

Q 0(ω) =

in the limit of low frequency

(7)

(8)

MSF: Q 0(ω) =

β⎞ 3⎛ De 2 ⎜α − ⎟ 2 1/2 2⎝ 10 ⎠ (1 + 4De ) (1 + 9De 2)1/2 (9)

corotational Maxwell: 1 De 2 Q 0(ω) = 4 (1 + 4De 2)1/2 (1 + 9De 2)1/2

(12)

The single-mode equations predicted overall curve shape well. Especially, the terminal region frequency scaling (Q0 ∝ ω2) and its relative magnitudes are predicted exactly by all constitutive equations.54 The reason for these good predictions is speculated to be related to the rule in multimode MSF fitting. The multimode MSF model is used to predict the MAOS behaviors of entangled chains. In this case, nonlinear master curves can be exactly anticipated under the assumption of terminal relaxation mode, where only relaxation times longer than the Rouse time scale contribute to nonlinearity in the MAOS region.70 In semidilute unentangled solutions, the longest Rouse relaxation time becomes the terminal relaxation time, which leads to usage of single-mode equations. Accordingly, the exact prediction of curve shape and terminal behavior is due to “single mode”. However, all single-mode predictions fails at De > 0.2 where the data show transition to the plateau behavior. For example, the MSF equation levels off at a plateau value of Q0,max = 0.034, which is the lowest value of all model equations used but 3.8 times larger than the experimental value of 0.009 (adjusted using 1.5 correction factor). The dilute rigid dumbbell model even gave the Q0,max value 12 times larger than the experimental value. Considering that PS300_100, the highest entangled solution in this study, has Q 0,max = 0.031, the difference of 0.025 between experimental value and the MSF model cannot be ignored. The above constitutive equations display a crossover point between linear viscoelastic moduli and have been applied to analyze the rheological behaviors of entangled polymer systems, except for the dilute rigid dumbbell model. In particular, the MSF model is based on Doi−Edwards tube theory, which contradicts no crossover of unentangled solutions. In the previous section, the unentangled solutions examined were found to follow the linear viscoelastic behavior predicted by the Rouse model. However, the Rouse model fails to predict nonlinear effects under MAOS flow because the upper-

De 2 2π (1 + De 2)1/2 (1 + 4De 2)1/2

at Z → ∞

(11)

fourth‐order fluid: Q 0(ω) = 0.126De 2 for b = 10

The corrected Q0 values are used in Figure 7. Each analytical solution is expressed as follows:29,54,71,72 Pom‐Pom:

9 14

(10) H


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Figure 9. Positional changes of terminal and MAOS relaxation times of PS300 series in linear and nonlinear master curves with increasing concentration. (a) At low concentration, MAOS relaxation time is larger than terminal relaxation time, but (b) with increasing concentration, the situation is reversed, and (c) MAOS relaxation time moves toward the crossover point of linear viscoelastic moduli.

nonlinear master curves (Figure 9). Interestingly, positional changes of terminal and MAOS relaxation times with increasing concentration were observed. At low concentration, MAOS relaxation time had a higher value than terminal relaxation time (Figure 9a), but this relationship was reversed with increasing concentration. Figure 9b shows that the reciprocal value of MAOS relaxation time was positioned at a slightly higher frequency than that of terminal relaxation time. Finally, MAOS relaxation time shifted to the crossover point of linear viscoelastic data in Figure 8c. Therefore, reversal of the magnitude relationship of two relaxation times occurred before reaching a concentration of 40 wt % in PS300 series. PS100 and PS183 series showed trends similar to that of PS300 series (not shown here). We plotted MAOS relaxation times as a function of terminal relaxation times for PS100, PS183, and PS300 series in Figure 10. Figure 10 shows that MAOS relaxation time has a higher value than terminal relaxation time at low terminal relaxation times, namely, low entanglement number. On the other hand, two points (unfilled symbols in Figure 10) obtained from melts display higher values for terminal relaxation times. Melt data at 190 °C could not be shifted to 22 °C using the WLF equation, and thus, they are indicated separately as unfilled symbols. The

