Viscoelasticity, Nonlinear Shear Start-up, and Relaxation of Entangled

Article pubs.acs.org/Macromolecules

Viscoelasticity, Nonlinear Shear Start-up, and Relaxation of Entangled Star Polymers Frank Snijkers,† Kedar Ratkanthwar,‡,§ Dimitris Vlassopoulos,*,†,⊥ and Nikos Hadjichristidis*,§,∥ †

Institute of Electronic Structure and Laser, FORTH, Heraklion 71110, Crete, Greece School of Chemical Sciences, Swami Ramanand Teerth Marathwada University, Vishnupuri, Nanded-431606, India § Department of Chemistry, University of Athens, Athens 15771, Greece ⊥ Department of Materials Science & Technology, University of Crete, Heraklion 71003, Crete, Greece ∥ Division of Physical Sciences & Engineering, KAUST Catalysis Center, Polymer Synthesis Laboratory, King Abdullah University of Science and Technology (KAUST), Thuwal, Kingdom of Saudi Arabia ‡

ABSTRACT: We report on a detailed rheological investigation of well-defined symmetric entangled polymer stars of low functionality with varying number of arms, molar mass of the arms, and solvent content. Emphasis is placed on the response of the stars in simple shear, during start-up, and for relaxation upon flow cessation. To reduce experimental artifacts associated with edge fracture (primarily) and wall slip, we employ a homemade cone-partitioned plate fixture which was successfully implemented in recent studies. Reliable data for these highly entangled stars could be obtained for Weissenberg numbers below 300. The appearance of a stress overshoot during startup with a corresponding strain approaching a value of 2 suggests that in the investigated shear regime the stars orient but do not stretch. This is corroborated by the fact that the empirical Cox−Merx rule appears to be validated, within experimental error. On the other hand, the (shear) rate dependent steady shear viscosity data exhibit a slope smaller than the convective constraint release slope of −1 (for linear polymers) for the investigated range of rates. The broadness of the stress overshoot reflects the broad linear relaxation spectrum of the stars. The initial stress relaxation rate, reflecting the initial loss of entanglements due to the action of convective constraint release in steady shear flow, increases with Weissenberg number. More importantly, when compared against the relevant rates for comb polymers with relatively short arms, the latter are slower at larger Weissenberg numbers. At long times, the relaxation data are consistent with the linear viscoelastic data on these systems.

I. INTRODUCTION

characterized polymers is on the other hand far less explored and understood. In particular, with the exception of a few data sets only, which are discussed below,14 systematic studies of shear rheology are essentially lacking. In order to have a reliable and versatile predictive toolbox for any entangled branched polymer, it is important to have reliable data for various welldefined architectures. It turns out that this remains an open challenge for at least three reasons. (i) The anionic highvacuum synthesis methods for the production of high-quality, well-characterized complex polymers are very involved15,16 and the resulting sample amounts very small, which limits the applicable techniques to study their behavior (for example, the use of capillary rheometry is impossible). (ii) Moreover, while the linear behavior can be measured with ease in commercial rotational rheometers, the nonlinear flow behavior is much more difficult to study due to flow instabilities of different

Entangled branched polymers represent a topic of great scientific and technological significance. In particular, understanding their rheology is central to linking processing performance to consumer products and to designing new materials with improved properties. Thanks to the substantial advances of the last two decades, the rheology of commercial branched polymers can be successfully predicted at molecular level, starting from their polymerization.1 Nevertheless, it is important to elucidate the role of exact molecular features (number, molar mass and distribution of branches) on the rheology of branched polymers. To this end, the role of wellcharacterized complex polymers is essential.2 In fact, the linear viscoelastic behavior of well-characterized polymer architectures is relatively well understood and the current versions of the tube models describe the relaxation behavior of star polymers,3,4 H polymers,5,6 combs,7−9 starcombs,10 and Cayley trees11−13 nearly quantitatively. The success is largely due to the synergy of anionic synthesis, characterization methods, rheology and modeling. The nonlinear flow behavior of well© 2013 American Chemical Society

Received: April 2, 2013 Revised: June 25, 2013 Published: July 8, 2013 5702

dx.doi.org/10.1021/ma400662b | Macromolecules 2013, 46, 5702−5713

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Scheme 1. General Reactions for the Synthesis of 4-Arm and 8-Arm Star Polyisoprenesa

a

8-SiCl: {Si[CH2CH2SiCl2(CH3)]4}. Red (outer part) represents polyisoprene chains and blue (inner part) polybutadiene chains.

origin, especially in shear flow.17−19 (iii) Even if anionic synthesis is the best means to obtain exact architectures, there are always shortcomings and the issue of architectural dispersity is emerging as an important one.6,20,21 In recent work, we have addressed the nonlinear shear response of well-characterized entangled polymers with comb architecture.22 A main finding of that work is that, in fast flows, the combs with relatively short arms behave qualitatively much like linear polymers since their arms relax very fast. The same conclusion was drawn by Ianniruberto and Marrucci for combs in elongational flows.23 It is therefore interesting to explore the role of the arms in more detail, and hence a systematic investigation of the simplest possible branched polymer architecture, i.e., monodisperse star polymers, is in order. We study symmetric entangled star polymers of low functionality, where each arm has an identical molar mass and in which the number of arms per star is small; hence, there is no effect of the core. We address points i and ii above: i.e., we investigate the nonlinear shear rheology of model stars in the melt and in solution using appropriate instrumentation to handle small sample quantities and to avoid flow instabilities. Specifically for stars, the linear relaxation behavior was found to be dominated by contour-length fluctuations of the arms, constraint release and the associated dynamic dilution,14,24,25 in sharp contrast to the response of linear monodisperse polymers, where reptation is the key mechanism. The relaxation of low-functionality stars is furthermore independent of the number of arms. Although not without outstanding issues, the star dynamics is rather well understood by now2,24 and the current versions of the popular generic tube models, such as the branch-on-branch model (BOB model), 4 capture the relaxation behavior near quantitatively. Turning now to the state-of-the-art of the nonlinear rheological behavior, there have been several studies on the stress relaxation of (mildly) entangled star polymer solutions.26−28 From a practical perspective, it can be stated

