Shear Thickening and Thinning Behavior of Hydrophobically Modifie

Article pubs.acs.org/Macromolecules

Nonlinear Rheology of Telechelic Associative Polymer Networks: Shear Thickening and Thinning Behavior of Hydrophobically Modified Ethoxylated Urethane (HEUR) in Aqueous Solution Shinya Suzuki,†,‡ Takashi Uneyama,† Tadashi Inoue,§ and Hiroshi Watanabe*,† †

Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Lintec Corporation, 5-14-42 Nishikicho, Warabi, Saitama 335-0005, Japan § Department of Macromolecular Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan ‡

ABSTRACT: Flow behavior was examined for a 1.0 wt % aqueous solution of hydrophobically modified ethoxylated urethane (HEUR; Mw = 4.6 × 104). In the linear viscoelastic regime, the solution exhibited single-Maxwellian behavior attributable to thermal reorganization of the transient network composed of strings of HEUR flower micelles. Under shear flow at intermediate shear rates γ̇ just above the equilibrium relaxation frequency 1/τ, the solution exhibited thickening characterized by monotonic increase of the viscosity growth function η+(t;γ̇) with time t above the linear η+(t) and by the steady-state viscosity η(γ̇) larger than the zero-shear viscosity η0. However, at those γ̇, the first normal stress coefficient growth function Ψ1+(t;γ̇) and its steady-state value Ψ1(γ̇) remained very close to the linear Ψ1+(t) and Ψ1,0 and exhibited no nonlinearity. In addition, the relaxation times of the viscosity and normal stress coefficient decay functions η−(t;γ̇) and Ψ1−(t;γ̇) measured after cessation of steady flow agreed with those in the linear regime. All these results suggested that the network strands were just moderately stretched to show no significant finite extensible nonlinear elasticity (FENE) effect and that the number density ν of the network strands was negligibly affected by the shear at γ̇ just above 1/τ. A simple transient Gaussian network model incorporating neither the FENE effect nor the increase of ν suggested that the thickening of η+(t;γ̇) and η(γ̇) and the lack of nonlinearity for Ψ1+(t;γ̇) and Ψ1(γ̇) could result from reassociation of the HEUR strands being in balance with the dissociation but anisotropically enhanced in the shear gradient direction. In contrast, at γ̇ ≫ 1/τ, η+(t;γ̇) exhibited overshoot above the linear η+(t) and then approached η(γ̇) < η0, whereas Ψ1+(t;γ̇) stayed below the linear Ψ1+(t) and approached Ψ1(γ̇) ≪ Ψ1,0 after exhibiting a peak. The relaxation of η−(t;γ̇) and Ψ1−(t;γ̇) after cessation of flow was considerably faster than that in the linear regime. These nonlinear thinning features at γ̇ ≫ 1/τ were attributable to the flow-induced disruption of the HEUR network (and decrease of ν).

1. INTRODUCTION Telechelic polymers consisting of hydrophilic main chain and hydrophobic end groups are associated with each other in water to form various structures. Hydrophobically modified ethoxylated urethane (HEUR) is one of such telechelic polymers. The main chain of HEUR is poly(ethylene oxide) (PEO), and short hydrophobic groups such as aliphatic alcohol, alkylphenyl,1 or fluorocarbons2−4 are attached to the chain ends through urethane groups. In aqueous solutions, the end groups of HEUR aggregate through the hydrophobic interaction and form micellar structures.5,6 At low concentration (that is still higher than the critical micellar concentration), the HEUR chains form so-called flower micelles. The core of this micelle is composed of hydrophobic end groups, and the corona is formed by hydrophilic PEO chains having the loop type conformation. As the concentration is increased, the number density of flower micelles increases, and the end groups of some chains are located in different cores to bridge the micelles. When the bridge fraction exceeds a percolation threshold, a huge network spreading throughout the whole solution is formed. The hydrophobic end groups are thermally detached from a core and eventually © 2012 American Chemical Society

attached to the same and/or other core. Because of this dissociation/association processes, the HEUR network is a temporal network relaxing in a finite time scale. Rheological properties of aqueous solutions of HEUR have been studied extensively.1−19 For example, Annable et al.5,6 found that concentrated HEUR solutions exhibit the singleMaxwellian relaxation in the linear viscoelastic regime. This behavior is attributed to the thermal reorganization (dissociation/ association) of the HEUR network, and effects of temperature, concentration, and type of the end groups on the basic rheological parameters, the relaxation time, and zero-shear viscosity have been also investigated.7−10 Despite the simplicity in the linear regime, rheological properties of the HEUR solutions in the nonlinear regime are quite complicated. In particular, nontrivial behavior has been noted for the steady-state shear viscosity. Concentrated HEUR solutions often exhibit shear thickening at intermediate shear rates and shear thinning Received: September 8, 2011 Revised: December 28, 2011 Published: January 10, 2012 888

dx.doi.org/10.1021/ma202050x | Macromolecules 2012, 45, 888−898

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(MDPDI), were purchased from Wako Pure Chemical Industries Ltd. and used without further purification. The synthesis was conducted in dehydrated tetrahydrofuran (THF; Guaranteed grade, Wako) containing 25 wt % of PEO and given amounts of HDOH and MDPDI (molar ratio PEO:MDPDI:HDOH = 3:4:2). The PEO chains were first extended through the condensation reaction with MDPDI at 60 °C for 2 h, and then the end-capping reaction between HDOH and MDPDI (attached at the ends of the extended PEO chains) was conducted at 60 °C for 24 h. Then, the reaction mixture was cooled to room temperature, diluted with excess THF, and poured into a large volume of THF/hexane mixture of 1/3 (w/w) composition. The HEUR sample was recovered as a precipitant in this mixture. The sample thus recovered was thoroughly dried in a vacuum oven at 40 °C. The HEUR sample was characterized with size-exclusion chromatography utilizing a column/pump system (HLC-8320 GPC EcoSEC, Tosoh) equipped with a refractive index monitor. The elution solvent was THF, and commercially available monodisperse PEO samples (Tosoh) were utilized as the elution standards. The weight-average molecular weight and polydispersity index of the HEUR sample, determined from the elution volume calibration with those standards, were Mw = 4.6 × 104 and Mw/Mn = 1.35, respectively. The material subjected to the rheological measurements was a 1.0 wt % aqueous solution of the HEUR sample in distilled water. Prescribed masses of water and HEUR were stirred for 24 h to prepare the solution. 2.2. Measurements. For the 1.0 wt % aqueous solution of HEUR, rheological measurements were conducted with laboratory rheometers, MCR-301 (Anton Paar) and ARES-G2 (TA Instruments). MCR-301 is a stress-controlled rheometer, whereas ARES-G2 is a straincontrolled rheometer. Dynamic measurements in the linear viscoelastic regime were made with MCR-301 in a cone−plate (CP) geometry (diameter d = 75 mm, cone angle θ = 1.0°) at several temperatures between 5 and 25 °C. The measurement at 25 °C was made also with ARES-G2 in a CP geometry (d = 25 mm, θ = 2.3°). The storage and loss moduli, G′(ω) and G″(ω) measured as functions of the angular frequency ω, obeyed the time−temperature superposition at low ω where the HEUR network exhibited the terminal relaxation (through its thermal reorganization). Those data were reduced at 25 °C. The viscosity and first normal stress coefficient growth functions after start-up of shear flow, η+(t,γ̇) and Ψ1+(t,γ̇), the steady-state viscosity and the steady state first normal stress coefficient, η(γ̇) = η+(∞,γ̇) and Ψ1(γ̇) = Ψ1+(∞,γ̇), and the viscosity and first normal stress coefficient decay functions after cessation of steady shear, η−(t,γ̇) and Ψ1−(t,γ̇), were measured at 25 °C with ARES-G2 in the CP geometry (d = 25 mm, θ = 2.3°) at several shear rates γ̇ between 0.05 and 100 s−1. η(γ̇) and Ψ1(γ̇) were measured also with MCR-301 in the CP geometry (d = 75 mm, θ = 1.0°).