convected Maxwell derivative in the model is codeformational, and thus it measures rate of change with respect to a coordinate system that translates, rotates, and deforms with the fluid.72,73 To further understand the MAOS behaviors of semidilute unentangled solutions, a new constitutive equation should be developed or a conventional constitutive equation modified. Crossover between Terminal and MAOS Relaxation Times (τL and τN). Intrinsic nonlinearity Q0(ω) sensitively reflects the terminal relaxation processes of polymer chains,25,27,29,30 and thus, the MAOS relaxation time (τN) was calculated from the point where elastic contribution of Q0 converges to zero, and simultaneously, the viscous contribution of Q0 is maximized (not shown here). At MAOS relaxation time, the Q0 value of unentangled solutions reaches the plateau value (0.006) for the first time, and entangled solutions display the maximum value of Q0 (≡ Q0,max). It has been reported that two relaxation times, i.e., terminal and MAOS relaxation times (τL and τN), almost coincide within experimental error and that they may be associated with similar molecular dynamics.25,29 In this section, MAOS relaxation time (τN) was systematically compared with terminal relaxation time (τL). First, we investigated PS300 series. To discuss in more detail, reciprocal values of terminal and MAOS relaxation times are indicated in linear and I


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molecular weight Mc from the scaling change in zero-shear viscosity versus molecular weight, and the other critical molecular weight Mc′ from the scaling change in steady-state recoverable compliance versus molecular weight.1 It has been reported in many references that these critical values are related, though not in a simple manner, and approximately Mc ≈ 2Me and Mc′ ≈ 4Me − 6Me.16,74−76 For monodisperse linear polymers, Mc′ indicates the transition from lightly or moderately entangled region to highly entangled region.76 In Figure 11, the crossover of two relaxation times occurs at the critical value of 4.28. The critical entanglement number obtained above might be related to the transition to highly entangled region. The correspondence between MAOS relaxation time obtained by MAOS tests and crossover point of G′ and G″ by SAOS tests is intriguing (Figure 9c). The Doi−Edwards constitutive equation assumes that relaxation occurs by reptation only and shows that terminal relaxation time is the reciprocal of crossover frequency in a linear master curve.17 The possibility of relaxation by reptation only has been previously suggested by Fetters et al.77,78 The Doi−Edwards model predicts that the zero-shear viscosity of an entangled polymer is proportional to M3 rather than M3.4 as observed experimentally due to some limiting processes, such as contour length fluctuation (CLF) and constraint release (CR). However, Fetters et al.77,78 suggested that the scaling exponent of experimental viscosity reverts to M3 beyond the reptation molecular weight Mr and that the effect of limiting processes vanishes. Pure reptation might occur at higher entanglement numbers, and thus, the crossover point in linear master curves might be the real terminal relaxation time of a pure reptation process. The reptation molecular weight of PS is about 624 kg/ mol,78 which is double that of PS300_100. Coincidence with the crossover point at 300 kg/mol, not at 624 kg/mol, is thought to be induced by sparse data between 2 and 3 rad/s in Figure 8c. What is more important is the possibility that MAOS relaxation time might reflect real terminal relaxation time. To explore this possibility, MAOS behaviors at higher entanglement number need to be further studied. Q0,max at Different Concentrations. Q0,max as a Function of Entanglement Number. The maximum Q0 value (Q0,max) was plotted as a function of polymer volume fraction, as shown in Figure 12. Q0,max values were obtained from plateau value at De > 0.4 for unentangled solutions and peak value near De = 1 for entangled solutions. The Q0,max of two melt states obtained at 190 °C was added to the plot since Q0,max is not a function of reference temperature. With increasing volume fraction, Q0,max

Figure 10. Comparison between terminal (τL) and MAOS (τN) relaxation times. At low entanglement number, the nonlinear relaxation time is larger than the linear relaxation time. The dashed line indicates the scaling exponent of 0.88.