that nonlinear stress relaxation measurements are convenient as they are somewhat less prone to instabilities and other artifacts, presumably due to the delay in the built-up of normal forces during short-time deformation at relatively small strains, although they are not without challenges such as slip.29−31 The results for the damping function for symmetric stars were found to be in agreement with the Doi−Edwards prediction for linear polymers. Using rheo-optical methods, Tezel et al.32,33 examined the validity of the convective constraint release (CCR) mechanism (proposed for entangled linear polymers)34 for the steady-state shear viscosity of star solutions. They concluded that, given some uncertainties, in general CCR works in roughly the same way for stars as for linear polymers. Menezes and Graessley35 reported on the start-up shear behavior of a well-defined entangled star solution. Because of the occurrence of instabilities, the nonlinear region they accessed was very limited, but they were able to capture the first normal stress reliably. For completion, we note that Ye and Sridhar36 studied the steady extensional viscosity of entangled star solutions using a filament stretching rheometer (FSR). The steady extensional viscosity initially decreased with strain rate, and at higher strain rates a region of extension thickening was observed. The results are qualitatively similar to those obtained with solutions of linear polymers. However, for star polymers the region of extensional thinning was much wider compared to the region for linear polymers, presumably due to the broad relaxation spectrum of the stars, reflecting a large difference between the longest relaxation time for renewal of chain conformations and the Rouse time of the entangled arms. Here, we explore the nonlinear shear flow behavior of entangled symmetric stars using a specially designed homemade cone partitioned-plate geometry that delays edge fracture problems and allows for more accurate nonlinear shear measurements.37 Possible wall-slip issues are reduced in this type of geometry due to the large normal stresses in the middle 5703


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Figure 1. Size exclusion chromatography traces of PI 4a-56k before (a) and after (b) fractionation. determined after passing PI through the SEC instrument which was calibrated with polystyrene (PS) standards. It relates to the fact that PS and PI have different hydrodynamic volumes (the key parameter in SEC).40 Indeed, our (unpublished) SEC experimental results from many anionically prepared PS and PI (of high 1,4 content) models revealed a constant ratio of the molar masses of 1.6, in a plot of molar mass versus elution volume. This is consistent with the well-known Mark−Houwink equation for the intrinsic viscosity of polymers as a function of molar mass M, [η] = KMα, with K and α being constants depending on the polymer−solvent system. For all samples, the polydispersity index was lower than 1.1. NMR spectra, generated with a Bruker 400-MHz instrument with CDCl3 at 25 °C, revealed that all PIs have a high 1,4-content (93 to 95%). Molar masses of the arms (Ma) and other characteristics are given in Table 1. The coding

region of the geometry where the viscoelastic properties are measured.38 The focus here is on start-up shear flow during a sudden imposition of a constant shear rate and then subsequent relaxation upon cessation of steady-state flow. The star polymers used and experimental details are described in section II. Then, the results are presented in section III, starting from linear viscoelasticity and continuing with nonlinear start-up and relaxation of shear stress. Section IV discusses the experimental results and compares against those available in the literature. Finally the main conclusions are summarized in section V.

II. MATERIALS AND METHODS II.1. Synthesis and Characterization of 4-Arm and 8-Arm Star Polyisoprenes. The synthesis of the 4- and 8-arm polyisoprenes (PI) was achieved by using anionic polymerization high vacuum techniques and chlorosilane chemistry (Scheme 1). All polymerizations and linking reactions were carried out in evacuated, n-BuLiwashed, and solvent-rinsed glass reactors. Reagents were introduced via break-seals and aliquots for characterization were removed by heatsealing of constrictions. Full details of the high vacuum techniques are given elsewhere.39 First, a narrow molecular weight linear living polyisoprene was prepared, in benzene at 25 °C, with sec-BuLi as the initiator. A small fraction was removed and terminated with degassed MeOH (arm of the star). The living polyisoprenyllithium was capped with a few units of butadiene (Bd) prior to reaction with the multifunctional chlorosilane compound. The 4-arm star was prepared by linking of the arms with tetrachlorosilane (SiCl4) and the 8-arm star was obtained with tetra(methyl dichlorosilylethyl)silane {Si[CH2CH2SiCl2(CH3)]4} (8-SiCl). About 10% (4-arm PI) to 50% (8-arm PI) excess of the living PI, end-capped with Bd, was used in order to force the linking reaction to completion. The linking reaction was monitored by comparison of the high molecular weight “star” peak and the low molecular weight “arm” peak of size exclusion chromatography (SEC) traces. The excess living polymer was terminated with degassed methanol. The polymers were extensively fractionated (solvent/non solvent: toluene/methanol) in order to remove the excess arm material. An example (4-arm PI, Mw = 56k) is given in Figure 1. All intermediates and final products were analyzed by SEC and nuclear magnetic resonance (NMR). SEC-experiments were performed at 25 °C with a Waters model 510 pump, a Waters model 410 differential refractometer, and three Styragel columns having a porosity range from 103 to 106 Å. The carrier solvent was a mixture of chloroform/triethylamine (95/5, v/v) at a flow rate of 1.0 mL/min. Polystyrene standards were used for calibration, the Mn was obtained after applying a correction coefficient of 1.6 for PI. The latter was

Table 1. Molecular Characteristics of the Investigated Polymersa sample code PI PI PI PI PI a

4a-56k 8a-56k 4a-103k 4a-103k 86% 4a-103k 62%

Ma [kg/mol]

MTOT [kg/mol]

Me [kg/mol]

Za

f

56 56 103 103 103

224 448 412 412 412

5.0 5.0 5.0 6.1 9.5

11.2 11.2 20.6 16.9 10.8

4 8 4 4 4

See text for definitions of the quantities.

indicates the chemistry (PI) followed by the number of arms (e.g., 4a) and finally the weight-averaged molar mass of a single arm (e.g., 56 kg/ mol). The first three samples in the table are melts, the two following samples are solutions prepared from sample PI 4a-103k. The solutions were prepared in the good solvent squalene (Sigma-Aldrich) using the cosolvent tetrahydrofuran (THF) (Sigma-Aldrich). After dissolution of the polymer, the volatile cosolvent THF was removed by evaporation under vacuum at room temperature for several days. The solutions also contained a tiny amount of antioxidant to improve their stability. The molar mass between entanglements Me (in kg/mol) in Table 1 is calculated from the plateau moduli of the melts or solutions as14 Me(ϕ) = (ϕρR(T + 273.2))/G0N(ϕ), with G0N(ϕ) the plateau modulus of the melt or solution (see discussion below), ϕ the volume fraction of polymer in the solution, ρ the mass density of the polymer (in kg/ m3; at 20 °C, ρ = 910 kg/m3),41 R the universal gas constant, and T the temperature (in °C). In addition to the weight-averaged molar mass of an arm, Ma, Table 1 lists the weight-averaged total molar mass of the star, MTOT, the number of entanglements per arm Za = Ma/Me and the number of arms f = MTOT/Ma. Note that MTOT was 5704