at higher rates. The mechanism of this thickening/thinning behavior of HEUR solutions has been investigated,5,9,11 but uncertainty still remains. The rheological properties of the concentrated HEUR solutions, the single-Maxwellian behavior in the linear regime, and the thickening/thinning under flow have been analyzed theoretically.20−34 One of the most frequently utilized models for telechelic polymers is the transient network model proposed by Tanaka and Edwards.20−23 (The transient network model was originally developed for polymeric systems by Green and Tobolsky24 and by Yamamoto,25 and it has been improved in later work.) Focusing on the association/dissociation process of a target chain in the solution, the Tanaka−Edwards transient network model naturally explains the single-Maxwellian relaxation. However, the original Tanaka−Edwards model does not explain the shear thickening behavior. Thus, several mechanisms such as shear-enhanced formation of the network strands and the finite extensible nonlinear elasticity (FENE) of the strands were introduced into the model to describe the thickening.26−37 Despite this improvement, it is still controversial if the thickening results from the FENE effect or the shear-enhanced strand formation, or other mechanism(s). Thus, we have further examined the thickening behavior of a model HEUR solution through rheological tests at various shear rates, γ̇. Specifically, we measured the viscosity and first normal stress coefficient growth functions η+(t;γ̇) and Ψ1+(t;γ̇) after start-up of shear flow, η(γ̇) (= η+(∞;γ̇)) and Ψ1(γ̇) (= Ψ1+(∞;γ̇)) in the steady state and the viscosity, and first normal stress coefficient decay functions η−(t;γ̇) and Ψ1−(t;γ̇) after cessation of steady flow. We found that the thickening seen for η+(t;γ̇) and η(γ̇) at intermediate γ̇ was associated with no nonlinearity of Ψ1+(t;γ̇) and Ψ1(γ̇) and that the relaxation times of η−(t;γ̇) and Ψ1−(t;γ̇) coincided with those in the linear regime. These features of Ψ1+(t;γ̇), Ψ1(γ̇), η−(t;γ̇), and Ψ1−(t;γ̇) suggest that the factors so far considered, the FENE effect and the shear-induced increase of the strand number density, are not important for the thickening of η+(t;γ̇) and η(γ̇) observed for our model HEUR solution. Thus, we analyzed the behavior with the aid of a simple transient Gaussian network model without these factors. Irrespective of the model details, this analysis showed that the thickening of η+(t;γ̇) and η(γ̇) and the lack of nonlinearity for Ψ1+(t;γ̇) and Ψ1(γ̇) could result from reassociation of the HEUR strands being in balance with the dissociation but anisotropically enhanced in the shear gradient direction, suggesting that the thickening is not always due to the FENE effect and/or the shear-induced strand formation. Details of these results are presented in this paper together with the thinning feature of the solution at high γ̇ and its mechanism (flow-induced disruption of the network).

3. RESULTS 3.1. Linear Viscoelastic Behavior. For the 1.0 wt % aqueous solution of HEUR, the storage and loss moduli data, G′(ω) and G″(ω), measured at several temperatures (5−25 °C) obeyed the time−temperature superposition (tTS) at low ω. Figure 2 shows the master curves of those data reduced at Tr = 25 °C. (The data obtained with the stress- and strain-controlled rheometers, MCR-301 and ARES-G2, agreed with each other.)

2. EXPERIMENTAL SECTION 2.1. Materials. HEUR having hexadecyl groups at the chain ends was synthesized with a conventional method.12 The structure of HEUR is shown in Figure 1. The chemicals utilized in the synthesis, poly(ethylene oxide) (PEO; Mw = 1.9 × 104, Mw/Mn = 1.1), hexadecanole (HDOH), and methylene diphenyl-4,4′-diisocyanate

Figure 1. Chemical structure of HEUR chain. 889


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Figure 2. Storage and loss moduli, G′ and G″, measured for the 1.0 wt % aqueous solution of HEUR reduced at 25 °C. Symbols represent the experimental data (after time−temperature superposition), and the thick curves show the results of fitting with the single-Maxwellian model.

Figure 3. Time−temperature shift factor aT for the 1.0 wt % aqueous solution of HEUR in the linear viscoelastic regime. The circles show the data, and the solid line represents the Arrhenius equation.

groups at the HEUR chain ends that stabilize the HEUR network. 3.2. Nonlinear Flow Behavior. For the 1.0 wt % HEUR solution at 25 °C, Figures 4 and 5 show the viscosity and first

The thick curves indicate the result of fitting the G′ and G″ data with the single-Maxwellian model:

G′(ω) = G0

ω2τ2 2 2

1+ωτ

,

G″(ω) = G0

ωτ 1 + ω2τ2

(1)

Here, G0 (= 15 Pa) and τ (= 0.45 s) are the high-frequency modulus and relaxation time, respectively. The fit is excellently achieved except at high ω (where tTS fails), as noted also in the previous studies.5,6 In fact, the G0 and τ values of our HEUR solution are close to the values reported for a similar HEUR solution.13 The single-Maxwellian behavior of the HEUR solution is attributed to the thermal reorganization (dissociation/association) of the transient network occurring at the time τ,20−23 and the deviation from this behavior seen at high ω reflects the motion within the network strand. (Since the activation process is different for the network reorganization and the intrastrand motion, tTS cannot be commonly achieved at low and high ω.) Consequently, G0 is related to the entropy elasticity of the strands, and the number density ν of active strands can be estimated as

ν=

G0 ≅ 3.7 × 1021 m−3 kBT

Figure 4. Shear viscosity growth function η+(t;γ̇) of the 1.0 wt % HEUR aqueous solution at 25 °C measured at various shear rates, γ̇/s−1 = 0.05, 0.8, 3, 10, 20, 50, and 100. The dashed curve represents the growth function η+(t;γ̇) in the linear viscoelastic regime (γ̇ → 0) evaluated from the G′ and G″ data.

(2)

where kB is the Boltzmann constant and T is the absolute temperature. This ν value is much smaller than the number density of the HEUR chains, ν0 = 1.8 × 1023 m−3, evaluated from the HEUR concentration (0.01 g cm−3) and molecular weight (Mn = 3.4 × 104): ν/ν0 = 0.021, which is close to the ν/ν0 ratio reported previously.13 Thus, the bridged sequence (string) of the flower micelles should behave as the active strand, although some fraction of those strings would be of loop type and not involved in the active strands. Additional information for the HEUR network can be found in Figure 3 where the natural logarithm of the shift factor, ln aT, associating to the G′ and G″ master curves is plotted against T −1. The well-known Arrhenius behavior, ln aT = Ea(T −1 − Tr−1)/R with R being the gas constant, is clearly noted, and the activation energy is evaluated to be Ea ≅ 88 kJ mol−1. This Ea value, close to the data reported for similar HEUR solutions,5,8 can be assigned as the association energy of the hexadecyl

Figure 5. First normal stress coefficient growth function Ψ1+(t;γ̇) of the 1.0 wt % HEUR aqueous solution at 25 °C measured at various shear rates, γ̇/s−1 = 0.8, 3, 10, 20, 50, and 100. The dashed curve represents the growth function Ψ1+(t;γ̇) in the linear viscoelastic regime (γ̇ → 0) evaluated from the G′ and G″ data.

normal stress coefficient growth functions after start-up of shear flow, η+(t;γ̇) ≡ σ+(t;γ̇)/γ̇ and Ψ1+(t;γ̇) ≡ N1+(t;γ̇)/γ̇2 with σ+ and 890


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overshoot, to a level above η+(t) (cf. Figure 4), and η(γ̇) is larger than η0 by ≅45% (cf. Figure 6). However, the Ψ1+(t;γ̇) and Ψ1(γ̇) data at γ̇ < 5 s−1 remain close to the linear Ψ1+(t) and Ψ1,0, as noted in Figures 5 and 6. Namely, the thickening of η+(t;γ̇) and η(γ̇) is associated with no nonlinearity of Ψ1+(t;γ̇) and Ψ1(γ̇). Both η(γ̇) and Ψ1(γ̇) begin to decrease on a further increase of γ̇ above 5 s−1, as seen in Figure 6. This thinning behavior is characterized by power-law relationships at high γ̇

N1+ being the shear stress and first normal stress difference. These η+(t;γ̇) and Ψ1+(t;γ̇) data were obtained with the straincontrolled rheometer, ARES-G2. The numbers indicate the shear rate γ̇ (s−1). For clarity of the plots, the data are shown only for representative γ̇ values. Figure 6 shows the corresponding steady-state viscosity and first normal stress coefficient, η(γ̇) (= η+(∞;γ̇)) and Ψ1(γ̇) (= Ψ1+(∞;γ̇)).