scaling exponent between the two relaxation times was 0.88 for solution data, which notably was less than 1. Therefore, if melt data are considered to be in line with solution data, the magnitude relationship of two relaxation times will be reversed at some entanglement number. Because increase of concentration leads to increase of the number of entanglement per chain, the terminal and MAOS relaxation times of PS300 series were plotted against entanglement number of solutions (Zsol) (Figure 11). Two

Figure 11. Terminal and MAOS relaxation times of PS300 series as a function of entanglement number in polymer solutions. Crossover between the two relaxation times is observed at (Zsol)c = 4.28.

relaxation times increased with an analogous tendency and crossover between PS300_35 and PS300_40. The last three points were fitted with a power-law type equation, and as a result, a critical entanglement number ((Zsol)c) of 4.28 was obtained from the intersection point of the two equations. At Zsol = (Zsol)c, the MAOS relaxation time coincided with terminal relaxation time. Based on the critical value, the relaxation time was divided into two regions. At Zsol < (Zsol)c, the MAOS relaxation time was longer than terminal relaxation time, indicating that the intrinsic nonlinear relaxation process might take longer than that expected based on the theory of linear viscoelasticity. At Zsol > (Zsol)c, intrinsic nonlinear relaxation preceded linear relaxation and occurred near the intersection of linear viscoelastic moduli at higher entanglement number. There are three different critical molecular weights obtained by different rheological criteria: the entanglement molecular weight Me from the plateau modulus of SAOS data, the critical

Figure 12. Experimentally determined maximum Q0 (= Q0,max) values as a function of volume fraction (ϕ) for PS100, PS183, and PS300. J


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For entangled solutions, Q0,max values were fitted with an logistic-type equation (y = a/(1 + bxc)) because Q0,max values exhibited an initial increase followed by saturation in log−log plot. The resultant fitting equation was as follows:

increased except for four points corresponding to unentangled solutions, which exhibited a constant Q0,max value of ∼0.006. This tendency suggests that all data can be superposed. Figure 13 displays a master curve of Q0,max when the x-axis is replaced by the number of entanglements per chain in solution

Q 0,max

⎧ 0.006 at Z sol [[mml:lt]] 1 ⎪ =⎨ 0.026 at Z sol ≥ 1 ⎪ sol −1.01 1 3.59( ) Z + ⎩

(13)

The Zsol of PS100_30 was used as a critical value for onset of entanglement. Equation 12 shows that the limiting value of Q0,max at high Zsol is 0.026, which is almost the same as the Q0,max of PS300_100 (0.022). This similarity is attributed to the lack of data points around Zsol = 10 and at higher Zsol. Nevertheless, the fitting line described the experimental data well (Figure 13). The experimental data were compared with two predictions based on reptation theory.70,79 The prediction results for Q0 behavior are described in Appendix 2. Pearson and Rochefort79 gave full analytical solutions of the Doi−Edwards model with independent alignment approximation (DE IA) under MAOS flow. The Q0,max predicted by their equations was indicated as 0.040 in Figure 13. The resultant value is rather different from 0.026 at infinite Zsol of eq 13. This deviation is removed when stretching effect of chains is introduced in the model as MSF. The MSF model gave Q0,max of 0.023, which is the same as the prediction of eq 13 at infinite Zsol. The MSF model introduces two parameters for predicting intrinsic nonlinear behavior in the MAOS region.80 The strain measure of DE IA model for linear polymers resulted in a constant value α = 5/21, which explains affine orientation of network strands. The MSF model considers an additional contribution of isotropic strand extension using stretching parameter β. By definition, the β value is fixed as 1 for linear chains.55 Thus, two parameters remain constant during MAOS measurements for linear chains. Successful multimode MSF predictions for experimental data told us that stretching effect should be considered for MAOS flow which is weak but still nonlinear flow.30,55,70 In analytical equations of the MSF model, Q 0 behavior depends predominantly on SAOS moduli and also on two parameters (α and β) (see eqs 19 and 20). The Q0 behavior depends solely on linear relaxation mechanism at fixed parameters for linear chains. Accordingly, any changes in linear relaxation processes (Rouse, CLF, and reptation, etc.) can be reflected by Q0. The prediction by MSF can be explained as follows: at infinite Zsol, pure reptation becomes a dominant linear relaxation mechanism. The relaxation motion change by pure reptation affects SAOS moduli and also Q0. Finally, the linear chains undergoing pure reptation quantitatively follow the prediction by DE IA model combined with stretching contribution. Thus, Q0,max changes in Figure 13 can be understood with the aid of linear viscoelastic data such as zero-shear viscosity as a function of molecular weight, and reversely, the Q0,max plot has the potential for analyzing polymer dynamics along with SAOS moduli. Consequently, Q0,max, which is a characteristic feature in the master curve of nonlinear data obtained by MAOS test, is quite different from plateau modulus (GN), which is a representative parameter in the master curve of linear rheological data obtained by SAOS test. Plateau modulus (GN) monotonically increases with increasing volume fraction or Zsol on a log−log scale irrespective of sample phase, such as unentangled or