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Figure 2. (a) Master curves of the dynamic moduli G′ and G″ as a function of angular frequency ω for the melts at 20 °C; PI 4a-56k (G′, ○; G″, ▽), PI 8a-56k (G′, blue ◇; G″, blue □), and PI 4a-103k (G′, red ○; G″, red ▽ for oscillatory data; and G′, red −; G″, red -- for creep converted data). (b) Master curves for the solutions at 20 °C; PI 4a-103k (G′, red ○; G″, red ▽ for oscillatory data; and G′, −; G″, -- for creep converted data), PI 4a103k 86% (G′, green ○; G″, green ▽), and PI 4a-103k 62% (G′, gray ○; G″, gray ▽). (c) Horizontal aT and vertical bT time−temperature shift factors as a function of temperature T at a reference temperature of 20 °C used to obtain the master curves; PI 4a-56k (aT ○; bT, ●), PI 8a-56k (aT, blue △; bT, blue ▲), PI 4a-103k (aT, red □; bT, red ■), PI 4a-103k 86% (aT, green ◇; bT, green ◆), and PI 4a-103k 62% (aT, gray ▽; bT, gray ▼). The three full lines are fits with the WLF equation of the horizontal shift factors of the melts (black), the PI 4a-103k 86% (green), and the PI 4a103k 62% (gray). ±0.01 °C was achieved with a Peltier system, under nitrogen atmosphere. However, the actual overall temperature control in the sample was not as good (closer to ±0.1 °C, which was still acceptable) due to air flow variations in the top part of the Peltier. Linearity of the creep measurements was ensured by performing tests at different stresses. Converting creep compliance versus time curves to the moduli as a function of frequency is well-known to be problematic as it is a mathematically ill-conditioned problem. There are several ways to do this, each with their own peculiarities,42−44 and here we have followed the approach of Pasquino et al.44 II.2.2. The Cone Partitioned-Plate (CPP). The study of nonlinear flow properties of polymer melts with a rotational rheometer is complicated by the occurrence of flow instabilities and other artifacts. Of particular concern for melts (with non-negligible second normal force) is edge fracture, which is shown to be directly associated with shear banding, i.e., nonlinear velocity profile in the melt.45,46 Following the idea of Meissner et al.47 and the subsequent extensive investigations of Schweizer et al.45,48−50 and Li and Wang,18 we constructed a so-called cone partitioned-plate (CPP) with temperature control for the strain-controlled ARES rheometer.37 The CPP allows us to perform nonlinear flow measurements unaffected by edge fracture up to higher strains and rates as compared to a regular cone and plate geometry (eventually, also for the CPP setup, edge fracture takes over, the effects are simply delayed). The technical realization and performance of our simple CPP on linear, monodisperse entangled polymers (PI and PS) was reported recently37 and the results for the start-up viscosity η+ were found to be identical to the results of Schweizer et al.50 on a similar sample. The CPP has also

determined either from the linear prepolymer molar mass and the star functionality, or by direct measurement using static light scattering. The results were identical within 2%. II.2. Methods. II.2.1. Rheological Measurements. All linear oscillatory and nonlinear rheological measurements have been performed on a strain-controlled ARES-rheometer (TA Instruments, USA) equipped with a force rebalance transducer (2KFRTN1). The linear measurements were always performed in a parallel plate geometry of 8 mm diameter with a temperature control of ±0.1 °C, and in a nitrogen environment to reduce the risk of degradation (achieved via an air/nitrogen convection oven and a liquid nitrogen Dewar). A temperature range of −60 to +80 °C was used, ensuring that the sample remained stable, and subsequently the data were shifted via time−temperature superposition. Details of the procedures for the linear measurements and the treatment of the data to obtain master curves can be found in previous publications.8,10,20,37 The nonlinear step-rate measurements were performed in a special cone partitioned-plate geometry, briefly motivated and discussed in the following section with reference to an earlier paper.37 Also the nonlinear measurements were performed in a controlled environment (under nitrogen) with a temperature control of ±0.1 °C. Finally, longtime creep measurements were performed for the melt with the highest molar mass of the arms (PI 4a-103k) to extend the accessed frequency range to lower values without the need to use high temperatures at which the polymer degrades. The creep measurements have been performed with the Physica MCR 501 (Anton Paar, Austria) stress-controlled rheometer using a homemade parallel plate geometry with diameter 8 mm as top plate. Temperature control of 5705


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Table 2. Viscoelastic Properties of the Investigated Polymers at TREF = 20 °Ca sample code PI PI PI PI PI

4a-56k 8a-56k 4a-103k 4a-103k 86% 4a-103k 62%

G0N × 10−5 [Pa] 4.4 4.4 4.4 3.1 1.4

η0 × 10−5 [Pa·s] 11.5 12.0 2360 110 2.0

± ± ± ± ±

JS0 × 105 [Pa−1]

τ0 [s]

τCross [s]

S.W.