η(γ̇) ∝ γ̇−0.97 ,

The data shown with unfilled and filled symbols were obtained with the stress- and strain-controlled rheometers, MCR-301 and ARES-G2, respectively. The data obtained with these rheometers agree with each other.38 Since the HEUR solution exhibits the single-Maxwellian G′ and G″ data (cf. Figure 2), the growth functions in the linear viscoelastic regime, η+(t) and Ψ1+(t), and the zero-shear viscosity and normal stress coefficient, η0 (= η+(∞)) and Ψ1,0 (= Ψ1+(∞)), can be analytically calculated with the aid of the 3-dimensional Maxwell model as

η0 = G0 τ

(3)

Ψ1,0 = 2G0 τ2

(4)

(for γ̇ ≥ 30 s−1)

(5)

Thus, the thinning at high γ̇ is characterized with γ̇-insensitive shear stress (σ ∝ γ̇0.03) and first normal stress difference (N1 ∝ γ̇0.02). It should be also noted that the thinning behavior is qualitatively different for η(γ̇) and Ψ1(γ̇). The thinning of η(γ̇) is associated with a transient overshoot of η+(t;γ̇) well above the linear η+(t) (cf. Figure 4), while the thinning of Ψ1(γ̇) is associated with no significant overshoot of Ψ1+(t;γ̇) (very weak overshoot, if any) above the linear Ψ1+(t;γ̇) (cf. Figure 5). As explained above, our 1.0 wt % HEUR solution exhibits characteristic thickening and thinning behavior commonly observed for solutions of telechelic polymers. The thickening and thinning obviously indicate that the HEUR network exhibits some structural change under shear. This change can be monitored through the stress decay after cessation of the steady shear. Thus, we examined the viscosity and normal stress coefficient decay functions, η−(t;γ̇) and Ψ1−(t;γ̇), with the straincontrolled rheometer, ARES-G2. As representative examples, the data measured for γ̇ = 3 and 20 s−1 (in the thickening and thinning regimes for η(γ̇)) are shown in Figures 7 and 8,

Figure 6. Steady-state shear viscosity η(γ̇) and steady-state first normal stress coefficient Ψ1(γ̇) measured for the 1.0 wt % aqueous solution of HEUR at 25 °C. The unfilled and filled symbols indicate the data obtained with the stress- and strain-controlled rheometers, MCR-301 and ARES-G2, respectively. Horizontal dashed lines indicate η0 and Ψ1,0 in the linear viscoelastic regime (γ̇ → 0) evaluated from the G′ and G″ data.

η+(t ) = G0 τ[1 − exp( − t /τ)],

Ψ1(γ̇) ∝ γ̇−1.98

Ψ1+(t ) = 2G0 τ2[1 − exp( − t /τ) − (t /τ) exp( − t /τ)],

where G0 (= 15 Pa) and τ (= 0.45 s) are the modulus and relaxation time determined for the G′ and G″ data. These η+(t) and Ψ1+(t) are shown with the dashed curves in Figures 4 and 5 and η0 and Ψ1,0 with the horizontal dashed lines in Figure 6. At shear rates well below the equilibrium relaxation frequency 1/τ (= 2.2 s−1), the linear viscoelastic flow behavior is observed in Figures 4 and 5. Namely, the η+(t;γ̇) data for γ̇ = 0.05 and 0.8 s−1 agree with the linear η+(t) within experimental uncertainty, and the Ψ1+(t;γ̇) data for γ̇ = 0.8 s−1 agree with Ψ1+(t). (At γ̇ = 0.05 s−1, N1+ was too small to give the Ψ1+(t;γ̇) data accurately, and thus those data are not shown in Figure 5. However, Ψ1+(t;γ̇) at such low γ̇ should agree with Ψ1+(t).) Correspondingly, the η(γ̇) and Ψ1(γ̇) data at γ̇ ≪ 1/τ agree with the linear η0 and Ψ1,0 (see Figure 6). On an increase of γ̇ from 1 s−1 (= 0.45/τ) to 5 s−1 (= 2.2/τ), the viscosity exhibits moderate thickening. For example, for γ̇ = 3.0 s−1, the η+(t;γ̇) data monotonically grow, without exhibiting

Figure 7. Shear viscosity and first normal stress coefficient decay functions, η−(t;γ̇) and Ψ1−(t;γ̇), measured for the 1.0 wt % HEUR aqueous solution presheared at γ̇ = 3.0 s−1 (in the thickening regime for η(γ̇)) at 25 °C. The solid curves indicate the linear η−(t) and Ψ1−(t) with adjustment made only for their initial values. For more details, see text.

respectively. The η−(t;γ̇) and Ψ1−(t;γ̇) data at short t unequivocally reflect the HEUR network structure under steady shear (just before cessation of shear). The initial values, η−(0;γ̇) and Ψ1−(0;γ̇), agreed with the steady-state values, η(γ̇) and Ψ1(γ̇), which lends support to this argument for the data at short t. In the linear regime, the decay functions are analytically expressed in terms of the time τ and modulus G0 associated 891


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The above decay behavior provides us with a clue for discussing the thickening and thinning behavior of the HEUR solution, as explained later. For this discussion, it is also informative to compare the behavior of wormlike micelles of surfactants formed in water with the behavior of the HEUR solution. Extensive studies35−37,39,40 revealed that the wormlike micelles of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) (1:1 molar ratio) exhibit the singleMaxwellian linear viscoelasticity very similar to that of the HEUR solution. Concentrated CTAB/NaSal (1/1) solutions under fast shear exhibit increases of both η+(t;γ̇) and Ψ1+(t;γ̇) to levels well above the linear η+(t) and Ψ1+(t), and this thickening behavior is attributable to the finite extensible nonlinear elasticity (FENE), i.e., stretch hardening of the wormlike micelles themselves, as reported by Inoue et al.36 The thickening behavior of the HEUR solution (Figures 4−6) is quite different: The thickening of η+(t;γ̇) and η(γ̇) of the HEUR solution is associated with the linear behavior of Ψ1+(t;γ̇) and Ψ1(γ̇) and thus not attributable to the simple FENE effect of the HEUR strands. Comparison of the thinning behavior of entangled polymers with that of the HEUR solution (at high γ̇ > 5 s−1) is also useful for elucidating the thinning mechanism in the HEUR solution. The thinning of entangled polymers due to strong shear orientation of the chains can be characterized by the power-law relationships:41,42

Figure 8. Shear viscosity and first normal stress coefficient decay functions, η−(t;γ̇) and Ψ1−(t;γ̇), measured for the 1.0 wt % HEUR aqueous solution presheared at γ̇ = 20 s−1 (in the thinning regime for both η(γ̇) and Ψ1(γ̇)) at 25 °C. The solid curves indicate the linear η−(t) and Ψ1−(t) with adjustment made only for their initial values. For more details, see text.

with the single-Maxwellian relaxation:

η−(t ) = η0 exp( − t /τ)

with η0 = G0 τ

(6a)

⎛ t⎞ Ψ1−(t ) = Ψ1,0⎜1 + ⎟ exp( − t /τ) ⎝ τ⎠ with Ψ1,0 = 2G0 τ2

η(γ̇) ∝ γ̇−0.82 and Ψ1(γ̇) ∝ γ̇−1.5 ± 0.05 for monodisperse linear chains (6b)