Figure 13. Master curve of Q0,max as a function of solution entanglement number. For unentangled solutions, Q0,max is constant at ∼0.006. It increases gradually from Zsol = 1 and seems to reach a limiting value. The resultant logistic-type fitting equation for the Q0,max of entangled solutions is shown in the plot. For comparison, several predictions obtained by Doi−Edwards (DE IA), molecular stress function (MSF), and semiempirical equation (from Cziep et al.29) are plotted together.

state (Zsol). Unentangled solutions exhibited a constant Q0,max value at Zsol < 1, and as Zsol was increased, Q0,max increased due to formation of entanglements, but the relationship between two parameters was not linear in the log−log plot. Q0,max seemed to increase linearly from Zsol = 1 to 4, but this increase started to decline after Zsol = 4. It rather seemed to converge at a limiting value. Although Cziep et al.29 demonstrated Q0,max ∝ (Zsol)0.35, it was observed that the relationship between the two parameters was not described by a simple power law. This behavior might be related to crossover between terminal and MAOS relaxation times, as shown in Figure 11. The MAOS relaxation time was always larger than terminal relaxation time at Zsol values less than the critical value of 4.28. After critical Zsol, MAOS relaxation time became smaller than terminal relaxation time and was shifted toward the reptation time defined in the tube model. Accordingly, as entanglement state moves from lightly or moderately entangled region to highly entangled region, the linear increment of Q0,max starts to slow down, and finally Q0,max can converge at high Zsol, where dominant linear relaxation mechanism is reptation for linear chains. The same result was drawn from a graph of reduced zero-shear viscosity data against molecular weight normalized with reptation molecular weight for various polymers (Figure 3 in Unidad et al.78). In this graph, slope changes indicated three different regions, that is, Rouse, reptation + CLF, and pure reptation regions. Thus, Q0,max behavior in Figure 13 can be understood in the same vein, and it is confirmed again that the master curve of Q0,max against Zsol responds sensitively to characteristic relaxation process in each region. In this respect, the starting point of increasing Q0,max from the baseline (0.006) in the plot might be indicative of the onset of interchain entanglement between monodisperse linear chains and used to determine the existence of entanglements in those samples in the same manner as slope change of Q0 master curve at De > 1. K


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Macromolecules entangled state,68 whereas Q0,max exhibits an approach to a limiting value on a log−log scale. Moreover, the plateau modulus (GN) is shifted onto a single master curve when it is plotted against volume fraction only, but a single master curve of GN fails against Zsol.68 Accordingly, Q0,max enables direct determination of the number of entanglements of linear chain. Comparison between Melt and Solution Data. A recent publication reported similar Q0,max behaviors for monodisperse linear homopolymer melts of different monomer types.29 This section compares our solution data with the melt data reported in other papers.25,29,30 The Q0,max data of various monodisperse linear homopolymers were added to the master curve of Q0,max shown in Figure 13. Figure 14 shows that the data sets obtained for melts and solutions of linear homopolymers superposed on the master curve within experimental error.