τR × 103 [s]

2.0 ± 0.1 2.4 ± 0.1

23 ± 1 30 ± 2 10000b 540 ± 80 14 ± 3

8.3 8.6 4000 200 4.6

8.8 11

2.5 2.5 8.5 1.5 0.21

0.1 0.1 50 9 0.2

4.9 ± 0.6 7±1

15 10

a

See text for definitions of the quantities. bOrder of magnitude estimation of the terminal relaxation time for sample PI 4a-103k based on the roughly constant ratio of order 3 between τCross and τ0 as found for the other samples. been used to study the nonlinear shear flow behavior of model combs with relatively short arms.22 In the present work we use the CPP setup to investigate the model stars in nonlinear shear deformations. Specifically, the CPP was composed of a standard 25 mm stainless steel cone with an angle of 0.1 rad, an Invar parallel plate of radius Rplate = 3 mm and an appropriately sized partitioned-plate part made of stainless steel, forming a gap of 0.15 mm. For the measurements, the sample overfilled the inner plate by about 80% in diameter and the radius used for measuring the torque was Rplate. This level of overfill did not substantially affect the linear viscoelastic data, conforming to earlier findings.37

mass of the arms. Both observations are in agreement with the literature.2,14,24,25 Table 2 lists an overview of the viscoelastic properties of the stars as obtained from the linear data in Figure 2. The values for the plateau moduli G0N(ϕ) are taken as value of the elastic modulus G′ at the minimum of the phase angle tan(δ) = G″/ G′. Note that the volume fraction of the polymer is nearly identical to its mass fraction here, due to the similarity in density of polyisoprene and squalene. The relative error on the concentration introduced in this way is in fact smaller than 2%. The data for the plateau modulus for the solutions obeys the usual scaling for good solvents14,52 G0N(ϕ) = G0N(1)ϕ2.3. The zero-shear viscosity η0 = limω→0(G″/ω) and the zero-shear recoverable compliance JS0 = (1/η02) limω→0(G′/ω2) are calculated from the linear oscillatory data. Their values are obtained as averages and standard deviations from three different estimations. First, (G″/ω) and (G′/ω2) are calculated, and then the zero-shear limits of these two functions are approximated by three different methods: simply taking the values at the lowest reliable frequency, fits with a Carreau-type model, X/XCarreau,0 = [1 + (τCarreau,0ω)2](nCarreau−1)/2 and a Crosstype model X/XCross,0 = 1/(1 + (τCrossω)nCross) with X the function of interest (being (G″/ω) or (G′/ω2)), XCross,0 and X Carreau,0 the fitted zero-shear limits for the function, τCross,τCarreau, nCross, and nCarreau, further fitting parameters, and ω the angular frequency.53 As the terminal regions were reached in most cases, all three estimates were rather close and error was limited. Only for the PI 4a-103k, we have not been able to get a reliable value for the recoverable compliance. The longest relaxation time τ0 listed in Table 2 was calculated as14 τ0 = JS0η0. It is used further on to define a characteristic Weissenberg number Wi0 = τ0γ̇, with γ̇ the shear rate, to enable comparisons between different polymers and quantify deviations from linear behavior. The inverse of the terminal crossover frequency τCross is shown as well. It is less prone to experimental error but its physical interpretation is less clear. The ratio between the two relaxation times is roughly constant over the different samples (3.0 ± 0.4). Since we have not been able to get a reliable value for the recoverable compliance for sample PI 4a-103k, we provide an order of magnitude estimation for the terminal relaxation time for sample PI 4a103k based on the mentioned ratio between τCross and τ0. Next column in Table 2 lists the spectral width (S.W.) defined as14 J0SG0N. It is a measure of the breadth of the relaxation spectrum. And finally, the last column lists the Rouse time τR for the stars using the relaxation time of an entanglement τE from Table 3 (see further) together with the number of entanglements of an arm Za from Table 1 in the usual way14 τR = τE(Za)2. The values for τR are in all cases below 10−2 s, i.e., orders of magnitude below the terminal relaxation times τ0. III.2. Modeling of the Master Curves with the Branchon-Branch Model (BOB). A further critical test of the sample

III. EXPERIMENTAL RESULTS III.1. Linear Viscoelastic Properties. The master curves for the dynamic moduli G′ and G″ as a function of angular frequency ω for the stars are shown in Figure 2, parts a and b, at a reference temperature of 20 °C. Figure 2a shows the master curves for the three melts and Figure 2b shows those for the two solutions and their parent melt. For the star with the longest arms (PI 4a-103k) and hence the slowest relaxation, we could not reach the terminal relaxation regime at the highest temperature of 80 °C, and hence we choose to perform longtime creep measurements in the linear regime and then convert the obtained creep compliance to dynamic moduli as explained before. The moduli as obtained via the conversion of the creep compliance are shown as full light gray lines in Figures 2a and 2b. The horizontal and vertical shift factors used to obtain the master curves are shown in Figure 1c. The vertical shift factors are determined from the change of thermal tension and density with temperature bT = (ρ(TREF)(TREF + 273.2))/(ρ(T)(T + 273.2)), with T the temperature (in °C) and with the temperature dependence of the density ρ (in g/cm3):41 ρ(T) = 0.918−5.34 × 10−4(T + 273.2) − 4.701 × 10−8(T + 273.2)2. The vertical shift factors are assumed to be identical for the melts and solutions, although for the solutions the use of the density shift is more obscure. The horizontal shift factors result from the least-squares fit as done by the Orchestrator software. The horizontal shift factors can be fitted with the WLFequation:51 log(aT) = (−C1(T − TREF))/(C2 + T − TREF), and with TREF = 20 °C the following parameters result for the melts: C1 = 5.3 and C2 = 138.1 °C. The respective values at the glass transition temperature of PI (Tg = −65 °C) are C1g = 6.7 and C2g = 53.1 °C. For the PI 4a-103k 86% solution, we obtain C1 = 4.6 and C2 = 134.8 °C, and for the PI 4a-103k 62% solution, we obtain C1 = 4.3 and C2 = 151.7 °C. By comparing the behavior of stars PI 4a-56k and PI 8a-56k in Figure 2a, we confirm that the linear relaxation of the stars is independent of the number of arms (the data are indistinguishable). On the other hand, comparison of the response of stars PI 4a-56k and PI 4a-103k in Figure 2a shows that the relaxation is clearly strongly dependent on the molar 5706