(7)

In addition, empirical rules relating the nonlinear quantities of those polymers, η(γ̇) and Ψ1(γ̇), to the linear quantities, Ψ1+(t),G′(ω), and the complex viscosity, η*(ω) ≡ {G″(ω) − iG′(ω)}/ω, have been proposed by Cox and Merz,43 Gleissle,44 and Osaki et al:45

The solid curves in Figures 7 and 8 indicates these linear decay functions with the initial values being adjusted for the nonlinearity, {η1(γ̇)/η0}η−(t) and { Ψ1(γ̇)/Ψ1,0}Ψ1−(t). In the thickening regime (Figure 7), these curves are close to the η−(t;γ̇) and Ψ1−(t;γ̇) data in particular at short t where the data reflect the HEUR network structure just before cessation of shear. A rapid initial decay would be observed for both η−(t;γ̇) and Ψ1−(t;γ̇) if the FENE-type nonlinear effect is significant in the steady shear state, and such a rapid decay, if any, can be detected with ARES-G2 (enabling the full cessation of flow within ∼0.025 s). In fact, the rapid initial decay (in the time scale of 0.2 s) is observed on cessation of fast shear, as explained later for Figure 8. However, no such rapid decay is observed on cessation of the steady shear flow at intermediate γ̇ in the thickening regime for η (cf. Figure 7). This result strongly suggests that the dissociation time of the HEUR network under shear agrees with τ in the linear regime and that the network structure under shear is not too much different from that at equilibrium. (The scatter of the Ψ1−(t;γ̇) data points at short t is mainly due to a mechanical noise in the shear-gradient direction on cessation of steady shear.) In contrast, in the thinning regime for η(γ̇) (Figure 8), the initial decay of the η−(t;γ̇) and Ψ1−(t;γ̇) data in the time scale of 0.2 s is considerably faster than that in the linear regime (solid curves). This result suggests that the HEUR network is largely disrupted, and the fragmented network strands are considerably stretched by the shear in the thinning regime to exhibit fast contraction process of the strands at t ≪ τ. The decay of η−(t;γ̇) and Ψ1−(t;γ̇) at longer t becomes as slow as that in the linear regime, possibly due to the thermal reorganization of the remaining network (that could also grow through association of the fragmented strands during the stress decay process).

Cox−Merz: Gleissle:

Osaki:

η(γ̇) ≅ |η*(ω)|ω=γ̇

(8a)

Ψ1(γ̇) ≅ [Ψ1+(t )]t = k / γ̇

(8b)

⎡ 2G′(ω) ⎤ Ψ1(γ̇) ≅ ⎢ ⎥ ⎣ ω2 ⎦ω=γ̇ / k ′

(8c)

Here, k and k′ are adjustable constants close to unity. These empirical rules hold for entangled polymers considerably well.43−46 The γ̇ dependence of η(γ̇) and Ψ1(γ̇) of the HEUR solution (eq 5) is considerably stronger that that for entangled polymers specified by eq 7. Furthermore, the above three empirical rules severely fail for the HEUR solution, as demonstrated in Figure 9, where k and k′ included in eqs 8b and 8c were set to be 1.55 and 1, respectively; compare large symbols. These results suggest that the thinning of the HEUR solution is not attributable to the simple shear orientation not associated with the network reorganization (dissociation and association under shear). At the same time, we also note that |η*(ω)| plotted against an adjusted (increased) angular frequency, ω = 4.6γ̇, agree with the η(γ̇) data at high γ̇; cf. small squares and large circles in top panel of Figure 9. Similarly, Ψ1+(t) and 2G′(ω)/ω2 plotted against an increased reciprocal time (t−1 = 2.7γ̇/k = 1.7γ̇) and an increased angular frequency (ω = 2.6γ̇/k′ = 2.6γ̇), respectively, agree with the Ψ1(γ̇) data at high γ̇; cf. small symbols and large circles in the bottom panel. These results could be 892


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Tanaka model in that range of γ̇. For example, Berret et al.3,14 conducted start-up flow experiments for HEUR solutions and attributed the thickening of the viscosity to the increase of the effective modulus due to the FENE effect. Pellens et al.11,15,16 reported that the stress-optical rule (SOR), being valid only in the absence of the FENE effect,42 fails for their HEUR solutions in the thickening regime, and thus the thickening is related to the FENE effect, although the reported increase of the η/η0 ratio is only by 1−2% (and thus the reported thickening is much weaker than that observed for our HEUR solution). A much stronger FENE effect has been confirmed for both viscosity and first normal stress coefficient of the wormlike micelles.36 However, our 1.0 wt % HEUR solution exhibits the thickening of the viscosity (η+(t;γ̇) and η(γ̇)) while allowing the first normal stress coefficient (Ψ1+(t;γ̇) and Ψ1(γ̇)) to stay in the linear regime (cf. Figures 4−6). In addition, the network strands being highly stretched (to a FENE level) under shear should exhibit fast contraction on cessation of shear, but no such fast process is observed for the η−(t;γ̇) and Ψ1−(t;γ̇) data of our HEUR solution in the thickening regime (Figure 7). These results lead us to conclude that the thickening of our HEUR solution is not primarily due to the simple FENE effect.49 In fact, this conclusion is in harmony with a simple but unambiguous analysis comparing the elastic energy of the HEUR strand and the association energy of the end groups of the HEUR chains, as explained in the Appendix. Of course, we cannot fully rule out a possibility that the FENE effect is coupled with a structural change of the network (anisotropic creation of network strands explained later), thereby contributing, to some extent, to the behavior of our HEUR solution, the thickening of the viscosity associated with linear behavior of the first normal stress coefficient. However, it should be emphasized that all data in Figures 4−7 strongly suggest that the simple FENE effect alone cannot explain the behavior of our HEUR solution. The contribution of the FENE effect (coupled with the change of the network) to this behavior, if any, can be examined through a test of SOR. Unfortunately, at this moment, rheo-optical data useful for this test are not available for our HEUR solutions. The rheo-optical measurement is now being attempted, and the results will be reported in our future paper. Concerning the above conclusion, we should note that the Koga−Tanaka model incorporates the FENE effect as one of the essential ingredients but does not always predicts thickening of both η(γ̇) and Ψ1(γ̇):30 The model includes two basic parameters A and g, with A representing the FENE contribution to the strand tension and g representing an effect of chain tension on the dissociation rate of the HEUR network. (The model reduces to a Gaussian transient network model if A = 0.) The nonlinear rheological behavior of the model including shear thinning and thickening are mainly determined by competition of the nonlinear elasticity (controlled by A) and the acceleration of the dissociation (controlled by g). The thickening of η(γ̇) and thinning of Ψ1(γ̇) could be simultaneously deduced from the Koga−Tanaka model for a specific combination of the parameters. For example, slight thickening of the η(γ̇) associated with slight thinning of Ψ1(γ̇) (both ∼10% in magnitude) reported in ref 11 is fairly well described by the model, although some deviation is noted for the prediction of Ψ1(γ̇); see Figure 10 of ref 30. However, the parameters giving this description are A = 5 and g = 0.16, the former giving a significant FENE effect in a case of the strand stretching. The FENE effect is not important for our HEUR solution, as explained above. Thus, if we apply the Koga−Tanaka

Figure 9. Test of validity of the empirical Cox−Merz, Gleissle, and Osaki rules for the 1.0 wt % HEUR solution at 25 °C.

related to the shear orientation of the fragmented (disrupted) network, as discussed later in more detail.