Comparison between Static Dilution and Dynamic Dilution. Static dilution and dynamic dilution affect the same final result to the tube diameters of backbone chains, but by different mechanisms.57 In static dilution (i.e., preparing a polymer solution), the tube diameter of a polymer chain is permanently widened by oligomer chains or small-molecule solvents, which enable the chain to move through the already dilated tube during the initial stage of relaxation. Thus, static dilution is a time-independent process. In dynamic tube dilution (DTD), which is a simplified version of the constraint release (CR) mechanism, already relaxed chains dilate the tube diameters of nonrelaxed chains. In particular, branched polymers do not relax their backbone chains before branches have relaxed due to the high friction generated by branching points.12 According to this hierarchy of relaxation motion, relaxed branches act as effective solvents for nonrelaxed backbones, and as a result, the backbone tube swells. Thus, DTD is a time-dependent process since backbone chains explore the gradually dilated tube as branches relax. The above description of dynamic dilution is completely valid in a socalled full-DTD only. For comb polymers with many branches, the hierarchical relaxations of branches and backbones are well separated, such that backbones have enough time to reach a CR-equilibrated state within the dilated tube prior to the terminal region.57 Thus, the framework of full-DTD is valid for such comb polymers. Hyun and Wilhelm27 attributed the Q0,max decrease shown by the backbone chains of comb polymers to reductions in the effective number of entanglements per backbone chain by relaxed branches within full-DTD picture. This explanation provokes the suggestion that Q0,max values of backbone chains in comb polymers might be identical to those of linear chains in solution state at the same effective number of backbone entanglements, with no dependence on architecture, because the effect of static dilution on tube dilation is equivalent to that of full-DTD. Kapnistos et al.12,13 suggested a method for calculating the effective number of entanglements per backbone chain by considering dangling ends of backbones. In the calculation, the two ends of a backbone chain are treated as additional branches, so that the effective part of the backbone shorter than the actual length moves along its primitive path. Under the assumption that branching points are evenly distributed along backbone chain, the molecular weight of the end part of a backbone becomes Mb,end = Mb/(q + 1), where Mb is the actual backbone molecular weight and q is the average number of branching points. Then, the effective length of the backbone chain is Mbdil = Mb − 2xcMb,end, where xc is the fractional length of the backbone-end portion varied by branch size. If branches are longer than backbone ends (Mbr > Mb,end), xc is equal to 1 because backbone ends can fully relax on the time scale of branch relaxation. On the other hand, if branches are shorter, xc becomes the ratio of molecular weight of a branch to backbone-end molecular weight (Mbr/Mb,end). Finally, the effective number of entanglements per backbone chain is calculated as

Figure 14. Comparison of Q0,max data for solutions obtained during the present study and previously published melt data. Zb represents the entanglement numbers of backbone chains in melts. All data set on a master curve and follow eq 13 within acceptable deviations.

The superposition in Figure 14 is an expected result. As mentioned in the Introduction, dynamic scaling in semidilute solutions follows the blob theory, in which correlation blobs behave as shorter chains of g monomers.61 If polymer chains of N monomers in θ solution are replaced by melt chains with N/ g correlation blobs as a rescaled monomer, semidilute solutions under the θ condition have the same molecular dynamics as melts. Universality in shape of nonlinear master curves was discussed for monodisperse entangled linear chain melts in several published studies.25,27,29,30 The semidilute θ solution data in Figure 6b displayed the same curve shape as melt counterparts. Figure 14 ascertains universality even for Q0,max for semidilute θ solutions and melts. Thus, we can speculate that for monodisperse linear polymer chains the MAOS behaviors of semidilute θ solutions are the same as those of melts at the same Z value and that the nonlinear master curve of Q0 depends only on the number of entanglements per chain. Moreover, using this universality, when MAOS test of a monodisperse linear polymer with unknown molecular weight returns a Q0,max > 0.006, it signifies that polymer chains are entangled. Recent experiments with linear polymer melts and solutions at the same Z have demonstrated that while their extensional rheology is drastically different due to different levels of alignment-induced reductions of monomeric friction coefficients, shear rheology, as determined by for example, SAOS, steady shear, and nonlinear start-up test, is identical.59,60 According to our results, MAOS test can be included in the category of shear rheology mentioned above.