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certain factor as indicated in the caption. The black lines in Figure 3 are the predictions of the BOB model.1,4,54 The parameters used in the BOB model (at a reference temperature of 20 °C) are a value of 0.43 MPa for the plateau modulus GN0 and 4.1 kg/mol for the molar mass between entanglements Me. For the BOB model the molar mass between entanglements follows from the plateau modulus according to Me(ϕ) = (4/ 5)(ϕρRT/G0N(ϕ)), i.e., including the 4/5 prefactor, unlike for the values in Table 1. For the p2-parameter we used a value of 1/40 and for the dynamic dilution parameter α we used 1 (these are the commonly values in the BOB model).4 All samples are modeled as being perfectly monodisperse. For the solutions, the solvent is considered by artificially introducing very small monodisperse linear polymers with molar mass of 410 g/mol, i.e., the molar mass of the solvent squalene. The remaining parameters in the model are taken as shown in Table 3, with τE the relaxation time of an entanglement, Ma the molar mass of an arm, and ϕ the volume fraction of polymer. Note that the value of 20 μs for τE for the melts is identical to the value used by Auhl et al.55 for linear polyisoprenes when comparing at the same reference temperature. We consider the obtained predictions very satisfactory, in agreement with the quality of the predictions in the literature for stars.4 The plateaus of the predictions are somewhat below the plateaus for the experimental data in all cases. This is a common problem of the tube-based modeling.4,56 Finally, note that for the polymer with the largest molar mass of the arms of 103 kg/mol, the value used in the model is slightly lower (100 kg/mol) and still the model predictions display a somewhat slower relaxation than the experimental data, and this for the melt as well as for the two solutions. Moreover, it could also be interesting to examine the possibility of some physics missing in the BoB model as the source of the slight disagreement between predictions and experimental data. Overall, however, the agreement is satisfactory and such subtle issues could be resolved with more detailed characterization of the stars and further analysis, which are beyond the scope of the present contribution. III.3. Nonlinear Start-up Shear Flow. In Figure 4, two examples are shown of start-up simple shear flow for samples PI 4a-56k and PI 4a-103k. The transient start-up viscosity η+ is depicted as a function of time t for different shear rates γ̇ . The

Table 3. Parameters Used in the Simulations with the BOB Model sample code PI PI PI PI PI

4a-56k 8a-56k 4a-103k 4a-103k 86% 4a-103k 62%

τE × 105 [s]

Ma [kg/mol]

ϕ

2.0 2.0 2.0 0.54 0.18

56 56 100 100 100

1 1 1 0.86 0.62

quality and accuracy of the measurements can be obtained by comparing the experimental results with predictions of one of the well-developed tube models. Figure 3 depicts the

Figure 3. Master curves of the dynamic moduli G′ and G″ as a function of angular frequency ω for the melts at 20 °C compared with predictions of the BOB model. PI 4a-56k, vertically shifted ×100 (experimental data, ○; BOB prediction, full black line). PI 8a-56k, vertically shifted ×100 (experimental data, blue △; BOB prediction, full black line). PI 4a-103k, vertically shifted ×10 (experimental data, red □ for oscillatory data and red line for creep converted data; BOB prediction, dashed black line). PI 4a-103k 86% (experimental data, green ◇; BOB prediction, dashed-dot black line). PI 4a-103k 62% (experimental data, gray ▽; BOB prediction, dotted black line).

experimental master curves of the dynamic moduli G′ and G″ as a function of angular frequency ω for all the investigated samples at 20 °C as in Figures 2a and 2b. To ensure a clear presentation, some of the curves are shifted vertically with a

Figure 4. Transient start-up shear viscosity η+ as a function of time t for the PI 4a-56k (a) and PI 4a-103k (b) (Table 1). The linear viscoelastic lines are shown as thick dark gray lines and are calculated from the linear data (Figure 1a). The shear rates are indicated in s−1 next to the lines as “imposed rate at measurement temperature ... shear rate shifted to reference temperature of 20°C”. The results for the melt PI 4a-56k were obtained at 20 °C directly, the results for the PI 4a-103k were obtained at 80 °C and shifted to 20 °C using the shift factors from the linear data (Figure 1c). 5707


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Figure 5. Steady viscosity scaled with zero-shear viscosity ηSTEADY/η0 (a), maximum peak-viscosity scaled with steady viscosity ηMAX/ηSTEADY (b), strain where the maximum occurs γMAX (c), and strain of the peak-broadness γPEAK BROADNESS (d) as a function of the characteristic Weissenberg number Wi0 for all samples. The black ○, blue △, red □, green ◇, and gray ▽ symbols are for the nonlinear data of the PI 4a-56k, PI 8a-56k, PI 4a103k, PI 4a-103k 86%, and PI 4a-103k 62%, respectively. In part a, the full black, blue, red, and dashed green and dark gray lines represent the respective complex viscosities (scaled with zero-shear viscosity) as a function of frequency (multiplied by τ0), as calculated from the linear data for the PI 4a-56k, PI 8a-56k, PI 4a-103k, PI 4a-103k 86%, and PI 4a-103k 62%, respectively. Error bars are present but often obscured by the symbols.

responsible, then it is not unexpected that star arms do not contribute to this undershoot in this range of Wi0. Apart from the undershoot, however, the generic findings of Figure 4 discussed here are qualitatively similar to the behavior of entangled linear polymer melts and solutions,35,37,59,60 and entangled comb polymer melts and solutions.22 However the scaling analysis of the data reveals specific features that signal the star response. To be able to compare the different data sets, several parameters are extracted from the start-up transient viscosity data: the steady-state shear viscosity ηSTEADY is calculated as average over the steady-state portion of the start-up curves. The standard deviation was always checked to make sure that reasonable values were obtained and the right portion of the curve was used. The standard deviation on the steady-state shear viscosity ηSTEADY calculated in this way was found to be negligible compared to other sources of error. As second quantity of interest, the maximum peak-viscosity ηMAX is extracted by simply taking the maximum of the transient startup curve. As third quantity, we extract the position (as strain) where the maximum peak-viscosity occurs γMAX, and finally, we quantify the broadness or width of the peaks γPEAK BROADNESS (as a strain) at half height, hence at a viscosity value of (ηMAX + ηSTEADY)/2. The latter two quantities are generally found to be more scattered. In many cases, it is more prone to error at low rates due to the torque resolution, while at high rates, the start-

linear viscoelastic line is shown as well (dark gray line). This line is obtained by direct transformation of the master curves (from Figure 1) by applying the Cox−Merz rule57 η(γ̇) = η*(ω)|ω=γ̇ in conjunction with the Gleissle rule58 η+(t) = η(γ̇)|γ̇=1/t. The start-up data should follow the LVE-line for low rates and strains. The data in Figure 4a for star PI 4a-56k was obtained at 20 °C directly. The data for star PI 4a-103k (Figure 4b) was obtained at 80 °C and then shifted to 20 °C using the shift factors from the linear data (Figure 2c) as indicated by the axes in Figure 4b. Start-up results for the other investigated samples are not shown here as transient viscosity versus time as there are no apparent qualitative differences. The key observation from Figure 4 is the progressive deviation of the transient viscosity data from the LVE envelope as the shear rate increases; this deviation is marked by a steady-state viscosity value below the LVE one. As the shear rate increases the timedependent viscosity exhibits an overshoot and reaches a maximum before eventually dropping to its steady-state value. Unlike our previous works on combs22 and linear polymers,37 here we did not observe an undershoot between the maximum and steady portions of the start-up curves at any of the experimental investigated rates. We note that whereas undershoots before steady-state have also been observed by Auhl et al.55 for linear monodisperse polymers and predicted by tubebased theory for linear polymers,59,60 their physical origin remains elusive. If recoil (following stretch) is primarily 5708