4. DISCUSSION 4.1. Test of Conventional Thickening Mechanisms for HEUR Solution. In the studies so far conducted for HEUR solutions, the shear thickening has been attributed to either the finite extensible nonlinear elasticity (FENE) of the shearstretched HEUR strands or the increase of the effective strand number density on shear-induced reorganization of the network (shear-enhanced strand formation). In principle, both mechanisms could lead to the thickening, and the origin of the thickening is still controversial. We here make a brief summary of these mechanisms and then test the validity of those mechanisms for our 1 wt % HEUR solution. The FENE concept, widely utilized for constitutive equations of polymers,47 was first introduced into the transient network model by Marrucci and co-workers.27,28 The model was further elaborated by Koga and Tanaka,30−32 who improved the expression of the network dissociation rate in the transient FENE network model but assumed the network reformation to occur randomly/isotropically in space. Indei33,34 showed that the transient FENE network model does not always predict the thickening because of competition between the stressenhancing FENE effect and the stress-suppressing dissociation effect. Nevertheless, in a considerably wide range of the shear rate γ̇ where the former overwhelms the latter, Koga−Tanaka model predicts the thickening of both viscosity and first normal stress coefficient.30 (Further details of this model are explained later.) Several experimental results were reported to be in favor of this FENE-induced thickening deduced from the Koga− 893


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model to our HEUR solution, the parameter A is to be set in a close vicinity of zero. Then, the other parameter g should be also set to be close to zero so as to reproduce the lack of thinning/ thickening of Ψ1(γ̇) in the thickening zone of η, as clearly noted from Figure 4 of ref 30. For that case, the Koga−Tanaka model reduces to the Green−Tobolsky type transient network model considering no effect of chain tension on the strand dissociation rate. It may be still possible to seek for a set of Koga−Tanaka parameters A and g in the vicinity of zero that might qualitatively reproduce the behavior of our HEUR solution. However, it does not seem to be fruitful to seek for such parameters (if any) because the FENE effect, one of the essential ingredients in that model, is not important for our HEUR solution. An attempt of finding such parameters is nothing more than parameter fitting without physical basis. Thus, for our HEUR solution examined in this study, we simply attempt to make a general discussion within the context of the Gaussian transient network model allowing anisotropic formation of the HEUR network strands under shear. This discussion is given in the next section. Now, we turn our attention to the conventionally considered second mechanism of thickening, the increase of the effective strand number density ν on shear-induced reorganization of the network. Some experimental data for HEUR solutions were reported to be not in accord to the FENE network model but support the model considering this increase of ν.17−19 For example, Tam et al.18 conducted shear experiments with parallel superposition of small-amplitude oscillation to find decrease and increase of the relaxation time and the characteristic modulus defined for the oscillation. This result led them to attribute the thickening to the increase of ν (due to incorporation of free micelles into the network). François et al.19 performed small-angle neutron scattering under shear and reported that the internal structure of the micellar core is not affected by flow in the thickening region. This shear insensitivity of the core structure is in harmony with the model considering the increase of ν. Nevertheless, this model results in simultaneous increases of the viscosity and normal stress coefficient, which does not match the behavior of our HEUR solution seen in Figures 4−6. Thus, neither the transient FENE network model nor the model considering the flow-induced increase of ν can explain the thickening of η and lack of nonlinearity of Ψ1 observed for our HEUR solution.49 Thus, in the next section, we analyze the behavior of a simple, transient Gaussian network model (without FENE/increase of ν) to discuss the thickening behavior of our HEUR solution. 4.2. Simple Transient Gaussian Network Model for Shear Thickening. The thickening behavior of our 1.0 wt % HEUR solution is governed by neither the FENE effect (stretch hardening) nor the shear-induced increase of number density ν of the active strands, as explained in the previous section. Thus, we assume that the HEUR strands are in the Gaussian state (not stretch hardened) and have the constant ν even under the shear in the η thickening regime. With this assumption, we can simply express the steady-state viscosity and normal stress coefficient in terms of the end-to-end vector r of the strands as47,48

η(γ̇) ≅ Ψ1(γ̇) ≅

3kBT ν r0 2 γ̇

∫ dr rxryψss(r, γ̇)

3kBT ν r0

2

γ̇

2

∫ dr (rx 2 − ry2)ψss(r, γ̇)

Here, kB and T are the Boltzmann constant and absolute temperature, respectively, and rx and ry indicate the components of r in the shear and shear gradient directions. ψss(r,γ̇) is the steady-state distribution function of r under the shear at γ̇. In eqs 9 and 10, all strands are approximated to have the same average size r0 at equilibrium. These strands have the Gaussian spring constant, 3kBT/r02, irrespective of their stretch ratio. In eqs 9 and 10, all nonlinearities emerge from the deviation of ψss(r,γ̇) from the equilibrium distribution function

⎛ ⎛ 3 ⎞3/2 3r 2 ⎞ ⎜ ⎟ ⎟ ψeq(r) = ⎜⎜ − exp ⎜ 2⎟ 2⎟ ⎝ 2r0 ⎠ ⎝ 2πr0 ⎠

(11)

This deviation of ψ ss (r,γ̇) is not trivial because the reorganization kinetics of the network should be affected by the shear flow. For description of this deviation in the thickening regime, we here adopt a simplified but analytically tractable version of the transient network model considering dissociation/association of the active, Gaussian strands having the constant number density ν in total. Namely, none of the thickening mechanisms explained in the previous section are incorporated in our model. In this model, the dynamics of the system is simply described by the probability distribution function of the active strands. Specifically, we focus on the following birthand-death type master equation50 for those strands after start-up of shear at γ̇:

∂ψ(r, t , γ̇) ∂ψ(r, t , γ̇) 1 = −γ̇ry − ψ(r, t , γ̇) ∂t ∂rx τ0 1 + ϕ(r, t , γ̇) τ0

(12)

Here, ψ(r,t,γ̇) is the probability distribution function of r of the strand at time t, τ0 is the characteristic time for the dissociation and association, and ϕ(r,t,γ̇) is the source function, i.e., probability distribution function of r of the newly created (associated) network strands at t. At equilibrium, we have ψ(r,t,γ̇=0) = ϕ(r,t,γ̇=0) = ψeq(r), with ψeq(r) being the equilibrium distribution function given by eq 11. Furthermore, τ0 is equal to τ for the single-Maxwellian relaxation in the linear viscoelastic regime (cf. eq 1). Here, a comment needs to be made for the source function, ϕ(r,t,γ̇). A reviewer for this paper regarded eq 12 to be physically not sound unless the molecular mechanism of the strand creation is specified. As a hypothesis, we can list candidates of this mechanism, for example, recombination of the dangling chains, conversion of the loops into bridges, splitting of superbridges into shorter bridge (a mechanism not considered in the Koga−Tanaka model30), and so on. However, an argument on the basis of such a specific mechanism is clearly dependent on the feature of the mechanism and is dependent too much on the assumption for the mechanism. In this study, we just focus on the anisotropy of the strand creation under shear flow, no matter what the underlying mechanism is. Under the shear flow, from the symmetry, all mechanisms listed above can lead to the strand creation process that is anisotropic and reasonably described by the γ̇-dependent anisotropic source function, ϕ(r,t,γ̇). This approach allows us to make a general discussion free from the assumption for the mechanism.