effective Z : Z bdil L

=

Mbdil

(ϕbdil)4/3 Memelt

⎞4/3 Mbdil ⎛ Mbdil ⎟⎟ = melt ⎜⎜ Me ⎝ Mb + qMbr ⎠

(14)


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apparent values displayed deviations except the point of largest Zb for comb polymers (PS275k-30-12k). Somewhat surprisingly, these deviations were decreased, and data agreement was enhanced when effective values were used instead of apparent values (Figure 15b). Remaining deviations might be due to

Table 2. Molecular Characteristics of Comb Polymers Investigated samples

Mb [kg/mol]

Mbr [kg/mol]

q

Zba

b Zdil b

PS275k-30-12k PS275k-25-26k PS275k-29-47k PpMS195k-14-15k PpMS195k-14-42k

275 275 275 195 195

12 26 47 15 42

30 25 29 14 14

5.21 3.15 1.47 3.27 1.36

4.46 2.62 1.25 2.34 0.97

a

Apparent Z calculated with no consideration of dangling end parts (eq 15). bEffective Z calculated using the method described by Kapnistos et al.12,13 (eq 14).

Table 2 lists calculated effective values (Mdil b ) and apparent values (Zb) for comparison. Apparent values for comparison were calculated as apparent Z : Zb =

Mb Memelt

4/3

( ϕ b)

⎞4/3 Mb ⎛ Mb = melt ⎜ ⎟ Me ⎝ Mb + qMbr ⎠

(15)

Note that the actual backbone length (Mb) was used in eq 15 without considering dangling end parts of backbone. As model comb systems, PS and PpMS (poly(p-methylstyrene)) used in Kempf et al.25 were selected. The backbone Q0,max values of these comb polymers were added to results of Figure 13 and are shown in Figure 15. It should be noted that apparent backbone chain was used in Figure 15a, and the effective entanglement number was used in Figure 15b. The use of

Figure 16. Experimental data for zero-shear viscosity as a function of the product of concentration and weight-average molecular weight (cMw).

Figure 17. (a) Normalized storage and loss moduli and (b) intrinsic nonlinearity Q0 predicted by DE IA and MSF models.

uneven distributions of branching points in real systems. In particular, PS275k-29-47k and PpMS195k-14-42k had Q0,max values corresponding to semidilute unentangled solutions of monodisperse linear chains. This is rationalized by their SAOS data, which showed few or virtually no entanglements (Rouselike behavior) of backbone chains with power law dependency of moduli (G′ and G″ ∝ ωn, n = 0.5) induced by DTD. The MSF model suggests that Q0 is made up of linear viscoelastic

Figure 15. Comparison of the Q0,max values of statically diluted linear chains and dynamically diluted backbones of comb polymers as a function of the number of entanglements per chain. For comb polymers, apparent value is used as the x-axis variable in (a) and effective value is used in (b). M


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These findings show that intrinsic nonlinearity Q0(ω) is highly sensitive to the characteristic relaxation processes of polymers. According to our results, Q0(ω) can be regarded a parameter that maximizes delicate changes observed in linear viscoelastic moduli. Thus, nonlinear behaviors under MAOS flow can cast light on unsolved problems about relaxation processes, such as reptation, contour length fluctuation (CLF), constraint release (CR), and others. To this end, the nonlinear behaviors of various well-defined polymers need to be further investigated and analyzed from the perspective of molecular dynamics. Four intrinsic nonlinear moduli can be defined in the MAOS region: two first-harmonic nonlinear moduli (G′31 and G″31) which indicate average intercycle nonlinearity of stress response to applied strain or strain rate, and two third-harmonic nonlinear moduli (G33 ′ and G33 ″ ) which affect intracycle nonlinearity by inducing stress curve distortion in elastic or viscous Lissajous plots.50,53,54 In addition, their sign changes carry physically meaningful interpretations.40,81 By definition, however, Q0 parameter combines two third-harmonic moduli and loses their sign information (eq 21).27,40 Therefore, further investigation is necessary about elastic and viscous part of Q0.