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Figure 6. Transient shut-down stress σ− divided by steady stress σSTEADY as a function of time t divided by terminal relaxation time τ0 for different shear rates for the PI 4a-56k (a) and PI 4a-103k (b) stars at 20 °C. The shear rates are indicated in s−1 next to the lines as “imposed rate... temperature shifted shear rate” as in Figure 4b. The results for the melt PI 4a-56k were obtained at 20 °C directly, the results for the PI 4a-103k were obtained at 80 °C and shifted to 20 °C using the shift factors from the linear data. Note that the horizontal axes scales are not identical in the two figures.

Figure 7. Initial relaxation rates 1/τ taken as average over the dimensionless time t/τ0 from 0.02 to 0.03 (a) and long-time relaxation rates 1/τ taken as average over the dimensionless time t/τ0 from 0.9 to 1.1 (b), as a function of characteristic Weissenberg number Wi0. The black ○, blue △, red □, green ◇, and gray ▽ symbols are for the nonlinear data of the PI 4a-56k, PI 8a-56k, PI 4a-103k, PI 4a-103k 86%, and PI 4a-103k 62%, respectively. The error bars are the standard deviations of the least-squares fits to the data in the mentioned time range. The ● symbols in (a) are the respective data obtained in ref 22 for entangled combs. The horizontal black line in (b) is the average value of 1.2 for the long-time relaxation rate.

master curve. This will be further discussed below in conjunction with literature data. Parts b−d of Figure 5 show the peak viscosity ηMAX normalized with steady viscosity ηSTEADY, the strain where the peak occurs γMAX, and the strain of the peak-broadness γPEAK BROADNESS as a function of the characteristic Weissenberg number Wi0. Also for these three quantities, the differences between the different samples are very small, especially when considering experimental error and the very small scale on the y-axes of the three figures. If anything, in Figure 5, parts c and d, one could discern a tiny systematic dependence on the number of entanglements of an arm for γMAX and γPEAK BROADNESS, with the overshoot becoming somewhat less broad and occurring at smaller strains as the number of entanglements of an arm increases. On the other hand, the more entangled the arms, the broader their relaxation spectrum14,24 (see also section III.1 above). However, small differences among few data sets and the experimental errors involved in the measurements preclude a more detailed analysis. For linear monodisperse polymers, Doi and Edwards61 predicted that ηMAX, when arising solely from

up time of the motor, combined with the resolution of the standard data acquisition of the ARES rheometer increase experimental error. We define the error on the strains here artificially by simply multiplying the shear rate with the maximum time resolution of the standard data acquisition of the ARES (0.01 s). In this paper, all reported data on stars is obtained at rates below 10 s−1. The steady-state shear viscosity ηSTEADY scaled with zeroshear viscosity η0 (Table 2) is shown as a function of the characteristic Weissenberg number Wi0 for all the samples in Figure 5a. The symbols represent the data obtained from the nonlinear step-rates, the lines are obtained from the linear oscillatory data by dividing the complex viscosity η+ by η0 and plotting this quantity as a function of frequency ω multiplied by τ0, following the empirical Cox−Merz rule:57 η(γ̇) = η*(ω)|ω=γ̇. The Cox−Merz rule seems to be followed rather well in all cases, with the nonlinear data being slightly below the linear data at the higher Weissenberg numbers in all cases. Within experimental error and for the rates considered here, all data for the two solutions and the three melts scale together on a single 5709


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shear viscosity data (normalized to the zero-shear viscosity), confirming the action of CCR which is the same for both architectures. However, as Wi0 increases above 10, the rate of the combs remains clearly slower, in contrast to the much faster increase of the inverse shear viscosity, and hence rate and inverse viscosity do not coincide anymore neither for stars nor for combs. Apparently, in this regime there is a complex interplay of dilution, friction due to branches and CCR, whose elucidation needs more work in the future. Assuming that no significant stretching takes place in this regime (see Figure 5c and section IV below), a tentative interpretation calls for a different action of CCR in the two architectures. Related to this, we expect that the orientation of the comb backbone (taking place at higher Wi0) has a slower relaxation (due to the lack of free ends), hence driving the comb initial relaxation rates to lower values. We also looked at the long-time relaxation. This is more difficult to analyze since the relaxation spectra for the stars are very broad and determination of a truly constant relaxation rate is often ambiguous, mainly due to the torque resolution. We managed to perform the analysis for only two samples and chose a dimensionless time region t/τ0 between 0.9 and 1.1 for samples following the analysis for the combs.22 The results for the long-time relaxation rates as a function of Wi0 are shown in Figure 7b and suggest that, within error the relaxation rates are shear rate independent. An average value of 1.2 ± 0.8 is found for the long-time relaxation rates, hence only slightly above terminal, exactly as for the combs.22 In this respect, it is encouraging that the relaxation data are consistent with the dynamic frequency sweep data of Figure 2 (section III.1).