(9)

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To analyze the behavior of η(γ̇) and Ψ1(γ̇), we need to obtain an explicit expression for the steady-state distribution function ψss(r,γ̇); cf. eqs 9 and 10. This ψss(r,γ̇), identical to ψ(r,t→∞,γ̇) determined by eq 12, is related to the source function in the steady state, ϕss(r,γ̇) = ϕ(r,t→∞,γ̇), as

γ̇ry

∂ψss(r, γ̇) ∂rx

+

1 1 ψss(r, γ̇) = ϕss(r, γ̇) τ0 τ0

(2) (4) b0,2,0(γ̇) = b2,0,0 {τ0γ̇}2 + b0,2,0 {τ0γ̇}4 + O({τ0γ̇}6)

From eqs 9 and 10 combined with eq 14 and eqs 16−21, η(γ̇) and Ψ1(γ̇) are calculated up to the lowest order of nonlinearity. The results are summarized as

η(γ̇) (3) (2) = 1 + {τ0γ̇}2[b1,1,0 + 2b2,0,0 ] η0

(13)

For relatively small γ̇ (where the thickening is actually observed for the HEUR solution), we can expand ψss(r,γ̇) and ϕss(r,γ̇) around their Gaussian forms at equilibrium, ψss(r,γ̇=0) = ϕss(r,γ̇=0) = ψeq(r). This expansion can be conveniently made by utilizing the Hermite polynomials as the basis:

+ O({τ0γ̇}4 )

∑

(4) − b0,2,0 ] + O({τ0γ̇}4 )

an , m , l(γ̇)2−(n + m + l)/2

(14)

∞

ϕss(r, γ̇) = ψeq(r)

∑

bn , m , l(γ̇)2−(n + m + l)/2

n,m,l=0

⎛ 3 rx ⎞ ⎛ 3 ry ⎞ ⎛ 3 rz ⎞ ⎟⎟Hl ⎜ × Hn⎜ ⎟ ⎟Hm⎜⎜ ⎝ 2 r0 ⎠ ⎝ 2 r0 ⎠ ⎝ 2 r0 ⎠

(15)

Here, an,m,l(γ̇) and bn,m,l(γ̇) are the expansion coefficients and Hn(x) is the nth-order Hermite polynomial.51 From eqs 13−15 with the aid of the recurrence formula, Hn+1(x) = 2xHn(x) − 2nHn−1(x) and dHn(x)/dx = 2nHn−1(x), we can find relationships between the coefficients an,m,l(γ̇) and bn,m,l(γ̇). For loworder coefficients necessary for calculating η(γ̇) and Ψ1(γ̇) up to the order of γ̇2 (up to the lowest order of nonlinearity), the relationships are summarized as

a1,1,0(γ̇) = b1,1,0(γ̇) + (τ0γ̇)[1 + 2b0,2,0(γ̇)]

(16)

a0,2,0(γ̇) = b0,2,0(γ̇)

(17)

a 2,0,0(γ̇) = b2,0,0(γ̇) + (τ0γ̇)b1,1,0(γ̇) + (τ0γ̇)2 [1 + 2b0,2,0(γ̇)]

(18)

The γ̇ dependence of these low-order coefficients can be found from simple consideration/analysis. First of all, the normalization condition for ψss(r,γ̇) and ϕss(r,γ̇) gives γ̇-independent coefficients, a0,0,0(γ̇) = b0,0,0(γ̇) = 1. Furthermore, from the symmetry of ψss(r,γ̇) and ϕss(r,γ̇) under simple shear field, an,m,l(γ̇) and bn,m,l(γ̇) should be nonzero only when both (n + m) and l are even integers. Finally, η(γ̇) and Ψ1(γ̇) are required to be even functions of γ̇ (to be invariant on reversal of the shear direction) and have finite nonzero values η0 and Ψ1,0 for the single-Maxwellian relaxation (cf. eqs 3 and 4) on a decrease of γ̇ toward zero. This requirement forces the low-order coefficients bn,m,l(γ̇) appearing in eqs 16−18 to have the following form of expansion with respect to γ̇ (unless γ̇ is too large to disturb the convergence): (3) b1,1,0(γ̇) = b1,1,0 {τ0γ̇}3 + O({τ0γ̇}5)

(19)

(2) (4) b2,0,0(γ̇) = b2,0,0 {τ0γ̇}2 + b2,0,0 {τ0γ̇}4 + O({τ0γ̇}6)

(20)

(22)

with Ψ1,0 = 2νkBT τ0 2

(23)

In the linear viscoelastic regime at low γ̇, our model gives the single-Maxwellian behavior associated with η0 and Ψ1,0 (= 2G0τ02 with G0 = νkBT) shown in eqs 22 and 23. In contrast, in the nonlinear regime, η(γ̇) deduced from the model exhibits thickening while Ψ1(γ̇) remains close to the linear Ψ1,0 if the (3) (2) coefficients satisfy the relationships b1,1,0 + 2b2,0,0 > 0 and (4) (4) (3) (2) b2,0,0 − b0,2,0 ≅ −(b1,1,0 + 2b2,0,0 ). Namely, the behavior of η(γ̇) and Ψ1(γ̇) experimentally observed for the HEUR solution at intermediate γ̇ can be qualitatively reproduced by the model that incorporates neither the FENE effect nor the shear-induced increase of ν. Concerning this behavior of the model, we should note in eqs 22 and 23 that η(γ̇) and Ψ1(γ̇) of our model are determined by the expansion coefficients b of the source function ϕss(r,γ̇). Namely, the strands are dissociated and associated under steady shear (to keep the constant ν), and the anisotropy of the orientation of the newly created (associated) strands, represented by ϕss(r,γ̇), determines the thickening/thinning behavior of η(γ̇) and Ψ1(γ̇). Specifically, the coefficient bn,m,l for ϕss(r,γ̇) with (n,m,l) = (1,1,0) corresponds to the mode of strand creation in the direction parallel to ex + ey, where ex and ey are the unit vectors in the shear and shear gradient direction, respectively. Similarly, the coefficients bn,m,l with (n,m,l) = (2,0,0) and (0,2,0) correspond to the mode of strand creation in the directions parallel to ex and ey, respectively. Thus, if the strands are preferentially created in the shear gradient direction rather than the shear direction and much less favorably in the direction in between (parallel to ex + ey), the coefficients (4) (2) (4) (3) could satisfy a relationship b0,2,0 > b2,0,0 > b2,0,0 > b1,1,0 (>0), thereby simultaneously fulfilling the condition for thickening of (3) (2) η(γ̇), b1,1,0 + 2b2,0,0 > 0, and the condition for the lack of (4) (4) (3) (2) nonlinearity of Ψ1(γ̇), b2,0,0 − b0,2,0 = −(b1,1,0 + 2b2,0,0 ). (For (2) (4) b2,0,0 and b2,0,0 having the same (n,m,l), the difference due to the (2) (4) expansion order (b2,0,0 > b2,0,0 ) is considered in this relationship.) The preferential strand creation in the shear gradient direction appears to be reasonable because the HEUR micellar cores are distributed anisotropically under shear and the strand is rather stabilized (not significantly stretched) if it is created in this direction and also because the already existing strands tend to be oriented out of that direction. (It is also worth mentioning that the anisotropy of the molecular mobility has been considered in the constitutive equation52 and molecular model.53 This anisotropy of mobility could also result in the preferential creation of the strands in the shear gradient direction.) Of course, our model does not specify the mechanism of the strand creation and the dynamics of the source function ϕ(r,t,γ̇), and it is too simple to reproduce the experimental

n,m,l=0

⎛ 3 rx ⎞ ⎛ 3 ry ⎞ ⎛ 3 rz ⎞ ⎟⎟Hl ⎜ × Hn⎜ ⎟Hm⎜⎜ ⎟ ⎝ 2 r0 ⎠ ⎝ 2 r0 ⎠ ⎝ 2 r0 ⎠

with η0 = νkBT τ0

Ψ1(γ̇) (3) (2) (4) = 1 + {τ0γ̇}2[b1,1,0 + 2b2,0,0 + b2,0,0 Ψ1,0

∞

ψss(r, γ̇) = ψeq(r)

(21)

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of such a model is considered to be an important subject of future work.