moduli, but its proportional factor is nonlinear (see eqs 18 and 19). The Rouse-like backbone behavior induced by DTD process influences SAOS data, and as a result, the backbone part of comb polymers has identical Q0,max values of linear chains at the same effective number of entanglements. This result confirms that the master curve of Q0,max obtained in Figure 13 can be used to quantify the effective entanglement numbers of backbone chains of branched polymers when Q0,max is known.

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CONCLUSION In this study, the nonlinear rheological behaviors of monodisperse linear polystyrene (PS) chains dissolved in θ solvent (DOP) at various concentrations were systematically investigated under medium oscillatory shear (MAOS) flow using FT rheology. Prepared semidilute solutions were classified as unentangled or entangled solutions based on the crossover point near the terminal region of linear master curves. The MAOS behaviors of two solution types were discussed in detail. The following notes could be presented from the nonlinear master curves. • In a dimensionless coordinate (Q0/Q0,max vs De), semidilute unentangled solutions displayed plateau behavior at De > 0.4, whereas semidilute entangled solutions displayed one peak near De = 1 followed by a scaling of Q0 with De−0.23. The transition from a plateau to decreasing Q0 might be indicative of the onset of entanglement. Moreover, the nonlinear curve shape was changed due to the existence of entanglements only and was unaffected by Zsol. • A comparison of terminal relaxation times calculated using linear viscoelasticity theory and MAOS relaxation times obtained from the starting point of plateau behavior or peak value in nonlinear master curves showed that a plot of relaxation times against Zsol was divided into two regions at the critical value, (Zsol)c = 4.28. At Zsol < 4.28, MAOS relaxation time was larger than the corresponding terminal relaxation time, whereas at Zsol > 4.28, the situation was reversed. The MAOS relaxation time became less than the terminal relaxation time and approached a crossover point corresponding to relaxation time defined by the tube model. • Q0,max values made a master curve against Zsol. Interestingly, unentangled solutions displayed a constant Q0,max (0.006), irrespective of Zsol, which might be due to the characteristic relaxation process of unentangled solutions, that is, Rouse-like relaxation with no interchain interaction and chain stretching. Q0,max increased with the onset of entanglement and seemed to converge to a limiting value as reptation process became dominant. • MAOS behaviors of semidilute solutions and melts were identical and followed well-known molecular dynamics because the shapes of their nonlinear master curves and Q0,max values were identical at the same Z values, indicating the universality of molecular dynamics between solutions and melts is even maintained under MAOS flow. • A comparison between static dilution and dynamic dilution suggested the possibility of characterizing the degree of backbone tube dilation in branched polymers using the static dilution effect of monodisperse linear chains. When the effective entanglement number of backbone chain was calculated under the DTD framework, the backbone Q0,max of comb polymer melts was superposed on that of linear chain solutions in the Q0,max−Z plot.

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APPENDIX 1. DETERMINATION OF ONSET OF ENTANGLEMENTS IN POLYMER SOLUTIONS The effect of entanglements on polymer rheology is reflected in the dependency of the zero-shear viscosity on molecular weight.1,16 The zero-shear viscosity increases proportionally to the molecular weight (η0 ∝ Mw) below the critical molecular weight Mc. Above Mc, the dependency on the molecular weight changes as η0 ∝ Mw3.4. The same context can be applied to polymer solutions laid in semidilute regime when the molecular weight is simply replaced by the product of concentration and molecular weight (cMw).16,82 In Figure 16, zero-shear viscosity data of the PS solutions used are plotted as a function of cMw. The solutions used in this study lie in transition region from unentangled state to entangled state. For comparison, we added zero-shear viscosity data of Gupta and Forsman83 to the plot. The black filled symbols represent zero-shear viscosity of 30 and 40 wt % PS solutions with different molecular weights dissolved in DOP. The reference data show that the scaling behavior of zero-shear viscosity changes at the constant cMw value. In other words, the onset of entanglements in polymer solutions corresponds to the critical value of ceMw, where ce is entanglement concentration.16,82 Accordingly, entanglement volume fraction is obtained as follows: ϕe =