orientation of the chains (without stretch), should occur at a strain of γMAX = 2.3. In experiments, one generally finds a value between 2 and 2.3 for γMAX.35,37,50 Here, for the stars, the values for γMAX as reported in Figure 5c are around the expected value for orientation and slightly below, for all samples. Indirectly, this suggests that for all investigated shear rates and for all samples, the stars are orienting without stretching at this range of Weissenberg numbers. Note that, for stars the terminal relaxation time and the stretch time are widely separated (see section IV for further discussion). III.4. Relaxation upon Cessation of Steady-State Shear Flow. The relaxation of stress from steady-state σ− at shear rate γ was monitored as a function of time t since the flow was stopped (γ = 0 s−1 for t ≥ 0 s). Figure 6 shows typical data for the transient shut-down stress σ− (divided by its steady-state value σSTEADY at time t = 0) as a function of time t for different shear rates for the two stars PI 4a-56k and PI 4a-103k. We note that, qualitatively, the observations are identical to those of Menezes and Graessley62 for solutions of linear polymers in the entangled regime and to those of Snijkers et al.22 for entangled comb polymer melts and solutions. First, the relaxation transients decrease monotonically with time for all rates. Second, the relaxation rate in the initial stages accelerates as the shear rate increases. And third, for long enough times after flow cessation, the relaxation rate becomes roughly independent of shear rate. In order to compare the different samples, the initial relaxation rates for the stress decay function were calculated by assuming a single exponential relaxation over a certain range of dimensionless times t/τ0 as σ−(t/τ0) = C exp(−t/τ0τ), with C an arbitrary constant and τ the relaxation time over the time region of interest. Therefore, the initial relaxation rate 1/τ could be obtained by fitting a straight line to the plot ln(σ−/ σSTEADY) vs t/τ0, with the only challenge the choice of the relevant dimensionless time range for the fit. To this end, some considerations were crucial. Because of instrumental constraints discussed earlier, fitting at times below 0.1 s was excluded. We chose a dimensionless time region t/τ0 between 0.02 and 0.03 for all samples, identical to the dimensionless time range we analyzed for the combs.22 Figure 7a depicts the results for these “initial” relaxation rates as a function of Wi0 for the investigated star samples with the error bars coming directly from the leastsquares fits. We observe that the initial relaxation rate increases with the characteristic Weissenberg number Wi0. All data fall onto a single master curve within experimental error, which suggests that regardless of the star functionality and molar masses of the arms, the orientation of the arm relaxes with a rate proportional to the inverse terminal relaxation time τ0 and hence it randomizes its orientation mainly via the usual linear relaxation mechanisms (arm retraction and constraint release), qualitatively identical to the observations for the combs.22 It appears that the initial relaxation rate increases with Wi0 because the initial steady-stress state of the entangled polymer at the relaxation experiment contains less entanglements compared to that at the beginning of the start-up experiment at constant shear rate, due to the action of CCR.22 At higher Wi0 the action of CCR is stronger and consequently the relaxation rate becomes faster. We also compare this behavior with that of combs22 in Figure 7a. Note that, in that case, the comb response is dominated by the backbone due to short (i.e., very fast retracting) arms.22 Clearly, for Wi0 < 10 the fast relaxation rate of both architectures is nearly identical. Moreover, in this regime, the data are identical to the inverse

IV. FURTHER DISCUSSION AND COMPARISON WITH THE LITERATURE Key observations in the presented data set is the near identity between the scaled quantities (in Figure 5) for the set of polymer star samples with systematically varying number of arms, molar mass of the arms, and solvent content. Especially the observation of a roughly constant strain at peak viscosity γMAX with a value of about 2 (Figure 5c) is interesting as it indicates that, in the investigated range of rates, we are not stretching the polymers, but just orienting them, even though we reach high characteristic Weissenberg numbers. As shown in the last column of Table 2, the values for the Rouse time τR for the stars are in all cases below 10−2 s, i.e., orders of magnitude below the terminal relaxation times τ0 (Table 2). In terms of characteristic Weissenberg number, this leads to an order of magnitude estimation for the range of values needed to stretch the stars in all cases (Wi0 > 104). In the performed experiments edge fracture sets in at Wi0 at least an order of magnitude lower, hence we could not reach the required rates for stretch. ̈ analysis is in full agreement with the work Note that this naive of Ye and Sridhar36 on extensional flow of star solutions discussed in the Introduction. A comparison with the data of Menezes and Graessley,35 which is to the best of our knowledge the only other data set available with start-up simple shear data of star polymers, shows mainly the severe problems with edge fracture, which limited their data to very low rates (Wi0 < 3). The data obtained with symmetric polybutadiene 4-arm stars having arm molar mass of 49.5 kg/mol, in solution at a concentration of 26.6% (i.e., between 4 and 5 entanglements per arm), is in excellent agreement with the data presented here (Figure 5). As stated in the Introduction, the focus of Menezes and Graessley was on 5710


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for these entangled stars, as they deviate from the −1 viscosityshear rate slope.34 CCR is proven to work for entangled linear polymers,34,37,63 but for stars there are subtle issues such as dynamic dilution, and hence deviations are not a surprise. In Figure 9, a comparison is made between the nonlinear data of the stars as presented in Figure 5, and the nonlinear start-up shear data of two linear polymers (black filled circles) measured earlier.37 The two linear polymers are monodisperse polymers of polyisoprene and polystyrene with molar masses of 60 kg/mol (Z ∼ 12) and 182 kg/mol (Z ∼ 11), respectively. As reported in ref 37, we found a master curve scaling for the nonlinear quantities reported in Figure 9 for the two linear polymers due to their similarity in the number of entanglements and hence we do not use distinct symbols for the two polymers in Figure 9. In Figure 9, the quantities are plotted as a function of the characteristic Weissenberg number Wi0 based on the longest relaxation time τ0. The values for τ0 for the linear polymers, calculated in the same way as for the stars described earlier, are 0.16 s at 10 °C for the PI sample and 0.33s at 180 °C for the PS. Note that the number of entanglements of the linear polymers Z is roughly identical to the number of entanglements of an arm Za as reported in Table 1 for the PI 4a-56k, PI 8a-56k, and PI 4a-103k 62%. Figure 9a shows that, despite the common onset of thinning at Wi0 ∼ 1, the linear polymers (black symbols) are stronger thinning and that their transition to a (seemingly) terminal thinning slope is more sudden, i.e., more narrow than for the stars, reflecting a broader relaxation spectrum for the stars. Concerning the overshoot, when looking at a fixed, relatively high Weissenberg number (say Wi0 = 10), the linear polymers have a stronger overshoot (Figure 9b), with a peak that is shifted toward larger strains (Figure 9c) and which are somewhat broader (Figure 9d). This might reflect the fact that, in contrast to stars, for linear polymers the stretch time τR is closer to the terminal relaxation time τ0. Actually, in terms of order of magnitude, the linear polymers are around a stretch Weissenberg number (i.e., a Weissenberg number based on the Rouse time) of 1 at the highest investigated rates,37 whereas for the stars considered here, we are in all cases still almost two decades too low to reach a stretch Weissenberg number of 1.36 Another comparison concerns the occurrence of a local minimum, i.e., undershoot, in the start-up viscosity curves between the maximum and the steady-state portions. As already mentioned, for the linear polymers we observed a very mild undershoot at the highest few rates,37 which is absent for the stars in all cases. Possibly, also the occurrence of the undershoot relates to stretch: The linear polymers are mildly stretching, while the stars are still far from stretching.