results quantitatively. In addition, it does not necessarily apply to all HEUR solutions so far examined in the literature: The FENE effect and shear-induced increase of ν could have been essential for some solutions.49 In particular, the FENE effect might be coupled with the anisotropic creation of network strands, thereby contributing, to some extent, to the thickening behavior of our HEUR solution. (A rheo-optical measurement testing this contribution is now being attempted.) Nonetheless, it should be emphasized that the thickening of η(γ̇) associated with no nonlinearity of Ψ1(γ̇) observed for our 1.0 wt % HEUR solution at relatively low γ̇ can be qualitatively explained without considering the FENE effect and the increase of ν. As far as the authors know, the anisotropy of the strand creation being balanced with the network disruption, the essence of our model, has not been considered explicitly in the transient network models. Our experimental results and the model analysis imply that the shear effect on the strand creation kinetics (anisotropic creation) can be an important factor for the nonlinearity in the thickening regime. A further study is desired for this anisotropic strand creation (and the coupling with the FENE effect discussed above) for full understanding of the thickening mechanism of HEUR solutions. 4.3. Mechanism of Shear Thinning. For our HEUR solution, both η(γ̇) and Ψ1(γ̇) exhibit thinning at high γ̇, as noted in Figures 4−6. The initial decay of η−(t;γ̇) and Ψ1−(t;γ̇) at those γ̇ is faster than that in the linear regime (Figure 8). In addition, the γ̇ dependence of η(γ̇) and Ψ1(γ̇) of the HEUR solution is stronger than that for entangled polymers (cf. eqs 5 and 7), and the empirical rules valid for those polymers (eq 8) do not hold for the HEUR solution (cf. Figure 9). These results suggest that the thinning of the HEUR solutions is attributable to the shear-induced disruption of the HEUR network, i.e., the decrease of the number density ν of the active strands. At the same time, eq 8 still holds for η(γ̇) and Ψ1(γ̇) in the power-law thinning region at high γ̇ given that the angular frequency ω and time t for the linear viscoelastic quantities involved in eq 8 are increased and decreased, respectively, as explained for Figure 9. This result could mean that the HEUR network is not only disrupted/fragmented (to have a shorter relaxation time) but also oriented under fast steady shear to exhibit the thinning, and the orientation of those fragmented network is somewhat similar to that of the polymers satisfying the empirical eq 8. Finally, we note that the thinning of η(γ̇) is associated with the transient thickening of η+(t;γ̇) above the linear η+(t) while the thinning of Ψ1(γ̇) is associated with no significant overshoot of Ψ1+(t;γ̇) (very weak overshoot, if any) above the linear Ψ+(t) (see Figures 4 and 5). These features of η+(t;γ̇) and Ψ1+(t;γ̇) appear to correspond to the anisotropic creation of the strands in the transient state, as similar to the anisotropic creation in the steady state in the thickening regime discussed earlier (although the transient hardening of the stretched strand, which could occur in a short time scale before disruption of the HEUR core, might contribute a little to those features). However, the model developed in the previous section cannot be applied to those η+(t;γ̇) and Ψ1+(t;γ̇) data because the dynamics of the source function, which can change with time in the transient state, is not specified in the model, and the expansion in the model is not valid (diverges) at high γ̇. A more refined model explicitly incorporating the dynamics of the source function, requiring no expansion and allowing a shear-induced change of ν, is desired for further studying the transient thickening followed by the steady-state thinning. Formulation

5. CONCLUDING REMARKS For the 1.0 wt % aqueous solution of HEUR containing the transient HEUR network and exhibiting the single-Maxwellian relaxation in the linear regime, we have examined the nonlinear thickening and thinning behavior under shear flow. At intermediate shear rates γ̇ just above the equilibrium relaxation frequency 1/τ, the solution exhibited thickening characterized by monotonic increase of the viscosity growth function η+(t;γ̇) above the linear η+(t) and by the steady-state viscosity η(γ̇) larger than η0. However, at those γ̇, the first normal stress coefficient growth function Ψ1+(t;γ̇) and the steady-state coefficient Ψ1(γ̇) exhibited no nonlinearity. In addition, the relaxation times of the viscosity and normal stress coefficient decay functions η−(t;γ̇) and Ψ1−(t;γ̇) obtained after cessation of steady flow agreed with those in the linear regime. These results, in particular the lack of nonlinearity of Ψ1+(t;γ̇) and Ψ1(γ̇), suggested that our HEUR network strands were just moderately stretched to show no significant FENE effect and that the number density ν of the network strands was negligibly affected by the shear at γ̇ just above 1/τ. A simple transient Gaussian network model incorporating neither the FENE effect nor the increase of ν suggested that the thickening of η+(t;γ̇) and η(γ̇) and the lack of nonlinearity of Ψ1+(t;γ̇) and Ψ1(γ̇) could result from anisotropy of creation of the HEUR strands attached to the network (although the FENE effect coupled with this anisotropic strand creation might contribute, to some extent, to the observed thickening behavior). The strand creation appeared to be enhanced in the shear gradient direction to result in both thickening of η+(t;γ̇) and η(γ̇) and lack of nonlinearity of Ψ1+(t;γ̇) and Ψ1(γ̇). In contrast, at γ̇ ≫ 1/τ, η+(t;γ̇) exhibited overshoot well above the linear η+(t) and then approached η(γ̇) < η0, whereas Ψ1+(t;γ̇) showed no significant overshoot (very weak overshoot, if any) to approach Ψ1(γ̇) ≪ Ψ1,0. In addition, the relaxation of η−(t;γ̇) and Ψ1−(t;γ̇) after cessation of flow was considerably faster than that in the linear regime. These nonlinear thinning features at γ̇ ≫ 1/τ can be mainly attributed to the flow-induced disruption/ fragmentation of the HEUR network (decrease of ν) associated with the shear orientation of the fragmented network. Finally, it should be emphasized that our HEUR chain is not unique in its structure (molecular weight and size of alkyl groups at the chain ends) and rather similar to the HEUR chains so far examined in the literature. However, the analysis of both viscosity and first normal stress coefficient data, not made in previous studies (except in ref 11 where the normal stress coefficient data were reported), led to the above conclusion. The results obtained in this study would contribute to better understanding of the behavior of HEUR solutions.

■

APPENDIX. ANALYSIS OF ELASTIC ENERGY It is informative to compare the elastic energy of the HEUR network strands Fel and the association energy of the end groups of those strands, Ea ≅ 88 kJ mol−1 (evaluated from the data in Figure 3), to test if the association is strong enough to allow the strands to be stretched into the FENE region. Fel increases with increasing end-to-end distance r of the strand. Thus, the end groups can sustain the strand stretching only in a region of r specified by Fel(r ) < Ea 896