ce M melt = c ρ Mw

(16)

where Mmelt is the critical molecular weight of the undiluted c (melt) polymer, which is approximately 2Mmelt e . In this study, the classification by eq 16 led to the same result as the classification by existence of a crossover point of SAOS data near the terminal region. For this reason, we called solutions with a crossover entangled solutions and solutions with no crossover unentangled solutions within this article.

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APPENDIX 2. Q0 BEHAVIOR PREDICTION USING CONSTITUTIVE MODELS BASED ON THE REPTATION THEORY If reptation is the only mechanism of relaxation, then Doi− Edwards theory gives linear viscoelastic moduli as follows:17 N


Macromolecules ∞

∑

′ (ω) = G0 G11

p ,odd ∞

∑

″ (ω) = G0 G11

p ,odd

De 2 1 p2 p4 + De 2

(17)

De p4 + De 2

(18)

■

=

⎤ β ⎞⎡ 3 ⎛⎜ 1 ⎟ G ′ (ω) − G ′ (2ω) + ′ (3ω)⎥ α− G11 ⎢ 11 11 ⎦ 4⎝ 10 ⎠⎣ 3

β⎞ 3 ⎛⎜ ⎟ G0 α − 4 ⎝ 10 ⎠

∞

∑ p ,odd

1 ⎛ De 2 4De 2 3De 2 ⎞ ⎜ 4 ⎟ − 4 + 4 2 2 2 p ⎝ p + De p + 4De p + 9De 2 ⎠

(19) ″ (ω) = G33 =

⎤ β ⎞⎡ 3 ⎛⎜ 1 ⎟ G ″ (ω) − G ″ (2ω) + ″ (3ω)⎥ α− G11 ⎢ 11 11 ⎦ 4⎝ 10 ⎠⎣ 3

β⎞ 3 ⎛⎜ ⎟ G0 α − 4 ⎝ 10 ⎠

⎛ De ⎞ 2De De ⎜ 4 ⎟ − 4 + 4 2 2 2 p + De p + 4De p + 9De ⎠ p ,odd ⎝ ∞

∑

(20)

Here, third-harmonic moduli depends on the difference (α − β/10) between the orientational effect (parameter α) according to the DE IA model and the stretching effect (parameter β) according to the MSF model. The parameter α is constant as 5/ 21 in both DE IA and MSF model. Thus, the parameter β controls overall level of third-harmonic moduli with β = 0 in DE IA model and β = 1 for linear chains in MSF model.55 Finally, intrinsic nonlinear Q0(ω) can be obtained by its definition as follows: Q 0(ω) =

G33′ 2(ω) + G33′′ 2(ω) G11′ 2(ω) + G11′′ 2(ω)

(21)

The predictions by DE IA and MSF models are presented in Figure 17. The Q0,max values in Figure 13 was adjusted by using 1.5 correction factor to match with inhomogeneous flow condition.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (K.H.). ORCID

Kyu Hyun: 0000-0001-5129-5169 Notes

The authors declare no competing financial interest.

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REFERENCES

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where G0 = (24/5π2)(ρkT/Ne) and De = ωτd. τd is disengagement time or terminal relaxation time in the Doi− Edwards theory. The Doi−Edwards model with independent alignment approximation (DE IA) and the molecular stress function (MSF) model give the same equation forms of third-harmonic Fourier moduli.55,70,79 ′ (ω) = G33

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ACKNOWLEDGMENTS

This research was supported by the Global Ph.D. Fellowship Program (2014H1A2A1015767) and the Basic Science Research Program (2015R1D1A1A09057413) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education. O


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Characterization of Dilution Effect of Semidilute Polymer Solution on

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