obtaining reliable normal stress data through the use of the specially stiffened rheometer.62 In Figure 8, a comparison is made between the steady-state shear data of Figure 5a with those of Tezel et al.33 Tezel et

Figure 8. Nonlinear steady shear data obtained on the stars and shown previously in Figure 5a compared to the steady shear data from Tezel et al.32 as a steady viscosity scaled with zero-shear viscosity ηSTEADY/η0 as a function of the characteristic Weissenberg number Wi0. The ○ symbols are a collection of all the nonlinear shear data from Figure 5a. The red ▲ and blue ■ symbols are for two of the samples from ref 33, coded star 2 and star 6, respectively. The full red and full blue lines are the predictions of the adjusted MLD model for star 2 and star 6 respectively; the dashed red and dashed blue lines are the predictions of the adjusted GLaMM model for star 2 and star 6, respectively. The full black line has a slope of −1 (CCR prediction) and serves to guide the eye.

al.32,33 obtained the experimental data using two-color birefringence in a Couette device and modeled the data with an adjusted GLaMM59 or MLD63 models. The adjustments of the two models relate to the fact that they turned the reptation mechanism off. Tezel et al.32,33 measured polybutadiene 4-arm stars dissolved in low molar mass linear polybutadiene. The data shown in Figure 8 is for the samples with coding Star 2 (in dark gray; Ma = 140 kg/mol, 13 wt %, yielding Za = 10) and Star 6 (in light gray; Ma = 130 kg/mol, 12 wt %, yielding Za = 8.6).33 The data shown in Figure 8 are the two most extreme data sets from Tezel et al.,33 as in a plot of ηSTEADY/η0 versus characteristic Weissenberg number Wi0 the other data sets fall in between these two. This comparison is relative as the experimental data and both models are in all cases set to start at viscosity ratio ηSTEADY/η0 = 1 (by tuning the η0-values individually, which have a rather large spread).33 However, such a representation helps performing a relative comparison of different stars. Furthermore, the longest relaxation times are the ones reported in the article, which are calculated differently from those in Table 2.33 Nevertheless, several observations can be made: (i) Optical measurements (based on the validity of the stress-optical rule) appear to be less accurate and lead to a significantly larger error on the experimental data as compared to mechanical data obtained with the CPP (data are much more scattered in Figure.8). (ii) A master curve scaling was not obtained, neither for the data, nor for the two models. (iii) The optical data and the models appear to exhibit somewhat stronger thinning; i.e., they are less broad in their spectrum than the data from the CPP. Slightly larger polydispersity (for star 2 of ref 33) and the approximate way in which CCR was modeled are the most likely reasons for this.32,33 (iv) The experimental data from CPP suggests that CCR may not hold

V. CONCLUSIONS In this work, we have presented a detailed experimental investigation of the linear and nonlinear rheology of welldefined symmetric entangled polymer stars of low functionality with varying number of arms, molar mass of the arms and solvent content. We have focused in particular on the response of the stars in simple shear, during start-up and relaxation upon flow cessation. To reduce experimental artifacts associated with edge fracture (primarily) and wall slip, which are unavoidable in nonlinear shear rheology, we have utilized a homemade conepartitioned plate fixture which was successfully implemented in recent studies with entangled linear and comb polymers. The following conclusions can be drawn: 5711


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Figure 9. Steady viscosity scaled with zero-shear viscosity ηSTEADY/η0 (a), maximum peak-viscosity scaled with steady viscosity ηMAX/ηSTEADY (b), strain where the maximum occurs γMAX (c), and strain of the peak-broadness γPEAK BROADNESS (d) as a function of the characteristic Weissenberg number Wi0 for all the star samples and the linear, monodisperse polymers from ref 37. The ● symbols are for the linear, monodisperse polymers from ref 37. (PI 60k and PS 182k), all other symbols are for the star samples (legend as in Figure 5). The black lines in part a are drawn to guide the eye. Error bars are present but often obscured by the symbols.

(1) For the star polymers considered here with up to about 20 entanglements per arm, reliable nonlinear data could be obtained with the CPP for characteristic Weissenberg numbers (based on the terminal time) below 300. (2) During start-up simple shear, the stress overshoot occurred nearly always at a strain approaching a value of 2, suggesting that in the investigated shear regime the star arms orient but do not stretch. (3) The validity of the empirical Cox−Merx rule was confirmed, within experimental error. On the other hand, the steady shear viscosity data exhibit a slope smaller than the CCR slope of −1 in the investigated range of rates. (4) The broadness of the stress overshoot increased with Weissenberg number. A relation with the broad linear viscoelastic relaxation spectrum of the stars is tempting, albeit not yet quantified. (5) The initial stress relaxation rate upon cessation of steady shear flow, reflecting the initial loss of entanglements due to the action of CCR in steady shear flow prior to relaxation, increased with Weissenberg number. Moreover, when compared against the relevant rate for comb polymers, the latter was clearly slower at Weissenberg numbers exceeding a value of about 10. This could be suggestive of a different action of CCR in stars and combs and a related slower relaxation of comb backbone

orientation compared to that of arms, apparently due to the lack of free ends. (6) The experimental data set presented here, i.e., linear viscoelasticity, stress start-up, and relaxation is selfconsistent and may be useful toward refining models for the nonlinear response of entangled star polymers.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: (D.V.) [email protected]. (N.H.) nikolaos. [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank R. Pasquino for help with the creep analysis and helpful discussions, as well as the reviewers of this work for their very contructive comments. Partial support from the EU (FP7 ITN DYNACOP Grant 214627) is gratefully acknowledged.

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REFERENCES

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