(A1) dx.doi.org/10.1021/ma202050x | Macromolecules 2012, 45, 888−898

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(7) May, R.; Kaczmarski, J. P.; Glass, J. E. Macromolecules 1996, 29, 4745. (8) Kaczmarski, J. P.; Tarng, M. R.; Ma, Z.; Glass, J. E. Colloids Surf., A 1999, 147, 39. (9) Ma, S. X.; Cooper, S. L. Macromolecules 2001, 34, 3294. (10) Barmar, M.; Ribitsch, V.; Kaffashi, B.; Barikani, M.; Sarreshtehdari, M.; Pfragner, J. Colloid Polym. Sci. 2004, 282, 454. (11) Pellens, L.; Corrales, R. G.; Mewis, J. J. Rheol. 2004, 48, 379. (12) Kaczmarski, J. P.; Glass, J. E. Langmuir 1994, 10, 3035. (13) Xu, B.; Yekta, A.; Li, L.; Masoumi, Z.; Winnik, M. A. Colloid Polym. Sci., A 1996, 112, 239. (14) Berret, J. F.; Séréro, Y. Phys. Rev. Lett. 2001, 87, 048303−1. (15) Pellens, L.; Vermant, J.; Mewis, J. Macromolecules 2005, 38, 1911. (16) Pellens, L.; Ahn, K. H.; Lee, S. J.; Mewis, J. J. Non-Newtonian Fluid Mech. 2004, 121, 87. (17) Tripathi, A.; Tam, K. C.; Mckinley, G. H. Macromolecules 2006, 39, 1981. (18) Tam, K. C.; Jenkins, R. D.; Winnik, M. A.; Bassett, D. R. Macromolecules 1998, 31, 4149. (19) François, J.; Maitre, S.; Rawiso, M.; Sarazin, D.; Beinert, G.; Isel, F. Colloids Surf., A. 1996, 112, 251. (20) Tanaka, F.; Edwards, S. F. J. Non-Newtonian Fluid Mech. 1992, 43, 247. (21) Tanaka, F.; Edwards, S. F. J. Non-Newtonian Fluid Mech. 1992, 43, 273. (22) Tanaka, F.; Edwards, S. F. J. Non-Newtonian Fluid Mech. 1992, 43, 289. (23) Tanaka, F.; Edwards, S. F. Macromolecules 1992, 25, 1516. (24) Green, M. S.; Tobolsky, A. V. J. Chem. Phys., A 1946, 14, 80. (25) Yamamoto, M. J. Phys. Soc. Jpn. 1956, 11, 413. (26) Wang, S. Q. Macromolecules 1992, 25, 7003. (27) Vaccaro, A.; Marrucci, G. J. Non-Newtonian Fluid Mech. 2000, 92, 261. (28) Marrucci, G.; Bhargava, S.; Cooper, S. L. Macromolecules 1993, 26, 6483. (29) Indei, T.; Koga, T.; Tanaka, F. Macromol. Rapid Commun. 2005, 26, 701. (30) Koga, T.; Tanaka, F. Macromolecules 2010, 43, 3052. (31) Koga, T.; Tanaka, F.; Kaneda, I.; Winnik, F. M. Langmuir 2009, 25, 8626. (32) Koga, T.; Tanaka, F.; Kaneda, I. Prog. Colloid Polym. Sci. 2009, 136, 39. (33) Indei, T. J. Non-Newtonian Fluid Mech. 2007, 141, 18. (34) Indei, T. Nihon Reoroji Gakkaishi (J. Soc. Rheol., Jpn.) 2007, 35, 147. (35) Van Egmond, J. W. Curr. Opin. Colloid Interface Sci. 1998, 3, 385. (36) Inoue, T.; Inoue, Y.; Watanabe, H. Langmuir 2005, 21, 1201. (37) Tirtaatmadja, V.; Tam, K. C.; Jenkins, R. D. Macromolecules 1997, 30, 3271. (38) With the stress-controlled MCR-301, the constant-rate start-up of flow could not be accurately achieved at short times because of a delay in the built-in stress−strain rate feedback loop, but the steady flow measurements at long times were made without any problem (cf. Figure 6). (39) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081. (40) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 354. (41) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980. (42) Graessley, W. W. Polymeric Liquids and Networks: Dynamics and Rheology; Garland Science: New York, 2008. (43) Cox, W. P.; Merz, E. H. J. Polym. Sci. 1958, 28, 619. (44) Gleissle, W. Two time-shear rate relations combining viscosity and first normal stress coefficient in the linear and nonlinear flow range, 8th Int. Cong. Rheol., Naples, 1980. (45) Osaki, K.; Watanabe, H.; Inoue, T. Nihon Reoroji Gakakishi (J. Soc. Rheol., Jpn.) 1998, 26, 49.

The FENE effect emerges if r in this region can have a value reasonably close to the full-stretch length of the strand, and vice versa. For this test, we can utilize the expression of Fel(r) deduced from the standard FENE model36

Fel(r ) = −

3nK RT ⎡ r2 ⎤ ⎥ ln⎢1 − ⎢⎣ 2 (nK bK )2 ⎥⎦

per 1 mole of strands

(A2)

Here, nK is the number of Kuhn segments per strand, bK is the Kuhn step length, and the product nKbK corresponds to the fullstretch length of the strand. The ratio λf = r/nKbK (a factor governing the r dependence of Fel in eq A2) specifies the relative stretch of the strand with respect to the full-stretch limit. Since Fel(r) for a given λf value increases with increasing nK, eq A1 can be satisfied in a wider range of λf to allow the FENE effect to emerge more easily for a smaller nK value. Thus, we here test eq A1 for an unrealistically small nK value, nK = 511 for individual HEUR chains: This nK value was obtained from the bK data of PEO,54 bK = 0.77 nm, and Mw (= 4.6 × 104) of the HEUR chain. The actual HEUR network strand is a bridged sequence (string) of the HEUR micelles (as explained for eq 2), and the actual nK value should be much larger than 511. Thus, the actual HEUR strand can exhibit the FENE effect much less easily compared to the extreme case examined below. Utilizing nK = 511, bK = 0.77 nm, and Ea = 88 kJ mol−1 in eq A2, we can specify the range of λf (range of r) where eq A1 is satisfied. The result is 1/2 ⎡ ⎛ ⎞⎤ 2 λ f < ⎢1 − exp⎜ − Ea⎟⎥ ⎢⎣ ⎝ 3nK RT ⎠⎥⎦

= 0.21 (at 25 °C)

(A3)

The maximum possible λf value, 0.21, is still too small to allow significant FENE effect to emerge for the elasticity of the HEUR chain, as can be clearly noted from comparison between Fel(λf) = 85.6 kJ mol−1 (eq A2) and the elastic energy estimated for Gaussian chains, Fel,G(λa) = {3nKRT/2}λf2 ≅ 85.2 kJ mol−1. The actual HEUR strands exhibit the FENE effect much less easily compared to individual HEUR chains examined above. Thus, this effect should have a negligible contribution to the actual HEUR solution, and the HEUR strands can be safely regarded as Gaussian strands.

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ACKNOWLEDGMENTS This work was supported by the Grant-in-Aid for Scientific Research (B) (grant 21350063) and by Grant-in-Aid for Young Scientists (B) (grant 22740273) from MEXT.

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REFERENCES

(1) Lundberg, D. J.; Glass, J. E.; Eley, R. R. J. Rheol. 1991, 35, 1255. (2) Xu, B.; Li, L.; Yekta, A.; Masoumi, Z.; Kanagalingam, S.; Winnik, M. A.; Zhang, K.; Macdonald, P. M. Langmuir 1997, 13, 2447. (3) Berret, J. F.; Séréro, Y.; Winkelman, B.; Calvet, D.; Collet, A.; Viguior, M. J. Rheol. 2001, 45, 477. (4) Calvet, D.; Collet, A.; Viguier, M.; Berret, J.; Séréro, Y. Macromolecules 2003, 36, 449. (5) Annable, T.; Buscall, R.; Ettelaie, R.; Whittlestone, D. J. Rheol. 1993, 37, 695. (6) Annable, T.; Buscall, R.; Ettelaie, R. Colloids Surf., A 1996, 112, 97. 897


Macromolecules

Article

(46) El-Kissi, N.; Piau, J. M.; Attané, R.; Turrel, G. Rheol. Acta 1993, 32, 293. (47) Larson, R. G. Constitutive Equations for Polymer Melts and Solutions; Butterworth-Heinemann: Oxford, 1998. (48) Fuller, G. G. Optical Rheometry of Complex Fluids; Oxford University Press: New York, 1995. (49) No Ψ1+(t;γ̇) and Ψ1(γ̇) data were shown in most of the previous reports for HEUR solutions, except in ref 11. Thus, we cannot rule out a possibility that the thickening of η(γ̇) reported therein was actually not due to the FENE effect and the flow-enhanced strand creation. Pellens et al.11 reported both η(γ̇) and Ψ1(γ̇) data for their HEUR solution: Their η(γ̇) data exhibited slight thickening (∼10% increase of η at the largest) and the Ψ1(γ̇) data exhibit slight thinning (∼10% decrease of Ψ1 at the shear rate for the maximum of η); see Figure 4 of ref 11. This thickening behavior of η, less significant in magnitude but qualitatively similar compared to the behavior of our HEUR solution (Figure 6), might be attributed to the anisotropic creation of HEUR network strands under shear rather than the FENE effect and/or the flow-enhanced strand creation. (50) Van Kampen, N. G. Stochastic Processes in Physics and Chemistry, 3rd ed.; Elsevier: Amsterdam, 2007. (51) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed.; Dover: New York, 1972. (52) Beris, A. N.; Edwards, B. J. Thermodynamics of Flowing Systems; Oxford University Press: Oxford, 1994. (53) Uneyama, T.; Horio, K.; Watanabe, H. Phys. Rev. E 2011, 83, 061802. (54) Aharoni, A. M. Macromolecules 1983, 16, 1722.

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Shear Thickening and Thinning Behavior of Hydrophobically Modifie

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